1 Introduction

Because Ni-based single crystal alloy (NSCA) operates under high temperature for a long time, their mechanical property degradation rate is greatly affected by the temperature, making it break easily. In recent years, there have been many air crashes caused by sudden breakage of aero-engine turbine blades during flight. Even the most mature CFM-56 aero-engine has also experienced similar accidents. This is mainly because the current detection and evaluation for the service status of turbine blades is at the late macro-defects stage, where the early predictions for the defect status of turbine blades can’t be made sufficiently, thus the effective predictive maintenance can’t be performed. NSCA is widely used in the manufacture of turbine blades for advanced aero-engines, and its defect status greatly affects the flight safety. Therefore, it is necessary to seek a new theoretical method to accurately evaluate the defect status of NSCA during service.

At present, the studies on the service defect status of NSCA at home and abroad mostly focus on the late macro-crack stage. Turbine blades are used for a long time under high temperature. Therefore, a large number of macro-cracks are generated inside and on the surface of the alloy material (Ref 1, 2), and internal cracks can be easily detected by ultrasonic longitudinal waves. However, the aero-engine has been unable to meet the flight safety requirements at this time. The nonlinear ultrasonic Lamb wave can penetrate into the interior of NSCA as well as its surface due to the property of high propagation efficiency, thus providing a new method for efficiently checking the defect status of NSCA. After serving for a while, the microstructure of NSCA produces significant nonlinear changes in the damaged area that mainly reflect in \(\Lambda\), average L, stress magnitude, Burt's vector, microcrack morphology, surface morphology, etc. (Ref 3), which are the main cause of nonlinear harmonics. The nonlinear characteristic parameter that characterizes the microstructure changes of NSCA can be constructed, by which the variation of the second harmonic (Ref 4,5,6,7,8) is characterized. Therefore, it is necessary to analyze the propagation law of the second harmonic.

The literatures (Ref 9,10,11,12,13,14,15) has extensively studied the nonlinear second harmonic of Lamb wave. Among them, literature (Ref 15) has summed up the analytical solution with universal adaptability. However, the analytical solution can’t reflect the theoretical relationship between the nonlinear ultrasonic Lamb wave and NSCA with high-temperature damage. Accordingly, this study experimentally measures the variation of the corresponding nonlinear characteristic parameter after the changes of the microstructure of NSCA, according to which the defect status of NSCA is quantitatively characterized. Then in view of the limitations of the experiments, molecular dynamics (MD) method is used to qualitatively describe the law and cause of the variation of nonlinear characteristic parameter.

2 Theoretical Analysis

As for the theoretical derivation of the U2f (second-harmonic amplitude) of the general Lamb wave, our research group has obtained some results (see literature (Ref 3)). According to the results in literature 3, the partial expressions related to the wave vector can be denoted as \(\vec{\user2{B}}_{{La{\text{mb}}}}\), which is:

$$\Lambda = \frac{{ - 5G^{3} b^{2} \left[ {4{\varvec{U}}^{{\left( {2f} \right)}} \Delta \varphi + \beta_{2} \vec{\user2{B}}_{{{\text{Lamb}}}} \sin \left( {\Delta \varphi } \right)} \right]}}{{24\vec{\user2{B}}_{{{\text{Lamb}}}} \sin \left( {\Delta \varphi } \right)\sigma F^{3} C_{11}^{2} \Omega L^{4} }}$$
(1)
$$L = \left\{ {\frac{{ - 5G^{3} b^{2} \left[ {4{\varvec{U}}^{{\left( {2f} \right)}} \Delta \varphi + \beta_{2} \vec{\user2{B}}_{{{\text{Lamb}}}} \sin \left( {\Delta \varphi } \right)} \right]}}{{24\vec{\user2{B}}_{{{\text{Lamb}}}} \sin \left( {\Delta \varphi } \right)\sigma F^{3} \Lambda C_{11}^{2} \Omega }}} \right\}^{\frac{1}{4}}$$
(2)

Formula (1) is the expression of the dislocation density Λ (Ref 16,17,18,19), and formula (2) is the expression of the dislocation chord length L. It can be found from the two formulas that \(\Lambda\) and L are both related to the mechanical properties of NSCA. In the formulas, G is the shear modulus of NSCA, C11 (Ref 20) is the second-order elastic constant of NSCA, and b is the Burgers vector of NSCA. ∆φ is the phase velocity matching degree of NSCA, which is determined by the elastic modulus, Poisson's ratio, density, etc. of the material. σ is the internal stress of NSCA, Ω is the conversion factor when strain occurs in NSCA, and F is the decomposition factor of the wave vector.

