Laser ultrasonic monitoring of reversible/irreversible modification of a real crack under photothermal loading

Abstract

We study the responses of laser-generated acoustic waves to localized reversible/irreversible modifications of microscopic asperities on crack surfaces during crack closure, which is an essential process in nonlinear photoacoustic/photothermal crack detection techniques. Our laser ultrasonics technique involves optical measurement of the transmission and mode conversion of the laser-generated surface acoustic waves caused by the crack. Reversible/irreversible modifications of asperities can be achieved via non-contact photothermal loading of the crack. Three photothermal loading cycles were realized in individual succession and were monitored using the laser ultrasonics technique at various experimental locations along the crack. In our experiments, each photothermal loading cycle includes multiple successive subcycles, in which the material is first heated and then cooled to its equilibrium temperature, thereby initiating local closing, followed by opening of the crack. Each subcycle is monitored twice using the laser ultrasonics technique, once each at the end of heating and cooling. Furthermore, each successive subcycle is accomplished at a higher power of heating laser than that of the previous subcycle. Significant differences in the peak-to-peak amplitude of the surface skimming longitudinal acoustic wave, which is excited by mode conversion of the Rayleigh wave by the crack, are revealed during the first cycle of photothermal loading. These differences clearly indicate a partial irreversibility of the mechanical processes occurring in the crack surfaces during subcycles of the first cycle. In a larger temporal scale, irreversible modification of crack surfaces is observed from the significant difference between the experimental results of the first photothermal loading cycle and the subsequent two cycles, whereas a reversible response of the crack surface to thermoelastic loading is observed from the similarity of the measurements accumulated during the two subsequent cycles.

Keywords

Laser ultrasonics acoustics rough surfaces in contact surface acoustic waves crack monitoring

Introduction

The presence and growth of cracks can lead to failure owing to fatigue.¹ More than 90% of failures of in-service metallic structures are caused by the growth of fatigue cracks. Therefore, non-destructive detection of fatigue cracks is of great importance in structural health monitoring (SHM).² Among the various non-destructive fatigue crack detection techniques, such as piezoelectric transducer–based and laser-based ultrasonic testing (UT),^3–8 magnetic particle testing,⁹ eddy current testing,⁹ X-ray tomography,¹⁰ and machine-vision¹¹ and image processing,¹² the laser-based ultrasonic testing (LUT) technique is extensively studied because of its multiple advantages; furthermore, LUT finds its application in manufacturing,^13–19 construction,^20,21 aviation and aeronautics,^22,23 and several other industrial fields.^13,14,24

The sensitivity of these conventional techniques,^23,25–30 which depend on linear property modifications of ultrasonic waves, has been reported to be insufficient to detect fatigue cracks until the cracks become visibly large;^29,31 this is because the changes caused in linear ultrasonic values by these cracks are not sufficiently large to be accurately measured using conventional linear ultrasonic techniques.^29,32

On the other hand, nonlinear acoustic^8,33–55 and vibro-thermography methods^56–58 have been proven to potentially overcome the limitations of linear UT techniques, because they depend on the nonlinear elastic/inelastic properties of cracks, which are known to be relatively different from those of artificial defects.⁴⁵ The nonlinear methods are extremely sensitive in characterizing non-perfectly contacting material surfaces, cracks and fatigue damage, by monitoring various nonlinear acoustic phenomena, such as harmonic generation,^{8,35,40,51,53–55} subharmonic generation,^35,36,40 parametric modulation,^37,41 modulation transfer,⁴² and acoustoelasticity.³⁹ The reported high sensitivity of the nonlinear acoustic methods for imperfect interfaces makes them a suitable candidate for early detection of cracks.

