Soft Soil Track Model
The Altair Soft Soil model provides a way to simulate the dynamic behavior of a track system on a surface that is compressible such as clay, dry sand, regolith, and icecovered snow.
Theoretical Approach
In the current track model, the deformations of the soil which eventually lead to the corresponding (normal and shear) stresses are considered as two independent effects. Specifically, the strength of the soil is considered in the normal direction (pressuresinkage relationship) as well as in the tangential direction (shear stressshear displacement relationship).
PressureSinkage Relationship
If a terrain is assumed to be homogeneous, its pressuresinkage relationship (the result of the platesinkage test) may take one of the forms which are shown in Figure 1, and it may be characterized by the following empirical equation proposed by Bekker. [1]
Where is pressure, is the width of the rectangular contact patch area, is sinkage, and , and , are pressuresinkage related parameters.
Soil Failure
There is a variety of criteria proposed for the failure of soils. One of the most widely used is the MohrCoulomb criterion which states that the maximum shear strength of the soil is:
Where is the apparent cohesion, is the normal stress, and is the angle of internal shearing resistance of the material. The aforementioned parameters can be derived with a shear test for different pressures, as shown in Figure 2:
TrackRoad Interaction
Contact Points
Within the track model, the number of contact points varies in the range 15 depending on the shape of the track link. More specifically, in the case of one (1) contact point, only a flat area is used for the description of the track link shape, whereas grousers are added for the cases of more than one (25) contact points. In Figure 3, three different examples of the track link shape are presented for the cases of two, three and five contact points, respectively.
Normal Stress
Using the pressuresinkage relationship proposed by Bekker (Equation (1)), the track link normal stress is provided by [2, 3]
where is the sinkage of the track link, while the definition of the parameter b is provided by the user. In particular, the width or the minimum dimension of the track link can be used for the definition of this parameter.
Furthermore, Equation (3) can be modified in order to account for the contribution of soil damping [24]. In that case, the normal stress is given by
where is the soil damping, is soil’s compression rate/velocity, and is the contact patch area.
Shear Stress
The shear stresses and are calculated using identical expressions [23], [5].
In the above equations, and denote the shear displacements in x and y direction, respectively. In addition, and represent the shear deformation modules, which are provided by the following equations:
where and are constant terms.
 ‘Soiltire interaction analysis for agricultural tractors: Modeling of traction
performance and soil damage’ by A. Battiato, 2014 [6]
 [m] for clay
 [m] for clay loam
 [m] for silty loam
 [m] for loamy sand
 ‘Terramechanicsbased analysis for slope climbing capability of a
lunar/planetary rover’ by K. Yoshida and G. Ishigami, 2004 [7]
 [m] for dry sand
 [m] for regolith simulant
 ‘Analysis of offroad tiresoil interaction through analytical and finite
element methods’ by H. Li, 2013 [4]
 [m]
 [m]
Moreover, according to Wong [8], based on experimental data collected, the value of ranges from 0.01 [m] for firm sandy terrain to 0.025 [m] for loose sand and is approximately 0.006 [m] for clay at maximum compaction. For undisturbed fresh snow, the value of varies in the range from 0.025 [m] to 0.05 [m].
Using the original Janosi approach, the maximum shear strength of the soil, provided by Equation (2), is substituted in Equations (5)(6) for the calculation of the shear stresses, resulting in the following expressions
However, due to the soil failure, the shear stress decreases above a certain value of sheardisplacement [3]. Herein, a simple approach is proposed in order to account for this effect, where the maximum shear strength of the soil constitutes a function of the shear–displacement [3]. Based on this, a modified Janosi approach is derived in the form
where corresponds to the value of the shear displacement where the soil failure starts and stands for the ultimate shear displacement, that is above this value the shear stress is unaffected by the shear displacement . Moreover, the parameter r denotes the maximum shear ratio .
In addition, the following expression must hold such that the maximum shear strength, given by Equation (11), corresponds to a continuous function
Grouser Effect
The proper modeling of the grouser forces is crucial for the track model since their contribution to the total forces of the track link can be very significant. It turns out that the soil–grouser interaction is very similar to the soil–blade interaction. Therefore, the model proposed by McKyes [9] is suitable for the proper modeling of this interaction. Within this work, the model proposed by McKyes [9] is used for the calculation of the grouser forces with some modifications [3].