Based on the above theoretical analysis, dislocations are the key factors that influence the second harmonic at the early service stage of NSCA blades, and the dislocation motion is the main form of microstructure evolution in early defect status. Qualitative or quantitative analysis of the change law of \(\Lambda\) is of great significance for analyzing the law and cause of the variation of nonlinear βk. The dislocations of the samples can be observed by means of transmission electron microscopy, according to which \(\Lambda\) of the samples can be qualitatively analyzed. Using \(\Lambda\) calculation software, \(\Lambda\) of a certain plane can be quantitatively calculated. However, the calculation of \(\Lambda\) is based on the two-dimensional plane, and \(\Lambda\) of the sample in different two-dimensional planes is quite different (During actual service process, various defects and complex stresses are distributed at different locations of turbine blades, resulting in the generation of different dislocations at different locations). Qualitative analysis or quantitative calculation of \(\Lambda\) in a two-dimensional plane has great limitations. Therefore, it is necessary to find a new method to calculate \(\Lambda\) of NSCA on a three-dimensional level. It is possible to solve this problem by a supercomputer.

At present, some achievements have been achieved by using MD to simulate the microstructural evolution of NSCA. The MD model of NSCA has been established, and the simulation of the model under various loads has been carried out. However, most of these studies have focused on the study of microstructure evolution and mechanical performance of the samples under different sizes, crystal orientations, temperatures, forms of loads and defects. There have been no reports on the quantitative calculation of \(\Lambda\) and L, especially using MD simulation to explain the law and cause of the variation of nonlinear βk. Therefore, the MD methods could be used to analyze these problems.

3 Experimental Details

3.1 Preparation of Samples

NSCA (DD6) is used as the experimental material in this study. The literatures (Ref 21,22,23,24,25,26,27,28,29,30,31,32) have carried out extensive research on the manufacturing process and mechanical properties of NSCA, according to which the solidification temperature of NSCA is between 1335 and 1386 °C, and melting temperature is between 1500 and 1600 °C. In order to actively manufacture high-temperature damaged samples, four starting samples were subjected to high-temperature experiments in the high-temperature furnaces at 1335, 1386, 1442 and 1495 °C respectively, then cooled quickly for maximum damage. For the purpose of comparison, there was another sample with no high-temperature damage.

3.2 Experimental Method

The combination of MD and nonlinear ultrasound experiments is used in this study. After sample preparation, they were excited by the RAM-5000-SNAP system. Appropriate excitation frequency and incident angle are the prerequisites for exciting Lamb wave. According to our previous work (Ref 33), the excitation frequency and incidence angle of NSCA are 2.45 MHz and 25°, respectively. Sandwich model is used in the MD simulation, and two types of models are established. One type is the MD simulation of NSCA with defects, and the other type is the MD simulation of the auxiliary analysis results of ultrasonic experiments.

4 Results and Discussion

4.1 MD Simulation of Defective NSCA

According to the above theoretical analysis results, it is clear that Λ and L are important factors affecting the second harmonic. By analyzing the change laws of Λ and L, the variation of characteristic parameter can be predicted. The location and size of the defects have a great impact on Λ and L of NSCA. At the same time, the change laws of Λ and L are very different under different temperatures. Since NSCA is served under high temperature for a long time, it is necessary to study the effects of defect size on Λ and L under different temperatures. MD simulations have been performed under three different temperatures of 300, 600 and 900 °C.

For the purpose of studying the influence of the micro-defect size on the characteristic parameter, a groove is excavated at the center on the upper surface of γ phase of the model, and the cross section of the groove is semicircular with a section radius of 0.5 nm. The simulation results of microstructure evolution of NSCA at 300 °C are presented in Fig. 1. In the model structure before excavation, the upper layer is γ phase, and the lower layer is γ' phase (Fig. 1a). As the groove continues to extend forward, the atomic motion near the groove is more severe, and a clear chaotic arrangement can be found. However, the Ni3Al radical of the γ' phase never diffuse into the γ phase, as shown in Fig. 1(b-l).

