## 3 Soil Thrust Assessment Method for Continuous Tracks  
### Existing Soil Thrust Assessment Method  
When an off-road tracked vehicle travels, an engine thrust is transmitted to the ground via continuous track, resulting in soil thrust on the bottom and side of the track. As the soil thrust is the shear force of the soil, it can be calculated from the integration of the shear stress over the shearing area on the bottom and side of the track system. Based on the theory of elasticity, [PERSON] [4] expressed the maximum values of soil thrust, as shown in Equation (1). \[F_{max}=wl\big{(}c+\sigma_{v}^{\prime}\tan\phi\big{)}+lh\bigg{(}2c+0.64\cot^{ -1}\frac{h}{w}\sigma_{v}^{\prime}\tan\phi\bigg{)}, \tag{1}\]  
where \(F_{max}\) is the maximum soil thrust, \(c\) and \(\phi\) are the cohesion and friction angle of the soil, respectively, \(\sigma_{v}\)' is the effective vertical stress acting on the bottom of the continuous track, \(w\) and \(l\) are the width and length of the soil-track interface, respectively, and \(h\) is the height of the grouser. Figure 4 illustrates the three different behaviors exhibited by soil thrust-slip displacement relationships: hardening, softening, and hump [3]. Hardening and softening behaviors occur mainly in loose and dense soils, respectively [25], and the hump behavior is mainly observed in muskeg mat. In this study, the soil thrust-slip displacement relationships proposed by [PERSON] and [PERSON] [11] and [PERSON] [12] were used to represent hardening (Figure 4a) and softening behaviors (Figure 4b), respectively, and are given in Equations (2) and (3), respectively. The soil thrust-slip displacement relationship proposed by [PERSON] and [PERSON] [13] and [PERSON] [14] was adopted to represent hump behavior (Figure 4c). It is given in Equation (4). \[F=F_{max}\{1-\exp(-j/K)\} \tag{2}\]  
\[F=F_{max}K_{r}[1+\{1/K_{r}(1-1/e)-1\}\exp(1-j/K_{w})][1-\exp(-j/K_{w})] \tag{3}\]  
\[F=F_{max}(j/K_{h})\exp(1-j/K_{h}) \tag{4}\]  
Here, \(j\) is the slip displacement, \(K\), \(K_{w}\), and \(K_{h}\) are the shear deformation constants (the shear displacement required to develop the maximum soil thrust), and \(K_{r}\) is the ratio of the maximum soil thrust (\(F_{max}\)) to the residual soil thrust (\(F_{res}\)) (\(F_{max}/F_{res}\)). \(F_{max}\) is determined using Equation (1), and \(F\) is the soil thrust when the slip displacement is \(j\). Figure 3: Schematic diagram of the vehicular load of an individual grouser and the block-shaped failure plane. [PERSON] [4] defined the slip ratio \(i\) as the ratio of the continuous track displacement \(x\) to the slip displacement \(j\) generated when the continuous track moved by \(x\) (\(i=j/x\)). In other words, if the slip ratio is 0 (i.e., \(j=0\)), all the displacements of the continuous track are converted to the displacement of the vehicle, and if the slip ratio is 1 (i.e., \(j=x\)), the vehicle is idling. In literature [2; 3; 4], the total slip displacement \(j\) generated by the continuous track is calculated as the product of the total distance covered by the continuous track along the ground (that is, length of the soil-track interface \(l\)) and the slip ratio \(i\) (i.e., \(j=il\)). The total soil thrust (\(F_{l}\)) is obtained by substituting the total slip displacement (i.e., \(j=il\)) into Equations (2)-(4). Equations (5)-(7) present the total soil thrust of the continuous track showing hardening, softening and hump behaviors, respectively, under a slip ratio of \(i\). \[F=F_{max}\{1-\exp(-il/K)\} \tag{5}\]  
\[F=F_{max}K_{r}[1+\{1/K_{r}(1-1/e)-1\}\exp(1-il/K_{w})][1-\exp(-il/K_{w})] \tag{6}\]  
\[F=F_{max}(il/K_{h})\exp(1-il/K_{h}) \tag{7}\]  
As described above, the relationship between the total soil thrust and slip ratio of the off-road tracked vehicle can be expressed using the ground characteristics and the geometries of the continuous track and grouser.