# Correlation between Coda Wave and Stresses in Uni-Axial Compression Concrete
## 5 Conclusions  
In this paper, the feasibility of measuring stresses in concrete by the CWI technique was studied and discussed. The correlation model of sound velocity and stress in concrete was established on the basis of acoustoelastic theory. Moreover, the development law of the coda wave in concrete was illustrated by experiments. The model with regressed parameters can describe the corresponding relation between the velocity change of the coda wave and stress well. The findings can also indicate that the coda wave formed by multiple scattering has a higher accuracy in detecting changes in the properties of the concrete material than does the direct wave method.  
**Author Contributions:** J.Z.: Establish the model for stress estimation by coda wave interference method; [PERSON]: Construct the research framework; H.-B.X.: Conduct the experimental study; [PERSON]: Write and revise the manuscript; [PERSON]: Assist to conduct the experimental study; W.W.: Edit the format of the manuscript.  
**Funding:** This research received funding from the China National Natural Science Foundation (Grant Nos. 51678030 and 51478072), the Science and technology program of Beijing Municipal Transportation Commission (Grant No. 2017-03-006-ZIJDZ), and the Transportation science and technology program of Hebei Province (Grant No. C18L00580).  
**Acknowledgments:** The authors appreciate LetPub (www.letpub.com) for its linguistic assistance during the preparation of this manuscript.  
**Conflicts of Interest:** The authors declare no conflict of interest.  
**Nomenclature**  
\(S\) acoustic slowness;  
\(V\) sound velocity;  
\(V_{pn}\), \(V_{p0}\) and \(V_{a}\)  
velocity of the waves propagating in the elastic matrix, concrete with cracks, and air,  
respectively;  
\(S_{pm}\), \(S_{p0}\), and \(S_{a}\)  
acoustic slowness in the elastic matrix, concrete with cracks, and air, respectively;  
\(\phi\) porosity of concrete;  
\(\sigma\) stress in concrete under an applied load;  
\(\epsilon_{i}(\sigma)\) total deformation of concrete;  
\(\epsilon_{i}^{e}\) deformation of elastic concrete matrix;  
\(\epsilon_{i}^{c}\) closed deformation of microcracks;  
\(\epsilon_{i}^{t}\) extended deformation of microcracks;  
\(E_{0}\) elastic modulus of concrete;  
\(G_{0}\) shear modulus of concrete;  
\(k\) elastic constant of concrete, \(k=\frac{3-v_{0}}{1+v_{0}}\);  
[PERSON]'s ratio of concrete;  
\(K_{\rm IC}\) fracture toughness of concrete;  
\(k_{\rm N}\) crack density parameter in unit volume concrete;  
\(a_{0}\) semi-major axis of concrete microcrack;  
\(b_{0}\) semi-minor axis of concrete microcrack;  
\(\gamma\) and \(\beta\) orientation parameters of crack;  
\(\sigma_{u}\) critical stress of crack closure or expansion;  
\(T\) proportional to the stress \(\sigma-\sigma_{u}\);  
\(U_{2}^{x}\) expansion displacement in the direction of the \(x\) axis;  
\(k_{\rm N}\) intensity of crack in concrete without an applied load;  
\(\epsilon_{i}^{c}\) closure deformation of concrete caused by stress \(\sigma\);  
\(\epsilon_{i}^{t}\) expansion deformation of concrete caused by stress \(\sigma\);  
\(\epsilon_{V}^{c}\) volume strain of the crack closure;  
\(\epsilon_{V}^{c}\) volume strain of the crack expansion;  
\(k_{\rm a}\), \(k_{\rm b1}\) and \(k_{\rm b2}\) undetermined coefficients;  
\(x_{1}\) density parameter of the horizontal equivalent cracks in concrete;  
\(x_{2}\) density parameter of the extended equivalent cracks in concrete;