Select the Linear elastic from the Contact Force Model list in the physics interface Force section to compute the contact forces using the linear elastic model. The two components of forces are discussed individually (Ref. 1).

The following diagram (left) shows two grains in contact with radii R1 and R2. As this is a soft-sphere model, there is a finite overlap between the grains in contact, although for illustrative purposes this overlap region has been greatly exaggerated.

Taking a closer look at the overlap region (right figure), define unit vectors in the tangential direction t and normal direction n. In 3D, there would be two orthogonal tangential directions t1 and t2. Let the normal displacement δn (SI unit: m) be the thickness of this overlap region. For intersecting spheres, the radius of the contact area is denoted a (SI unit: m). For two grains in contact with positions qi and qj (SI unit: m), the normal direction is

where kn is the normal elastic stiffness coefficient and cn is the normal damping coefficient. In this model, these two coefficients are known and constant. Nc is the number of neighboring grains, and vr is the relative velocity between colliding grains at the contact point and is given by

where ω is the rotational velocity of the grain. The normal damping coefficient cn is calculated as

and en is the coefficient of restitution in normal direction.

where kt is the tangential elastic stiffness coefficient and ct is the tangential damping coefficient. Similar to normal force, these two are known and constant. The tangential damping constant is calculated as

et is the coefficient of restitution in tangential direction.

where vt is the tangential component of vr, which is obtained by subtracting normal component vn from vr:

Thus, Equation 3-7 and Equation 3-8 can be rewritten as

The calculation of the tangential displacement, δt, is dependent on time history of the physical contact between two grains. When a new physical contact happens at time t0 between two grains, δt is zero and is calculated as

The two steps in the calculation ensure that δt is in the contact plane. At the end of contact δt, is set to zero.

where qw is the contact point between grain and wall.

Substituting all of the above equations in Equation 3-9 and Equation 3-10, the normal and tangential components of the contact force between grain and wall are calculated.