Fig. 1
figure 1

Simulation results of microstructure evolution of NSCA with the extension of groove at 300 °C: (a) the length is 0 nm; (b) the length is 0.1 nm; (c) the length is 0.2 nm; (d) the length is 0.7 nm; (e) the length is 3 nm; (f) the length is 6 nm; (g) the length is 9 nm; (h) the length is 12 nm; (i) the length is 15 nm; (j) the length is 18 nm; (k) the length is 21 nm; (l) the length is 24 nm

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The simulation results of dislocation motion trend of NSCA at 300 °C is shown in Fig. 2, which corresponds to the microstructure evolution in Fig. 1. Obviously, there is no dislocations generating before excavation, as shown in Fig. 1(a). When the groove extends to 0.1 nm, a large number of complex dislocations are apparent in the model (Fig. 2b), and the dislocation density reaches a maximum during the extension process. According to formula (1), when the dislocation density reaches a maximum, the corresponding second-harmonic amplitude also reaches the biggest value. As the trenches continue to expand, some dislocations begin to disappear and the dislocation density begins to decrease, which finally maintains at 5.0 e16/m2-5.2 e16/m2. In the whole process of groove excavation, the square dislocation network is seldom affected by the groove, hence its structure is complete, as shown in Fig. 2(c-l).

Fig. 2
figure 2

Simulation results of dislocation motion trend of NSCA with the extension of groove at 300 °C: (a) the length is 0 nm; (b) the length is 0.1 nm; (c) the length is 0.2 nm; (d) the length is 0.7 nm; (e) the length is 3 nm; (f) the length is 6 nm; (g) the length is 9 nm; (h) the length is 12 nm; (i) the length is 15 nm; (j) the length is 18 nm; (k) the length is 21 nm; (l) the length is 24 nm

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The simulation results of microstructure evolution of NSCA at 600 °C are displayed in Fig. 3. As the temperature increases, the atomic motion is more intense. It can be found that part of the Ni3Al radical of the γ' phase diffuses into the γ phase (Fig. 3a-l). The microstructural defect statuses are more serious compared with those at 300 °C. Since the degree of defect status can affect the change of the second-harmonic amplitude, the second-harmonic amplitude and characteristic parameter are significantly different at 600 °C.

Fig. 3
figure 3

Simulation results of microstructure evolution of NSCA with the extension of groove at 600 °C: (a) the length is 0 nm; (b) the length is 0.1 nm; (c) the length is 0.2 nm; (d) the length is 0.7 nm; (e) the length is 3 nm; (f) the length is 6 nm; (g) the length is 9 nm; (h) the length is 12 nm; (i) the length is 15 nm; (j) the length is 18 nm; (k) the length is 21 nm; (l) the length is 24 nm

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The simulation results of dislocation motion trend of NSCA at 600 °C is shown in Fig. 4, which corresponds to the microstructure evolution in Fig. 3. Overall, as the temperature increases, there is obvious plastic damage in the microstructure, and the dislocation density increases first and then decreases (Fig. 4a-l). The change law of dislocation density at 600 °C is consistent with that at 300 °C, while the microstructure damage at 600 °C is more serious than that at 300 °C. When the groove extends to 0.1 nm, the dislocation motion is the most intense (Fig. 4b). As the groove continues to extend forward, the dislocation motion tends to be stable, and the dislocation density tends to be moderate, which finally maintains at 4.3 e16/m2-4.9 e16/m2. In the whole process of groove excavation, the square dislocation network is greatly affected by the groove, and the larger the groove extension, the more serious damage to the square dislocation network structure. By comparing Fig. 4 with Fig. 2, the increase in temperature has an effect on dislocation motion, resulting in the lower dislocation density in Fig. 4 compared that in Fig. 2 at the steady stage. It can be found from formula (1) that after the temperature rises, the dislocation density becomes smaller, as well as the second-harmonic amplitude.

