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 (
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 [
The traditional ultrasonic technique for residual stress measurement applies the acoustoelasticity theory [
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 [
Applying the nonlinear property of ultrasonic to characterize the stress state gets more and more attention. Bartoli et al. [
Most of those studies use the guided or Rayleigh waves to detect the stress of specimens, while the nonlinear property of
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 is
If the ultrasonic has fixed driving frequency, wave number, and propagation distance, the ultrasonic nonlinear coefficient is proportional to
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.
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
The mechanism of
Due to the fact that the stress would not only influence the wave velocity [
The material used in this research is standard grade 45 steel with yield limit of 355 MPa which is highquality carbon structure steel with a nominal chemical composition of C, 0.42~0.50%; Si, 0.17~0.37%; Mn, 0.50~0.80%; Cr, ⩽0.25%; Ni, ⩽0.30%; and Cu, ⩽0.25%. Dogbone specimens are machined from a rolled steel plate; the dimension of specimens is displayed in Figure
Dimension and shape of grade 45 steel specimen.
In order to ensure the specimens with different stress state, the prestress loading method is applied. The specimens are fixed on the electronic universal testing machine, WDW3100, and different prestress states could be obtained. The specimens are used to conduct the tensile test with a loading rate of 0.5 mm/min. Because of the fact that the yield limit of grade 45 steel is 355 MPa, in some actual engineering application, the measured residual stress would be bigger than the yield stress for the service components and parts. In order to really simulate the residual stress states of components in actual engineering application, the prestress for the tested specimen changes from 0 MPa to 400 MPa, where the stress, 400 MPa, is a little more than the yield limit of grade 45 steel, 355 MPa. The loading test is interrupted every 20 MPa without specimens unloaded, where the full load time is 150 s; the nonlinear ultrasonic experiments are performed at the full load time to record the receiving signals, while the traditional ultrasonic experiments are conducted every 40 MPa to measure the wave velocity, because the variation of wave velocity is negligible when the stress variation is less than 26 MPa [
A schematic diagram of nonlinear ultrasonic experiment based on
Experimental setup for the ultrasonic measurement based on
A tone burst is generated by the RITEC RAM5000 system with a highpower gated amplifier, where the length of tone burst is 30 cycles and 70% of the maximum output level of the amplifier is selected to make a reliable nonlinear ultrasonic experiment. The amplified highvoltage signal passes through a 50 ohm load and lowpass filter to suppress the transient behavior due to the mismatch in electrical impedance between the amplifier and transducers. The receiving signals are recorded by an oscilloscope (Tektronix MDO3022), where the sampling rate is 100 MHz, and averaged 256 times to increase the signal to noise ratio and then transferred to a computer for further signal processing.
In the nonlinear ultrasonic experiment, the
When choosing the incident frequency of ultrasonic, it not only considers the influence on detection depth but also takes into account the waveform of receiving signal of receiver in nonlinear ultrasonic experiment. Because the center frequency of incident transducer is 2.25 MHz, when the incident frequency changes from 1 MHz to 5 MHz, the waveform and amplitude of receiving signal recorded by oscilloscope are compared. It is found that when the incident frequency is 2.2 MHz, the waveform of receiving signal is relatively better and the amplitude is the biggest. Therefore, based on the above analysis, the incident frequency of ultrasonic is selected at 2.2 MHz, and the detection depth of residual stress for established system is 2.5 mm.
A typical receiving signal in nonlinear ultrasonic experiment is shown in Figure
The time domain of receiving signals in nonlinear ultrasonic experiment
The frequency spectrum of receiving signals in nonlinear ultrasonic experiment
In the nonlinear ultrasonic experiments, the propagation distance of
The relationship between the propagation distance and the relative nonlinear coefficient.
It can be seen from Figure
The ultrasonic system shown in Figure
Furthermore, in order to compare the detected minimum stress of nonlinear and traditional ultrasonic methods, the stress detection resolution is studied. On the basis of 80 MPa prestress of specimens p4 and p5, when the prestress is 120 MPa, 110 MPa, 100 MPa, 90 MPa, and 85 MPa, respectively, where the stress curve is shown in Figure
The prestress curve for stress detection resolution study.
In the nonlinear ultrasonic measurement, the reliability of measurement should be firstly verified, because the experimental equipment, the coupling medium, and transducers would bring in the nonlinearity. If the wave number and propagation distance are fixed in (
Second harmonic amplitude versus the square of fundamental amplitude for the input voltage increasing from 300 V to 400 V.
As shown in Figure
Three specimens, p1, p2, and p3, with different prestress status are used to conduct the nonlinear ultrasonic experiments. According to the frequency spectrum of receiving signals, the relative nonlinear coefficient is estimated. The relationships between the relative nonlinear coefficient and prestress for three specimens are shown in Figure
Relationship between the relative nonlinear coefficient and prestress when stress increases from 0 MPa to 400 MPa
Relationship between the deformation state and prestress when stress increases from 0 MPa to 400 MPa
As can be seen in Figure
In Figure
The normal stressstrain curve for grade 45 steel.
It also can be found from Figure
The microstructures of specimens enduring different prestress are considered the main reason to cause the variation of the ultrasonic nonlinearity. The optical microscope is applied to observe the microstructure of grade 45 steel specimens enduring stress of 100 MPa, 200 MPa, 300 MPa, and 400 MPa as shown in Figure
Microstructures of grade 45 steel specimens when the prestress is (a) 100 MPa, (b) 200 MPa, (c) 300 MPa, and (d) 400 MPa.
When the prestress of three specimens is 0 MPa, 40 MPa, 80 MPa, 120 MPa, 160 MPa, 200 MPa, 240 MPa, 280 MPa, 320 MPa, 360 MPa, and 400 MPa, specimens are also used to conduct the traditional ultrasonic experiment to calculate the wave velocity of
The relationship between the wave velocity and prestress value.
As shown in Figure
Due to the fact that the relative nonlinear coefficient and wave velocity could reflect the stress level of material, the sensitivities of nonlinear and traditional ultrasonic methods for stress estimation are compared. The sensitivity parameter is defined as
Sensitivity parameters of the relative nonlinear coefficient and wave velocity for stress.
Prestress (MPa)  Sensitivity parameter 1 (%)  Standard deviation 1 (×10^{−6})  Sensitivity parameter 2 (%)  Standard deviation 2 

