# Correlation between Coda Wave and Stresses in Uni-Axial Compression Concrete
## 2 Correlation Model of Sound Velocity and Stress in Concrete
### Sound Velocity Change Induced by Uni-Axial Stress  
By analyzing the deformation of a crack under uni-axial stress, it is found that the axial closure deformation of the crack is much more than the horizontal deformation. At the same time, the secondary cracks in the extended stage are along the axial direction. Then the acoustoelastic properties of concrete can be described as follows: the sound velocity change caused by the closure volume deformation of the primary microcracks is equivalent to that caused by a set of horizontal cracks, and the sound velocity change caused by the extended volume deformation of the microcracks is equivalent to that caused by a set of vertical cracks. Moreover, the equivalent cracks have the same properties as the original cracks. It should be noted that the concrete cracks described here are equivalent cracks and the deformation of concrete cracks is not analyzed from the viewpoint of the microscopic mechanism. If the density parameter of the horizontal equivalent cracks in the concrete is \(x_{1}\), it can be approximately expressed as follows:  
\[x_{1}=k_{\rm N}\frac{\epsilon_{\rm V}^{\rm f}}{\phi} \tag{13}\]  
where \(k_{\rm N}\) represents the crack density of the concrete without loading. In the same way, the density parameter of the extended equivalent cracks \(x_{2}\) can be expressed as follows:  
\[x_{2}=k_{\rm N}\frac{\epsilon_{\rm V}^{\rm f}}{\phi} \tag{14}\]  
Set a group of horizontal opening oval cracks in the concrete element with length \(l\). The long- and short-axes of the crack are denoted as \(2a\) and \(2b\), respectively, as shown in Figure 3. Without considering the evolution of microcracks, for the convenience of modeling, we simplified the microcrack shape as a rectangular with a size of \(2a\times 2b\). Suppose the sound wave passes through the concrete element with equal probability density. If the sound wave encounters cracks, it will bypass the cracks and propagation along the boundary of the concrete matrix. When the sound wave passes through the element from the direction perpendicular to the long-axis direction of the crack, the probability of passing through a microcrack is \(\frac{2a}{l}\). At this time, the average propagation path length of the sound wave is \(l+2a\), the probability of not passing through the crack is \(\frac{l-2a}{l}\), and the average propagation path length of the sound wave is \(l\). Then, the increment of the acoustic slowness caused by microcracks is:  
\[\Delta S_{1}=S_{1}-S_{pm}=\frac{\frac{2a}{l}\frac{l+2a}{V_{pm}}+\frac{l-2a}{l} \frac{l}{V_{pm}}}{l}-S_{pm}=\frac{4a^{2}}{l^{2}}S_{pm} \tag{15}\]  
Figure 3: Concrete contains a set of horizontal microcracks. where \(S_{1}\) denotes the acoustic slowness caused by the microcrack when the sound wave passes through the element in the short-axis direction of the crack. In the same way, when the sound wave passes through the element in the direction parallel to the long-axis of that crack, the increment of the acoustic slowness caused by the microcrack can be expressed as:  
\[\Delta S_{2}=\frac{4b^{2}}{l^{2}}S_{\text{pm}} \tag{16}\]  
By introducing \(k_{\alpha}\) to represent the shape correction coefficient of an elliptical crack in a rectangular crack, and substituting Equations (13) and (14) into Equations (15) and (16), respectively, we can obtain increments of the acoustic slowness in the axial and the horizontal directions due to the crack closure or expansion deformation, as shown in Equation (17). \[\left\{