Advances in Materials Science and Engineering

Volume 2017 (2017), Article ID 6274054, 11 pages

https://doi.org/10.1155/2017/6274054

## Feasibility of Residual Stress Nondestructive Estimation Using the Nonlinear Property of Critical Refraction Longitudinal Wave

^{1}Institute of Light Industry and Food Engineering, Guangxi University, Nanning 530004, China^{2}College of Mechanical Engineering, Guangxi University, Nanning 530004, China^{3}Hunan Province Key Laboratory of Safety Design and Reliability Technology for Engineering Vehicle, Changsha University of Science & Technology, Changsha 410004, China^{4}School of Automobile and Transportation, Guangxi University of Science and Technology, Liuzhou 545006, China

Correspondence should be addressed to Han-Ling Mao and Han-Ying Mao

Received 7 June 2017; Revised 9 October 2017; Accepted 22 October 2017; Published 14 November 2017

Academic Editor: Donato Sorgente

Copyright © 2017 Yu-Hua Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Residual stress has significant influence on the performance of mechanical components, and the nondestructive estimation of residual stress is always a difficult problem. This study applies the relative nonlinear coefficient of critical refraction longitudinal () wave to nondestructively characterize the stress state of materials; the feasibility of residual stress estimation using the nonlinear property of wave is verified. The nonlinear ultrasonic measurements based on wave are conducted on components with known stress state to calculate the relative nonlinear coefficient. Experimental results indicate that the relative nonlinear coefficient monotonically increases with prestress and the increment of relative nonlinear coefficient is about 80%, while the wave velocity only decreases about 0.2%. The sensitivity of the relative nonlinear coefficient for stress is much higher than wave velocity. Furthermore, the dependence between the relative nonlinear coefficient and deformation state of components is found. The stress detection resolution based on the nonlinear property of wave is 10 MPa, which has higher resolution than wave velocity. These results demonstrate that the nonlinear property of wave is more suitable for stress characterization than wave velocity, and this quantitative information could be used for residual stress estimation.

#### 1. Introduction

Residual stress is an inherent stress which keeps the stress balance in the inner material when the mechanical components are unaffected by external strength. The service properties of mechanical components, such as fatigue life and strength, could be considerably influenced by residual stress, and this would result in considerable expenditure in repair and maintenance of components. Therefore, the effective estimation of residual stress is very important for mechanical components.

Residual stress measurement methods could be categorized into destructive and nondestructive methods [1]. The slice and contour methods are destructive, and the blind-hole and deep-hole methods are destructive too. The destructive methods belong to the category of stress release and would cause the damage of components, which is fatal and must be avoided [2], while nondestructive methods are more widely applied in recent years, such as X-ray diffraction technique [3, 4], thermoelastic stress analysis (TSA), and ultrasonic method [5]. TSA is a noncontacting and sensitive experimental stress analysis technique [6, 7], which provides full-field stress data over the surface of a cyclically loaded specimen or component [8]. Robinson [7] gave significant attention to the effect of plastic deformation on the thermoelastic constant and the influence of the mean stress on the thermoelastic response in stainless steel and aluminium. In [9], it was shown that mean stresses significantly influenced the TSA results for titanium-based alloys and nickel-based alloys. And, in [10], the mean stress sensitivity was established for both titanium and nickel alloys. Zhukovskii and Gokhman [11] proposed the method of determining the residual stresses in metallic sheets, which consisted in finding the strain field induced by central point heating and calculating the field of residual stresses using a functional relation between a linear thermal expansion coefficient and residual stresses. The stress diagram of the whole component or a certain part can be obtained, which only requires few surface preparations and does not need further data processing. However, this technique merely provides information on the surface stress field in structures. The ultrasonic method has the characteristics of high resolution, high penetration, and no harm to human body; therefore it is one of the most promising nondestructive measuring methods.

