## Abstract

Nonlinear Lamb waves including second harmonic and acoustic-radiation-induced quasi-static components (QSC) have a potential for accurately evaluating early-stage fatigue damage. Most previous studies focus on second-harmonic-based techniques that require phase velocity matching and are hard to isolate interferences from ultrasonic testing systems. The aforementioned requirement and deficiency limit applications of the second-harmonic-based techniques. In this study, a QSC-based technique of low-frequency Lamb waves is proposed for early-stage fatigue damage evaluation of metal plates, which does not need to require phase velocity matching and can remove interferences from ultrasonic testing systems. Both in simulations and in experiments, the primary Lamb wave mode at a low frequency that meets approximate group velocity matching with the generated QSC is selected. In finite element simulations, different levels of material nonlinearities by changing the third-order elastic constants are used to characterize levels of fatigue damage. Numerical results show that the magnitude of the generated QSC pulse increases with the levels of fatigue damage. Early-stage fatigue damage in aluminum plates with different fatigue cycles is further experimentally evaluated. The generated QSC pulse is extracted from received time-domain signals using the phase-inversion technique and low-pass digital filtering processing. The curve of the normalized relative acoustic nonlinearity parameter versus the cyclic loading number is obtained. Numerical simulations and experimental results show that the early-stage fatigue damage in aluminum plates can effectively be evaluated using the QSC generated by low-frequency Lamb waves.

## 1 Introduction

Metal plate-like structures have been widely used in many practical fields such as aerospace and nuclear engineering. Under repeated mechanical loading in practical applications, microstructural damage such as fatigue damage [1–4], plasticity damage [5,6], and micro-cracks [7–11] would easily appear in metal plate-like structures. These early-stage damage incidents can gradually evolve into macro-damage like crack or small local rupture, and then weaken the integrity of structures used, and finally lead to structural failures and irreparable results. Therefore, it is significant to develop an effective evaluation method for the early-stage damage of metal plate-like structures.

Various ultrasonic methods have been developed for microstructural damage detection in metal structures over the last few decades. These developed methods include non-destructive evaluation (NDE) methods based on traditional linear ultrasound, nonlinear ultrasound, acoustic emission, etc. Compared with the traditional linear ultrasonic methods, nonlinear ultrasonic methods are more sensitive to various types of microstructural damage. Therefore, more and more studies of NDE focus on nonlinear ultrasonic methods including second-harmonic generation [1–14], mixing-frequency effect [15–19], and acoustic-radiation-induced quasi-static component (QSC) [20,21]. It is known that Lamb waves have the advantage of long propagation distances with less attenuation in plate-like structures and are sensitive to various types of damage [22]. For effectively evaluating the early-stage damage in plate-like structures, the nonlinear Lamb wave-based technique that combines the advantages of nonlinear ultrasound and Lamb waves has attracted more and more attention. Specifically, damage evaluation methods based on second-harmonic generation of Lamb waves have been developed for over two decades. Deng [23–25] theoretically investigated the nonlinear effect of Lamb wave propagation in plate-like structures. Two necessary conditions (phase velocity matching and non-zero energy flux) for generation of the cumulative second harmonic of Lamb wave propagation are respectively stated by Deng [24] and de Lima et al. [26]. Based on the aforementioned research findings, the characteristics of generated second-harmonic Lamb waves have been widely and deeply studied. Many numerical simulations [13,14] and experiments [1–6] aimed at the early-stage damage detection of plate-like structures based on second-harmonic Lamb waves have also been implemented. However, due to the fact that the two necessary conditions for the generation of the cumulative second harmonic must be satisfied, the primary Lamb wave mode and the driving frequency that can be used for cumulative second-harmonic generation are very limited. Additionally, interferences from the second harmonics induced by ultrasonic testing systems are very difficult to avoid [27], which are easy to cause inaccurate test results. Therefore, these deficiencies will limit applications of the second-harmonic Lamb waves-based methods in practice.

