# Assessing Voided Reinforced Concrete by Numerical Modelling of Impact-Generated Rayleigh Waves
## 4 Results of Simulations and Discussion
### Correlation with Velocity Index  
The velocity index, \(I_{\rm V}\) is plotted against the density of voids in the concrete model, as shown in Figure 11, using the proposed formula given below. \[I_{V}=\frac{V_{Defected}}{V_{Sound}} \tag{9}\]  
where \(V_{\rm Defected}\) and \(V_{\rm Sound}\) are the velocities of R-waves in the voided (defect) and unvoided (sound) models, respectively. Both \(V_{\rm Defected}\) and \(V_{\rm Sound}\) were obtained by calculating the gradient value of the linear regression for wave propagating distance versus propagation time plots, which is obtained by extracting the data acquired from each of the sensors for all the cases modelled and simulated. In general, the velocity index is found to decrease with void density, giving a good linear correlation in all cases covered in this study. For the cases of 5 kHz excitation frequency, the change in velocity index against void density is found to be more significant than that of the 50 kHz cases. This is most likely due to the longer wavelength given by the 5 kHz excitations, resulting in higher penetration depths of the R-waves and the waves being more easily distorted by voids. In general, from the analysis of the numerical results, it can be confirmed that defects in concrete in the form of internal voids increase R-wave propagation time because of the resulting distortion. The analysis also revealed that waves excited with a dominant frequency of 50 kHz do not seem to be highly sensitive to the change in void density; this is evidenced by the velocity index results for all three void location cases, which barely change against void density, as opposed to those from the 5 kHz and 25 kHz cases, which show much more significant drops. Whilst it is widely agreed that the sensitivity of elastic waves against defects increases with their frequency, it is also commonly known that the R-wave penetration depth in any material is governed by wavelength in a positive correlation. This means that the penetration depth of R-waves normally decreases with an increase in their frequency. As confirmed from the simulation and analysis in this study, R-waves generated with a dominant frequency of 50 kHz could not propagate to the depth where the waves would be significantly distorted by a void. Additionally, it is known that R-wave amplitude decays exponentially with depth ([PERSON], 2003) [31]. At the depth of the void block in the models studied herein, the simulated R-waves did not seem to possess sufficient energy to \"interact\" with the voids, and the sensors still detected them at the concrete surface. Figure 10: CWT plots for wave data with dominant excitation frequency of 5 kHz for **(a)** 10% and **(b)** 100% voided cases acquired at Sensor 4; and **(c)** 10% and **(d)** 100% voided cases acquired at Sensor 7. As R-waves propagate in the concrete model, wave diffraction occurs as the waves reach the voided area. Energy is dissipated due to wave scattering and absorption by the medium, as visualised in the simulation exemplified in Figure 12. Reduced wave energy is apparent due to distortion generally, which also causes a delay in wave travel time. The key observations obtained from simulations, supported by the data analysis, confirm the contribution of the voided area to the reduction in R-wave velocity. The data analysis also suggests that the R-wave velocity index is a good indicator for estimating void density. The analysis suggests a promising potential for the velocity index to serve as an effective estimator for voided areas; however, in order to fully validate this assertion, it becomes necessary to delve deeper into understanding the various mechanisms contributing to the reduction in the R-wave velocity index. This would entail scrutinising potential correlations beyond linear relationships and exploring the influence of various environmental and structural factors on the velocity index. Such investigations are the basis of our next stage of study and are beyond the scope of the current paper.