The new frequency components, which are generated in the acoustic spectrum because of the interaction with cracks of powerful ultrasound generated by piezoelectric transducers, have been demonstrated to be detected optically.^43,59 However, this method has the disadvantages of the conventional ultrasonic techniques because the contact between the test sample and the transducer could be nonlinear as well. Furthermore, fast scanning or remote measurement using these techniques is difficult because of the contact characteristic of transducers (although non-contact transducers have been developed, the liftoff distance and the detection bandwidth are limited). Furthermore, this ultrasonic method was proposed to be improved via modulation of crack rigidity by inducing thermoelastic stresses through the absorption of pulsed laser radiation.⁴⁴ This modulation is transferred into the variation of reflectivity of the surface acoustic waves, which is measured by the conventional ultrasonic technique. If the modulation frequency is smaller than that of the ultrasonic waves, the observed nonlinear acoustic phenomenon of frequency-mixing could be classified as acoustoelasticity,⁴⁵ in which the modification of material elasticity is caused by thermoelastic stress instead of static pressure. This approach was later advanced by applying the laser radiation not only for crack modulation but also for generation of ultrasound. The frequency-mixed components of the spectrum are detected by in-contact accelerometers^60–64 and contactless optical techniques, such as interferometry and beam deflection.⁶¹ Furthermore, the contrast of photoacoustic imaging of cracks is improved because of the strong dependence of optoacoustic conversion efficiency on the state of the crack.⁶⁵ Moreover, all-optical studies of thermoelastic crack modulation have been realized using pulsed ultrasound generated by a sub-nanosecond laser.^66,67 It has been reported that detecting the parametric modulation of reflected and transmitted acoustic pulses is a sensitive technique for evaluating crack modifications.

All the abovementioned crack detection models and techniques based on nonlinear acoustic approaches depend on repeated modulation of crack states between “open” and “closed” states, that is, on “breathing”(tapping/clapping) of cracks introduced by mechanical or photothermal loading. Most theoretical studies on crack modulation simplify the crack surfaces using two smooth surfaces, and thus do not consider the real physical microscopic structure of the crack surfaces.^38,65,68 However, in actuality, two different characteristics of crack surfaces are inevitable: case 1 asperities usually exist on the crack surfaces; thus, only some regions, and not the entire surface, are in contact;³⁸ case 2 when a crack is closed by modulation, some asperities are irreversibly modified (broken). These characteristics possibly not only change the crack surfaces under modulating loading but also significantly modify the state of contact between the crack faces which consequently modify the nonlinear acoustic phenomena.⁶⁹ For case 1, studies on crack modulation based on the assumption of purely elastic reversible contact between asperities of rough surfaces have been conducted.^70,71 On the contrary, to date, only a few studies have been focused on irreversible deformation of asperities, that is, on case 2, because of the difficulties faced by conventional methods in accessing crack surfaces.⁶⁹

In this study, localized irreversible microscopic modification in crack surfaces is monitored by utilizing the responses of laser-generated and laser-detected acoustic waves to laser-induced crack modulation. Through upgrading and improving of the experimental approaches developed in previous works,^66,67for the first time, monitoring of the macroscopic behavior of crack motion, as well as microscopic modifications of crack surfaces (reversible/irreversible modifications of asperities) by optically inducing and probing the acoustoelastic phenomena, has been enabled. A pulsed laser was used to generate surface skimming longitudinal waves and Rayleigh-type surface acoustic waves in a cracked black glass, and a continuous wave laser was used to detect them in the beam deflection configuration. In addition, a quasi-continuous laser was used to locally heat the glass at different positions along the crack and induce crack closure via photothermal loading. Three photothermal loading cycles were realized in individual succession, and were monitored using the laser ultrasonics technique at different locations along the crack. Each photothermal loading cycle in our experiments included multiple successive subcycles, in which the material was first heated and then cooled to its equilibrium temperature, thereby initiating local closing, followed by opening of the crack. Each subcycle is monitored twice using the laser ultrasonics technique: first at the end of heating and second at the end of cooling. Each successive subcycle is accomplished at a power of heating laser higher than that of the previous one. At the same position of the crack, the peak-to-peak amplitudes of both the surface wave and the longitudinal wave that was mode-converted from the surface wave are recorded in the transmission configuration.