Fig. 4
figure 4

Simulation results of dislocation motion trend of NSCA with the extension of groove at 600 °C: (a) the length is 0 nm; (b) the length is 0.1 nm; (c) the length is 0.2 nm; (d) the length is 0.7 nm; (e) the length is 3 nm; (f) the length is 6 nm; (g) the length is 9 nm; (h) the length is 12 nm; (i) the length is 15 nm; (j) the length is 18 nm; (k) the length is 21 nm; (l) the length is 24 nm

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The simulation results of microstructure evolution of NSCA at 900 °C are shown in Fig. 5. As the temperature rises to 900 °C, the atomic motion is more severe. Under the influence of the groove, the microstructure is damaged obviously (Fig. 5a-l). The microstructure damage at 900 °C is greater than that at 300 and 600 °C. In the case of increased structural damage, the corresponding second-harmonic amplitude also produce significant differences.

Fig. 5
figure 5

Simulation results of microstructure evolution of NSCA with the extension of groove at 900 °C: (a) the length is 0 nm; (b) the length is 0.1 nm; (c) the length is 0.2 nm; (d) the length is 0.7 nm; (e) the length is 3 nm; (f) the length is 6 nm; (g) the length is 9 nm; (h) the length is 12 nm; (i) the length is 15 nm; (j) the length is 18 nm; (k) the length is 21 nm; (l) the length is 24 nm

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The simulation results of dislocation motion trend of NSCA at 900 °C is shown in Fig. 6, which corresponds to the microstructure evolution in Fig. 5. In general, the most serious degree of plastic damage is at the center of the model, which is significantly greater than that at 300 and 600 °C. Under the influence of high temperature, the dislocation motion is relatively intense at first, and then tends to moderate (Fig. 6a-l). When the groove extends to 0.7 nm, the dislocation motion is the most intense (Fig. 6c), and the corresponding dislocation density is also the largest. As the groove continues to extend forward, the dislocation motion tends to be stable. The dislocation density tends to moderate at the late stage of groove excavation, which finally maintains at 1.11e16/m2-1.43e16/m2. In the whole process of groove excavation, the square dislocation network is most affected by the groove. Moreover, the larger the groove extension, the more serious damage to the square dislocation network structure, which even disappears at the late stage of the excavation. Accordingly, temperature is the main factor affecting the dislocation motion when the excavation groove extends uniformly. The higher the temperature, the bigger the dislocation density, and the smaller the second-harmonic amplitude.

Fig. 6
figure 6

Simulation results of dislocation motion trend of NSCA with the extension of groove at 900 °C: (a) the length is 0 nm; (b) the length is 0.1 nm; (c) the length is 0.2 nm; (d) the length is 0.7 nm; (e) the length is 3 nm; (f) the length is 6 nm; (g) the length is 9 nm; (h) the length is 12 nm; (i) the length is 15 nm; (j) the length is 18 nm; (k) the length is 21 nm; (l) the length is 24 nm

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Based on the above simulation results, the variation curves of stress magnitude as a function of excavation length under different temperatures are shown in Fig. 7. According to formula (1), the magnitude of stress is proportional to the amplitude of second harmonic, hence the variation of stress can reflect the change law of second-harmonic amplitude. The relationship between the stress magnitude and the excavation length at 300 °C is shown in Fig. 7(a). It can be found that the stress increases first and then decreases with the rise of excavation length, and the highest value is 58 nN. The relationship between the stress magnitude and the excavation length at 600 °C is shown in Fig. 7(b). Obviously, the change law between the stress magnitude and the excavation length is the same as that at 300 °C, but the intermediate stress is generally lower than that at 300 °C. The relationship between the stress magnitude and the excavation length at 900 °C is shown in Fig. 7(c). It is clear that the stress at 900 °C is the lowest. Based on formula (1), the lower the stress, the lower the second-harmonic amplitude, while the higher the temperature, the lower the corresponding second-harmonic amplitude and characteristic parameter.

