## 2 Theoretical Background  
### Longitudinal Critically Refracted Wave  
Methods for measuring residual stress using ultrasonic waves are based on the acoustoelastic effect, a relationship between the acoustic wave velocity and stress in engineering materials. Recently, this technique has mainly been applied using longitudinal critically refracted (LCR) waves traveling parallel to the material surface. In this situation, the waves are sensitive to changes in residual stress inside a material. As shown in Figure 1, an LCR wave is a wave that does not disappear completely, is concentrated near the surface, and moves along the surface at high speed. This occurs when the ultrasonic longitudinal wave incident on the material refracts at an angle greater than or equal to the first critical angle (\(\theta_{i}\)).  
[PERSON] and [PERSON] [22] verified the sensitivity of velocity changes to each wave propagating in different directions in tensile and compressive load tests. As a result, it was determined that the wave traveling along the direction of the stress field showed the most pronounced sensitivity to the stress, and the longitudinal wave showed the greatest change in travel time according to deformation.  
The velocity of a longitudinal plane wave propagating parallel to the load in an initially isotropic body subjected to a homogeneous triaxial strain field is related to the strain \(\alpha_{1}\) by Equation (1) [22]:  
\[\rho_{0}V_{11}^{2}=\lambda+2\mu+(2l+\lambda)\theta+(4m+4\lambda+10\mu)\alpha_{1} \tag{1}\]  
Figure 1: Excitation of the LCR wave through transmitter and wedge [35].  
where, \(\rho_{0}\) is the initial density, \(V_{11}\) is the speed of the wave propagating parallel to the load, \(\lambda\) and \(\mu\) are Lame or second-order elastic constants, \(l\) and \(m\) are [PERSON]'s third-order elastic constants, and \(\theta\) is the sum of components of the homogeneous triaxial principal strains. For a given state of uniaxial stress, \(\theta=\alpha_{1}+\alpha_{2}+\alpha_{3}=\varepsilon+(-\
u\times\varepsilon)+ (-\
u\times\varepsilon)\), where \(\varepsilon\) is the strain in line with the load direction, and \(\
u\) is [PERSON]'s ratio.  
Equation (1) can be reduced to Equation (2) [22]:  
\[\rho_{0}V_{11}^{2}=\lambda+2\mu+\bigg{\{}4(\lambda+2\mu)+2(\mu+2m)+\
u\mu\bigg{(} 1+\frac{2l}{\lambda}\bigg{)}\bigg{\}}\varepsilon \tag{2}\]  
The relative changes in wave velocity due to axial strain may be calculated under the assumption that the relative changes are small [22]:  
\[\frac{dV_{11}/V_{11}}{d\varepsilon}=2+\frac{(\mu+2m)+\
u\Big{(}1+\frac{2l}{ \lambda}\Big{)}}{\lambda+2\mu} \tag{3}\]  
Relative sensitivity can be calculated through Equation (3), which is consistent with the acoustoelastic constant \(L_{11}\) for the LCR wave. The change in stress can be calculated by applying a stress-strain relationship in one dimension to an elastic solid. Equation (3) can be rearranged to calculate the stress change in terms of TOF as in Equation (4) [24; 36]:  
\[d\sigma=\frac{E(dV_{11}/V_{11})}{L_{11}}=\frac{E}{L_{11}t_{0}}dt \tag{4}\]  
where, \(\sigma\) is the stress, \(E\) is the elasticity modulus, and \(t_{0}\) is the time for the wave to travel through its path in the material without stress.  
The elasticity modulus can be calculated by substituting the density of the material in Equation (5) [37]:  
\[E=\frac{\rho V_{l}^{2}(1+v)(1-2v)}{(1-v)} \tag{5}\]  
where, \(\rho\) is the density of the material, and \(V_{l}\) is the longitudinal velocity in the material.  
[PERSON]'s ratio can be obtained through Equation (6) [37]:  
\[v=\frac{1-2\big{(}V_{l}^{2}-V_{l}^{2}\big{)}}{2-2\big{(}V_{t}^{2}-V_{l}^{2} \big{)}} \tag{6}\]  
where, \(V_{t}\) is the transverse velocity in the material.