# Correlation between Coda Wave and Stresses in Uni-Axial Compression Concrete
## 2 Correlation Model of Sound Velocity and Stress in Concrete
### Crack Deformation under Uni-Axial Stress  
The deformation of concrete under uni-axial stress is decomposed into the linear elastic deformation of the matrix, as well as the closed deformation and extended deformation of the microcracks. The axial strain and horizontal strain can be expressed as [23]:  
\[\left\{ \begin{array}{l}\varepsilon_{1}(\sigma)=\varepsilon_{1}^{\rm e}+ \varepsilon_{1}^{\rm c}+\varepsilon_{1}^{\rm f}\\ \varepsilon_{2}(\sigma)=\varepsilon_{2}^{\rm e}+\varepsilon_{2}^{\rm c}+ \varepsilon_{2}^{\rm f}\end{array} \right. \tag{4}\]  
where \(\varepsilon_{i}(\sigma)\) is the total deformation of the concrete; \(\varepsilon_{i}^{\rm e}\) is the deformation of the concrete matrix; \(\varepsilon_{i}^{\rm c}\) is the closed deformation of the microcracks; and \(\varepsilon_{i}^{\rm f}\) is the extended deformation of the microcracks. Here, \(i=1,2,\dots\) in which 1 and 2 represent the axial and horizontal directions, respectively. The expressions of relative displacements of the cracks in the axial and horizontal directions can be derived based on the fracture mechanics theory. Suppose there is an elliptical crack in an infinite element loaded with a uniform compressive stress \(\sigma\), as shown in Figure 2.  
The displacement field around the crack can be expressed as shown in Equation (5):  
\[\left\{ \begin{array}{l}2G_{0}U=\frac{\sigma}{4}(k+1)b_{0}\cos\beta-\frac{ \sigma}{4}(k+1)(a_{0}+b_{0})\cos(2\gamma+\beta)\\ 2G_{0}V=\frac{\sigma}{4}(k+1)b_{0}\sin\beta-\frac{\sigma}{4}(k+1)(a_{0}+b_{0}) \sin(2\gamma+\beta)\end{array} \right. \tag{5}\]  
Figure 1: Sound velocity equivalent model of concrete with microcracks under the zero stress state. Figure 2: Close of a compression-shear crack. For the crack in Figure 1, the axial and horizontal displacements caused by the closure of the crack can be expressed by the relative displacement between points A-A' and points B-B'. The displacement of points A and B caused by the stress \(\sigma\) can be expressed as shown in Equation (6). \[\left\{ \begin{array}{l}U_{A}^{y}=U_{A}\sin\gamma+V_{A}\cos\gamma=\frac{k+1}{8 C_{0}}a_{0}(\sin\gamma\cos\beta_{1}+2\sin\beta_{1}\cos\gamma)\sigma\\ U_{B}^{x}=U_{B}\cos\gamma-V_{B}\sin\gamma=-\frac{k+1}{8C_{0}}a_{0}\cos\beta_{2 }\cos\gamma\sigma\end{array} \right. \tag{6}\]  
Thus, the displacement caused by closure of the crack can be expressed as:  
\[\left\{ \begin{array}{l}U_{1}^{y}=\frac{k+1}{4C_{0}}a_{0}(\sin\gamma\cos \beta_{1}+2\sin\beta_{1}\cos\gamma)\sigma\\ U_{2}^{x}=\frac{k+1}{4C_{0}}a_{0}\cos\beta_{2}\cos\gamma\sigma\end{array} \right. \tag{7}\]  
Suppose the intensity of the crack in the element is \(k_{\rm N}\); then, the deformation of the element caused by stress \(\sigma\) can be calculated by Equation (8). \[\left\{ \begin{array}{l}\varepsilon_{1}^{\rm c}=\frac{k+1}{2\pi C_{0}}k_{\rm N }a_{0}\sigma\int_{0}^{\frac{\pi}{2}}(\sin\gamma\cos\beta_{1}+2\sin\beta_{1} \cos\gamma)\mathrm{d}\gamma\\ \varepsilon_{2}^{\rm c}=-\frac{k+1}{2\pi C_{0}}k_{\rm N}a_{0}\sigma\int_{0}^{ \frac{\pi}{2}}(\cos\beta_{2}\cos\gamma)\mathrm{d}\gamma\end{array} \right. \tag{8}\]  
For the displacement caused by crack growth, the axial and horizontal displacements of a crack under stress \(\sigma\) can be expressed as:  
\[\left\{ \begin{array}{l}U_{1}^{y}=\frac{8\sqrt{2}\tan\gamma}{\pi E_{0}}+\frac {32T^{3}}{\pi^{2}K_{L}^{x}E_{0}a_{0}\cos\gamma}\\ U_{2}^{x}=\frac{8\sqrt{2}T}{\pi E_{0}}-\frac{32T^{3}}{\pi^{2}K_{L}^{x}E_{0}a_{0 }\sin\gamma}\end{array} \right. \tag{9}\]  
The deformation of the element induced by crack growth can be expressed as shown in Equation (10):  
\[\left\{ \begin{array}{l}\varepsilon_{1}^{\rm f}=\frac{2}{\pi}\int_{0}^{\frac {\pi}{2}}(\frac{8\sqrt{2}Tk_{\rm N}\tan\gamma}{\pi E_{0}}+\frac{32T^{3}k_{\rm N }}{\pi^{2}K_{L}^{x}E_{0}a_{0}\cos\gamma})\mathrm{d}\gamma\\ \varepsilon_{2}^{\rm f}=-\frac{2}{\pi}\int_{0}^{\frac{\pi}{2}}(\frac{8\sqrt{2} Tk_{\rm N}}{\pi E_{0}}+\frac{32T^{3}k_{\rm N}}{\pi^{2}K_{L}^{x}E_{0}a_{0}\sin \gamma})\mathrm{d}\gamma\end{array} \right.