The traditional ultrasonic technique for residual stress measurement applies the acoustoelasticity theory [12], which is based on the finite deformation of continuum mechanics to study the relationship between the stress state and wave velocity of ultrasonic. Based on the acoustoelasticity theory, the sensitivity of different types of ultrasonic to stress is explored. In Cartesian coordinate system, seven kinds of equations of propagation velocity and stress in solids are established; taking the derivative of wave velocity to stress, the stress-sensitive coefficients of seven kinds of ultrasonic can be obtained [13]. When the longitudinal wave propagates along the stress direction, the stress-sensitive coefficient is 8.085, which is much higher than six other sensitive coefficients, 1.025, 0.503, 0.505, 2.495, 0.153, and 0.153, respectively. Based on above analysis, the longitudinal wave propagating along the stress direction is the most sensitive to stress; that is to say, the longitudinal wave is the most sensitive to tangential stress. When the longitudinal wave propagates along the tangential stress, it is named as wave. Because the wave has the highest sensitivity to tangential stress, the wave based on the acoustoelasticity theory is applied to measure the residual stress [14, 15]. However, Bray [16] found that when the stress variation was below 26 MPa, the variation of transmit time or wave velocity of ultrasonic was not obvious. On the other hand, the variation of wave velocity caused by stress is very small; for example, the change of wave velocity is only about 0.1% in aluminium and 0.01% in steel induced by 100 MPa stress variation. Therefore, the accurate measurement of residual stress based on the acoustoelasticity theory is often a difficult task.

Because of the earlier performance, degradation and dislocation structures would not cause obvious change in macroscopic properties of ultrasonic wave, such as attenuation and wave velocity. Nevertheless, the accumulation of dislocations would cause the distortion of ultrasonic; higher harmonics are generated when a monochromatic ultrasonic propagates through the medium. The nonlinear ultrasonic technique has shown the ability to evaluate the fatigue damage [17] and plastic deformation [18] of metal material which has a close relationship with the microstructure evolution of material. The dislocation string model [19] and the dislocation dipole model [20] establish the relationship between the ultrasonic nonlinear coefficient and stress. Those studies provide the theoretical foundation of applying the nonlinear properties of ultrasonic to characterize and evaluate the stress of material.

Applying the nonlinear property of ultrasonic to characterize the stress state gets more and more attention. Bartoli et al. [21] applied the nonlinear guided wave for stress monitoring in prestressing tendons for posttensioned concrete structures. Liu et al. [22] used the nonlinear Rayleigh wave to detect the residual stress in shot-peened aluminum plates; the nonlinear ultrasonic parameter was sensitive to residual stress. Kim et al. [23] proposed the use of the nonlinear resonant ultrasonic spectroscopy (NRUS) for the stress state monitoring of concrete.

Most of those studies use the guided or Rayleigh waves to detect the stress of specimens, while the nonlinear property of wave for stress measurement has been little researched. Therefore, in this paper, a preliminary study is performed to investigate the dependency between the stress state and the nonlinear property of wave. Firstly, the ultrasonic measure system based on the wave is established to conduct the nonlinear ultrasonic experiments for prestress specimens and the relative nonlinear coefficient is calculated. Then the relationship between the relative nonlinear coefficient and prestress is established and analyzed. At last, the sensitivity and resolution of nonlinear ultrasonic methods for stress measurement are detected. Based on above results, the dependency between the prestress state and the relative nonlinear coefficient is investigated; the possibility of residual stress estimation based on the nonlinear property of wave is verified.

#### 2. Theoretical Background

##### 2.1. Ultrasonic Nonlinear Coefficient

When a pure sinusoidal wave propagates through the nonlinear solid medium, higher harmonics are generated due to the nonlinearity of medium. The ultrasonic nonlinear coefficient is generally defined to characterize and evaluate the nonlinearity of medium, and the expression iswhere and are the wave number and propagation distance of ultrasonic, and represent amplitudes of fundamental wave and second harmonic wave, respectively, in the frequency spectrum of receiving signals in nonlinear ultrasonic experiments.

If the ultrasonic has fixed driving frequency, wave number, and propagation distance, the ultrasonic nonlinear coefficient is proportional to . Therefore, for the convenience of computing in this study, the relative nonlinear coefficient is defined as

If the stress state of metal material varies, the elastic constants would be changed and the variation of ultrasonic nonlinear coefficient would occur. Therefore, the ultrasonic nonlinear coefficient could be applied to characterize the stress state of metal material in theory.

##### 2.2. The Testing Principle of L_{CR} Wave

When a longitudinal wave propagates from the medium with lower wave velocity to the other medium with faster wave velocity, according to Snell’s law, there is an incident angle making the refraction angle of longitudinal wave equal to 90°. In this case, the incident angle is called for the first critical angle and the longitudinal wave with 90° refraction angle is named for the wave. Taking the organic glass wedges, transducers, and grade 45 steel specimens as an example, as shown in Figure 1, the calculation formula of the first critical angle is , where and represent the longitudinal wave velocity in organic glass and grade 45 steel. Due to the fact that m/s and m/s, the first critical angle is .