Hertz-Mindlin (no slip) Model

The Hertz-Mindlin (no slip) contact model 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 work of Mindlin-Deresiewicz (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 defined as:

F_{n} = \frac{4}{3} E^{*} \sqrt{R^{*}} δ_{n}_{\frac{3}{2}}^{_{\frac{3}{2}}}

$F_{n} = \frac{4}{3} E^{*} \sqrt{R^{*}} δ_{n}^{_{\frac{3}{2}}}$

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

\frac{1}{E^{*}} = \frac{(1 - v_{i}^{2})}{E_{i}} + \frac{(1 - v_{j}^{2})}{E_{j}}

$\frac{1}{E^{*}} = \frac{(1 - v_{i}^{2})}{E_{i}} + \frac{(1 - v_{j}^{2})}{E_{j}}$

\frac{1}{R} = \frac{1}{R_{i}} + \frac{1}{R_{j}}

$\frac{1}{R} = \frac{1}{R_{i}} + \frac{1}{R_{j}}$

With Eⁱ, vⁱ, Rⁱ, and E^j, v^j, R^j, being the Young’s Modulus, Poisson's ratio, and radius of each sphere in contact. Additionally, there is a damping force, F_n^d, defined as:

m^{*} = {(\frac{1}{m_{j}} + \frac{1}{m_{i}})}^{_{^{^{- 1}}}^{_{^{^{- 1}}}^{^{- 1}}}}

$m^{*} = {(\frac{1}{m_{j}} + \frac{1}{m_{i}})}^{^{_{^{^{- 1}}}}}$

Where m^* is the equivalent mass, v_n^rel is the normal component of the relative velocity, and β and S_n (the normal stiffness) defined as:

β = \frac{- ln e}{\sqrt{{ln}^{2} e + π^{2}}}

$β = \frac{- \ln e}{\sqrt{\ln^{2} e + π^{2}}}$

S_{n} = 2 E^{*} \sqrt{R^{*} δ_{n}}

$S_{n} = 2 E^{*} \sqrt{R^{*} δ_{n}}$

With the Coefficient of Restitution e, the tangential force, F_t, depends on the tangential overlap δ_t and the tangential stiffness S_t.

F_{t} = - S_{t} δ_{t}

$F_{t} = - S_{t} δ_{t}$

with

S_{t} = 8 G^{*} \sqrt{(R^{*} δ_{n})}

$S_{t} = 8 G^{*} \sqrt{(R^{*} δ_{n})}$

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

$F_{t}^{d} = - 2 \sqrt{\frac{5}{6}} β \sqrt{S_{t} m^{*}} v_{t}^{\begin{array}{l} \to \\ r e l \end{array}}$

Where v_t^rel is the relative tangential velocity. The tangential force is limited by the Coulomb friction μ_sF_n, where μ_s is the Coefficient of Static Friction.