Appendix - Summary of Sinkage and Shearing Approaches

Bekker for penetration forces:(1)

p = (\frac{k_{c}}{b} + k_{φ}) D^{n} - C \dot{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)

τ = (c + p \tan φ) (1 - e^{- \frac{j}{k}})

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)

τ_{\max} = c + p \tan φ

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)

τ_{\max} = {\begin{array}{l} c + p \tan φ & j < j_{\max} \\ (c + p \tan φ) e^{\frac{j_{\max} - j}{k_{1}}} & j_{\max} \leq j \leq j_{u} \\ (c + p \tan φ) r & j > j_{u} \end{array}

where:

j_max = the maximum shear displacement which affects increasing the shear stress
j_u = the ultimate shear displacement
there is no effect on the shear stress while increasing the shear displacement above this point, j_u>j_max
k₁ = constant; r = the maximum shear ratio, 1>r>0

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

(5)

r = e^{\frac{j_{\max} - j_{u}}{k_{1}}}

Thus:

(6)

k_{1} = \frac{j_{\max} - j_{u}}{l n r}

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

Where:

(7)

τ * = \frac{τ}{c + p \tan φ}; j * = \frac{j}{k}; j_{\max}^{*} = \frac{j_{\max}}{k} = 10; j_{u}^{*} = \frac{j_{u}}{k} = 17; r = 0.5