Experimental Analysis for Nonlinear Time Reversal Imaging of Damaged Materials

Pierre-Yves LE BAS, Koen VAN DEN ABEELE, K.U.Leuven Campus Kortrijk, Kortrijk, Belgium

Serge DOS SANTOS, Thomas GOURSOLLE, LUSSI, Université François-Rabelais de Tours; GIP Ultrasons, Blois cedex, France

Olivier BOU MATAR, European Associated Laboratory , Ecole Centrale de Lille, Villeneuve d'Acsq, France

Abstract. Laboratory studies were performed in order to determine and enhance the possibility to localize microdamage based on a combination of classical time reversal techniques and nonlinear elastic wave response. The use of emitters having a more or less chaotic shape is investigated. Its benefit is clearly found in breaking the symmetry of the problem, leading to an unambiguous retrofocusing. Further, several methodologies for Nonlinear Time Reversal Imaging are proposed, and some of them are illustrated with experimental results for either surface imperfections or body defects. These preliminary experiments confirm that the harmonisation of time reversal techniques with pre- or post-treatment in terms of nonlinear wave phenomena brings about a significant new challenge in the future development of high class imaging tools.

Introduction

Driven by the request from industry for sensitive quantification and localization of microstructural damage, researchers from all over the world have developed innovative techniques that explicitly interrogate the material’s micromechanical behavior and its effect on wave propagation by investigating the amplitude dependence of macroscopically observable properties [1-12]. Such techniques are termed Nonlinear Elastic Wave Spectroscopy (NEWS) techniques. The basis of all NEWS techniques is to measure and analyze macroscopic signatures resulting from a local violation of the linear stress-strain relation at the microscale. This violation of Hooke’s constitutive law can be attributed to a reversible nonlinear dependence of stress on strain, or to a non-uniqueness of the relation depending on the stress or strain rate [13-15]. Several NEWS techniques have been developed to probe the existence of damage induced nonlinearity (e.g., delaminations, microcracks or weak adhesive bonds) by investigating the generation of harmonics and intermodulation of frequency components, the amplitude dependent shift in resonance frequencies, the nonlinear contribution to attenuation properties, the phase modulation, the slow dynamic conditioning and relaxation phenomena, and so on. The concept of NEWS-based methods is that the internal damage can be measured directly with the instantaneous detection of an increase in the nonlinearity parameters. The direct connection between nonlinearity and microdamage in metals and composites is established without doubt in several preliminary experimental case-studies. Tests performed on a wide variety of materials subjected to different microdamage mechanisms of mechanical, chemical and thermal origin, have shown that the sensitivity of such nonlinear methods to the detection

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of microscale features is far greater than that obtained with linear acoustical methods [1-10]. Following the laboratory studies using NEWS techniques, one can underline two important principles: 1) the macroscopically observed nonlinear signatures originate from zones with microdamage and micromechanical nonlinear stress-strain relations; and 2) the nonlinear signatures are most efficiently generated at those locations where the strain within the sample is prevailing. These two principles can be used as the basis for new microdamage visualization techniques based on nonlinear material properties. In particular, we will focus here on the recent efforts in the development of a ‘nonlinear’ version of the well-known Time Reversal (TR) technique.

The basic premise of Time Reversed Acoustics (TRA) is the following [16-17]: If the wavefield can be known as a function of time on some boundary surrounding a given region, then it can also be found at every point inside that region at previous times by using the wave equation with time running backwards. In other words, the result of a time-reversal process is that the waves recorded on the boundary are focused back in space and time on the acoustic sources, or on the scattering targets inside the region that were acting as sources. One of the benefits of TR is that it enables us to locate strong scatterers (voids and interfaces with high impedance contrast) which are hidden inside a region. For more delicate scatterers such as zones of microdamage, weakened bonds or tiny defects, the sensitivity of classical TR has been found to be quite low. The main reason for this is that the principle used in TR applications is based on linear time reversal, and relies on the wave propagation changes due to the linear scattering at inhomogeneities in the material, i.e., reflections, refractions, mode conversions. Our experience with NEWS techniques have demonstrated that microdamage is first of all a process of nonlinear scattering giving rise to the creation of higher harmonics, rather than to “linear” scattering effects. So, from this point of view, the classical TR procedure should be modified in such a way that the main signal treatment is concentrated on the nonlinear components of the signals. One of the alternatives to classical TR consists in selecting only the nonlinear/harmonic energy contained in the response signals and returning merely this part back into the medium by the time reverse mirror. Doing so, the time reversed signal will focus on the microdamaged area, which is the zone where the harmonics were created, while “linear scatterers” will not show up at all.

