## 2 Theoretical Background  
Stress measurement by ultrasonic methods is based on the linear variation of the velocity of an ultrasonic wave with the stress. This relationship is denoted as an acoustoelastic effect within the elastic limit, and it provides the flight time variation of the ultrasonic wave with the stress [26]. As indicated in Figure 1, the LCR method uses a special longitudinal bulk wave, which propagates parallel to the surface and penetrates below the surface. Reflection or refraction occurs when an ultrasonic wave propagates from one medium to another. For example, when a longitudinal wave is transmitted from a polymethyl methacrylate (PMMA)  
Figure 1: Generation of LCR waves. PMMA: polymethyl methacrylate. wedge to steel (Figure 1), shear and longitudinal waves are both created in the steel. The propagation angles of the waves follow [PERSON]'s law:  
\[\frac{C_{L1}}{\sin\alpha_{CR}}=\frac{C_{L2}}{\sin\beta_{L}}=\frac{C_{S2}}{\sin \beta_{S}} \tag{1}\]  
where \(C_{L1}\) is the longitudinal velocity in the PMMA and \(C_{L2}\) and \(C_{S2}\) are the longitudinal and shear velocities in the steel, respectively. As suggested by [PERSON]'s law, when the incident angle \(\alpha_{CR}\) increases from \(0^{\circ}\) to \(27.8^{\circ}\), the refraction angle \(\beta_{L}\) reaches \(90^{\circ}\) (i.e., first critical angle), and the wave is called the L\({}_{\text{CR}}\) wave. The velocity of the L\({}_{\text{CR}}\) waves propagating parallel to the load direction correlates with the strain (\(\sigma\)) according to the equation below:  
\[\rho_{0}V_{11}^{2}=\lambda+2\mu+(2l+\lambda)\theta+(4m+4\lambda+10\mu)\sigma_{1} \tag{2}\]  
where \(\rho_{0}\) represents the initial density and the first and the second subscript of the velocity (\(V\)) indicate the propagation and polarization direction of the wave, respectively. \(V_{11}\) is the speed of the L\({}_{\text{CR}}\) wave traveling parallel to the applied stress; \(\lambda\) and \(\mu\) are both second-order elastic constants (Lame constant); \(l\) and \(m\) represent the third elastic constants (Murnaghan constant); \(\theta=\sigma_{1}+\sigma_{2}+\sigma_{3}\), and \(\sigma_{1},\sigma_{2}\), and \(\sigma_{3}\) are the homogeneous triaxial principal strains. When dealing with uniaxial stress conditions, we know that \(\sigma_{1}=\varepsilon\), \(\sigma_{2}=\sigma_{3}=-v\times\varepsilon\) is the strain along the normal direction (i.e., parallel to the propagation direction), and \(v\) is the Poisson's ratio. With these values, Equation (2) becomes:  
\[\rho_{0}V_{11}^{2}=\lambda+2\mu+[4(\lambda+2\mu)+2(\mu+2m)+v\mu(1+\frac{2 \lambda}{\mu})]\cdot\varepsilon \tag{3}\]  
A list of materials and their elastic constants are given in Table 1[27]. The relative sensitivity measures the degree to which the velocity varies with the strain. This parameter is estimated by Equation (4), where \(K\) is a dimensionless constant for L\({}_{\text{CR}}\) waves, which can be evaluated during a uniaxial tensile test or calculated from Equation (4). For anisotropic elastomers, the independent second- and third-order elastic constant are 21 and 56, respectively. For isotropic elastomers, there are two independent second-order elastic constants (Lame constants \(\lambda\) and \(\mu\)) and three independent third-order elastic constants (Murnaghan constants \(l\), \(m\), and \(n\)) [28]. \[\frac{\
icefrac{{\alpha_{V1}}}{{\alpha}}/{\
u_{11}}}{d\varepsilon}=2+\frac{ \mu+2m+v\mu(1+\
icefrac{{2l}}{{\lambda}})}{\lambda+2\mu}=K \tag{4}\]  
As shown in Equation (4), the acoustoelastic constant \(K\) is related to the Lame constant and Murnaghan constant. From Equation (4), it can be deduced that the calculation formula of \(K\) is  
\[K=\frac{-2V_{0}(3\lambda+2\mu)}{(\frac{4\lambda+10\mu+4m}{\mu}+\frac{2-3 \lambda-10\mu-4m}{\lambda+2\mu})\text{L}} \tag{5}\]  
where \(V_{0}\) is the velocity of the material and \(L\) is the travel distance for the ultrasonic wave. For steel, \(V_{0}=5790\) m/s and \(L=37.6\) mm.