# Stress Estimation Using the Acoustoelastic Effect of Surface Waves in Weak Anisotropic Materials
## Theory  
### Stress Estimation Method in Isotropic Material  
The acoustoelastic effect of surface wave in isotropic materials is expressed as follows,  
\[\frac{V_{\theta}-V_{R}}{V_{R}}=\frac{K_{1}+K_{2}}{2}(\sigma_{1}+\sigma_{2})+ \frac{K_{1}-K_{2}}{2}(\sigma_{1}-\sigma_{2})\cos 2\theta \tag{1}\]  
which represents the relationship between the surface wave velocity change and the principal stresses [9]. Here, \(V_{R}\) is the reference velocity of the surface wave in the unstressed state, \(V_{\theta}\) is the surface wave velocity in the stressed state in the \(\theta\) direction, and \(\theta\) is the angle between the surface wave propagation direction and the principal stress \(\sigma_{1}\). \(\sigma_{1}\) and \(\sigma_{2}\) are principal stresses, and \(K_{1}\) and \(K_{2}\) are the acoustoelastic coefficients.  
The acoustoelastic coefficients \(K_{1}\) and \(K_{2}\) represent the linear proportional coefficients between the velocity change rate and stress when the surface wave propagates in the stress direction and in the direction perpendicular to the stress direction under the uniaxial stress condition, respectively. That is, when \(\sigma_{1}=\sigma\) and \(\sigma_{2}=0\), the coefficient can be expressed as follows:  
\[K_{1}=\frac{V_{0}-V_{R}}{V_{R}}\frac{1}{\sigma} \tag{2}\]  
\[K_{2}=\frac{V_{90}-V_{R}}{V_{R}}\frac{1}{\sigma} \tag{3}\]  
where, \(V_{0}\) is the surface wave velocity in the direction of stress and \(V_{90}\) is the velocity in the direction perpendicular to the stress.  
To determine the three terms (\(\sigma_{1}\), \(\sigma_{2}\), \(\theta\)) in Equation (1), three-directional surface wave velocities can be used. As an example, using the velocities in the three-directions with a \(45^{\circ}\) difference, as shown in Figure 1, the following three equations can be obtained.  
\[\frac{V_{\theta}-V_{R}}{V_{R}}=\frac{K_{1}+K_{2}}{2}(\sigma_{1}+\sigma_{2})+ \frac{K_{1}-K_{2}}{2}(\sigma_{1}-\sigma_{2})\cos 2\theta \tag{4}\]  
\[\frac{V_{\theta-45}-V_{R}}{V_{R}}=\frac{K_{1}+K_{2}}{2}(\sigma_{1}+\sigma_{2}) +\frac{K_{1}-K_{2}}{2}(\sigma_{1}-\sigma_{2})\cos 2(\theta-45) \tag{5}\]  
\[\frac{V_{\theta-90}-V_{R}}{V_{R}}=\frac{K_{1}+K_{2}}{2}(\sigma_{1}+\sigma_{2}) +\frac{K_{1}-K_{2}}{2}(\sigma_{1}-\sigma_{2})\cos 2(\theta-90) \tag{6}\]  
In the above equations, \(V_{\theta}\), \(V_{\theta-45}\), and \(V_{\theta-90}\) are surface wave velocities propagating in the \(\theta\), \(\theta-45^{\circ}\), and \(\theta-90^{\circ}\) directions, respectively, in the stressed material, and \(V_{R}\) is the reference velocity in the unstressed state. From Equations (4)-(6), the principal stresses, \(\sigma_{1}\) and \(\sigma_{2}\), and the angle \(\theta\) can be determined as follows:  
\[\sigma_{1}=\frac{1}{2}\bigg{[}\frac{1}{K_{1}+K_{2}}\bigg{(}\frac{V_{\theta}}{V _{R}}+\frac{V_{\theta-90}}{V_{R}}-2\bigg{)}+\frac{1}{K_{1}-K_{2}}\bigg{(}\frac {V_{\theta}}{V_{R}}-\frac{V_{\theta-90}}{V_{R}}\bigg{)}\cdot\frac{1}{\cos 2\theta} \bigg{]} \tag{7}\]\[\sigma_{2}=\frac{1}{2}\bigg{[}\frac{1}{K_{1}+K_{2}}\Big{(}\frac{V_{ \theta}}{V_{R}}+\frac{V_{\theta-90}}{V_{R}}-2\Big{)}-\frac{1}{K_{1}-K_{2}}\Big{(} \frac{V_{\theta}}{V_{R}}-\frac{V_{\theta-90}}{V_{R}}\Big{)}\cdot\frac{1}{\cos 2 \theta}\bigg{]} \tag{8}\] \[\theta=\frac{1}{2}\bigg{[}\tan^{-1}\bigg{(}\frac{\frac{V_{\theta} }{V_{R}}+\frac{V_{\theta-90}}{V_{R}}-2\frac{V_{\theta-90}}{V_{R}}}{\frac{V_{ \theta}}{V_{R}}-\frac{V_{\theta-90}}{V_{R}}}\bigg{)}\bigg{]} \tag{9}\]  
Using these three equations, when the acoustoelastic coefficients (\(K_{1}\), \(K_{2}\)) and the reference surface wave velocity (\(V_{R}\)) in the unstressed condition are known, the principal stresses (\(\sigma_{1}\), \(\sigma_{2}\)) and the angle (\(\theta\)) can be estimated using the three-directional measurements of the surface wave velocity.