In recent years, the QSC generated by ultrasonic wave propagation in elastic media with weak quadratic nonlinearity has attracted extensive attention. Qu et al. [20] and Nagy et al. [21] investigated the generation of the QSC pulse by the propagation of a longitudinal wave tone burst in an elastic medium with quadratic nonlinearity and found that the generated QSC pulse exhibits a flat-top shape with magnitude proportional to the propagation distance. Sun et al. [28–30] further investigated the generation of the QSC of primary Lamb wave propagation in an elastic plate and proposed an experimental approach for evaluating the local plastic damage. The QSC generated by primary wave propagation is essentially a kind of secondary wave. Previous studies have shown that the cumulative effect of the generated QSC occurs at any frequency of all Lamb wave modes [28], which is different from the generation of the cumulative second harmonic of Lamb wave propagation where the phase velocity matching must be required [23–26]. Wan et al. [31] investigated the QSC generation of primary Lamb wave propagation in an elastic plate with quadratic elastic nonlinearity from a numerical perspective. Recently, Deng [32] proposed an experimental approach to directly detect the generated QSC pulse of a primary longitudinal wave tone burst in elastic media. Gao et al. [33] reported the experimental observation of the generated QSC pulse of the specific Lamb wave mode whose group velocity is equal to that of the S0 mode at zero frequency.

Early fatigue damage refers to the initiation and development of microscopic cracks in a material under cyclic loading, which occurs before the macroscopic failure or visible damage of the material [34]. It is a critical phenomenon in the fatigue behavior of materials, as it can significantly reduce the fatigue life of a component or structure. Evaluation of early-stage fatigue damage is an important issue for structural health monitoring. Most previous studies mainly focused on the methods using the second harmonic of Lamb wave propagation. By comparison, the investigations on the QSC-based method are relatively few although the generated QSC pulse has lower propagation attenuation and is immune to the effect of the instrument-induced nonlinearity [35]. Since there are only two Lamb wave modes (A0 and S0) at low-frequency range and the signal processing for low-frequency Lamb waves is relatively simple, this paper focuses on analyzing the QSC generation of low-frequency Lamb waves and then applies it for evaluation of early-stage fatigue damage in metal plates. The rest of this paper is organized as follows. Theoretical analyses are briefly introduced in Sec. 2. Finite element (FE) simulations are presented in Sec. 3. Experimental investigations are performed in Sec. 4, and conclusions are drawn in Sec. 5. The main contribution of this paper is to provide a method for evaluation of the early-stage fatigue damage using the QSC generated by low-frequency Lamb waves.

## 2 Generation of Quasi-Static Component Pulse

When a one-dimensional (1D) primary longitudinal wave tone burst with carrier frequency *f* propagates in an elastic medium with quadratic elastic nonlinearity, the second-order bulk forces with a finite duration will be induced and distributed along the propagation path of the primary wave tone burst. These second-order finite-duration bulk forces can be regarded as the driving sources that generate the corresponding secondary-wave pulses include the second-harmonic pulse (whose carrier frequency is 2*f*) and the QSC pulse (whose carrier frequency is 0*f*) [20,21,35]. The generated second harmonic has drawn considerable attention in NDE and structural health monitoring [1–14]. In the last decade, the effect of the QSC generation is also getting more and more attention. Qu et al. [20] and Nagy et al. [21] theoretically analyzed the generation of the QSC pulse of the 1D primary longitudinal wave tone burst propagating in an isotropic and homogeneous medium without wave dispersion and attenuation. Under the condition of prescribed displacement boundary, the QSC displacement pulse generated by the 1D primary longitudinal wave tone burst propagating in an isotropic and homogeneous medium can be expressed as (*βU*^{2}*ω*^{2}*z*/8*c*^{2})*P*(*t* − *z*/*c*) [20], where *β* is the acoustic nonlinearity parameter, *U* is the primary wave displacement amplitude, *ω* = 2*πf* is the angular frequency, *P*(*t*) is the modulation window function of the 1D primary longitudinal wave tone burst, and *z* and *c* are, respectively, the propagation distance and the longitudinal wave velocity. It is apparent that the magnitude of the generated QSC pulse increases proportionally with the propagation distance *z* and the acoustic nonlinearity parameter *β*. Besides, the generated QSC pulse exhibits a time-domain shape similar to the envelope of the primary longitudinal wave tone burst. Under the case of the 1D longitudinal wave propagation, the accumulation of the generated QSC pulse is only related to the propagation distance *z* when the parameter *β* is prescribed.