A significant difference between the experimental results during the first photothermal loading cycle and during the two subsequent photothermal loading cycles (following the first cycle), and a similarity between the results of the latter two photothermal loading cycles, can be observed in our experiments. In a short time, significant differences in the peak-to-peak amplitudes of the mode-converted acoustic signal measured after each subcycle of heating and cooling during the first cycle of photothermal loading are observed. The following hypotheses are introduced to explain these observed phenomena. First, the important difference between the experimental results observed during the first cycle of photothermal loading of the crack and those observed during the two subsequent loading cycles is caused by irreversible modification (break/destruction) of some asperities distributed on the crack surfaces, when two crack surfaces move toward each other during the heating phase of the heating/cooling subcycles. Second, the absence of modifications in the experimental results during the latter two photothermal cycles indicates the elastic behavior of the crack surfaces, that is, reversibility of the phenomenon under study.

The first hypothesis is corroborated by comparing two atomic force microscopy (AFM) images of the crack obtained before and after its first closure cycle induced by localized laser heating, which confirms irreversible modification of the sample surface; the images are obtained at the region of laser-induced heating. The approach proposed in this work presents a means to study various modifications of crack surfaces caused by crack opening/closing without breaking the sample, thereby providing a possibility to non-destructively and more comprehensively study the origins of acoustical nonlinearities of cracks.

The almost similar experimental data observed during the second and third loading cycles indicate the dominance of reversible processes and appeals to an analogy with various end-point memory effects under stress loading of materials.^72–74 For example, the so-called Kaiser effect⁷² constitutes almost a complete absence of acoustic emission effect until the load imposed on the material exceeds the previously applied load, which earlier caused significant acoustic emission. In our experiments, we do not measure the acoustic emission caused by irreversible modifications (break) of asperities at the crack surfaces in the process of thermoelastic loading of cracks, and instead probe the irreversible modifications of cracks by measuring the interaction of surface acoustic waves with the cracks. These measurements reveal the following manifestation of the end-point memory of the crack: the surface acoustic wave transmission remains almost unchanged across the crack after the first thermoelastic loading cycle, when all the successive loading cycles are identical. This is probably because most asperities, which could be broken by the same maximal laser-induced thermoelastic stress applied during each cycle, are prematurely broken during the first loading cycle.

The publications indicate that a significant interest exists in the SHM community toward crack evaluation techniques based on the principles of nonlinear acoustics and parametric modulation of acoustic waves.^{2,8,53–55,59,68,75} Thus, we believe that the results reported in this research, which deepen the understanding of these phenomena and reveal a possible influence of irreversible modification of crack asperities on these phenomena, would be of great interest for the SHM community. In addition, some laser ultrasonics techniques utilized in this research have potential applications in actual SHM.

The non-contact localized optical loading used in this article can address the limitations of the conventional mechanical contact loadings, such as the potential ability to break the sample or introduce other defects, and can be applied in multiple crack detection techniques. For example, it has been frequently reported that modulated photothermal loading can be used in crack detection using acoustoelasticity.^61–64 In addition, it can be applied to distinguish surface-breaking cracks from other surface structures or attachments.⁷⁶ Furthermore, in this study, the reported non-monotonic evolution of peak-to-peak amplitude of the mode-converted signal at the end of cooling time intervals after heating at various heating powers can be potentially used to evaluate the width of the crack in SHM.^64,66,67 Furthermore, optical generation and detection of acoustic waves presented in this study can be easily scanned over the surface of the sample; thus, linear laser ultrasonic techniques, such as time-of-flight diffraction (TOFD)⁷⁷ and scanning laser source,²⁹ can be realized when required in combination with the laser-induced thermoelastic loading of the crack. These techniques have been proven to be beneficial for crack detection, and numerous applications have been reported in many fields of SHM.

This article is organized as follows: the “Experimental setup and test experiments” section presents the complete laser ultrasonic experiment setup, test sample to be tested, and a description of the experiment; the “Monitoring of cyclic photothermal loading of the crack” section presents the experimental procedure, results, and the corresponding physical interpretations; finally, the article is concluded with a summary of this research in the “Conclusion” section.