The concept of merging the benefits of NEWS and Time Reversal has a large potential of application in ultrasonic imaging, particularly in Nondestructive Testing (NDT), and it has been the subject of several investigations since last five years, both from an experimental and a numerical point of view [18-25]. TR and NEWS can be used either with NEWS as a pre-treatment for the TR procedure (NEWS-TR), or with NEWS as a post-treatment for high energy retrofocused signals obtained by linear TR (TR-NEWS). Both methodologies can be implemented in a scanning system for adequate localization of microdamage.

In this paper we will concentrate on certain experimental and practical issues about time reversal techniques, and illustrate the TR and NEWS combination with a few examples. A companion article focuses more on numerical simulations related to Nonlinear TR imaging [26].

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1. Methodology

1.1 Enhancement of TR results using chaotic cavity transducers

One of the drawbacks of TR techniques is the construction of an adequate time reverse mirror. The more information is stored and reversed, the better the retrofocalisation will be. Besides, if the symmetry of the TR principle is broken due to a deviation of the linear physical laws, it may lead to a retrofocusing of energy in another place than expected, i.e. on a virtual source. In addition, the breaking of the initial pulse symmetry can hinder experimental observations with spatial and temporal artifacts leading to diffraction patterns in space (Fresnel zones) and in time (side lobes).

Nevertheless, when dealing with (solid) engineering materials TR, has an important advantage in that it works extremely well in heterogeneous media, actually better than in homogeneous ones. This surprising property is remarkably adapted to the problematic of complex medium analysis, such as layered or composites plates with various geometry. Multiple scattering in transmission experiments or multiple reflections in waveguides, instead of being a hindrance, actually improve the focusing, to below the diffraction limit of about half of a wavelength. This reduces the number of receivers needed to obtain a good reversal quality. Wave distortions caused by large-scale defects can be compensated by time-reversal mirrors because there is no entropy (information) loss as a result of the diffraction. The TR technique is based on the reversibility of acoustic propagation and on the invariance of the Green function with respect to its spatial arguments, which implies that the time-reversed version of an incident pressure field naturally refocuses on its source [17]. The great importance of TR in various applications is the ability to efficiently focus an ultrasonic beam, regardless of the position of the initial source and regardless of the heterogeneity of the propagation medium. Without use of TR, the problem of efficient focusing of propagation fields in heterogeneous objects could be a critical issue for many promising applications[27-32].

As part of our preliminary study to use TR for solid (and plate like) media, we investigated the quality of a single channel TR using a combination of a PZT disk as emitter and a laser vibrometer as receiver. The test sample was an aluminium plate of 150 by 26 by 4 mm. We investigated three cases. In the first case the PZT disc was positioned directly on the sample. In the second and third case a chaotic cavity was used as an intermediate coupling medium between the PZT and the sample. The cavity was made of either plexiglass or steel. We performed the following TR experiment for each ‘transducer’: with the transducer positioned near a corner of the aluminum plate, we generated a pulsed signal, and recorded the out-of plane vibration signal at a fixed location using a laser vibrometer, away from the central axis (5mm from the edge). Subsequently, the recorded signal was sent back repeatedly from the original transducer in time reversed mode. The laser vibrometer finally scans the area of the original signal recording position to determine the level of vibration near the focus. Figure 1a shows the vibration level in a line scan along the width of the plate, crossing the original recording position, for each case considered. One clearly notices the focusing of energy due to a constructive interference at x=5mm. However, for the plain PZT disk, without chaotic cavity, a symmetric image is obtained. An important observation is that the ‘phantom’ image disappears when a chaotic cavity is used. The reason for this is that the cavity is breaking the symmetry of the problem and therefore helps to localize unambiguously the true origin of the signal. Indeed, for systems with a high degree of symmetry, simple reverberating properties would lead to the concentration of virtual sources on a pattern with dimensions correlated to size of the sample (or the transducer) resulting in spatial diffraction figures. Figure 1b illustrates the

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frequency dependence of the TR retrofocused signal. As expected the spatial focus is better for higher frequencies since the wavelength, and thus the diffraction limit, reduces accordingly. In Figure 1d, a 2D scan of the rectangular area indicated in Figure 1c shows the nice spatial focusing at 1MHz as obtained with one of the chaotic cavity transducers.

