## 4 Verification on the Mesoscale Equivalent Model
### Model Verification with Uniaxial Compression Tests
#### 4.1.2 Results and Discussion  
The numerical failure modes of specimens under different amplification factors \(n\) are shown in Figure 12. For the cases of \(n=1\), \(3\), \(4\), \(5\), the specimens exhibit ductile shearing failure modes, where a main oblique crack develops from the end of the specimen to the middle in the direction of about \(45\) degrees, which is the same as the typical experimental failure mode shown in Figure 12g. However, for the case of \(n=6\) and \(n=7\), the failure mode is quite different from the other cases and the experimental result, where the damage occurs firstly in the middle of the specimen and is approximately concentrated in the middle, which displays as brittle splitting failure mode to a certain content. Shearing failure is a type of ductile failure, while splitting failure belongs to brittle failure. Although the volume percentage of steel fibers is the same, the failure mode tends to change from ductile failure to brittle failure with the increase of the amplification factor. The difference between failure modes under different amplification factors indicates that steel fibers greatly influence the ductility of UHPC.  
Figure 13 shows the uniaxial compressive curves with different amplification factors \(n\) and the average full stress-strain curve obtained from uniaxial compression tests. There is little difference in the ascending branches of stress-strain curves obtained from the numerical simulations under different amplification factors, and all the ascending branches are in good agreement with the test data. As for the descending branches of the stress-strain curves, the numerical results of \(n=1\), \(3\), \(4\), and \(5\) are in good accord with the test data. Nevertheless, for the cases of \(n=6\) and \(n=7\), the numerical descending branches drop more sharply than the other cases and have much lower residual strength, which presents a great difference from the experimental result. If taking the area under the full stress-strain curve of uniaxial compression as an index to characterize the toughness of specimens, Figure 13 shows that the toughness decreases with the increase of the amplification factor.  
Figure 12: Failure mode of specimens: (**a**) \(n=1\); (**b**) \(n=3\); (**c**) \(n=4\); (**d**) \(n=5\); (**e**) \(n=6\); (**f**) \(n=7\); (**g**) \(n=8\).  
The variation of the toughness index caused by the change in amplification factor shows that steel fiber influences the toughness of UHPC significantly.  
Steel fibers improve the ductility and toughness of UHPC by the bridging effect in the matrix. According to the above comparison and discussion, the bridging effect gradually weakens with the increase in the amplification factor. In the present study, the mesoscale equivalent model can accurately simulate the uniaxial compression behavior of UHPC when the amplification factor is lower than 5. However, with further increases in the amplification factor, the mesoscale equivalent model loses its validity. This phenomenon may be because the equivalence of the resultant interfacial shearing force of steel fibers, as shown in Equation (11), is destroyed due to the excessive amplification in the fiber dimension. The equivalent treatment on fibers is based on the assumption that there are enough original fibers in any direction to synthesize a certain number of amplified fibers. When the amplification factor is too large, such as \(n=6\) or \(n=7\), the number of original fibers in some directions may not be enough to synthesize sufficient amplified fibers, leading to the uneven distribution of amplified fibers and causing Equation (11) to fail. Therefore, to ensure the accuracy of the simulation, it is recommended that the amplification factor should not be greater than 5.