Fig. 7
figure 7

The stress magnitude as a function of excavation length under different temperatures: (a) 300 °C; (b) 600 °C; (c) 900 °C

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The variation curves of the number of dislocations as a function of excavation length are shown in Fig. 8. At the initial stage, the number of dislocations is the most at 900 °C, and the maximum number of dislocations generated by the model is 137 during the whole process of groove excavation. The main reason is that under the action of high temperature, atoms are more likely to obtain energy, atomic motion is more intense, and it is easier to form dislocations. However, the number of dislocations drops sharply at the late stage, which remains at 19-28. At the steady stage, the model produces the largest number of dislocations at 300 °C. There are significant differences in the number of dislocations produced by the model at three different temperatures, but the change laws and stresses are consistent.

Fig. 8
figure 8

The number of dislocations as a function of excavation length

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The variation curves of dislocation length of each typical dislocation as a function of excavation length under different temperatures are shown in Fig. 9-12. At 300 °C (Fig. 9), the dislocation length of each typical dislocation increases first, then decreases, and finally tends to be stable. When the temperature reaches 600 °C (Fig. 10), the variation of the length of each typical dislocation with the excavation length is the same as that at 300 °C. But at the steady stage, the total dislocation length of the dislocations at 600 °C is smaller than that at 300 °C. As the temperature further increases to 900 °C (Fig. 11), the dislocation motion is more severe, and the number of dislocations at the initial stage is large. While at the steady stage, the number of dislocations decreases sharply and the length of dislocation chord decreases. Figure 12 shows the curve of average chord length of the dislocations with the excavation length under different temperatures. It can be found that the higher the temperature, the lower the average chord length, and the smaller the corresponding second-harmonic amplitude.

Fig. 9
figure 9

The dislocation length of each typical dislocation as a function of excavation length under 300 °C

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Fig. 10
figure 10

The dislocation length of each typical dislocation as a function of excavation length under 600 °C

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Fig. 11
figure 11

The dislocation length of each typical dislocation as a function of excavation length under 900 °C

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Fig. 12
figure 12

The average dislocation chord length as a function of excavation length under different temperatures

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The variation curves of dislocation density as a function of excavation length are shown in Fig. 13. Comparing the MD simulation results at 300, 600 and 900 °C, it can be found that the higher the temperature, the lower the dislocation density. Combined with formula (1), it can be concluded that the higher the temperature, the lower the amplitude of corresponding second harmonic.

Fig. 13
figure 13

The dislocation density as a function of excavation length

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4.2 Ultrasonic Experiments

The change law of characteristic parameter has been studied through the ultrasonic experiments. The waveform diagrams of the sample at 26 °C (sample without high-temperature damage) are shown in Fig. 14. The present original signal is given in an oscillography. In addition, signal in frequency domain is also given in a spectrum. As shown in Fig. 14, the transmitted signal is very simple, and an obvious harmonic signal is generated in the received signal. The Lamb wave can be regarded as a composite wave of longitudinal and transverse waves. The particle point is an elliptical trajectory motion, and an elliptical waveform can be found from the waveform diagram. When the frequency is 2.05 MHz, the nonlinear ultrasonic Lamb wave signal generated is the strongest, which has obvious accumulation effect, fully verifying the conditions that the excited Lamb wave meets. Meanwhile, the fundamental frequency Lamb wave corresponds to the amplitude of 19.684. French mathematician Fourier said that nonlinear load is the main cause of nonlinear harmonics. The nonlinear factors of Lamb wave propagation in NSCA include material geometric nonlinearity, mechanical property nonlinearity and microstructure evolution nonlinearity. The ratio of U2f (second-harmonic amplitude) to the square of fundamental frequency Lamb wave amplitude is named as the βk (Ref 34). It can be found from the frequency domain signal diagram that when the frequency is 4.1 MHz, the U2f is 0.0665. The surface roughness of the sample at room temperature (26 °C) is measured, and its three-dimensional topography is displayed in Fig. 15. The roughness (Ra) of the sample is automatically calculated to be 0.0021 um, which is because the sample does not undergo any damage under room temperature, the value of roughness is on the nanometer scale. At this time, the βk is 1.72 e−4.