Figure 1: Single channel TR experiment on a small aluminium plate. The signal used in the TR procedure was recorded at x=5mm a) Retrofocused energy as function of the position along the width of the plate for a plain and two chaotic cavity transducers b) Retrofocused energy as function of the position along the width of the plate for a chaotic cavity transducer operating at three different frequencies c) Set-up d) 2D scan of the retrofocused energy in the rectangular area indicated in figure 1c

1.2 NL-TR methodology

As mentioned in the introduction, it is necessary to adjust the classical TR methodology in such a way that it favors the reconstruction or localization of the nonlinear properties of the medium. One option is to use a high-pass filter on the response signals and returning only the harmonic energy back into the medium by the time reverse mirror. This way, the reinjected signals will interfere constructively on the microdamaged area, which is the zone where the harmonics were created. The harmonic filter assures that only the “nonlinear” scatterers show up, while the “linear scatterers” do not emerge at all. An alternative filtering procedure is based on the fact that the phase inversion of a pulsed excitation signal (180° phase shift) will lead to the exact phase inverted response signal within a linear medium. However, this is not the case in a nonlinear (or microdamaged) material due to the generation of harmonics. We can take advantage of this observation by adding the responses from two phase-inverted pulses (positive and negative) and sending back the sum at the receivers. We call this operation “phase-coded pulse-sequence (PC-PS) filtering” or “pulse inversion filtering”. The sum guarantees that all information on the linear scatterers is filtered out of the signal before time reversing it. Doing so, only the relevant information on the local nonlinearities is reversed and sent back into the material. Again, the filtered energy will interfere constructively on its sources, i.e. the microdamaged zone. Other options may be based on a band-pass filter around the sum- or difference-frequency components (f2±f1) generated as a result of an excitation of a sample at two fundamental

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frequencies f1 and f2. In this methodology called NEWS-TR, NEWS is used as a pre-treatment for TR.

The above mentioned supplementary filtering processes introduced between the acquisition and the time reversal of signals can be of great help to locate ‘nonlinear’ scatterers buried inside a medium. When it is of interest to locate damage near the surface of an object, it is possible to follow a slightly different Nonlinear TR methodology. Colleagues of the 'nonlinear elasticity of materials' group at Los Alamos National Laboratory and Artann Laboratories recently suggested an enhancement of the time reversal procedure to pinpoint surfacial nonlinear scatterers by using NEWS analysis as a post-treatment after the TR process [18,20,21,23,24]. The first step in the procedure is the analogue of classical TR for elastic waves: By measuring the response signals arriving from a single or different sources using a laser vibrometer at a fixed point and returning the received signals from the original transducers in time reversed mode, a high level of energy can be focused locally in time and space on the surface. In the second step, the nonlinear content in the retrofocused signal is analyzed: if a nonlinear scatterer is located at the laser position close to the surface, the harmonic content will be substantial. If, on the other hand, the retrofocusing is performed at an intact position, there will be no sign of nonlinearity. By scanning the surface of the object and repeating the retrofocusing procedure and subsequent nonlinear postprocessing point by point, it is possible to determine the extent of the damage at the surface. In addition to harmonic analysis of the retrofocused signal, one could also perform a pulse inversion (PCPS) filtering, use the intermodulation of two frequencies, investigate the side-lobe energy levels, or use phase modulation analysis.

The different ‘Nonlinear TR’ methodologies are summarized in Figure 2. Note that the term ‘Nonlinear-TR’ can be misleading in the sense that not the TR process is nonlinear, but the pre- or post-treatment of the signals is based on nonlinear wave phenomena.

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Retro-Focusingon the defect

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Figure 2: Diagram of the methodologies for combining Time Reversal techniques and Nonlinear Elastic Wave Spectroscopy. Left: NEWS-TR : News used as pre-treatment for TR; Right: TR-NEWS : NEWS used as post-treatment for TR.

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2. TR-NEWS : NEWS used as post-treatment for TR

In this section, we start by giving two examples of a NL-TR imaging process using classical TR followed by a NEWS analysis to investigate the surface of a plate like structure. The object is a surrogate of an airplane wing panel consisting of an 2mm thick aluminum sheet with stringers connected by rivets. The area of interest is a small zone near a rivet from which a tiny and barely visible crack with a length of about 2 mm has originated during a cyclic tension fatigue test (Figure 3).

