## 2 Lattice Discrete Particle Model
### Fiber Modeling and Crack-Bridging Mechanism  
Concrete requires reinforcement for the purposes of enhancing its tensile capacity, as well as ductility since it is brittle under tensile loading. Fiber-reinforced concrete (FRC) has a complex interrelation between fibers and the confining concrete matrix at the mesoscale that controls it. The tensile brittleness of concrete generates the early initiation and propagation of cracks, while fibers dispersed within the matrix constrain the cracking phenomenon through a crack-bridging mechanism. The influence of fibers on the concrete behavior can be modeled in the Lattice Discrete Particle Model with Fibers (LDPM-F) that extends the parent LDPM with discrete, randomly distributed, and randomly oriented fibers in the concrete mesostructure [21]. Once a crack is formed, bridging fibers restrain its growth by creating a bridging force \(P_{f}\), which is a function of the slip of the fiber concerning the crack opening--the slip is denoted as \(v\). Fibers carry the load by bridging the crack when a crack is formed. The crack-bridging force \(P_{f}\) of a fiber provided for LDPM is a function of its slippage \(v\), which describes the relative displacement between the fiber and the matrix at the crack interface. The bridging force can be divided into the debonding phase and the pull-out phase, each of which is defined by some governing equations. In the debonding phase (\(v<v_{d}\)), the fiber-matrix bond is progressively lost with the opening of the crack, and the fiber begins to slip. The semi-empirical expression gives the bridging force in this context:  
\[P(v)=\left[\frac{\pi^{2}E_{f}d_{f}^{3}}{2}(\tau_{0}v+G_{d})\right] \tag{1}\]  
where E\({}_{f}\) denotes the elastic modulus of the fiber, \(d_{f}\) denotes the fiber diameter, \(\tau_{0}\) denotes the bond stress, \(G_{d}\) denotes the debonding fracture energy, and \(v_{d}\) denotes the critical slippage at which complete debonding occurs. This equation captures the increase in load resistance as the slip progresses, up to the point where complete debonding occurs. Following debonding, the pull-out stage (\(v>v_{d}\)) initiates, where frictional resistance and mechanical interlocking (e.g., hooked or waved fibers) govern the residual load-carrying capacity. This stage is defined by the relationship:  
\[P(v)=p_{0}\bigg{(}1-\frac{v-v_{d}}{L_{e}}\bigg{)}\bigg{(}1+\frac{\beta(v-v_{d })}{d_{f}}\bigg{)} \tag{2}\]  
where \(P_{0}=\pi L_{e}d_{f}\tau_{0}\) denotes the maximum pull-out force, \(L_{e}\) denotes the embedment length of the fiber within the matrix, and \(\beta\) denotes a dimensionless parameter that controls the slip-hardening (\(\beta>0\)), slip-softening (\(\beta<0\)), or neutral (\(\beta=0\)) behavior during the pull-out process. As the fiber continues to slip out of the matrix, the bridging force gradually decreases until the fiber is fully extracted, unless fiber rupture occurs before complete pull-out. Figure 1 illustrates the relationship between load and slippage for different values of \(\beta\). Figure 1: Load–slippage relationship for different values of \(\beta\), demonstrating the influence of \(\beta\) on the mechanical response [22]. In addition, other micromechanical effects govern the fiber-matrix interaction during pull-out, which are accounted for in LDPM-F. These are spalling at the crack interface, snubbing effects, and the Cook-Gordon mechanism. Spalling refers to the local debonding and damage of the matrix in front of the fiber exit point, which decreases the effective embedment length. This effect is modeled by reducing the embedment length as a function of the matrix's bridging force and tensile strength. The snubbing effect is when fibers are at an angle to the crack plane, resulting in additional resistance due to the bending of the fiber around the matrix corner. An exponential increase factor typically models it as a function of the fiber angle and snubbing coefficient.