Fig. 14
figure 14

Lamb wave excitation test for non-damaged sample

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Fig. 15
figure 15

Three-dimensional topography of NSCA at room temperature

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The sample is subjected to a Lamb wave excitation test after a high temperature of 1335 °C. The obtained waveform diagrams are shown in Fig. 16. Similarly, it can be found that a significant harmonic signal is generated in the received signal, and its waveform is elliptical. The signal of Lamb wave is the strongest, and the amplitude of the corresponding primary frequency lamb wave is 19.589. As shown in the frequency domain signal diagram, the amplitude of the second harmonic is 0.1619. After the ultrasonic test, the surface roughness of the sample is measured, as shown in Fig. 17. At this time, the Ra of the sample is calculated to be 0.202 um. It is due to the fact that the sample experiences significant plastic damage under high-temperature conditions, the roughness value enters the micrometer scale, and the βk is 4.22 e−4.

Fig. 16
figure 16

Lamb wave excitation test after the sample is subjected to high temperature at 1335 °C

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Fig. 17
figure 17

Three-dimensional topography of NSCA at 1335 °C

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According to formula (1), it can be found that subjected to high temperature, the second-harmonic amplitude of NSCA is affected by the \(\Lambda\), L, stress, phase velocity mismatch, etc. When the experimental temperature increases, the atom moves violently, and a huge number of atoms gain the energy and enter the interstices between the atoms to become interstitial atoms, resulting in the generating of plenty of point defects and line defects (dislocations). Due to the experimental system that keeps in the higher temperature and then rapidly cools, the defects are largely preserved, leading to an increase in the internal stress rise point of the material and the values of average stress. Furthermore, there is a short-term high temperature progress during solidification, thus the elastic modulus, Poisson's ratio and density of NSCA (DD6) can’t change apparently. Accordingly, the velocities of longitudinal wave and transverse wave that determined by the above three parameters can’t also change, and eventually the phase velocity matching degree remains the same. Compared with NSCA that were not subjected to high temperature, the stress and \(\Lambda\) of the material increase significantly after high temperature. Therefore, the U2f of NSCA increases significantly.

The waveform diagrams of the sample with high-temperature damage at 1386 °C are displayed in Fig. 18. Compared with the sample in Fig. 16, the temperature is increased by 51 °C, and a distinct elliptical harmonic signal can also be generated. As shown in the frequency domain signal diagram, the amplitude of the second harmonic corresponding to the frequency of 4.9 MHz is 0.3054, which increases by 0.1435 in comparison with that in Fig. 16, and the βk also increases by 3.75 e−4. Accordingly, the U2f changes mainly due to the influence of temperature. This is because the increasing temperature leads to an increase of \(\Lambda\), which causes the U2f to increase. Similarly, the surface roughness of the sample is measured after the ultrasonic test, the three-dimensional topography is shown in Fig. 19. At this time, the Ra of the sample is calculated to be 0.48 um, and the βk is 7.97 e−4.

Fig. 18
figure 18

Lamb wave excitation test after the sample is subjected to high temperature at 1386 °C

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Fig. 19
figure 19

Three-dimensional topography of NSCA at 1386 °C

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The waveform diagrams of the sample with high-temperature damage at 1442 °C are presented in Fig. 20. Due to the high-temperature condition (1442 °C) is close to the melting point of NSCA, the material state is between solidification and melting, while it still can receive a distinct elliptical signal, and the harmonic signal is weaker than that in Fig. 18. As shown in Fig. 20, the U2f is 0.2066, and the βk is 5.39 e−4. This phenomenon should be attributed to a decrease in phase velocity matching degree. In addition, the waveform amplitude is not the maximum, indicating that the phase velocity no longer matches. At this time, the surface roughness of the sample is calculated to be 3.58 um, and the three-dimensional topography is shown in Fig. 21.

Fig. 20
figure 20

Lamb wave excitation test after the sample is subjected to high temperature at 1442 °C

