# Residual Stress in Laser Welding of TC4 Titanium Alloy Based on Ultrasonic laser Technology
## 3 Experimental Process
### Ultrasonic laser for Measurement of Residual Stress in Laser Welding
#### 3.2.2 Determination of Acoustoelastic Coefficients  
To measure residual stress with an ultrasonic laser, we first performed a calibration experiment, and the acoustoelastic coefficients of the material were obtained. The acoustoelastic coefficients were obtained via online pre-stress loading. One end of the TC4 base material specimen was fixed to the optical platform (Thorlabs Nexus, Thorlabs Inc., Newton, NJ, USA) with bolts. The constraints were considered as fixed boundary conditions. The other end of the specimen was connected with a weight using a cable, and the different tension stresses were achieved by changing the weight. Based on the [PERSON] principle, the tensile stress field with uniform distribution was obtained far from the loading zone. Firstly, the velocity of surface wave \(v_{0}\) was measured in the base material specimen without loading. Then, the specimens under uniform tensile stress were measured. Notably, the loading stress should be less than the yield strength of the TC4 base material to avoid plastic deformation. The velocities, \(v_{x}\) and \(v_{y}\), were determined parallel and perpendicular to the principal stress direction, respectively. In the two cases, the velocities were measured using the same experimental program. The ultrasonic wave velocity was determined by travel time and distance. The travel time \(\Delta t\) was recorded with an oscilloscope, and the results are presented in Figure 2. The distance \(\Delta l\) between the excitation location and the detecting location was measured with a Vernier caliper. Two channels of the oscilloscope were used in the experiment. The yellow curve record the ultrasonic vibration signal from the laser Doppler vibrometer. The surface wave signal was obvious, which satisfies the surface wave characteristics. The blue curve recorded the synchronous time reference signal from the photoelectric detector. The travel time \(\Delta t\) between the excitation location and  
Figure 1: Experimental system: (**a**) ultrasonic laser and (**b**) pre-stress loading apparatus. Figure 2: Travel time of surface waves in the base material without stress. The change in distance is \(\Delta l=12.82\) mm, Channel 1 is the vibration signal, Channel 2 is the time reference signal, a and b are the cursors for reading travel time. the detection location was determined by the cursor. The measurement error of the ultrasonic wave propagation time and distance was solved using a linear fitting experiment. The slope of the linear fitting curve is \(v_{0}\), where \(v_{0}=2795.2\) m/s is the velocity of the surface wave, as shown in Figure 3.  
The pulsed laser acting on the surface of a material produces ultrasonic longitudinal waves, shear waves, and surface waves. The propagation velocities of these waves are related to the density and elastic constants of the material [33]. According to solid mechanics theory [34], the equations can be expressed as:  
\[C_{S}=\sqrt{\frac{E}{\rho}}\sqrt{\frac{1}{2(1+\mu)}} \tag{5}\]  
\[C_{R}=\frac{0.87+1.13\mu}{1+\mu}C_{S} \tag{6}\]  
where \(C_{S}\) and \(C_{R}\) are the shear wave velocity and surface wave velocity, respectively; \(E=110\) GPa is the elastic modulus; \(\mu=0.34\) is the Poisson ratio; and \(\rho=4.5\) g/cm\({}^{3}\) is the density of the TC4. The theoretical solution of surface waves velocity \(v^{\prime}_{0}=2826.4\) m/s can be obtained using Equations (5) and (6). The error of the theoretical and experimental results was 1.1%. Then, different stress fields were applied to the specimen via pre-stress loading. At each uniaxial tensile loading stage, the propagation velocity of the surface wave, perpendicular and parallel to the loading direction, was measured. By step-by-step loading of the TC4 specimen, the relationship between surface wave velocity ratio and tensile stress are represented by the curves.