In the first experiment, similar to the work of Ulrich et al [23], we use two high frequency signals to excite the medium, and analyse the intermodulation of the retrofocalised signals point-by-point on a line perpendicular to the crack. For the excitation, we use simple PZT disks epoxied to a chaotic cavity and glued onto the aluminum sheet. First a 1MHz signal (f1) is produced at the first source. The response is recorded at a fixed location using the laser vibrometer. Then a 200kHz signal (f2) is sent from second source. Again the laser vibrometer records the signal at the fixed position. Both recorded signals are then time reversed and reemitted from their corresponding original transducers at exactly the same time. In fact, the system performs two times a one channel time reversal, but with an appropriate external synchronisation which also allows to conduct averaging. Doing so, the time reversal principle makes sure that both signals arrive at the same time at the fixed point where the laser picks up the out of plane vibration.. The intermodulation at the focused signal in time is then analyzed in terms of the sum- and difference-components (A(f1-f2) and A(f1+f2)) in the spectrum. This procedure is repeated for all points on a line crossing the flaw position. For an intact location we expect no intermodulation, for a microdamaged zone the intermodulation can be quite high. The nonlinearity signatures contained in the sum- and difference frequencies are visualized in Figure 3 as function of the distance to the crack. At the position of the crack, the intermodulation signature is evidently much larger than elsewhere. The contrast is about a factor of 10.

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Figure 3: Results of a time reversal experiment extended with nonlinear frequency intermodulation analysis, for a cracked wing panel: Point-by-point measurement of the high frequency intermodulation signatures in the retrofocalized time reversed signals as function of the distance to the crack. The frequencies applied are 200kHz and 1MHz. The intermodulation signature corresponds to the sum- and difference-frequency components at the retrofocusing.

The second TR-NEWS implementation is related to the previous one. In fact, in [33], it was shown that the amplitude of a probe wave can be influenced by a superimposed low frequency vibration of the sample. This idea can be transferred to a TR methodology by investigating, point-by-point, the amplitude change of the time reversed signal due to a superimposed low frequency vibration of the sample. We tested this idea experimentally on the same wing panel as in the first example (Figure 4a). The procedure consists of several

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steps, which are repeated point-by-point along a line crossing the flaw. First, we excite the panel with a 1MHz signal from a transducer consisting of a PZT disk mounted on a chaotic cavity. The local response at a fixed position is recorded by a laser vibrometer without the low frequency vibration being present. In the next step, the low frequency vibration is activated. The frequency was tuned to the first flexural mode of the wing panel (60 Hz), and its vibration is supposed to activate the crack. Meanwhile, the 1MHz response signal is returned to the transducer and time reversed (a single channel reversal). Using appropriate synchronization it is possible to record the time reversed signal at different phases during the low frequency vibration. Again the synchronization allows for adequate averaging. The peak-to-peak amplitudes of the TR signals are stored. Finally, the variation in the amplitude of the TR signal during the 60 Hz cycle is deduced from a sinusoidal fit of the recorded data (Figure 4b). This value is clearly a measure of the nonlinear behavior, since the probe amplitude variation due to the low frequency vibration is assumed to be zero at an intact location and non-zero at a microdamaged zones. This procedure is repeated for a point-by-point line scan crossing the crack. The behavior of the amplitude modulation as function of the distance to the flaw is illustrated in Figure 4c. Again, we see a clear indication of the crack position.

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Figure 4: Results of a time reversal experiment extended with low frequency amplitude modulation analysis, for a cracked wing panel: a) Experimental set-up; b) Low frequency modulation (60 Hz) of the amplitude of the retrofocalized TR signal at a fixed position near the crack; c) Point-by-point evolution of the low frequency amplitude modulation in the retrofocused TR signals as function of the distance to the crack.