The above briefly analyzes the QSC generation of the 1D primary longitudinal wave propagating through an elastic medium with weak elastic nonlinearity. However, due to the dispersion and multimode features of Lamb wave propagation, the QSC generation of the primary Lamb wave tone burst will exhibit greater complexity. It is known that the QSC can be generated from an arbitrary Lamb wave mode at any frequency without the requirement of phase velocity matching and that the time-domain shape of the generated QSC pulse is closely associated with the group velocity difference between the primary Lamb wave and the S0 mode at zero frequency [28,31,33,35,36]. The brief analysis of the QSC generation of primary Lamb wave tone burst will be given in the following contents.

Analogous to the analysis process of the 1D longitudinal wave tone burst, the second-order finite-duration driving sources (including bulk forces in the interior of the plate and surface tractions on the upper and lower surface of the plate) will be induced and distributed along the propagation path of the primary Lamb wave tone burst (with carrier frequency *f* and duration *τ*), whose carrier frequency can be zero frequency (i.e., 0*f*) or double the fundamental frequency (i.e., 2*f*) [36]. Here, we only consider the effect of the second-order finite-duration driving sources with carrier frequency 0*f*, and no longer consider that with carrier frequency 2*f* [35,36].

First, the case of group velocity matching (i.e., $cg[f]=cg[0]$) is analyzed, where $cg[f]$ and $cg[0]$ are, respectively, the group velocities of the primary Lamb wave and the S0 mode at zero frequency. For ease of analysis, the propagation path of the primary Lamb wave tone burst is discretized as a series of equally spaced points, like that shown in Fig. 1. When the primary Lamb wave tone burst passes through an arbitrary point (e.g., the *i*th point *z*_{i}), the corresponding QSC pulse (denoted by $Ai[0]$) will be generated due to the second-order finite-duration driving sources with carrier frequency 0*f* [36]. The same phenomenon will take place in its adjacent point; i.e., the QSC pulse (denoted by $Ai+1[0]$) will be generated when the primary Lamb wave tone burst passes through the (*i* + 1)th point *z*_{i+1}. This physical process will continue until the primary Lamb wave tone burst arrives at the end point *z* = *z*_{N}. Due to the group velocity matching (i.e., $cg[f]=cg[0]$), all the QSC pulses $Ai[0]$ (*i* = 0, …, *N*) generated at each point will simultaneously arrive at *z* = *z*_{N}. The total QSC pulse generated by the primary Lamb wave tone burst propagating through the path from *z*_{0} to *z*_{N} can be determined as $A[0](t)=\u2211i=1i=NAi[0]$, as schematically shown in Fig. 1 [36]. For the case of the exact group velocity matching, it should be pointed out that the time-domain shape of the generated QSC pulse *A*^{[0]}(*t*) is similar to the time-domain envelope of the primary Lamb wave tone burst and that the magnitude of *A*^{[0]}(*t*) (i.e., *A*^{[0]}), as well as the integrated amplitude of *A*^{[0]}(*t*) (i.e., the area of the trapezoid $P1P2P3P4\xaf$ in Fig. 1), will grow linearly with the propagation distance [31,33,36].

