Normal Interaction Force of Two Particles

Available identifiers

force component acting normally at the contact interface between two particles

Defining Formula: $\begin{matrix} 𝑭_{i j}^{N} & = (k_{i j}^{N} δ_{i j} + d_{i j}^{N} {\dot{δ}}_{i j}) 𝒏_{i j} \\ δ_{i j} & = ⟨ 𝒙_{i} - 𝒙_{j}, 𝒏_{i j} ⟩ \\ {\dot{δ}}_{i j} & = ⟨ 𝒗_{i} - 𝒗_{j}, 𝒏_{i j} ⟩ \\ r_{i j} & = (r_{i} + r_{j}) / 2 \\ k_{i j}^{N} & = E_{N} π r_{i j} / 2 \\ d_{i j}^{N} & = D_{N} 2 \sqrt{k_{i j}^{N} m_{i j}} \\ m_{i j} & = \frac{m_{i} m_{j}}{m_{i} + m_{j}} \\ 𝒏_{i j} & = \frac{𝒙_{i} - 𝒙_{j}}{‖ 𝒙_{i} - 𝒙_{j} ‖} \end{matrix}$
$D_{N}$ symbol represents Damping Coefficient
$E_{N}$ symbol represents Young Modulus
$𝑭_{i j}^{N}$ symbol represents Interaction Force
$𝒗_{i} \in ℝ^{3}$ symbol represents Particle Velocity
$r_{i j}$ symbol represents Particle Radius
$𝒏_{i j} \in ℝ^{3}$ symbol represents Unit Normal Vector
$𝒙_{i} \in ℝ^{3}$ symbol represents Particle Position
$k_{i j}^{N}$ symbol represents Normal Stiffness
$m_{i}$ symbol represents Particle Mass