# The Hertz-Mindlin (no slip) Model

Hertz-Mindlin (no slip) is the default model used in EDEM due to its accurate and efficient force calculation.

In this model, the normal force component is based on the Hertzian contact theory (Hertz 1882). The tangential force model is based on the Mindlin-Deresiewicz work (Mindlin 1949) (Mindlin and Deresiewicz 1953). Both normal and tangential forces have damping components, where the damping coefficient is related to the coefficient of restitution as described in (Tsuji, Tanaka and Ishida 1992). The tangential friction force follows the Coulomb law of friction model, such as (Cundall and Strack 1979). The rolling friction is implemented as the contact independent directional constant torque model, such as (Sakaguchi, Ozaki and Igarashi 1993).

In particular, the normal force, F_{n}, is a function of normal overlap δ_{n} and is defined as:

The equivalent Young’s Modulus E*, the equivalent radius R* are defined as:

with E^{i}, v^{i}, R^{i}, and E^{j}, v^{j}, R^{j}, being the Young’s Modulus, Poisson ratio, and Radius of each sphere in contact. Additionally, there is a damping force, F_{n}^{d}, given by:

is the equivalent mass, v_{n}^{rel} is the normal component of the relative velocity, and β and S_{n }(the normal stiffness) is defined as:

With e the coefficient of restitution, the tangential force, F_{t}, depends on the tangential overlap δ_{t} and the tangential stiffness S_{t}.

with

Here, G* is the equivalent Shear modulus. Additionally, tangential damping is defined as:

where v_{t}rel is the relative tangential velocity. The tangential force is limited by Coulomb friction μ_{s}F_{n,} where μ_{s }is the coefficient of static friction.

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