Next, the case of group velocity mismatching (i.e., $cg[f]\u2260cg[0]$) will be analyzed. Figure 2 illustrates the physical process of generation of the QSC pulse of the primary Lamb wave tone burst under the case of $cg[f]<cg[0]$. A similar analysis under the case of $cg[f]>cg[0]$ will no longer be repeated. Compared with the case of $cg[f]=cg[0]$, the only difference is that the QSC pulses generated at each point *z*_{i} (*i* = 0, …, *N*) no longer simultaneously arrive at the endpoint *z* = *z*_{N}. Specifically, there is a time delay $(zi+1\u2212zi)(1/cg[f]\u22121/cg[0])$ between the QSC pulse generated at *z* = *z*_{i} and that generated at *z* = *z*_{i+1}. Similarly, the total QSC pulse of the primary Lamb wave tone burst is equal to the summation of the QSC pulses $Ai[0]$ (*i* = 0, …, *N*) generated at each point *z*_{i}. Clearly, the duration of the generated QSC pulse *A*^{[0]}(*t*) is from $t=(zN\u2212z0)/cg[0]$ to $t=\tau +(zN\u2212z0)/cg[f]$ (see Fig. 2) [36]. Figure 2 schematically illustrates that the time-domain shape of the generated QSC pulse *A*^{[0]}(*t*) received at *z* = *z*_{N} exhibits a trapezoidal shape $(P1P2P3P4\xaf)$. Although the magnitude of the generated QSC pulse *A*^{[0]}(*t*) (i.e., *A*^{[0]} in Fig. 2) is no longer linearly proportional to the propagation distance, it is found that the integrated amplitude of *A*^{[0]}(*t*) (i.e., the area of the trapezoid $P1P2P3P4\xaf$ in Fig. 2) still increases linearly with the propagation distance under the case of $cg[f]\u2260cg[0]$ [31,36].

Accompanying propagation of the primary Lamb wave tone burst, the process of generation of the QSC pulse has been schematically analyzed under the cases of both $cg[f]=cg[0]$ and $cg[f]\u2260cg[0]$. Generally, the shape of the generated QSC pulse is closely associated with the difference between $cg[f]$ and $cg[0]$. When the duration of the primary Lamb wave tone burst (i.e., *τ* in Figs. 1 and 2) is given, the larger the difference between $cg[f]$ and $cg[0]$ is, the longer the duration of the generated QSC pulse is (see Fig. 2). Considering the fact that the QSC component of primary Lamb wave propagation is associated with the third-order elastic constants [28,31,33], the damage level of material can be evaluated by measuring the generated QSC pulse of primary Lamb wave tone burst [35]. In practical applications, when a low-frequency piezoelectric transducer is used to directly detect the generated QSC pulse, it is suggested to select a primary Lamb wave whose group velocity is equal to or close to that of the S0 mode at zero frequency [33,35]. In this case, by appropriately adjusting the duration (or cycle number) of the primary Lamb wave tone burst, the generated QSC pulse can be effectively detected using the given low-frequency piezoelectric transducer [32,33].

## 3 Finite Element Simulations

To validate the earlier analyses concerning the effect of the QSC pulse by primary Lamb wave tone burst, FE simulations will be implemented with the commercial FE modeling software (comsol multiphysics^{®}, Burlington, MA), where the hyperelastic material module is used. The material parameters of the aluminum plate examined are listed in Table 1, where *ρ* is the mass density, *λ* and *μ* are the Lame's constants, and *l*, and *m* and *n* are the third-order elastic constants used to define the “nonlinearity” (NL) of material. Based on the material parameters given in Table 1, dispersion curves of phase and group velocities of Lamb waves in a 2.64 mm thick aluminum plate are numerically calculated and shown in Fig. 3. It can readily be seen that there are only two Lamb wave modes (S0 and A0) at the low-frequency range, and only the S0 mode at a low frequency approximately satisfies group velocity matching with the QSC (namely the S0 mode at zero frequency) while the A0 mode does not.

ρ (kg/m^{3}) | λ (GPa) | μ (GPa) | l (GPa) | m (GPa) | n (GPa) |
---|---|---|---|---|---|

2700 | 51.08 | 26.32 | −250 | −330 | −350 |

ρ (kg/m^{3}) | λ (GPa) | μ (GPa) | l (GPa) | m (GPa) | n (GPa) |
---|---|---|---|---|---|

2700 | 51.08 | 26.32 | −250 | −330 | −350 |

As shown in Fig. 4, a two-dimensional FE model of an aluminum plate is set to be 2.64 mm in thickness and 3000 mm in length, which is long enough to avoid wave reflection by the right-side end. The upper and lower surfaces of the given plate are set to be stress-free, and the fixed boundary condition (BC) is applied on the right-side end to eliminate the rigid displacement. A prescribed displacement BC is horizontally applied to the left-side end of the given plate as the excitation signal, which is the Hanning-windowed sinusoidal tone burst consisting of 30 cycles with a central frequency of *f* = 0.265 MHz and an amplitude of 1 × 10^{−4} mm. It is noteworthy that the excitation of prescribed displacement BC is asymmetrical along the thickness direction, which means that both the symmetric (S0) and antisymmetric (A0) modes can be generated simultaneously. The FE elements size is set to be 0.2 mm, which is smaller than the value of *λ*_{min}/20, where *λ*_{min} is the shortest wavelength of interest. The time-step is set to be 0.01 *μ*s, which is smaller than 1/(20*f*_{max}), where *f*_{max} is the largest frequency of interest.

