## 4 Results and Discussion  
### Amplitude Analysis  
Figure 7 illustrates the amplitudes of received signals with different frequencies (60 kHz and 100 kHz). It is noticeable that the stage stress has a considerable influence on the amplitude of sensing signals for both frequencies. The amplitudes change non-monotonically with incremental stress. The amplitudes of response signals first decrease sharply for early steps of load, and then vary irregularly. It can be seen that a critical point occurred at a stress level of about \(0.3\sigma_{\mathrm{max}}\), where a sudden drop occurs to the amplitudes. An explanation is that apparent damage occurs after \(0.3\sigma_{\mathrm{max}}\). On the whole, the development of amplitudes of the stress wave versus incremental load showed a downward trend. According to previous research of Gu [9] and Equation (3) [9], provided that concrete is homogeneous material; the amplitude would decrease monotonously with the development of concrete strength and elastic modulus in unloaded state. This can be verified by Figure 8, which illustrates the elastic modulus development versus stress. It should be noted that elastic modulus after 18 MPa was not obtainable, since that plastic characteristic started to appear. Further, the elastic modulus values are only approximate values, which was obtained by non-standard test method. Noteworthy is the fact that concrete is a kind of heterogeneous material, especially in the hydration  
Figure 5: Response signal.  
Figure 6: Typical pulse drive (CH2) and sensing (CH1) signals [22].  
process. From the monitoring results, it can be seen that the effect of load on micro-crack propagation is quite evident in the early age of concrete. The amplitude of the harmonic stress wave can be expressed as  
\[A=\frac{1}{\omega}(\frac{4I^{2}}{E\rho})^{\frac{1}{4}} \tag{3}\]  
where \(A\), \(E\), \(\rho\), \(w\), and \(I\) are the harmonic amplitude of stress waves, elastic modulus, density of concrete, the average power of the harmonic response over a period, and angular frequency, respectively.  
The intrinsic characteristics of concrete can account for this phenomenon. From the microcosmic view, concrete is a kind of nonhomogeneous material, normally comprised of mortar, aggregate, and interfacial transition zone (ITZ). While with the increase of the stress, the micro-cracks in ITZ began to develop, leading to the drop of the amplitude. It should be pointed out that concrete is a heterogeneous material, and there exist initial defects in different concrete specimens, which also bringing in some development difference among wave amplitudes. Additionally, it is difficult to make sure the specimens completely axially compressed. As concrete is a brittle material, if some part of the specimen is in tension, the test results may deviate. Last but not least, when the measurement was performed, it can happen that a random noise signal interferes with the target signals. To conclude, although the amplitude results fluctuate locally, the general trend is consistent. For instance, with regard to C1, the deviation of the amplitude in 100 kHz can happen. Furthermore, for the health  
Figure 8: Elastic modulus development versus applied stress.  
Figure 7: Amplitude development with different stress. (**a**) C1; (**b**) C2.  
monitoring of practical engineering structures, the variation of amplitudes may result from change of the bearing load rather than damage in concrete.