## 4 Verification on the Mesoscale Equivalent Model
### Model Verification with Uniaxial Compression Tests  
#### 4.1.1 Numerical Model  
The typical mesoscale finite element model for prismatic specimens under uniaxial compression is shown in Figure 9. The specimen is placed on a fixed loading plate, and the load is applied to the specimen by imposing displacement on the moving loading plate. An automatic surface-to-surface contact algorithm considering the friction effect  
Figure 8: Free-body diagram of the infinitesimal segment of a fiber.  
is adopted to simulate the contact behavior between the loading plate and the specimen. The solid element SOLID164 is used for modeling the matrix of the specimen and loading plates. The steel fibers are modeled by beam element BEAM161. The keyword \"CONSTRAINED_BEAM_IN_SOLID (CBIS) [25] in LS-DYNA is adopted to simulate the bond-slip behavior between steel fibers and matrixes.  
For different amplification factors \(n\), the dimensions and number of steel fibers in the mesoscale equivalent model are different, and the effect of the amplification factor on the numerical needs to be investigated. In this study, numerical simulations on uniaxial compression with different amplification factors \(n\) (\(n=1\), \(3\), \(4\), \(5\), \(6\), \(7\), \(n=1\) means no equivalence treatment on steel fibers) are performed. Figure 10 shows the distribution of steel fibers in a specimen with different \(n\), where the number of steel fiber elements decreases fast with the increasing \(n\). The critical parameters of the CBIS algorithm for the mesoscale equivalent model with different \(n\) are listed in Table 2.  
Figure 10: Distribution of steel fibers in matrix with different \(n\): (**a**) \(n=1\), 146,913 beam elements; (**b**) \(n=3\), 5442 beam elements; (**c**) \(n=4\), 2296 beam elements; (**d**) \(n=5\), 1176 beam elements; (**e**) \(n=6\), 681 beam elements; (**f**) \(n=7\), 429 beam elements.  
Figure 9: Typical mesoscale FE model for uniaxial compression.  
Without adding steel fibers, the UHPC matrix could be regarded as high-strength concrete, modeled by the RHT model [26] in the present study. The critical RHT failure surface parameters \(A_{fail}\) and \(N_{fail}\) are determined by fitting the triaxial compression test data of 140 MPa HSC in Reference [27], as shown in Figure 11, where \(A_{fail}\) and \(N_{fail}\) are 1.78 and 0.35, respectively. The other critical parameters are self-defined according to Refs. [10; 28; 29]. Due to the stiffness of loading plates in practice being much larger than that of the specimen, the loading plates are modeled as a rigid body with the material model *MAT_RIGID. The isotropic and kinematic hardening material model *MAT_PLASTIC_KINEMATIC (*MAT_003) is chosen to build the steel fibers. Parameters of material models used in the uniaxial compression simulations are listed in Table 3.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \(d\) & \(l\) & \(G_{S}\) & \(S_{max}\) & \(EXP\) \\ \hline
0.2\(\cdot\)n mm & 13\(\cdot\)n mm & 2393\(\cdot\)n MPa & \(1.25\times 10^{-3}\) & 0.2 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Critical CBIS algorithm parameters under different \(n\).  
Figure 11: Determination of the parameters \(A_{fail}\) and \(N_{fail}\).