Experimental setup and test experiments

The experiment setup is schematically shown in Figure 1. Furthermore, laser pulses of wavelength $λ_{G} = 532 nm$ for acoustic wave generation are obtained by frequency-doubling—in the KTP crystal—of the pulses from an Nd:YAG laser (CryLaS DSS1064-450, wavelength $λ = 1064 nm,$ pulse duration $τ = 1.5 ns,$ single pulse energy $E_{s} = 650 μ J$ , repetition frequency $f_{R} = 200 Hz$ ). This pump laser beam is focused onto a circular spot of diameter $d_{G} ~ 10 μ m$ on the sample surface. The optical radiation for heating the sample is provided by a CW laser (Changchun New Industries, MGL-N-532A, wavelength $λ_{H} = 532 nm$ ). It passes through a wave plate and a polarizing beam splitter (PBS) for power adjustment, and it is then focused at a circular spot of diameter $d_{H} ~ 100 μ m$ on the sample surface. The beam deflection technique⁷⁸ is used for probing the acoustic waves propagating through the sample surface using a CW laser (LuxXplus 638-100, wavelength $λ_{D} = 638 nm$ ). The probe laser passes through a set of mirrors, wave-plates, and a lens, following which it is focused onto a circular spot of diameter $d_{D} ~ 10 μ m$ on the sample surface. The reflection is split into two equal portions by a mirror set, which are sent to the two inputs of a balanced photodetector (New Focus, 1607, 3 dB bandwidth: 650 MHz). The detected signal is displayed and recorded by an oscilloscope (RIGOL, DS4024, bandwidth: 200 MHz). The sample used in this study is a lime–soda–silica glass plate (thickness ~2.8 mm) that strongly absorbs light (optical penetration depth: approximately $300 μ m$ ).⁶⁶ A single crack of length ~10 mm is generated by thermal shock. The orientation of the crack surface is almost normal to both the edges and both surfaces of the plate. The crack is generated by first locally heating the sample near the sample edge by candle light, followed by cooling to room temperature by removing the candle light and manually waving the sample in air. The shape of the heating area is similar to a semi-elliptical shape. Only a single crack that is almost normal to both surfaces of the sample can be generated. The crack penetrates through the entire plate thickness. However, because the length of the crack is ~10 mm, and the width of the sample is ~50 mm, the crack does not reach the other edge, as illustrated in Figure 2.

To identify different modes of laser-generated acoustic waves, the TOFD⁷⁷ method is employed. The generation laser spot (green spot in Figure 2(a)) is initially at the same side of the crack as the detection laser spot (indicated by the red point “D” in Figure 2(a)). The generation laser spot then scans along the path represented by dashed lines in steps of $~ 66 μ m,$ and stops after 30 steps at the opposite side of the crack (represented by the green point “G” in Figure 2(a)). The B-scan image and the acoustic signals detected in reflection and transmission configurations are shown in Figure 3.

The TOFD measurement directly identifies the modes of detected acoustic waves by analyzing their propagating velocities, which are defined by the slopes of lines connecting the arrival times of the acoustic modes at the plane (coordinate, time). The detected velocities of the longitudinal wave and the surface acoustic wave are approximately 5800 m/s and 3000 m/s, respectively. All the waves identified in Figure 3 are listed in Table 1.

Table 1.

Description of detected acoustic signals.

Detected signal notation	Detected signal description
L	Direct surface skimming longitudinal wave
R	Direct Rayleigh wave
rL	Surface skimming longitudinal wave reflected by the crack
rR	Rayleigh wave reflected by the crack
rL-R	Reflected surface skimming longitudinal wave mode-converted from the Rayleigh wave
tR	Rayleigh wave transmitted through the crack
tL-R	Transmitted surface skimming longitudinal wave mode-converted from the Rayleigh wave

Previous studies have demonstrated that a crack can be closed by constant heating induced by laser irradiation.^66,67 In this study, experiments are conducted in the transmission configuration at different positions $h$ along the crack to monitor its internal modification after each closure (see Figure 2(b)). All the three spots (for generation, detection, and heating) are adjusted to be along one line that is perpendicular to the crack, as shown in Figure 2(b). The heating spot is centered on the crack, whereas generation and detection spots are located at different sides of the crack at a distance of ~1.3 mm. The laser-generated acoustic waves propagating through the crack can be observed at the detection point.

