Damage Detection of Closed Crack in a Metallic Plate Using Nonlinear Ultrasonic Time Reversal Method

Initial cracks in metallic structures incline to be closed at rest. Such incipient damage generally fails to be detected and located with traditional linear ultrasonic techniques because ultrasonic waves penetrate the contact area of the closed crack. In this paper, an imaging algorithm based on nonlinear ultrasonic time reversal method is proposed to detect closed cracks in aluminumplates. Two surface-bonded piezoelectric transducer arrays are used to generate, receive, and reemit ultrasonic wave signals. The closed crack is simulated by tightening a bolt on the aluminum plate. By applying large amplitude excitation voltage on the PZT transducers, the closed crack could be opened and closed. The transmitted waves recorded by PZT array contain nonlinear components, the signals are time reversed and emitted back, and the tone burst reconstructions are achieved. The linear reciprocity and the time reversibility break downdue to the presence of the nonlinear components.The correlation coefficient between the original excitation signal and the reconstructed signal is calculated to define the damage index for individual sensing path and is used to develop an imaging algorithm to locate the closed crack on the plate.The experimental results demonstrate that incident wave signals and their reconstructed signals can be used to accurately detect and locate closed cracks.


Introduction
In mechanical, aerospace, and civil infrastructures, metallic components made of aluminum are ubiquitous.Failures in metallic structures are often caused by closed cracks developed due to fatigue.In practice, closed cracks usually cannot be avoided.Thus, there is a need in nondestructive inspection for detection of such incipient damage.The closed crack stays in contact and presents the same as linear damage at rest, and only when the excitation overcomes a threshold, the closed crack can be opened and produces nonlinear response, so it is hard to detect closed crack with traditional linear ultrasonic technology.Since the nonlinearity signatures are much more sensitive to small damage features than the measurement of the variations of linear elastic properties [1], it is a very promising field of research to detect nonlinear scatterers in metallic structures using nonlinear ultrasound.However, the nonlinear response is usually small and hard to be detected, and hence imaging the closed crack is a challenging problem.Recently, more and more attentions had been drawn to the damage detection with application of time reversal wave signals.This technique uses the reconstruction property of the time reversal procedure; that is, an original wave can be reconstructed at its source point if its forward wave recorded at another point is time reversed and emitted back to the source point.Time reversal method has been used to focus Lamb wave energy to detect flaws or damages in plates by Ing and Fink [2,3].Park    spot.Although the Lamb wave time reversal technique has been attempted experimentally and shows its effectiveness for detecting certain types of damage, the experiment of imaging closed crack using Lamb wave time reversal has not been reported yet.
This paper starts with a description of the time reversal method.The main part covers the use of the time reversal nonlinear ultrasonic wave signals method for detecting and imaging damage areas with closed crack in aluminum plate.Experimental studies are carried out to examine the proposed method.Finally, this paper concludes with a brief summary.

Time Reversal Process of Wave Signals
The principle of the time reversal process in a twodimensional plate is illustrated in Figure 1, where a tone burst is applied to transducer A functioning as a transmitter (step 1), activating a wave signal that is captured by transducer B acting as a receiver (step 2); the captured signal is time reversed in the time domain (step 3) and reapplied to transducer B; then, the wave signal at transducer A is recorded and time reversed as a reconstructed signal of the original one (step 4).
Based on Figure 1, when a tone burst signal  A () at a central frequency  is applied to transducer A, it activates a wave signal that propagates within the plate.The wave energy generated by transducer A is defined by where  A () is the electromechanical coefficient of transducer A. The wave signals captured by transducer B can be defined by where () is the structural transfer function of the plate from A point to B point. B () is the electromechanical coefficient of transducer B. If  B () =  A () when the same transducers are used for actuator and sensor, then When  B () is time reversed in time domain and applied to transducer B following the same procedure, a wave signal at transducer A is captured: If there is no defect along the wave propagation path between transducers A and B, then, theoretically,   A () is similar to  A () in the time domain, and the shape of the reconstructed signal   A () should be the same as the original input signal  A () after normalization.
But the time reversal process is complicated owing to the multimodal characteristics of Lamb waves.When PZT A is excited with a tone burst signal, multimode signals are received by PZT B. In Figure 1, the narrowband input frequency is selected so that only the first symmetric  0 and antisymmetric  0 modes are generated.When the response signal is reversed in the time domain and reemitted to PZT B, each signal associated with the  0 or  0 mode at PZT B creates both  0 and  0 modes producing a total of four modes in the reconstructed signal.After superposition of signals, the reconstructed signal consists of the main packets in the middle and two side packets around the main mode wave packet.Finally, the shape of the main packets will be practically the same as the original input signal.
Because the time reversibility of waves is fundamentally based on the linear reciprocity of the system [7,8], the linear reciprocity and the time reversibility break down if there exists any source of wave distortion due to wave scattering along the wave path.A nonlinear scatterer in the wave path results in a modification in the wave's central frequency so that () is no longer a scalar, and the shape of the reconstructed   A () after wave interaction with damage will differ from that of wave due to wave distortion.With the original wave signal  A () and the reconstructed wave signal   A () after time reversal in the time domain, a damage index (DI) can be defined by the difference between the two signals.Hence, nonlinear type damages such as closed cracks in metallic structures and delaminations in composite structures could be detected by comparing the discrepancy between the original input signal and the reconstructed signal.

