Case I:
A slip ratio due to the critical inclination angle is
denoted by 
, then it can be defined as:
(1)
If 
represents a slip ratio due to
the critical Longitudinal slip ratio, then it can be defined as:
(2)
If a slip ratio due to the critical slip angle is denoted by
, then it can be determined as
(3)
when 
•The term “critical” stands for the maximum value which allows an elastic deformation of a tire during pure slip due to pure slip ratio slip angle, or inclination angle.
•Whenever any slip ratio becomes greater than its corresponding critical value, an elastic deformation no longer exists, but instead complete sliding state represents the contact condition between the tire tread base and the contact surface.
A nondimensional slip ratio 
is determined as:
(4)
where,:
A nondimensional contact patch length is determined as:
A modified later friction coefficient 
is evaluated as:
(5)
where, 
is the available friction as
determined by the friction circle.
•To determine the longitudinal force, the lateral force, and the self-aligning torque, considers two sub-cases separately.
•The first case is for the elastic deformation state, while the other is for the complete sliding state without any elastic deformation of a tire.
•Specifically, if all of slip ratios are smaller than those of their corresponding critical values, then there exists an elastic deformation state, otherwise there exists only complete sliding state between the tire tread base and the terrain surface.
1. Elastic Deformation State: 
, 
and 
In the elastic deformation state,
the longitudinal force 
, the
lateral force 
, and three
components of the self-aligning torque are written as functions of the elastic
stiffness and the slip ratio as well as the normal force and the friction
coefficients, such as:
(6)
(7)
(8)
(9)
(10)
where,
is the offset between the wheel plane
center and the tire tread base, 
is set to zero if it is negative.
is the portion of the
self-aligning torque generated by the slip angle 
. 
and 
are other components of the
self-aligning torque produced by the longitudinal force, which has an offset
between the wheel center plane and the tire tread base, due to the slip angle
and the inclination
angle 
, respectively. The
self-aligning torque 
is
determined as combinations of 
, 
and 
.
2. Complete Sliding State: 
, 
and 
In the complete sliding state, the longitudinal force, the later force, and three components of the self-aligning torque are determined as functions of the normal force and the friction coefficients without any elastic stiffness and slip ratio as:
(11)
(12)
(13)
(14)
(15)