Monitoring of cyclic photothermal loading of the crack

Experimental procedure

At each experiment location, the measurement includes three successive photothermal loading cycles. Each photothermal loading cycle includes 35 subcycles of periodic crack heating followed by crack cooling, as described below.

The heating laser is first turned off, and the transmitted acoustic signals “tR(cool)” and “tL-R(cool)” at $P_{Heat} = 0 mW$ are recorded. All the signals are averaged 128 times. The acquisition time for a single measurement is less than 1 s.

After the previous acquisition is complete, the heating laser is turned on at $P_{Heat} = 10 mW$ , which irradiates the crack locally. After a period of 30 s, which is the time required for the crack to reach its equilibrium state, the transmitted acoustic signals “tR(heat)” and “tL-R(heat)” at $P_{Heat} = 10 mW$ are recorded.

After completing the previous recording of signals, the heating laser is turned off. After 30 s, which is the time required for the crack to reach its equilibrium state, the transmitted acoustic signals “tR(cool)” and “tL-R(cool)” at $P_{Heat} = 10 mW$ are recorded.

The steps 2 and 3, which constitute a single photothermal loading subcycle, are repeated by increasing the heating laser power in steps of 10 mW up to $P_{Heat} = 350 mW$ .

After performing all the abovementioned steps, which constitute a single photothermal loading cycle, we wait for 5 min and repeat the same photothermal loading cycle twice. The locations of the sample and the laser focusing spots were fixed during the entire measurement process. The maximum heating laser power is selected as $P_{Heat} = 350 mW$ , because earlier experiments^66,67 have indicated that a heating laser power of $P_{Heat} = 350 mW$ is sufficient for complete crack closure in a similar sample, and that no damage/softening of glass occurs because of the temperature variation induced by photothermal loading.

Experimental results

The experiment is conducted at several locations along the crack. Figure 4 shows the peak-to-peak amplitudes of all detected acoustic signals at $h ~ 6 mm$ recorded during three photothermal loading cycles. Furthermore, “tR(heat)” and “tL-R(heat)” denote the detected peak-to-peak amplitudes of transmitted surface acoustic wave signals and mode-converted wave signals, respectively, at the end of heating time intervals contributing to each heating/cooling subcycle (experimental procedure 2 from subsection “Experimental procedure” of the “Monitoring of cyclic photothermal loading of the crack” section conducted repeatedly by increasing the heating laser power). Furthermore, “tR(cool)” and “tL-R(cool)” denote the detected peak-to-peak amplitudes of transmitted surface acoustic wave signals and mode-converted wave signals, respectively, at the end of cooling time intervals contributing to each heating/cooling subcycle (experimental procedure 3 from subsection “Experimental procedure” conducted repeatedly by increasing the heating laser power).

In Figure 4(a), it can be seen that, in the first photothermal loading cycle, “tR(heat)” increases continuously, whereas “tL-R(heat)” decreases continuously with the increase in heating laser power. The first phenomenon has been reported earlier.⁶⁶ Although the second phenomenon has been reported as well,⁶⁶ it was reported for the “tR-L” mode and not for the “tL-R” mode, that is, for the Rayleigh wave mode-converted from the incident longitudinal skimming bulk wave on the crack. Both these phenomena are supposed⁶⁶ as indications of crack closure caused by continuous laser heating under the condition of partial contact between the opposite surfaces of the crack via the surface asperities initially. Under this assumption, the contact between the crack faces increases continuously with the increase in crack closure caused by laser-induced heating. It has been hypothesized⁶⁶ that better contact between the crack faces leads to better transmission of the surface Rayleigh waves and poorer mode conversion of the surface skimming longitudinal wave into the Rayleigh surface wave.