Damage Index and Imaging Method
The damage index can be defined using the characteristics of wave signals in the time domain.In the time domain, the differences between the original incident wave signal . ., V   } of each actuator-sensor path can be measured quantitatively by the correlation coefficient, defined as Before correlation, both  A () and   A () are normalized.The value of  V  ,V   () defines the degree of similarity between the two signals  A () and   A ().When each signal is perfectly approximated by the other,  V  ,V   () = 1.The damage index (DI) is defined as In this way, the higher the DI, the higher the possibility of the existence of damage or the closer the damage to the wave path, as defined by the time reversibility of the input wave signal.
With the existence of damage at a location near the wave path, the effect of the DI is confined with a normal distribution function, which means that the effect of the DI is maximized on the wave path and decreases gradually as the distance from the wave path increases [9]: In (7),  and  were set to 0 and 40 mm, respectively.The value of  was determined in an elliptical zone for each sensing path as  = (  +   )/  − 1, where   is the distance between the actuator and sensor for the th sensing path and   and   are the distances between a grid to the actuator and the sensor, respectively.
Assuming that there are  sensing paths for damage identification from the sensor network, the probability of the presence of damage at position (, ) can be written as [10]  (, ) = where  (,) is the probability of the presence of damage at position (, ) for the th sensing path and DI  and   () are the damage index defined in ( 6) and ( 7) for the th sensing path and the effect of the damage index for the th sensing path on the presence of damage in the position, respectively.This process was applied to each individual path, and (, ) at each grid was obtained for the inspection area.By combining all the paths, the existence of damage was highlighted by an area with high values of (, ).2(a)).An arbitrary-function generator (Agilent33522A) and a highvoltage amplifier (TEGAM2350) are used to control the signal transmitted and an oscilloscope (Agilent D50-X3014A) is used for recording the sensing signals.The geometry of the plate with the PZT array and damage positions is shown in Figure 3. Park et al. [4] discussed the effect of boundary reflections of waves on the time reversal process (TRP).The numerical simulations and experimental tests indicated that the boundary reflection had no substantial effect on the reconstructed wave signal.Through studying a free boundary and with wave-absorbing material around the edges, Song et al. [10] found that the main wave pack was almost the same and the side bands differ slightly after the time reversal process was applied.In this paper, the reflections from structural boundaries are not considered by choosing appropriate time window in the wave signals.
Because of the characteristics of the closed crack, with low amplitude voltage excitation that could not overcome the threshold, the closed crack behaves as a linear scatterer.While high amplitude voltage is applied, once the threshold of the closed crack is overcome, the closed crack can be opened and the nonlinear response is produced.The experiment is conducted as follows.A 3.5 counts 100 kHz Hanning window modulated sine tone burst is used as the excitation signal.First, an amplitude of 50 V peak-to-peak excitation is chosen to actuate one row of PZT transducers (number 1-number 8), one at a time.The eight transducers (number 9-number 16) record the signals.Each recorded signal is time reversed by using appropriate time window.Before reemitting, the timereversed signals are normalized to the same amplitude, so that the contribution of the backpropagated field is approximately the same from each transducer; the backpropagated wave field is recorded in the receiver positions and analyzed.Then, a larger amplitude of 150 V peak-to-peak excitation is applied with the same above steps to record the responses.

Experimental Results
. In this study, signals extracted from three paths (P7-16, P6-12, and P2-12) in the aluminum plate are selected for demonstration.Figure 4 shows the experimental results for the time reversal of the Lamb wave on the plate of the path P7-16 with 50 V peak-to-peak excitation and 150 V peak-to-peak excitation.This path is away from the block and the bolt.The forward waves are captured after propagation of 424 mm by PZT16 (Figures 4(a), 4(b)).(Note: the first wave packet shown in the experimental forward wave is the E/M coupling.)Figures 4(c 6)).It indicates that this path is not influenced by the closed crack and linear scatterer and it can be treated as intact path.
The experiment results of the path P6-12 for the time reversal of the Lamb wave propagating are shown in Figure 6.The block is in this path and the bolt is away from it.In order to extract the nonlinear contributions from the received signal of the path P2-11 with 150 V excitation, we need to quantify the difference between the received signal and a linear reference signal.The linear reference signal is

Figure 1 :
Figure 1: A sketch map of the Lamb wave time reversal process [4].

Figure 5 :
Figure 5: Original tone burst and reconstructed tone burst after time reversal procedure (path P7-16): (a) 50 V excitation and (b) 150 V excitation.Blue line: original and red line: reconstructed.

Figure 7 :
Figure 7: Original tone burst and reconstructed tone burst after time reversal procedure for path P6-12: (a) 50 V excitation, (b) 150 V excitation.Blue line: original and red line: reconstructed.
) and 4(d) show the reconstructed signals which are normalized.The main packets of the reconstructed and the original tone burst signals are shown in Figures 5(a) and 5(b).The reconstructed signals with low amplitude and high amplitude of excitation all achieved a similarity of 98% to the original tone burst signals (according to (

Figures 6 (
Figures 6(a) and 6(b) show the waves captured after propagation of 420 mm by PZT12 with low and high excitations, respectively.The reconstructed waves are shown in Figures 6(c) and 6(d); the main wave packets in the reconstructed wave resemble well the original tone burst.Figures 7(a) and 7(b) show the reconstructed and the original tone burst signals for the two excitations; the similarity of the reconstructed signals to the original tone burst signals is the same as the intact path (P7-16); it indicates that the linear scatterers had not broken down the linear reciprocity and the time reversibility; the existence of linear damage would not influence the detection of closed crack.

Table 2 :
DI of different paths.
4.1.Experimental Setup.The overall test configuration of this study is shown in Figure2.A 1000 mm * 1000 mm * 2 mm aluminum plate is adopted to evaluate the method presented in this paper.A sensor network is created consisting of 16 circular PZTs (APC 850) with a distance of 60 mm between the elements.Elements (P1 to P16), 6.35 mm in diameter and 0.25 mm in thickness, were surface mounted on the aluminum plate.Their positions are listed in Table1.A steel block with the diameter and height of 15 mm and