## 5 Summary and Conclusions  
Four different types of asphalt concretes (AC-13, SMA-13, AC-20, and AM-20) were used for density, dynamic modulus, and ultrasonic tests. The dynamic moduli of these AC samples were caculated based on ultrasonic wave velocity, density, and Poisson's ratio. A two-dimensional numerical modeling of ultrasonic wave propagation through asphalt conretes was performed using commercial software package ABAQUS and comparisons were made between simulated and measured results of wave velocity. The influence of varying dynamic moduli and density on wave propagation was also analyzed by changing the values of the dynamic modulus and the density alternatively. The influence of varying location and sizes of cracks on ultrasonic wave propagation was also analzed. Based on the analysis, the following conclusions can be drawn:  
* Simulated wave velocities under caculated dynamic moduli were very close to experimentally measured wave velocities., with relative errors ranging from 0.57% to 9%. However, there was a huge difference between the simulated results of the wave velocities under measured dynamic moduli and experimentally measured wave velocities, with relative errors ranging from 21% to 31%. This was due to the fact that real AC samples exhibited heterogeneity, while the models used in the simulation process were homogenous in nature. Second, the values of the measured dynamic moduli were lower in comparison with the calculated dynamic moduli of asphalt concretes.
* A strong relationship between the dynamic moduli and the simulated ultrasonic wave velocities was observed; increased dynamic modulus lead to increased ultrasonic wave velocity. The simulated wave velocity decreased with the increase in density, but this decrease was not large and wave velocity tended to become stable at some point. This shows that density has less influence on ultrasonic wave propagation as compared with dynamic modulus.
* The crack location and size were significantly affected by the ultrasonic wave propagation mode. Models with the same crack size but different locations showed different displacment in comparison with each other. Similarly, models with the same location but different crack sizes exhibited a decrease in displacement with the increase of crack size.  
Figure 10: (**a**) Time-displacement curves with a crack size of 30 mm at distances of 1/3, 1/2, and 2/3 and 40 mm crack size at a distance of 2/3 from the pavement surface. (**b**) Comparison of displacements with a crack size of 30 mm and of 40 mm at distances 1/3, 1/2, and 2/3 from the pavement surface.  
**Author Contributions:** The authors confirm contribution to the paper as follows: Study conception and design: W.Z., S.H, and M.A.A.; data collection: J.B. and D.Z.; analysis and interpretation of results: W.G. and Q.L.; draft manuscript preparation: M.A.A. and W.G. All authors reviewed the results and approved the final version of the manuscript.  
**Funding:** This research was funded by the National Science Foundation of China grant number 51608266, the Research Fund Mechanism of Simultaneous Deformation of Earth-Subgrade-Pavement Intergraded Design Method with grant number 8521002546, and the research funded by the Qilu Transportation Group grant number 2018851.  
**Conflicts of Interest:** The authors declare no conflicts of interest.