## 3 Mesoscale Equivalent Model for UHPC
### Equivalent Treatment on Steel Fibers  
Since the basic reason for the ultra-high computational cost of the original mesoscale model is the huge number of steel fibers, the key to establishing a mesoscale equivalent model is to reduce the number through the equivalent treatment of steel fibers. Steel fibers play a bridging role in UHPC, which limits the deformation and crack development in the matrix. This kind of bridging effect is reflected by the interaction between steel fibers and matrixes; the essence is the transmission of the interfacial shear force. Therefore, the bridging effect can be equivalent as long as the interfacial shear force can be guaranteed to be unchanged when the equivalent treatment on steel fibers is performed [23; 24]. The specific implementation procedure of the equivalent treatment on steel fibers is as follows.  
The first step is to amplify steel fibers geometrically under the condition that the aspect ratio of fibers remains unchanged and replace the original fibers with amplified fibers.  
\[d_{af}=nd_{f},\qquad L_{af}=nL_{f} \tag{3}\]  
where \(d_{af}\) and \(L_{af}\) are the diameter and length of the amplified fiber, respectively, and \(n\) is the geometric amplification factor. According to the premise that the volume percentage of steel fibers \(V_{f}\) remains unchanged, the number of fibers \(N_{af}\) after equivalent treatment is determined as:  
\[N_{af}=\frac{V_{f}V}{\pi{(d_{af})}^{2}L_{af}}=\frac{V_{f}V}{\pi{(nd_{f})}^{2}( nL_{f})}=\frac{1}{n^{3}}N \tag{4}\]  
At the same time, due to the fibers being uniformly and randomly distributed, the number of fibers in any direction is the same, where the number is labeled as \(N^{\kappa}\) for original fibers and \(N^{\kappa}_{af}\) for amplified fibers \(N^{\kappa}_{af}=N^{\kappa}/n^{3}\).  
For a single original steel fiber embedded in the matrix, as shown in Figure 8, an increment of the axial force in an infinitesimal segment of fiber is:  
\[dP=\tau_{0}\cdot\tau d_{f}\cdot dl \tag{5}\]  
where \(\tau_{0}\) is the actual interfacial shear stress, the interfacial shear force, i.e., the axial force of the fiber, can be computed as:Based on the equivalence of interfacial shear stress distribution [22], Equation (6) is simplified to:  
\[F=\ \ \tau\cdot\pi d_{f}\cdot L_{e} \tag{7}\]  
Similarly, the interfacial shearing force of a single amplified fiber is:  
\[F_{af}=\ \ \tau\cdot\pi(nd_{f})\cdot nL_{e}=n^{2}F \tag{8}\]  
Then the resultant interfacial shearing force of the original fibers in a direction can be computed as:  
\[F^{\kappa}=\ \ N^{\kappa}F=N^{\kappa}\tau\cdot\pi d_{f}\cdot L_{e} \tag{9}\]  
and the resultant force of amplified fibers in the same direction is:  
\[F^{\kappa^{\prime}}_{af}=N^{\kappa}_{af}\tau\cdot\pi(nd_{f})\cdot nL_{e}=\frac {1}{n}F^{\kappa} \tag{10}\]  
However, as Equation (10) shows, \(F^{\kappa^{\prime}}_{af}\) is not equal \(F^{\kappa}\). Therefore, the second step is to modify the interfacial shear modulus \(G_{\mathrm{s}}\) to be \(G_{sat}=nG_{\mathrm{s}}\) and then the final resultant interfacial shearing force of amplified fibers is:  
\[F^{\kappa}_{af}=N^{\kappa}_{af}(n\tau)\cdot\pi(nd_{f})\cdot nL_{e}=F^{\kappa} \tag{11}\]  
Equation (11) indicates that the interfacial shearing force of equivalent steel fibers and original steel fibers are ensured to be equal, which means the bridging effect of steel fibers is unchanged and then demonstrates that the equivalent treatment on steel fibers is theoretically reasonable. Ultimately, the mesoscale equivalent model for UHPC with steel fibers is developed based on the equivalent treatment of fibers.