# Appendix - Summary of Sinkage and Shearing Approaches

Bekker for penetration forces:(1)
$p=\left(\frac{{k}_{c}}{b}+{k}_{\phi }\right){D}^{n}-C\stackrel{˙}{D}$
Where:
• D = the sinkage of the track link in the direction perpendicular to the link surface
• p = pressure
• b = track width
• C = damping per unit area coefficient
• kc, kΦ, n = empirically determined constants
Janosi shear force:(2)
$\tau =\left(c+p\mathrm{tan}\phi \right)\left(1-{e}^{-\frac{j}{k}}\right)$
Where:
• τ = shear stress
• j = shear displacement
• c = cohesion
• Φ= angle of internal friction
• k = empirically determined constant
According to the Janosi approach, the shear stress increases while increasing the shear displacement. The maximum shear stress is:(3)
${\tau }_{\mathrm{max}}=c+p\mathrm{tan}\phi$

However, if a value is above a certain value of the shear displacement, then the shear stress is decreased. This is due to the soil failure changing the soil parameters (c and Φ). It's recommended to use a simple approach, where the shear displacement affects the maximum shear stress and not the soil parameter. The term for the proposed approach is a modified Janosi approach.

(4)
${\tau }_{\mathrm{max}}=\left\{\begin{array}{lll}c+p\mathrm{tan}\phi \hfill & \text{ }\text{ }\text{ }\hfill & j<{j}_{\mathrm{max}}\hfill \\ \left(c+p\mathrm{tan}\phi \right){e}^{\frac{{j}_{\mathrm{max}}-j}{{k}_{1}}}\hfill & \hfill & {j}_{\mathrm{max}}\le j\le {j}_{u}\hfill \\ \left(c+p\mathrm{tan}\phi \right)\text{\hspace{0.17em}}r\hfill & \hfill & j>{j}_{u}\hfill \end{array}$
where:
• jmax = the maximum shear displacement which affects increasing the shear stress
• ju = the ultimate shear displacement
• there is no effect on the shear stress while increasing the shear displacement above this point, ju>jmax
• k1 = constant; r = the maximum shear ratio, 1>r>0

For τmax to be a continuous function, the following relationship should be maintained:

(5)
$r={e}^{\frac{{j}_{\mathrm{max}}-{j}_{u}}{{k}_{1}}}$

Thus:

(6)
${k}_{1}=\frac{{j}_{\mathrm{max}}-{j}_{u}}{ln\text{\hspace{0.17em}}\text{\hspace{0.17em}}r}$

A plot of non-dimensional shear stress τ* versus non-dimensional shear displacement j* is provided in the following figure.

Where:

(7)
$\tau *=\frac{\tau }{c+p\mathrm{tan}\phi }\text{ };\text{ }j*=\frac{j}{k}\text{ };\text{ }{j}_{\mathrm{max}}^{*}=\frac{{j}_{\mathrm{max}}}{k}=10\text{ };\text{ }{j}_{u}^{*}=\frac{{j}_{u}}{k}=17\text{ };\text{ }r=0.5$