Our first extension of the previously used technique^66,67 involves measurements not only after each heating phase of the heating/cooling loading subcycles but also after each cooling phase. The measurements after each cooling phase provide additional information on the reversibility of the considered photo-induced thermoelastic processes. For example, strong modifications of both “tR(cool)” and “tL-R(cool)” signals with the increase in heating power for each subcycle during the first photothermal cycle (Figure 4(a)) are directly caused by the quasi-eq1 state of crack modification after each heating/cooling subcycle of loading, that is, it is a direct indication of irreversible processes. On the contrary, the absence of any significant modifications of “tR(cool)” and “tL-R(cool)” signals with the increase in heating power for each subcycle during the second and third photothermal loading cycles (Figure 4(b) and (c), respectively) indicates that the crack returns to the same quasi-equilibrium state after each heating/cooling subcycle of loading, that is, it is not modified. This is a direct indication of reversibility of thermoelastic processes in each subcycle of crack loading during both the second and third photothermal loading cycles.

Our second extension of the previously used technique^66,67 involves measurements during multiple photothermal loading cycles. A comparison of the data accumulated in the first and second cycles of photothermal loading (Figure 4(a) and (b), respectively) indicates the transition from irreversible heating/cooling subcycles (Figure 4(a)) to reversible subcycles (Figure 4(b)). Similarly, a comparison of the data accumulated in the second and third cycles of photothermal loading (Figure 4(b) and (c), respectively) indicates that, starting from the second cycle, the cycles of photothermal loading are mostly reversible. Starting from the second cycle, the photothermal loading cycles are similar not only in the amplitudes of signals measured after the cooling phase of the heating/cooling subcycle, that is, “tR(cool)” and “tL-R(cool),” but also in the dynamics of signals measured after the heating phase of these cycles, that is, “tR(heat)” and “tL-R(heat).” For example, peak-to-peak amplitudes of the signal “tR(heat)” in both second and third cycles (Figure 4(b) and (c), respectively) show saturation starting with the critical power of 180 mW, followed by an increase in amplitude starting from a power of $P_{Heat} ~ 280 - 300 mW .$

More detailed interpretation of our experimental results, which are presented in Figure 4, is provided in the “Discussion” section.

AFM observations

To corroborate the presented explanation, two AFM observations are conducted. The first AFM topography image was obtained before the first photothermal loading of the crack by heating laser, and the results are shown in Figure 5(a) and (c). After a series of laser-induced heating/cooling subcycles, the second AFM topography image was obtained at a similar surface area, and the results are shown in Figure 5(b) and (d). It must be noted that the two AFM observation areas—before and after laser-induced photothermal loading—do not strictly overlap but are within the area of heating laser spot.

In both Figure 5(a) and (b), a step in height between the opposite crack surfaces can be observed. The images indicate that the crack is rather irregular in the scale of a few micrometers, especially in the area shown in Figure 5(a). The diameter and the height of the AFM tip (ScanAsyst-Fluid+ from Bruker) apex are ~4 nm and $5 μ m$ , respectively, which provide more than sufficient spatial resolution for imaging the “morphology” of the rather wide crack in our experiments.

