# Residual Stress in Laser Welding of TC4 Titanium Alloy Based on Ultrasonic laser Technology
## 2 Theoretical Basis  
The main effect of stress on the propagation of ultrasonic waves in a material is the variation in wave velocity of propagation [23]. Another effect, to a lesser extent, is the variation in the amplitude of the ultrasonic wave, which can be specified according to their attenuation [24]. Generally, the relative change in the velocity caused by stress is proportional to the latter, and the proportional coefficient is a parameter dependent on the material, which is called the acoustoelastic coefficient [25]. When the stress is located near the surface, rather than having the wave propagating through the thickness of the material, the energy of the wave can be directed along the surface between the transmitter and the receiver. Among waves that propagate near the surface, the surface wave is particularly attractive because it propagates without radiation loss and the energy is concentrated within a layer of about one wavelength thickness under the surface. Some studies have shown the sensitivity of these waves to stress in different materials [26; 27]. Since the depth of penetration of the surface wave into the solid is approximately one wavelength, it is possible to measure stress gradients using surface waves at different frequencies [28]. The first theory of surface waves in elastic materials with homogeneous deformation was developed by [PERSON] and [PERSON] for waves propagating along one of the principal axes of stress [29]. The acoustoelastic theory effect for ultrasonic surface wave was reviewed in [PERSON] et al. [2 ]5. Only the final results are provided here. Considering the study of the laser welding residual stress of an isotropic and uniform sheet, compared with the other two directions, the stress along the thickness direction was very small. We approximated \(\sigma_{z}=0\)\(\sigma_{z}=0\) and the model can be simplified to the plane stress state. In this paper, the surface wave propagation on the free surface of an elastic and isotropic media defined by normal plane coordinates (\(x\),\(y\)) is considered. Based on the discussion of the acoustoelastic effect of surface wave conducted by [PERSON] [31], when the displacement caused by the surface wave propagation is infinite, the relative change in the propagation velocity can be expressed as a function of two nonzero surface stresses, \(\sigma_{x}\) and \(\sigma_{y}\), as follows:  
\[\frac{\Delta v_{x}}{v_{0}}=\frac{v_{x}-v_{0}}{v_{0}}=A_{x}\sigma_{x}+A_{y} \sigma_{y} \tag{1}\]  
\[\frac{\Delta v_{y}}{v_{0}}=\frac{v_{y}-v_{0}}{v_{0}}=A_{x}\sigma_{y}+A_{y} \sigma_{x} \tag{2}\]  
where \(v_{0}\) is the velocity of the surface acoustic wave with no residual stress present; \(v_{x}\) and \(v_{y}\) are the surface wave velocities in the presence of (\(\sigma_{x},\sigma_{y}\)) in the \(x\) and \(y\) directions, respectively; \(\sigma_{x}\) and \(\sigma_{y}\) are the principal stresses; and \(A_{x}\) and \(A_{y}\) are the acoustoelastic coefficients of the surface ultrasonic wave that depend on not only the stress direction but also on the propagation direction of the wave. The acoustoelastic coefficients can be evaluated according to the mechanical parameters of the material [32]. Considering the difference between experiment and theory, in order to enhance the measurement precision, a pre-stress loading method is proposed to calibrate the coefficients in the experiment. To achieve stress inversion, the coefficients \(A_{x}\) and \(A_{y}\) must be determined in advance. For uniaxial tensile stress fields, only the pre-stress \(\sigma_{x}\) is applied; the coefficients \(A_{x}\) and \(A_{y}\) can be obtained from the calibration experiment.