## 6 Conclusions  
In this work, a mesoscale equivalent model is first developed to numerically investigate the dynamic response of UHPC subjected to projectile impacts in a more refined and efficient way. Experiments on UHPC subjected to uniaxial compression and projectile impacts are conducted and are used to validate the developed model. Relying on the mesoscale equivalent model, the influence of projectile striking velocities, UHPC compression strengths, and volume percentages of steel fibers on the DOP is numerically investigated, and a modified Young equation for predicting the DOP is proposed. The main conclusions can be drawn as follows:  
1. The equivalent treatment on steel fibers is to amplify the size of the fibers and the interfacial shearing modulus between fibers and the matrix by \(n\) times synchronously. The interfacial shearing force is analytically proven to be equal to that before the equivalent treatment conducted on steel fibers, demonstrating that the equivalent treatment on steel fibers is theoretically reasonable. The mesoscale equivalent model for UHPC with steel fibers is successfully developed based on the equivalent treatment on fibers. When the amplification factor of steel fiber is lower than 5, the proposed model can accurately simulate the uniaxial compression behavior of UHPC specimens. However, when the amplification factor is greater than 5, the model cannot well characterize the ductility and toughness of UHPC.
2. When the amplification factor of steel fibers is lower than 5, the mesoscale equivalent model can accurately reproduce the failure mode and stress-strain curve of the UHPC specimens under the uniaxial compression. The computational cost of the numerical simulations of penetration experiments decreases rapidly with the increase of the amplification factor. With an amplification factor of 5, the maximum relative error between the numerical results of the cater diameter and penetration depth and experimental results is 11.7%, indicating that the mesoscale equivalent model has high accuracy.
3. The mesoscale equivalent model provides a more refined and in-depth perspective into numerically investigating the response of UHPC subjected to projectile impacts. The numerical investigation of the DOP shows that the DOP increases exponentially with the increase of the projectile striking velocity. The decreasing relationships between the DOP and the compression strength and volume percentage of steel fibers can be described by the power function and quadratic polynomial, respectively. Steel fibers exert a bridging effect by limiting the deformation of matrix elements to improve the penetration resistance of UHPC, but there is a limit to reducing the DOP by adding more steel fibers.
4. Based on the simulated data of the DOP, a modified Young equation is proposed for predicting the DOP of UHPC targets subjected to projectile impacts. The maximum relative error between the modified equation and experimental data is 13.9%, showing the proposed equation has high accuracy.  
Future studies will focus on establishing a mesoscale equivalent model for UHPC with fibers in different shapes and materials or numerically investigating the dynamic response of UHPC under blast loadings based on the mesoscale equivalent model.  
Conceptualization, J.A., J.Y., W.W. and Y.L.; methodology, J.A. and J.Y.; software, J.A.; validation, J.A., J.Y., W.W. and Y.L.; formal analysis, J.A., W.W. and Y.L.; investigation, J.A.; resources, J.Y. and W.W.; data curation, J.A.; writing--original draft preparation, J.A.; writing--review and editing, J.Y., W.W. and Y.L.; visualization, J.A.; supervision, J.Y. and W.W.; project administration, W.W.; funding acquisition, J.Y. and W.W. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.