The simulation results are shown in Fig. 5. The time-domain signals of in-plane displacement detected by the probe located at the upper surface are shown in Fig. 5(a), where the probe is 900 mm, 960 mm, and 1020 mm away from the left-side end. The first time-domain signals detected at different distances (e.g., *z* = 900 mm and 1020 mm) have a time delay of Δ*t*_{1} = 22.7 *μ*s, through which the corresponding group velocity is calculated to be Δ*z*/Δ*t*_{1} = 5.286 km/s (Δ*z* = 120 mm), which is very close to the theoretical group velocity of the S0 mode at 0.265 MHz (5.274 km/s). Similarly, the group velocity of the second time-domain signal can be calculated to be Δ*z*/Δ*t*_{2} = 3.080 km/s (Δ*t*_{2} = 38.96 *μ*s), which is very close to the theoretical group velocity of the A0 mode at 0.265 MHz (3.052 km/s). Thus, it is convinced that the first and second time-domain signals shown in Fig. 5(a) are respectively the S0 and A0 modes at 0.265 MHz.

It is known that the phase-inversion technique [36] can be applied to effectively extract the even-order harmonics (including QSC and second harmonic generated [20]) from the original time-domain signals. Specifically, the superposition of the two time-domain signals of in-plane displacement obtained from 180-deg out-of-phase inputs results in the cancellation of the primary term and doubles the even-order harmonic term since even-order harmonic terms have a square relationship with the input signal [20]. At a given probe point (e.g., *z* = 900 mm, 960 mm, or 1020 mm), the superposition of two time-domain signals incident with the input phase of 0deg and 180deg remains the even-order harmonics, as shown in Fig. 5(b). Here, a low-pass digital filtering processing (0–100 kHz) is applied to the signals of Fig. 5(b) to extract the time-domain signals of the QSC generated, as shown in Fig. 5(c). The group velocity of the QSC pulses in Fig. 5(c) is calculated to be 5.333 km/s, which is very close to the theoretical group velocity of the S0 mode at zero frequency (5.394 km/s) with an error of 1.13%. This just goes to show that the carrier of the generated QSC pulse is exactly the S0 mode at zero frequency.

Through the results shown in Figs. 5(a)–5(c), it can readily be found that the magnitude of the generated QSC pulse due to the S0 mode at 0.265 MHz is far more pronounced than that due to the A0 mode at 0.265 MHz; the latter cannot almost be observed. This should be attributed that the group velocity of the A0 mode at 0.265 MHz clearly does not match that of the S0 at zero frequency [36]. Moreover, due to approximate group-velocity matching between the S0 mode at 0.265 MHz and the S0 mode at zero frequency (see Fig. 3(b)), Fig. 5(c) shows that the maximum magnitude of the QSC pulse almost linearly increases with the propagation distance, which is consistent with the theory analysis shown in Sec. 2.

Here, a scale coefficient (*α* > 1) is introduced to conveniently describe changes in the level of material damage. When the early-stage material damage takes place, the third-order elastic constants increase from (*l*, *m*, *n*) to (*αl*, *αm*, *αn*), through which the level of material damage can be conveniently described only using one scale parameter *α* [36]. Figure 5(d) shows the time-domain waveforms of the generated QSC under different scale coefficients *α* detected by the probe at *z* = 900 mm. It can be observed that the maximum magnitude of the QSC pulse increases proportionally with the scale coefficient. In other words, the magnitude of the QSC pulse increases with the level of early-stage fatigue damage. The FE simulation results lay a foundation for the subsequent experimental evaluation of early-stage fatigue damage using the QSC generation of the low-frequency S0 mode.

