## 1 Introduction  
Residual stresses are created during most manufacturing processes involving thermal treatment, deformation, or other operations required to reshape materials or change their properties. In addition, residual stresses may exist in the raw materials before processing or may arise during in-service loading operations [1]. The properties of engineering materials and their structural components (e.g., distortion, fatigue life, corrosion resistance, dimensional stability, and brittle fracture) can be greatly influenced by residual stress [2; 3]. The effects of residual stress on these properties increase the repairing/restoring cost. Thus, residual stress analysis is an indispensable design tool when working with structural elements or evaluating their reliability under actual service conditions. Residual stress evaluation methods can be destructive, semidestructive, or nondestructive [4]. The main destructive and semidestructive techniques include contour, sectioning, hole-drilling, ring-core, and deep-hole methods [5; 6; 7; 8; 9], among others. However, these methods are complicated, time-consuming, and will cause damage to the material. Typical nondestructive methods include neutron diffraction, X-ray, ultrasound, and magnetic-based techniques [10; 11; 12]. Neutron diffraction requires several minutes to over an hour and the measurements are costly. The X-ray method is harmful to human health and sensitive to the direction of detection. The magnetic-based method is only applied to measure residual stresses of ferromagnetic materials. The ultrasonic method is advantageous because it is portable, not expensive, harmless to operators, and provides real-time results. In recent years, ultrasound has been widely used in defect detection, residual stress measurement, and many other applications. There are numerous types of ultrasonic testing methods for determining residual stresses in engineering materials, including ultrasonic longitudinal wave, ultrasonic shear wave (and their combination with the former), ultrasonic surface wave, ultrasonic guided wave, and ultrasonic critically refracted longitudinal (LCR) wave methods [13; 14; 15]. The latter method has been demonstrated to possess the highest sensitivity to strain among the ultrasonic wave methods [16]. Many researchers have worked on detecting residual stress by LCR waves. [PERSON] et al. [17] analyzed the temperature effect and found that the LCR waves propagating in the stress direction is more sensitive than those propagating in the thickness direction of materials. [PERSON] et al. [18] proved that it is feasible to use the neural network to improve the accuracy and reliability of residual stress measurement by the ultrasonic method. The results of [PERSON] et al. [19] showed that longitudinal residual stresses were increased by using a clamp during the welding of stainless steel plates. [PERSON] et al. [20] proposed the method of calibration of a stress coefficient to improve detection precision. [PERSON] et al. [21] discussed the effect of different test directions on the result of surface wave velocity measurement. [PERSON] et al. [22] studied the effects of initial stress \(d\sigma_{0}\) and microstructure on the stress coefficient \(K\) and flight time \(t_{0}\) in stress-free conditions. [PERSON] et al. [23] discussed the optimal step length and grain size for residual stress measurement by LCR waves. [PERSON] et al. [24] proved the effectiveness of LCR waves by comparing the finite element simulation method with the LCR experimental method. [PERSON] et al. [25] presented the velocity variations of LCR waves induced by orthotropic anisotropy and internal stresses. Among the above references, several uncertain factors involved in measuring residual stress with LCR wave have been analyzed, such as temperature, neural network, clamping effect, stress coefficient, test direction, microstructure, and grain size.