# Stress Estimation Using the Acoustoelastic Effect of Surface Waves in Weak Anisotropic Materials
## 1 Introduction  
Monitoring of applied or residual stress is important for quality control of industrial parts manufacturing and health management of structures. The conventional methods of measuring stress include mechanical methods such as hole-drilling and nondestructive techniques such as X-ray diffraction (XRD) and acoustoelastic method. However, mechanical methods are complicated and not suitable for online monitoring. X-ray diffraction has the disadvantage of being harmful to the human body and requires separate pretreatment of the specimen. Unlike these approaches, the acoustoelastic method is harmless and has advantages that it can be applied to real products without pre-treatment and it is suitable for online monitoring. The acoustoelastic method utilizes the propagation velocity of an elastic wave, which changes with the stress state of a medium [1]. The acoustoelastic effect appears in all kinds of ultrasonic waves including longitudinal, transverse and surface waves. For many applications, however, surface stresses are needed and in this case, surface waves are suitable since they only penetrate to a depth of approximately one wavelength. In addition, the transmitter and receiver of the surface waves are located on the same plane so that it is possible to perform inspection using only one side. Furthermore, it is not necessary to know the exact thickness of the applied object to measure the wave velocity. These features enhance the application of surface waves in the field. Accordingly, various studies on stress estimation using acoustoelastic effect of surface wave have been conducted. In particular, manystudies have been carried out to analyze stresses in weld [2] or rail specimen [3; 4]. In those studies, however, the direction of the dominant stress is already known, so it is only possible to determine the presence of the stress or to estimate the stress in the known direction of the dominant stress. Meanwhile, there is also a method using longitudinal critical refraction (Lcr) waves, which have advantage of high sensitivity to stress [2; 5]. However, the Lcr wave is only applicable in contact manner because it uses the critical refraction at the contact interface between the wedge and the test object. On the other hand, surface waves are easy to extend in a non-contact manner such as laser ultrasonic technology. Early studies regarding the acoustoelastic effect of surface wave were conducted to identify the linear relationships between stress and surface wave velocity in isotropic material [6; 7]. However, even if a material is initially isotropic, changes of the micro-structure occur during material production and processing, which may result in anisotropic properties. In this case, no matter how weak the anisotropy is, the use of the isotropic theory causes a large error in stress estimation and therefore cannot be applied. Only a very limited number of studies have investigated on the acoustoelastic effect in anisotropic material. [PERSON] et al. [8] and [PERSON] and [PERSON] [9] proposed a model that separates the contribution of the anisotropic effect and the contributions of the acoustoelastic effect to evaluate how the surface wave velocity changes from the isotropic state when a stress is applied in a weakly anisotropic material. Although, this method has not been experimentally verified, it is worthwhile to analyze the effect of anisotropy. However, it is very inconvenient for actual application, because it is necessary to know in advance how the actual elastic modulus of the material differs from the isotropic elastic modulus before the stress is applied. Another study by [PERSON] et al. [10] proposed a general solution for plane wave propagation in a symmetry plane of an orthorhombic, biaxial stressed anisotropic material. However, it does not provide a direct solution to the case of using surface waves.