Ground pressure. Nominal pressure p = W / A. A track of width b and ground length L gives A = 2 b L; a tyre has A = Wwheel / pcontact, where pcontact is about 1.1 times the inflation pressure while the tyre carcass is stiffer than the soil. A farm tyre at 100 kPa and a tracked carrier at 45 kPa both look "light", but the sinkage depends on how the pressure is spread.
Pressure-sinkage (Bekker, 1956). p = (k_c/b + k_phi) z^n for a plate of width b. Inverse: z = (p / (k_c/b + k_phi))^(1/n). Narrow contacts see the kc/b term: that is why a narrow wheel sinks more than a wide track at the same pressure. For a rigid wheel of diameter D and load W: z = [3W / (b (3-n) (k_c + b k_phi) sqrt(D))]^(2/(2n+1)).
Compaction resistance. Pushing the soil down costs energy per metre of travel: R_c = b * integral p dz = b (k_c/b + k_phi) z^(n+1) / (n+1) for each wheel or track. Bulldozing resistance (soil pushed ahead, Reece form, simplified) R_b = b (c z K_c + 0.5 rho g z^2 K_gamma) is added when the wheel is sunk deeper than a few percent of its diameter. Tyre carcass hysteresis is a small term, f_h = 0.008 + 0.9/p_t, a rough heuristic. Rolling resistance coefficient is fr = R / W.
Soil strength and thrust. Mohr-Coulomb: tau_max = c + p tan(phi). Shear stress grows with slip displacement j (Janosi-Hanamoto): tau = tau_max (1 - exp(-j/K)). Integrating over a patch of length l at slip i gives the gross thrust H = (A c + W tan(phi)) [1 - (K/(i l)) (1 - exp(-i l / K))]. Net drawbar pull is DP = H - R - W sin(theta); tractive efficiency is (DP/H)(1 - i). A vehicle bogs when DP is not positive at any slip below 100 %, or when the chassis or hubs sit on the soil.
Tyre pressure. Lowering the inflation pressure enlarges A, which lowers p and therefore sinkage and compaction resistance. This is why central tyre inflation systems are used on soft ground. Very low pressure can unseat the tyre bead and raises hysteresis: the model keeps the heuristic but you should not drop below manufacturer limits.
Seasonal "rasputitsa". In spring thaw and autumn rains the loam saturates. In this model cohesion c falls from 22 to 2 kPa, friction angle from 32 to 9 degrees and kphi by a factor of 50 as the moisture index goes 0 to 1 (interpolated, illustrative curves, not measured soil data).
Multiple passes. Sinkage grows as zN = z1 N0.3 and the compaction resistance of pass N is the increment of b p dz between zN-1 and zN. Rear wheels that follow the front ones run in the rut (one extra pass). Real cyclic soil behaviour is more complicated.
Weight transfer. For two-axle wheeled vehicles, the rear axle share grows by (h/wheelbase)(W sin(theta) + towed load)/W when climbing or pulling. Tracks are treated as uniformly loaded.
Simplifications. Quasi-static steady state, flat uniform soil, rigid-wheel / plate patch shapes, nominal track pressure, no suspension dynamics, no steering, soil parameters of textbook order of magnitude (dry sand, stiff clay, snow after Wong; loam is a blend made for this lab). Half-tracks and skid-steer turning are not modelled. Use for understanding trends, not for predicting a real machine.
Sources in plain text: M. G. Bekker, "Theory of Land Locomotion" (1956) and "Introduction to Terrain-Vehicle Systems" (1969); J. Y. Wong, "Theory of Ground Vehicles"; Z. Janosi and B. Hanamoto (1961); A. R. Reece (1964).
Physics teaching model of generic civilian farm, logistics and rescue vehicles. Not affiliated with any manufacturer; no weapons or military content.