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Fig. 21
figure 21

Three-dimensional topography of NSCA at 1442 °C

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The waveform diagrams of the sample with high- temperature damage at 1495 °C are displayed in Fig. 22. It can be found that when the frequency is 4.1 MHz, the amplitude of the second harmonic is 0.1838. Since NSCA is melted between 1500 and 1600 °C, its damage degree increases under the high temperature of 1495 °C, and its mechanical properties also change greatly. The elastic modulus of NSCA (DD6) is tremendously affected by temperature. The higher the temperature, the more energy the atom obtains, and the greater the kinetic energy. When the atom gains sufficient kinetic energy, its motion is very intense, and its ability to get out of balance is strong. The existed research results show that the elastic modulus decreases sharply when the temperature is close to the melting point of the material. When the elastic modulus changes greatly, the velocities of longitudinal wave and shear wave of NSCA change correspondingly. Since the phase velocity dispersion curve and the group velocity dispersion curve are determined by the longitudinal wave velocity, shear wave velocity and frequency-thickness product of the material. After the wave speed changes, the relevant dispersion curve changes accordingly, and the phase velocity matching is naturally reduced. The closer the temperature is to the melting point, the smaller the related amplitude of second harmonic. When the temperature is 1495 °C, the sample undergoes significant plastic deformation, and its surface roughness is bound to produce very large changes. The test result is shown in Fig. 23, and the value of Ra is 8.9 um. At this time, the βk is 4.79 e−4.

Fig. 22
figure 22

Lamb wave excitation test after the sample is subjected to high temperature at 1495 °C

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Fig. 23
figure 23

Three-dimensional topography of NSCA at 1495 °C

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Based on the above analysis, the variation curve of βk with the roughness is shown in Fig. 24. It can be found that the value of βk increases first and then decreases. When the temperature increases from room temperature to 1386 °C, the value of βk increases from 1.72 e−4 to 7.97 e−4 due to the rise of \(\Lambda\) and stress. When the temperature rises to 1495 °C, the material is close to the melting state, and the elastic modulus decreases rapidly, leading to great changes in the velocities of longitudinal wave and shear wave. Therefore, the phase velocity matching degree which plays a major role in the βk decreases rapidly, making the value of βk reduce from 7.97 e−4 to 4.79 e−4. The sample is affected by the increasing temperatures, its plastic deformation is getting larger and larger, thus making the increase in the corresponding roughness. While the value of βk increases first and then decreases. The main reason is that when the plastic deformation is small, the internal stress and dislocation density increase rapidly, leading to the βk increases significantly. At the late stage, due to the large plastic deformation, the surface roughness of the sample is obviously enlarged, which affects the propagation mode of Lamb wave, resulting in a significant decrease in the excitation efficiency of Lamb wave, and the corresponding characteristic value decreases.

Fig. 24
figure 24

The variation curve of βk of the samples with the surface roughness

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After the excitation tests, the high-temperature damaged samples are analyzed by field emission scanning electron microscope (FESEM), as shown in Fig. 25. The defect status of the sample under high-temperature condition of 1335 °C is displayed in Fig. 25(a). There are plenty of microcracks generating under the action of high temperature accompanied by a few of precipitates. The defect status of the sample under high-temperature condition of 1386 °C is shown in Fig. 25(b). Obviously, a large number of precipitates and microcracks exist on the surface of the sample. The defect status of the sample under high-temperature condition of 1442 °C is presented in Fig. 25(c). With the rise of temperature, the microcrack density increases, and some microcracks expand and cross each other, and the crack depth and length grow greatly. The defect status of the sample under high-temperature condition of 1495 °C is shown in Fig. 25(d). As the temperature approaching the melting state of NSCA, the microcrack density increases continuously and is honeycomb-like. Based on the above analysis, the depth, length and density of the cracks increase with the rise of temperature, and different crack patterns correspond to different defect statuses. The βk changes with temperature, which can quantitatively evaluate the defect status of NSCA.

Fig.25
figure 25

Surface state of the sample under different high temperature conditions: (a) 1335 °C; (b) 1386 °C; (c) 1442 °C; (d) 1495 °C

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4.3 Aided Analysis of MD Method

The MD simulation of NSCA with defects from 300 °C to 900 has achieved certain research results. As for nonlinear ultrasonic experiments, MD simulations under different temperatures are carried out, and the defect statuses of the calculated model under different high temperatures are shown in Fig. 26. In the MD simulation calculation results at 1386 °C, the length of dislocations is relatively large (Fig. 26a-b), and the corresponding maximum dislocation length is 1846 Å. In the simulation calculation results at 1442 °C (Fig. 26c-d), the length of the dislocations decreases, and the maximum value of the dislocation length is 1794 Å. As the temperature rising to 1495 °C, the model is in a molten state (Fig. 26e-f), and the length of the dislocations does not change much. The above results show that after the temperature increasing to 1442 °C (NSCA enters the molten state), the change of the dislocation length is no longer sensitive to temperature, and the main factor that affects the βk changes.