Another way to interpret TR results for imaging purposes is to analyse the level of the side-lobes generated by the TR process versus the position of the laser scanning. An experiment illustrating this idea was performed on a rectangular sample of 12cm×2,5cm×1 cm with a crack of 1 cm length (figure 5). A 600 kHz tone burst excitation was applied at the left side of the sample, and the coda signal of 2ms duration (corresponding to 12 meters of propagation in the steel sample!) was recorded and time reversed in order to produce a

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33 μs focused TR signal. Due to the multiple reflections at the boundaries, the high number of crack excitations, generated by each propagation of the wave through the crack, result in the fact that TR process is not perfectly obtained. We observed that, the closer the laser is to the crack, the higher is the number of side lobes. Imaging of the crack can be performed by evaluating the energy ratio between the focused signal and the side lobes.

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Figure 5: (a) TR focused signal efficiency versus position of the laser spot in a scanning process near a crack. Note that the compression in time of the TR signal depends on the distance of the laser to the crack. (b) Imaging of the crack can be performed by evaluating the energy ratio between the focused signal and side lobes.

3. NEWS used as pre-treatment for TR imaging

When defects are located deep inside an object, it is generally impossible to apply classical time reversal techniques to locate the defect, unless it produces a high level of reflecting energy. One of the techniques that could work however, is to first filter the signals before time reversing them. The filtering technique should be such that most of the linear wave propagation contribution is eliminated. As mentioned in section 1, several options for NEWS filtering as pre-treatment for TR imaging are possible.

In the following two examples, we verified the NL-TR methodology using PC-PS or pulse inversion filtering. The first experiment was performed on parallelepiped of PMMA glass (used for cockpit windows) containing two laser induced cracks in the center of the sample. We considered several steps in the imaging procedure. First, we excited the sample successively using a pulsed signal (center frequency 140 kHz) with 0 and 180° phase shifts, and we recorded the respective response signals at a fixed location on the surface using a broadband transducer (80-800 kHz). Subsequently, we added up the two response signals, and sent the resulting time history in reversed mode back from the

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receiver position (single channel reversal). A laser vibrometer was then used to measure the signals along a line at the surface across the (deeply buried) crack location. Finally, using standard ultrasonic tomography, the signals were used for imaging a 2D section of the parallelepiped in the direction of the laser scan. Figure 5 shows the repartition of energy at a certain time in the reconstruction. One clearly observes a focal point located at a distance of less than half a wavelength from the center of the cracks. The result is not perfect, but considering it is performed using a one channel reversal and limited information for the reconstruction algorithm, it is encouraging.

Figure 6: Result of a time reversal experiment using a pre-treatment of the signals by means of a PCPS or pulse inversion filtering for PMMA glass with laser induced cracks. Using standard imaging techniques, it is possible to observe a retrofocalisation of the filtered energy near the cracked area

In an other example, we used a similar set-up as in the previous example, consisting of two Panametrics transducers of 1MHz and 2.25 MHz central frequencies and a laser vibrometer. The first transducer is sending out 1 MHz waves: direct and inverted pulses to be used for the subsequent PCPS filtering action. The pulses are received by the second transducer with the higher center frequency. Then, the responses are added and the resulting nonlinear part is time reversed and sent back into the medium by the 2.25 MHz transducer. A scan with the laser vibrometer shows that the focusing maxima is near the damaged region, as expected. Some spectra of the responses are given in Figure 7, demonstrating a significant difference in the harmonic components with respect to the position.

Figure 7: Result of a time reversal experiment using a pre-treatment of the signals by means of a PCPS or pulse inversion filtering for a cracked aluminum plate. Left: experimental set-up; Right: Spectrum of the signal acquired by the Polytec laser near the crack (in blue), on the crack (on green) and near the central hole where a crack has been detected with classical analysis.

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- Reception 0 & 180° - TR Excitation PCPS signal

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Conclusion

Recent work has proven that Nonlinear Elastic Wave Spectroscopy techniques are superior to linear acoustic techniques in diagnosing early stage damage processes in various materials. One of the current challenges is to exploit and extend the successful results of NEWS techniques to defect localization/imaging. In this paper, we demonstrated by means of laboratory experiments that sensitive defect localization can be achieved by adapting the methodology of Time Reversal in combination with nonlinear wave processes. Several methodologies were considered, implementing NEWS as a pre- or post-treatment for TR procedures. We believe that the combination of TR and NEWS will greatly enhance the diagnostic capabilities of both TR and NEWS techniques at many scales.

Acknowledgments

The authors gratefully acknowledge the support of the Flemish Fund for Scientific Research. (G.0206.02, G.0257.02, and G.0554.06), and the European FP6 Grant AERONEWS (AST-CT-2003-502927).

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