## 4 Experimental Examinations

### 4.1 Experiment Preparation.

Ultrasonic experiments on aluminum plates are also implemented to validate the effectiveness of evaluation for early-stage fatigue damage of metal plates using the QSC of low-frequency Lamb waves. As is well known, under given fatigue loading conditions, there is a corresponding relationship between the fatigue (or damage) state and the number of cycles of loading [37]. Therefore, this paper represents the damage state in terms of the number of cycles under the given fatigue loading conditions. To obtain aluminum plates with different fatigue levels, fatigue tests are conducted on 2.64-mm-thick aluminum plate specimens, as schematically shown in Fig. 6, where the geometric parameters are set to be *H*_{1} = 100 mm, *H*_{2} = 50 mm, *L*_{1} = 336 mm, *L*_{2} = 130 mm, and *R* = 50 mm. All fatigue tests are conducted at room temperature under a laboratory environment using a 100 kN servohydraulic testing system (MTS 809.10). At first, a static tensile test on an intact specimen is carried out to obtain the yield stress of aluminum. The force–displacement curve is shown in Fig. 7, through which the yield stress is found to be around 210 MPa. Then, the nine intact specimens like that given in Fig. 6 are prepared. One intact specimen is used as the benchmark, while the other eight ones are subjected to fatigue loading under different cycles to bring in different levels of fatigue damage. Fatigue tests are conducted under the stress-control mode using a sinusoidal waveform with a frequency of 20 Hz, and the specimens are subjected to cyclic tensile stress with a maximum load of 13.2 kN and a minimum load of 1.32 kN with a stress ratio of 0.1. The cyclic numbers of fatigue loading of the eight damaged specimens are set to be 30, 50, 70, 90, 110, 130, 150, and 170 thousand, respectively.

The QSC-based damage detection experiments are then performed on the specimens. Based on the FE results in Sec. 3, the S0 mode at 0.265 MHz is selected as the primary wave mode. The experimental system of Lamb waves is shown in Fig. 8. Same as the FE simulations in Sec. 3, the excitation signal applied on the narrowband longitudinal piezoelectric transducer *T*_{x} with a central frequency of 0.5 MHz (Model A413, Panametrics Inc., Waltham, MA) is a 30-cycle Hanning-windowed sinusoidal tone burst voltage, amplified by a high-power gated amplifier (Model SNAP 5000, Ritec Inc., Warwick, RI). The tone-burst voltages generated by a high-power gated amplifier pass through the high-power adjustable attenuators to suppress the transient behavior of the radio frequency amplifier. A signal sampler with 40 dB attenuation connected to the digital oscilloscope (Model DPO3014, Tektronix Inc., Beaverton, OR) is used to sample the excitation signal applied on *T*_{x}. The ultrasonic signals received by the broadband longitudinal piezoelectric transducer *R*_{x} with a central frequency of 0.5 MHz (Model V413, Panametrics Inc.) are fed into the digital oscilloscope for further post-processing. Both *T*_{x} and *R*_{x} are respectively installed on two oblique plexiglass wedges with the same oblique angle *θ*. The oblique angle *θ* is determined by Snell’s law, i.e., $\theta =sin\u22121(cL/cp[S0])$, where *c*_{L} is the longitudinal wave velocity of the plexiglas wedge (*c _{L}* = 2720 m/s) and $cp[S0]$ is the phase velocity of the desired S0 mode at 0.5 MHz $(cp[S0]=5355m/s)$. Thus, the oblique angle

*θ*is calculated to be 30.1 deg. The spatial separation

*L*between

*T*

_{x}and

*R*

_{x}is set to be 192 mm.

### 4.2 Results and Discussions.

When the phases of the Hanning-windowed sinusoidal tone burst voltages applied on *T*_{x} are set to be 0 and *π*, the corresponding ultrasonic signals received by *R*_{x} are shown in Fig. 9 (denoted by *u*^{[0]} and *u*^{[π]}), where their peak values are denoted by $A1[S0]$. In order to isolate the waves reflected from the plexiglas wedges and to obtain pure wave packets, only the time-domain signals within the time range of 40−180 *µ*s are extracted and used for further post-processing.

