Tangential Interaction Force of Two Particles

Available identifiers

force component acting tangentially at the contact interface between two particles

Defining Formula: $\begin{matrix} 𝑭_{i j}^{T, p r e} & = & - k_{i j}^{T} 𝝃_{i j} - d_{i j}^{T} {\dot{𝝃}}_{i j} \\ 𝐢𝐟 ‖ 𝑭_{i j}^{T, c o n s} ‖ & \leq & μ ‖ 𝑭_{i j}^{N} ‖ 𝐭𝐡𝐞𝐧 𝑭_{i j}^{T} = 𝑭_{i j}^{T, p r e}, \\ 𝐞𝐥𝐬𝐞 𝑭_{i j}^{T} & = & μ ‖ 𝑭_{i j}^{N} ‖ 𝒕, \\ 𝐰𝐡𝐞𝐫𝐞: \\ 𝑭_{i j}^{T, c o n s} & = & - k_{i j}^{T} 𝝃_{i j}, \\ 𝝃_{i j} & = & 𝒙_{C_{j i}} - 𝒙_{C_{i j}}, \\ {\dot{𝝃}}_{i j} & = & ⟨ 𝒗_{i} - 𝒗_{j}, 𝒕 ⟩ 𝒕, \\ 𝒕 & = & \frac{𝝃_{i j}}{‖ 𝝃_{i j} ‖}, \\ k_{i j}^{T} & = & k_{i j}^{N} / 1.2, \\ k_{i j}^{N} & = & \frac{π}{2} \frac{r_{i} + r_{j}}{2} E_{N}, \\ d_{i j}^{T} & = & 2 D_{T} \sqrt{k_{i j}^{N} \frac{m_{i} m_{j}}{m_{i} + m_{j}}} . \end{matrix}$
$𝑭_{i j}^{T} \in ℝ^{3}$ symbol represents Tangential Interaction Force of Two Particles
$𝒕$ symbol represents Unit Tangent Vector
$𝒗_{i} \in ℝ^{3}$ symbol represents Particle Velocity
$𝒙_{C_{i j}} \in ℝ^{3}$ symbol represents Contact Point Of Particles
$D_{T}$ symbol represents Damping Coefficient
$E_{N}$ symbol represents Young Modulus
$k_{i j}^{T}$ symbol represents Tangential Stiffness
$k_{i j}^{N}$ symbol represents Normal Stiffness
$m_{i}$ symbol represents Particle Mass
$r_{i}$ symbol represents Particle Radius