Solid Bond Model: Kelvin Model

Bond Properties

L_b = |O_2 - O_1|

L_{init} = L_b(t=0)

A_b = \pi \cdot R_b^2

I = \frac{\pi \cdot R_b^4}{4}

J = \frac{\pi \cdot R_b^4}{2}

Material Properties

G = \frac{E}{2(1+\nu)}

Contact Vector

\vec{r}_c = O_2 - O_1

\vec{r}_n = \frac{O_2 - O_1}{|O_2 - O_1|}

Velocities

\vec{v}_{rel} = \vec{v}_2 - \vec{v}_1 - \frac{\vec{\omega}_1 + \vec{\omega}_2}{2} \times \vec{r_c}

\vec{\omega}_{rel} = \vec{\omega}_1 - \vec{\omega}_2

\vec{v}_{rel,n} = \vec{r}_n (\vec{r}_n \cdot \vec{v}_{rel})

\vec{\omega}_{rel,n} = \vec{r}_n (\vec{r}_n \cdot \vec{\omega}_{rel})

\vec{v}_{rel,t} = \vec{v}_{rel} - \vec{v}_{rel,n}

\vec{\omega}_{rel,t} = \vec{\omega}_{rel} - \vec{\omega}_{rel,n}

\Delta\vec{\delta}_t = \vec{v}_{rel,t} \cdot \Delta t

\Delta\vec{\delta}_{\omega n,b} = \vec{\omega}_{rel,n} \cdot \Delta t

\Delta\vec{\delta}_{\omega t,b} = \vec{\omega}_{rel,t} \cdot \Delta t

Force and Moment in Normal Direction

\vec{F}_{n,b} = \vec{r}_n \times \frac{(L_b - L_{init})}{L_{init}} \cdot E \cdot A_b

\vec{F}_{n,diss} = \vec{v}_{rel,n} \cdot \frac{(L_b - L_{init})}{L_{init}} \cdot E \cdot \eta \cdot A_b

If MAX_DAMP_RATIO ($R$) is set and $(|\vec{F}_{n,diss}| > R \cdot |\vec{F}_{n,b}|)$, the dissipative force is capped by $ R \cdot |\vec{F}_{n,b}|$ as:

\vec{F}_{n,diss} = \frac{\vec{F}_{n,diss}}{|\vec{F}_{n,diss}|} \cdot R \cdot |\vec{F}_{n,b}|

\vec{M}_{n,b} = T \cdot \vec{M}_{n,b} - \Delta\delta_{\omega n,b} \cdot \frac{J}{L_{init}} \cdot G

Force and Moment in Shear Direction

\vec{F}_{t,b} = T \cdot \vec{F}_{t,b} - \frac{\Delta\vec{\delta_t}}{L_{init}} \cdot G \cdot A_b

\vec{F}_{t,diss} = T \cdot \vec{F}_{t,diss} -\vec{v}_{rel,t} \cdot \frac{\Delta\vec{\delta_t}}{L_{init}} \cdot G \cdot \eta \cdot A_b

If MAX_DAMP_RATIO ($R$) is set and $(|\vec{F}_{t,diss}| > R \cdot |\vec{F}_{t,b}|)$, the dissipative force is capped by $ R \cdot |\vec{F}_{t,b}|$ as:

\vec{F}_{t,diss} = \frac{\vec{F}_{t,diss}}{|\vec{F}_{t,diss}|} \cdot R \cdot |\vec{F}_{t,b}|

\vec{M}_{t,b} = T \cdot \vec{M}_{t,b} - \Delta\delta_{\omega t,b} \cdot \frac{I}{L_{init}} \cdot E

Total Force

\vec{F}_{n,tot} = \vec{F}_{n,b} - \vec{F}_{n,diss}

\vec{F}_{t,tot} = \vec{F}_{t,b} - \vec{F}_{t,diss}

Breakage Criteria

\frac{{F}_{n,tot}}{A_b} + \frac{{M}_{t,b} R_b}{I} \ge \sigma_{max}

\frac{{F}_{t,tot}}{A_b} + \frac{{M}_{n,b} R_b}{J} \ge \tau_{max}

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Model Parameters

• CONSIDER_BREAKAGE - Allow (1) or prohibit (0) the breakage of bonds.
• MAX_DAMP_RATIO - The parameter limits the magnitude of the damping (viscous) force relative to the elastic (spring) force. This prevents the damping force from becoming unrealistically large, which could destabilize the simulation or produce nonphysical results. It ensures that the damping remains a correction, not the dominant force. For example, a value of 0.5 means the damping force will be capped at 50% of the elastic force magnitude. Set to 0 to disable capping.

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Symbols

Symbol	Description
$A_b$	Cross-sectional area of the bond $[m^2]$
$E$	Young's modulus $[Pa]$
$F_{n}, F_{t}$	Force in normal and tangential direction $[N]$
$G$	Shear modulus $[Pa]$
$I, J$	Moments of inertia $[m^4]$
$L_b$	Current bond length $[m]$
$L_{init}$	Initial bond length $[m]$
$M_{n,b}, M_{t,b}$	Normal and tangential moment of the bond $[N \cdot m]$
$O_1, O_2$	Centers of contact partners $[m]$
$r_1, r_2$	Particle radii $[m]$
$r_b$	Bond radius $[m]$
$r_c$	Contact vector $[-]$
$r_n$	Normal vector $[-]$
$T$	Transformation matrix (rotation to global system) $[-]$
${v}_{rel,n}, {v}_{rel,t}$	Relative velocity of particles in normal and tangential direction $[m/s]$
$\delta_t$	Tangential overlap $[m]$
$\Delta\delta_t$	Tangential displacement in current step $[m]$
$\Delta\delta_{\omega n,b}, \Delta\delta_{\omega t,b}$	Increment of displacement (in current step) between particles in contact point due to the rotational velocities $[m]$
$\eta$	Dynamic viscosity $[Pa \cdot s]$
$\nu$	Poisson ratio $[-]$
$\sigma$	Normal strength $[Pa]$
$\tau$	Tangential strength $[Pa]$
$\omega_1, \omega_2$	Rotational velocities of particles $[rad/s]$