# 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
\begin{array}{l}\Delta S_{\text{p,1}}^{\text{c}}=4k_{\alpha}x_{1}a_ {0}^{2}S_{\text{pm}}=4k_{\alpha}k_{\text{N}}\frac{\epsilon_{\text{J}}}{4}a_{0 }^{2}S_{\text{pm}}=4k_{\alpha}k_{\text{N}}\frac{\epsilon_{\text{J}}}{\phi}a_{ 0}^{2}S_{\text{pm}}\\ \Delta S_{\text{p,1}}^{\text{f}}=4k_{\alpha}x_{2}b_{0}^{2}S_{\text{pm}}=4k_{ \alpha}k_{\text{N}}\frac{\epsilon_{\text{J}}}{\phi}b_{0}^{2}S_{\text{pm}}=4k _{\alpha}k_{\text{N}}\frac{\epsilon_{\text{J}}}{\phi}b_{0}^{2}S_{\text{pm}}=4 k_{\alpha}k_{\text{N}}\frac{\epsilon_{\text{J}}}{\phi}b_{0}^{2}S_{\text{pm}}\\ \Delta S_{\text{p,2}}^{\text{c}}=4k_{\alpha}x_{1}b_{0}^{2}S_{\text{pm}}=4k_{ \alpha}k_{\text{N}}\frac{\epsilon_{\text{J}}}{\phi}b_{0}^{2}S_{\text{pm}}=4k _{\alpha}k_{\text{N}}\frac{\epsilon_{\text{J}}}{\phi}b_{0}^{2}S_{\text{pm}}=4 k_{\alpha}k_{\text{N}}\frac{k_{\text{B}}(\sigma-\sigma_{\text{u}})+k_{\text{B}}(\sigma- \sigma_{\text{u}})}{\phi}a_{0}^{2}S_{\text{pm}}\end{array} \right. \tag{17}\]  
where \(\Delta S_{\text{p,1}}^{\text{c}}\), \(\Delta S_{\text{p,1}}^{\text{f}}\), \(\Delta S_{\text{p,2}}^{\text{c}}\), and \(\Delta S_{\text{p,2}}^{\text{f}}\) represent the increment of the acoustic slowness caused by the crack closure and expansion in the axial and horizontal directions, respectively. According to Equation (3), Equation (18) is rewritten into the form of a slowness ratio. \[\left\{\begin{array}{l}\frac{\Delta S_{\text{p,1}}^{\text{c}}}{S_{\text{p, 0}}}=\frac{4k_{\alpha}k_{\text{N}}\frac{k_{\alpha}k_{\text{N}}}{\phi}a_{0}^{2 }S_{\text{pm}}}{(1+0.57\phi)S_{\text{pm}}}=\frac{4k_{\alpha}k_{\text{N}}}{ \phi(1+0.57\phi)}a_{0}^{2}k_{\alpha  
Axial relative acoustic slowness in the crack expansion stage:  
\[\frac{\Delta S_{1}}{S_{\text{p0}}}=-\frac{\Delta S_{\text{p,1}}^{c}}{S_{\text{p0 }}}+\frac{\Delta S_{\text{p,1}}^{f}}{S_{\text{p0}}}=-k_{\text{c1}}\sigma_{\text {u}}+k_{\text{f11}}(\sigma-\sigma_{\text{u}})+k_{\text{f12}}(\sigma-\sigma_{ \text{u}})^{3} \tag{20b}\]  
Horizontal relative acoustic slowness in the crack closure stage:  
\[\frac{\Delta S_{2}}{S_{\text{p0}}}=-k_{\text{c2}}\sigma \tag{20c}\]  
Horizontal relative acoustic slowness in the crack expansion stage:  
\[\frac{\Delta S_{2}}{S_{\text{p0}}}=-\frac{\Delta S_{p,2}^{c}}{S_{\text{p0}}}+ \frac{\Delta S_{p,2}^{f}}{S_{\text{p0}}}=-k_{\text{c2}}\sigma_{\text{u}}+k_{ \text{f21}}(\sigma-\sigma_{\text{u}})+k_{\text{f22}}(\sigma-\sigma_{\text{u}} )^{3} \tag{20d}\]  
Because the acoustic slowness and sound velocity are reciprocal, it is easy to obtain the following relation:  
\[(1+\frac{\Delta S_{i}}{S_{\text{p0}}})(1+\frac{\Delta V_{i}}{V_{\text{p0}}})= 1(i=1,2) \tag{21}\]  
Changing the acoustic slowness to the sound velocity, we can then obtain the following. \[\frac{\Delta V_{1}}{V_{\text{p0}}}=\frac{1}{1-k_{\text{c1}}\sigma}-1\] (22a) Axial relative sound velocity in the crack extended stage: \[\frac{\Delta V_{1}}{V_{\text{p0}}}=\frac{1}{1-k_{\text{c1}}\sigma_{\text{u}} +k_{\text{f11}}(\sigma-\sigma_{\text{u}})+k_{\text{f12}}(\sigma-\sigma_{\text {u}})^{3}}-1\] (22b) Horizontal relative sound velocity in the crack closed stage: \[\frac{\Delta V_{2}}{V_{\text{p0}}}=\frac{1}{1-k_{\text{c2}}\sigma}-1\] (22c) Horizontal relative sound velocity in the crack extended stage: \[\frac{\Delta V_{2}}{V_{\text{p0}}}=\frac{1}{1-k_{\text{c2}}\sigma_{\text{u}} +k_{\text{f21}}(\sigma-\sigma_{\text{u}})+k_{\text{f22}}(\sigma-\sigma_{\text {u}})^{3}}-1 \tag{22d}\]