The phase-inversion technique [36] is used to extract the QSC. Figure 10(a) shows the averaged time-domain signal $u\xaf=(u[0]+u[\pi ])/2$, where the quadratic nonlinear signals including QSC and second-harmonic components are retained, while the fundamental frequency component is almost eliminated. Next, the averaged time-domain signal $u\xaf$ is processed by a low-pass digital filtering processing (0−200 kHz) to extract the QSC, as shown in Fig. 10(b). It can be observed that the QSC pulse is generated mainly in the duration of the primary Lamb wave signal, which is consistent with the theoretical prediction and FE simulation. Here, the integrated amplitude defined by $A\xaf[QSC]=\u222bt1t2E[QSC](t)dt$ is used to quantify the efficiency of the QSC generation, where the function *E*^{[QSC]}(*t*) represents the time-domain envelope of the QSC pulse (see in Fig. 10(b)) [33].

Early-stage fatigue damage will affect the elastic nonlinearity of material and then affect the efficiency of the QSC generation of primary Lamb wave propagation, characterized by the value of $A\xaf[QSC]$ [33]. Furtherly, the relative acoustic nonlinearity parameter of QSC is defined by $\beta [QSC]=A\xaf[QSC]/(A1[S0])2$, and it can be used to evaluate the level of early-stage fatigue damage [33]. In order to investigate the change trend of linear $(A1[S0])$ and nonlinear (*β*^{[QSC]}) parameters on the level of damage, the same ultrasonic measurements like Figs. 9 and 10 are repeatedly conducted for one intact specimen and eight damaged ones. For each specimen, the values of $A1[S0]$ and *β*^{[QSC]} can readily be obtained and normalized with respect to the intact specimen. Figure 11 shows the data of the normalized $A1[S0]$ and *β*^{[QSC]} versus the number of cyclic loading. It can be seen from Fig. 11 that the fitting curve of the data of *β*^{[QSC]} shows a mountain shape. The relative acoustic nonlinearity parameter *β*^{[QSC]} first increases with the number of cyclic loading and then reaches the maximum when the number of cyclic loading is around 130 thousand. After that, *β*^{[QSC]} decreases with the number of cyclic loading, which may be attributed to the reduction of the precipitates volume fraction and the dislocation density after such fatigue cycles [38]. Compared with the change trends of *β*^{[QSC]} and $A1[S0]$, the latter behaves in a small range of change with the number of cyclic loading, which means that the QSC-based method is more sensitive to early-stage fatigue damage than linear ultrasound.

It can be also seen from Fig. 11 that there is a certain extent of discreteness of *β*^{[QSC]} as well as $A1[S0]$, which may be attributed to the difference of acoustic coupling conditions both of *T*_{x} and *R*_{x} in repeated measurements. The experimental results show that the relative acoustic nonlinearity parameter has a significant response upon the level of early-stage fatigue damage, where the acoustic nonlinearity parameter increases with increasing fatigue cycles while the amplitude of primary waves barely changes. The experimental results show that the QSC-based method of low-frequency Lamb waves has the potential for accurately evaluating early-stage fatigue damage of metal plates.

## 5 Conclusions

This paper focuses on evaluating the early-stage fatigue damage in metal plates using generated QSC of low-frequency Lamb waves. First, the generation and propagation mechanism of the generated QSC of primary Lamb wave propagation are schematically analyzed. Second, the characteristic changes of the generated QSC with respect to different levels of early-stage fatigue damage are studied numerically. The FE simulation results show that when the group velocity of the primary S0 mode approximately matches that of the S0 mode at zero frequency, the magnitude of the generated QSC pulse increases linearly with changes in the third-order elastic constants. On this basis, ultrasonic measurements are conducted on the aluminum specimens with different levels of fatigue damage induced by continuous cyclic fatigue loading. The experimental results show that the changing trend of the normalized relative acoustic nonlinearity parameter of QSC is sensitive to levels of fatigue damage induced by cyclic loading. This paper validates the potential that the QSC-based method of low-frequency Lamb waves can be used to effectively evaluate the levels of early-stage fatigue damage in metal plates.

## Funding Data

The study is funded by The National Natural Science Foundation of China (Grant Nos. 12134002, 12074050, 52005058, and 11834008).

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The data sets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.