## 2 Lattice Discrete Particle Model
### Fiber Modeling and Crack-Bridging Mechanism
The Cook-Gordon mechanism can simulate the premature debonding of fibers in front of an advancing crack tip, considering an additional slip component that affects the crack opening. These interactions can be represented as:
* Spalling length. \[s_{f}\ =\ \frac{P_{f}Nsin\left(\frac{\theta}{2}\right)}{k_{sp}\sigma_{t}d_{f} cos^{2}\left(\frac{\theta}{2}\right)}\] (3) where \(P_{f\text{N}}\) denotes the normal part of the bridging force, \(\theta\) denotes the angle between the embedded fiber and the normal fiber orientation of the facet, \(\sigma_{t}\) denotes the tensile strength of the matrix, and \(k_{sp}\) denotes an empirical coefficient that can be calibrated. * Snubbing effect adjustment. \[P_{f}=\exp\left(k_{sn}\phi_{f}^{\prime}\right)P(v)\] (4) where \(\phi_{f}^{\prime}\) is the angle between the fiber in the matrix and deflected fiber after spalling for the same fiber, and \(k_{sn}\) is a coefficient to be calibrated. * Cook-Gordon additional slip. \[v_{cg}=\frac{4k_{cg}p_{f}}{\pi d_{f}^{2}E_{f}}\] (5) where \(k_{cg}\) denotes the Cook-Gordon coefficient of the premature slip due to initiated cracks. * Fiber rupture condition. The fiber load must not exceed the rupture strength, with a demand that: \[\sigma_{f}=\frac{4p_{f}}{\pi d_{f}^{2}}\leq\sigma_{u,f}\exp(-k_{rup}\phi_{f}^{ \prime})\] (6) where \(k_{rup}\) denotes the material parameter and is the ultimate tensile strength of the fiber. When this condition fails, the fiber load is set to zero. * Crack-bridging force calculation. For an embedded fiber at the two sides of a crack opening \(\omega\), the crack-bridging force is expressed as: \[P_{f}=P_{f}n_{f}^{\prime}\] (7) where the bridging segment satisfies: \[\left\|W^{\prime}\right\|=2s_{f}+v_{s}+v_{l}\] (8) where \(v_{s}\) and \(v_{l}\) are the slip components of the fiber's embedment lengths. * Pull-out resistance condition. The resisting pull-out forces on the two sides must be equal:  
\[P_{f}=P(v_{s})\text{exp}\Big{(}k_{sn}\phi_{f}^{\prime}\Big{)}=P(vl)\text{exp} \Big{(}k_{sn}\phi_{f}^{\prime}\Big{)} \tag{9}\]  
An iterative solution procedure is needed due to the nonlinearity in the computation of the fiber bridging force caused by the spalling length and slip response's dependence on the evolving crack opening. The procedure begins with a starting approximation of the fiber force, p\({}_{f}\), from the previously computed value. From this estimate, the spalling length sf and effective embedment lengths are calculated, which define the bridging segment's geometry. Compatibility between the slip value and the crack opening is enforced by solving for relative slip components, \(v_{s}\) and \(v_{l}\), with [PERSON]'s method. This numerical approach iteratively adjusts the slip values to satisfy equilibrium conditions between the pull-out resisting forces on both sides of the crack. Finally, the computed fiber bridging force is validated using the fiber rupture criterion to ensure structural safety once the convergence occurs. Where the calculated force is greater than the ultimate tensile strength of the fiber, the adjustment is made accordingly to prevent overestimation of the bridging capacity. These micro-scale interactions collectively control the macro-scale toughness and ductility of fiber-reinforced concrete. Fibers, especially steel and synthetic macro-fibers, enhance the ability of the concrete to withstand energy during cracking by restraining crack propagation and increasing the work needed for total separation. The ensuing multi-cracking, as opposed to the single dominant crack in plain concrete, enhances the post-cracking load-carrying capacity and the structural response under tensile and flexural loading. This energy absorption capacity is even enhanced under cyclic loading, with fibers bridging opening and closing cracks, leading to typical hysteresis loops. The area of the hysteresis loop is proportional to the energy dissipation capacity of the material, a basic property for seismic uses.