40  1.9299  5.99  0.0019  0.413 
80  7.3764  6.61  0.0279  0.369 
120  10.1532  4.979  0.0392  0.42 
160  13.2785  7.084  0.0554  0.582 
200  18.9297  9.46  0.0649  0.537 
240  23.8714  6.54  0.1197  0.328 
280  40.058  8.165  0.1349  0.43 
320  56.1403  7.993  0.1664  0.516 
360  71.7694  6.064  0.2021  0.484 
400  87.2034  5.315  0.2075  0.576 
As shown in Table
In the process of cumulative plastic deformation, the microstructure variation of material is mainly the formation of dislocation and the increase of dislocation density. Studies have shown that the dislocation structures evolution and microplastic deformation would not induce obvious change in macroscopic properties of ultrasonic, for example, the attenuation and wave velocity; therefore, the sensitivity of wave velocity for stress variation is relatively small as shown in Table
Next, the stress detection resolution of nonlinear and traditional ultrasonic method is analyzed. On the basis of 80 MPa prestress of specimens, when the prestress is 120 MPa, 110 MPa, 100 MPa, 90 MPa, and 85 MPa, respectively, the nonlinear and traditional ultrasonic measurements are conducted to calculate the relative nonlinear coefficient
The experimental result of stress detection resolution for grade 45 steel.
Parameters  Values  


40  30  20  10  5 

2.6483  1.7991  1.2674  0.5928  0.0653 
Error 1 (×10^{−6})  4.7425  5.3217  3.9451  5.854  6.3418 

1  0.6793  0.4786  0.2238  0.0247 

1.2476  0.8938  0.0736  0.0541  0.0622 
Error 2  0.4328  0.3946  0.5037  0.5565  0.4932 

1  0.7164  0.059  0.0434  0.0498 
As shown in Table
In this paper, the nonlinearity property of
With increasing prestress level, the relative nonlinear coefficient increases slowly before 200 MPa and fortifies rapidly after 300 MPa, and the increment of the relative nonlinear coefficient is about 80%, which is closely related to the cumulative plastic deformation caused by microstructure evolution of materials. The wave velocity decreases only about 0.2%, and that is because the dislocation structures evolution and microplastic deformation would not induce the significant change in wave velocity.
The sensitivity of nonlinear ultrasonic method for stress is much higher than traditional ultrasonic technique. And the stress detection resolution of nonlinear ultrasonic measurement based on the
Because the feasibility of residual stress estimation based on the nonlinearity property of
The authors declare that there are no conflicts of interest regarding the publication of this article.
This work was supported by the National Natural Science Foundation of China (Grants nos. 51365006 and 51445013), the Natural Science Foundation of Guangxi (Grant no. 2016GXNSFAA380119), the project of Guangxi Key Laboratory for Manufacturing Systems and Advanced Manufacturing Technology (Grant no. 1404515S05), and the Research Foundation of Education Bureau of Hunan Province (Grant no. 15A008).