Fig. 26
figure 26

Simulation results of the defect statuses of calculated model under different high temperatures: (a) The temperature is 1386 °C, and the simulation time is 100 ps. (b) The temperature is 1386 °C, and the simulation time is 1000 ps. (c) The temperature is 1442 °C, and the simulation time is 100 ps. (d) The temperature is 1442 °C, and the simulation time is 1000 ps. (e) The temperature is 1495 °C, and the simulation time is 100 ps. (f) The temperature is 1495 °C, and the simulation time is 1000 ps

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The variation curves of \(\Lambda\) with the simulation time under different temperatures are shown in Fig. 27. The black curve is the simulation result at the temperature of 1386 °C. It can be found that \(\Lambda\) increases with the rise of high-temperature action time, but the increase amplitude is small. The corresponding maximum \(\Lambda\) is 4.8163 e16 /m−2. Since NSCA is in a state of solidification and melting at 1386 °C, the degradation of mechanical properties presents a cliff-like decline, and the elastic modulus also undergo a large change, which can affect the velocities of longitudinal wave and shear wave of the material. The red curve is the simulation result at the temperature of 1442 °C. \(\Lambda\) increases as the time of high-temperature action rising, but the increase amplitude is small. The corresponding \(\Lambda\) increases from 2.8125 e16/m−2 to 4.6806 e16/m−2. Obviously, the red curve is entirely below the black curve, revealing that \(\Lambda\) tends to decrease with the increase in temperature. According to formula (1), when \(\Lambda\) reduces, the value of βk also decreases, which fully explains the phenomenon in the experiments that the corresponding U2f and βk decrease after the temperature rising from 1386 °C. The blue curve is the simulation result at the temperature of 1495 °C, which is clearly distinguished from the black and red curves and entirely under them. The high-temperature action time has little effect on \(\Lambda\), whose value is very small. The corresponding \(\Lambda\) increases from 2.974 e15/m−2 to 4.8528 e15/m−2 with a small increase in magnitude. The above results show that as the temperature increasing, the change of \(\Lambda\) is no longer sensitive to temperature, and the phase velocity matching degree have a greater effect on βk.

Fig. 27
figure 27

Variation curves of \(\Lambda\) with simulation time under different temperatures

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The variation curves of average L with the simulation time under different temperatures are shown in Fig. 28. Overall, the average L decreases with the increase in temperature, which is similar to the change in \(\Lambda\). The results show that at the early and middle stages of high-temperature damaged samples, \(\Lambda\) and L are the key factors affecting the second harmonic. While at the late stage of damaged samples, the main reason that influences the second harmonic is the decrease in elastic modulus.

Fig. 28
figure 28

Variation curves of L with simulation time under different temperatures

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5 Conclusions

This study establishes an expression of \(\Lambda\), which relates the theoretical relationship between \(\Lambda\) and Lamb wave. Meanwhile, two types of MD simulations of NSCA are established. The conclusions are as follows:

(1) The MD simulation of NSCA with defects is studied. The results show that the dislocation density increases first and then decreases with the increase in the groove length. In the range of 300-900 °C, the higher the temperature, the lower the dislocation density at the steady state of the model, and the lower the corresponding second-harmonic amplitude.

(2) The propagation characteristics of samples subjected to high temperature at 1442 and 1495 °C are greatly affected by surface roughness. Because the temperatures approach the melting point of NSCA, the elastic modulus decreases greatly, the plastic deformation and the surface roughness increase obviously, which seriously affects the propagation mode of Lamb wave, leading to a significant decrease in the excitation efficiency of Lamb wave, and eventually the corresponding U2f and βk decrease.

(3) The results of ultrasonic experiments are analyzed auxiliarily by MD method. After the temperature exceeded 1386 °C, the higher the temperature, the lower the dislocation density, and the lower the corresponding second-harmonic amplitude.

(4) The preparation quality of NSCA could be evaluated by combining the MD method and the characterization of surface roughness.