Multipass Effect
 Normal Stress

For the multipass effect, the response of the soil to repetitive normal load needs to be established. Specifically, the mathematical description of the normal stress must be modified in cases of existing precompaction of the soil. As shown in Figure 6, the normal stress will be comprised initially from an elastic part , which is equal to the elastic (unloading) sinkage that has already been created due to the interaction of the soil with a track link. Then, the pressuresinkage relationship continues according to Equation (3). Finally, an unloading elastic part is encountered.
One part of the induced soil deformation is elastic (elastic sinkage), and the remaining part (plastic sinkage) is irreversible. The elastic part is provided by the equation
where is the soil elastic stiffness.
 Shear Stress

A similar approach to the normal stress is also employed for the description of the multipass effect for the shear stress. Specifically, the soil shear stress, is initially, formulated as a function of the shear displacement by using the equation
until certain (maximum) values of the shear stress and shear displacement are derived. Subsequently, a linear unloading/loading curve is created based on the maximum values of the shearrelated parameters, as depicted in Figure 7. The induced plastic shear displacement of the soil is provided by the expression
where is the (current) maximum value of the shear displacement and AS constitutes a userdefined parameter that describes the shear plasticity of the soil . In particular, for , the whole induced shear displacement of the soil is elastic, whereas for the shear displacement is entirely plastic.
Apparently, this linear curve describes the shear stressshear displacement relationship for values of the shear displacement in the range . On the contrary, for , the initial curve, given by Equation (14), is again used. Lastly, it should be noted that a similar approach is also employed for the description of the multipass effect for the grouser force by using the parameter AM instead of the parameter AS [3].
MotionSolve Output Requests
 Soil force in the global direction with respect to the reference marker for each link.
 Soil normal force in the link local coordinate for each link.
 Sinkage, longitudinal slip, and lateral slip for each link.
 Total soil force for each wrap.
 Total soil force for all links in all wraps.
The force output for contacts can be created for all the contacts using the Entity set option in the output request. The output for forces like tension or bending force can be measured on the respective joint and or force. The mid marker in the link can be used as a reference marker for the force.
References
[1] M.G.Bekker, Introduction to terrainvehicle systems. part i: The terrain. part ii: The vehicle. Michigan Univ Ann Arbor, 1969.
[2] D. Rubinstein and R. Hitron, “A detailed multibody model for dynamic simulation of offroad tracked vehicles,” Journal of Terramechanics, 41(23), 163173, 2004.
[3] D. Rubinstein and J. L. Coppock, “A detailed singlelink track model for multibody dynamic simulation of crawlers,” Journal of Terramechanics, 44(5), 355364, 2007.
[4] HaoLi, “Analysis of OffRoad TireSoil Interaction through Analytical and Finite Element Methods,” Technischen Universität Kaiserslautern, 2013.
[5] Z.Janosi, “The analytical determination of drawbar pull as a function of slip for tracked vehicles in deformable soils,” in Proc. of 1st Int. Conf. of ISTVS, 1961.
[6] A.Battiato, “Soiltyre interaction analysis for agricultural tractors: modelling of traction performance and soil damage,” 2014.
[7] K. Yoshida, N. Mizuno, G. Ishigami, and A. Miwa, “Terramechanicsbased analysis for slope climbing capability of a lunar/planetary rover,” in 24th Int. Symp. on Space Technology and Science, 2004.
[8] J.Y.Wong, Theory of ground vehicles. John Wiley & Sons, 2008.
[9] E.McKyes, Agricultural engineering soil mechanics. Elsevier, 2012.
[10] J.Y.Wong, Terramechanics and offroad vehicle engineering: terrain behaviour, offroad vehicle performance and design. Butterworthheinemann, 2009.
[11] J.Y.Wong and A.R.Reece, “Prediction of rigid wheel performance based on the analysis of soilwheel stresses Part I. Performance of driven rigid wheels,” J. Terramechanics, vol. 4, no. 1, pp. 81–98, 1967.
[12] I. Genya, M. Akiko, N. Keiji, and Y. Kazuya, “TerramechanicsBased Model for Steering Maneuver of Planetary Exploration Rovers on Loose Soil,” J. F. Robot., vol. 7, no. PART 1, pp. 81–86, 2015, doi: 10.1002/rob.