# Correlation between Coda Wave and Stresses in Uni-Axial Compression Concrete
## 4 Discussion  
From the perspective of microcrack deformation, this paper proposed a model to simulate the relation between the sound velocity and stress in concrete, and the relevant parameters in the model  
Figure 11: Comparison between the proposed model and experimental data of the axial sound velocity ratio. Figure 12: Comparison between the proposed model and experimental data of the horizontal sound velocity ratio. were fitted using the CWI analysis method. Finally, the specific expression of the relation between the sound velocity and uni-axial stress of concrete was obtained. In practical engineering, to obtain the stress \(\sigma\) or stress increment \(\Delta\sigma\) in a concrete structure, the following steps are required:  
1. Conduction of field test or laboratory experiment on unloaded concrete to obtain the initial waveform of the coda wave;
2. Calibration or regression of the parameters in Equation (22);
3. Conduction of field test on loaded concrete structures to obtain the waveform of the coda wave;
4. Analysis of the obtained coda wave by the wave expansion method to determine the relative change between the sound velocity \(\Delta V/V_{p0}\);
5. Determine the stress \(\sigma\) (or stress increment \(\Delta\sigma\)) using Equation (22) with calibrated parameters. Conventional ultrasonic testing methods usually use the duration of direct wave propagation in concrete to calculate the wave velocity; however, this approach has a low accuracy due to two main reasons: (1) the measurement of the duration of direct waves in concrete is easily affected by the timing error of the instrument; and (2) it is difficult to overcome the obstacles caused by the multiple components and inhomogeneity of concrete, and the energy will attenuate rapidly. The coda wave formed by multiple scattering can overcome these defects, and the accuracy of the calculation is much higher than that of the traditional methods. CWI technology calculates the wave velocity through the processing of the whole waveform along the time axis, which can avoid the error of point measurements and achieve high-precision and more stable results. At the same time, in complex media, the coda wave is better developed than that in homogeneous materials. Therefore, it is more suitable for the detection of structural changes in concrete. In this paper, the sensitivity calculated by the CWI analysis method reached \(\mathrm{d}V/\mathrm{V}=0.00095/\mathrm{MPa}\). However, it is notable that the measurement of CWI is the average effect of the stress on the path of sound wave propagation, while the spatial resolution is weak. Moreover, the temperature and humidity can also influence the wave velocity, resulting in error of stress estimation. The authors suggest that the results of CWI should be corrected by considering the temperature and humidity in the model in future research. To establish a quantitative relation between the change of the coda wave and stress (or stress increment) in concrete structures, crack was selected to be the medium between the stress and coda wave. For the method used to describe the influence of stress on concrete cracks, both the closure and expansion of existing cracks were considered. The fundamental principle of the method was referred from [PERSON]'s study in 1965 [PERSON] (1965), which is a classical method adopted by many researches. In the proposed model describing the crack propagation, concrete was treated as a linearly elastic material, like rocks. Pores in concrete were simplified as ellipses, and they would close or expand while loads were applied on the material. The influence of cracks and concrete material properties on the sound velocity would be independent with each other, which would lead to a much more simplified and logical derivation of the theoretical formula, as compared to some other concrete crack models, such as the fictitious crack model and adhesive crack model.