# Stress Estimation Using the Acoustoelastic Effect of Surface Waves in Weak Anisotropic Materials
## Theory
### Stress Estimation Method in Weakly Anisotropic Material  
The method mentioned above is based on the theory for an ideal isotropic material, in which the surface wave velocity in the unstressed condition is constant regardless of the direction of propagation. However, the actual material is not always produced in a perfectly isotropic condition and has anisotropic properties. In this case the surface wave velocity is directionally dependent, and the isotropic method cannot be directly applied. As will be shown later in the numerical simulations, a very large error occurs even in the case of a very small anisotropy.  
To compensate for this anisotropic effect, in this study, Equation (1) for the isotropic case was modified as follows,  
\[\frac{V_{\theta}-V_{\theta,R}}{V_{\theta,R}}=\frac{K_{1}+K_{2}}{2}(\sigma_{1}+ \sigma_{2})+\frac{K_{1}-K_{2}}{2}(\sigma_{1}-\sigma_{2})\cos 2\theta \tag{10}\]  
where \(V_{R}\) in Equation (1) is replaced with \(V_{\theta,R}\) which is the surface wave velocity in the \(\theta\) direction in the unstressed state. This equation represents the rate of change of the surface wave velocity before and after applying stress in the \(\theta\) direction.  
Then, the acoustoelastic coefficients \(K_{1}\) and \(K_{2}\) can be rederived as follows under the uniaxial stress condition (\(\sigma_{1}=\sigma\) and \(\sigma_{2}=0\)):  
\[K_{1}=\frac{V_{0}-V_{0,R}}{V_{0,R}}\frac{1}{\sigma} \tag{11}\]  
\[K_{2}=\frac{V_{90}-V_{90,R}}{V_{90,R}}\frac{1}{\sigma} \tag{12}\]  
Figure 1: Schematic of the relationships of the angles between the principal stress direction and three-measuring directions in \(45^{\circ}\) increments, in which \(\theta\) is the angle from a measuring direction to the principal stress (\(\sigma_{1}\)) direction, and \(V_{\theta}\), \(V_{\theta-45}\), \(V_{\theta-90}\) are surface wave velocities propagating in the \(\theta\), \(\theta-45^{\circ}\), and \(\theta-90^{\circ}\) directions, respectively.  
where \(V_{0,R}\) and \(V_{90,R}\) are the reference surface wave velocities parallel and perpendicular to the stress direction in the unstressed state, respectively. As a result, Equations (11) and (12) replace \(V_{R}\) in Equations (2) and (3) with the initial directional velocities, \(V_{0,R}\) and \(V_{90,R}\), respectively.  
Next, for stress estimation using three-directional measurements of the surface wave velocity in the same way as isotropic theory, we derive the following three equations from Equation (10),  
\[\sigma_{1}=\frac{1}{2}\bigg{[}\frac{1}{K_{1}+K_{2}}\bigg{(}\frac{V_{\theta}}{V _{\theta,R}}+\frac{V_{\theta-90}}{V_{\theta-90,R}}-2\bigg{)}+\frac{1}{K_{1}-K _{2}}\bigg{(}\frac{V_{\theta}}{V_{\theta,R}}-\frac{V_{\theta-90}}{V_{\theta-90,R}}\bigg{)}\cdot\frac{1}{\cos 2\theta}\bigg{]} \tag{13}\]  
\[\sigma_{2}=\frac{1}{2}\bigg{[}\frac{1}{K_{1}+K_{2}}\bigg{(}\frac{V_{\theta}}{ V_{\theta,R}}+\frac{V_{\theta-90}}{V_{\theta-90,R}}-2\bigg{)}-\frac{1}{K_{1}-K _{2}}\bigg{(}\frac{V_{\theta}}{V_{\theta,R}}-\frac{V_{\theta-90}}{V_{\theta-9 0,R}}\bigg{)}\cdot\frac{1}{\cos 2\theta}\bigg{]} \tag{14}\]  
\[\theta=\frac{1}{2}\bigg{[}\tan^{-1}\bigg{(}\frac{\frac{V_{\theta}}{V_{\theta,R }}+\frac{V_{\theta-90}}{V_{\theta+90,R}}-2\frac{V_{\theta-45}}{V_{\theta-45,R }}}{\frac{V_{\theta}}{V_{\theta,R}}-\frac{V_{\theta-90}}{V_{\theta-90,R}}} \bigg{)}\bigg{]} \tag{15}\]  
where, \(V_{\theta,R}\), \(V_{\theta-45,R}\), and \(V_{\theta-90,R}\) are the reference surface wave velocities in the \(\theta\), \(\theta-45^{\circ}\) and \(\theta-90^{\circ}\) directions in the unstressed state, respectively, and \(V_{\theta}\), \(V_{\theta-45}\), and \(V_{\theta-90}\) are the surface wave velocities in the stressed state. As a results, Equations (13)-(15) replace the reference velocity \(\mathrm{V_{R}}\) in the unstressed condition in Equations (7)-(9) with the initial velocity in each measurement direction.