Discussion

Physical nature of the irreversible processes

We hypothesize that the irreversible processes that cause the modification of the quasi-equilibrium state of the crack revealed by the measurements of “tR(cool)” and “tL-R(cool)” signal amplitudes presented in Figure 4(a) are the processes of asperities destruction on the crack surfaces. Several studies have demonstrated that a crack can be closed with the thermoelastic stress induced by the absorption of intensity-modulated^61–64 or quasi-continuous laser irradiation.^66,67 Moreover, the AFM topography of a similar crack surface (see Figure 7 in the previous publication)⁶⁷ reported earlier, which was obtained by the destruction of a similar sample, revealed asperities of different heights on the crack faces. The heights of some asperities are comparable to the crack width, that is, up to several hundreds of nanometers. Thus, we introduce a hypothesis stating that the modification of signals (Figure 4(a)) is caused by the irreversible modification (destruction) of some asperities distributed on the crack surfaces when two crack surfaces move toward each other during the heating phases of the heating/cooling subcycles. The inelastic/plastic modifications of surface asperities are caused by each heating/cooling subcycle during the first photothermal loading cycle because each successive subcycle is realized at a higher heating power than that of the previous subcycle; thus, smaller asperities could be modified because the average distance between the crack faces (crack local opening/width) becomes progressively smaller at the end of each successive heating phase. However, as the heating laser powers of the second and the following cycles of photothermal loading in the heating/cooling subcycles do not exceed those applied during the first photothermal loading cycle, the irreversible modification of surface asperities stops, because all asperities that are accessible at the fixed maximum heating powers are prematurely modified and mostly exhibit reversible elastic modifications under further heating/cooling loading subcycles. Therefore, the experimental results presented in Figure 4(b) and (c) indicate highly reversible cyclic processes. In addition, an analogy of our hypotheses to the interpretations of the Kaiser effect in acoustic emission⁷² is evident. The only difference lies in the experimental probing method of the reversibility/irreversibility of the stress-loading processes.

These hypotheses can be additionally supported by the AFM results shown in Figure 5. The most insightful conclusions could be derived by comparing the data presented in Figure 5(c) and (d), which present the sample surface profiles measured by AFM along two directions normally crossing the crack at two different points. The topography profiles in Figure 5(c) indicate that no gap/opening between two crack surfaces can be observed by AFM before laser-induced heating, whereas those in Figure 5(d) reveal the appearance of gaps between the opposite crack surfaces after laser-induced heating (marked by ellipses in Figure 5(d)). These facts support the proposed hypothesis on the irreversible modification of the crack. In addition, the possibility of photothermal-induced crack opening or crack growth is considered to be highly unlikely because of the following reasons:

(a) Relative to the initial state of the crack, the thermoelastic loading applied by us introduces only compressive stresses on the crack. Local laser-induced heating is not expected to generate tension forces that could increase the distance between the crack faces (crack opening).

(b) The experiments were conducted without heating the crack tip region. Thus, the observed separation between the crack surfaces after laser heating is not expected to be caused by crack growth, that is, increase in crack length toward the other edge of the plate. In addition, most reports indicate that periodic heating of the cracks in glass,⁷⁹ as well as stationary heating,^80,81 heals the cracks rather than initiating crack growth and opening. Therefore, even if the heating in our experiments reaches the crack tip, it should induce crack closure and not crack opening.

Thus, we consider that the obtained AFM images not only explicitly confirm the hypothesis of irreversible crack modification during the first cycle of laser-induced thermoelastic loading of the crack but also implicitly confirm the hypothesis that irreversible modifications are caused by the destruction of asperities on the crack surfaces.

Physical nature of the non-monotonic variation of acoustic signals with the increase in photothermal loading, which causes irreversible modifications of the crack

Here, we discuss the interpretation of the dependence of acoustic signal amplitudes on the continuously increasing heating power, as shown in Figure 4. As discussed above, the physical phenomena occurring during the first photothermal loading cycle (Figure 4(a)) include irreversible modifications of the crack, and are importantly different from those occurring during the second and third cycles (Figure 4(b) and (c), respectively). Thus, they should be interpreted separately.

For the physical explanation of signal evolutions in Figure 4(a), we propose a qualitative model that considers the first approximation for the difference between the process of the wave transmission across the crack without mode conversion and the transmission processes accompanied by mode conversion, as well as for the possible irreversible modifications of the crack. Our qualitative model is based on the following arguments on the dependence of the ability of the crack to transfer and mode-convert the acoustic waves on the separation between the crack faces. We assume that, generally, reducing the distance between the opposite surfaces of the crack improves the transmission of acoustic waves across the crack. If the crack is completely and perfectly closed, the wave transmission is $100 %$ because transmission occurs in the absence of crack. On the contrary, if the crack is completely open, that is, there is no contact between the opposite surfaces of the crack via the asperities, the wave transmission does not occur ( $0 %$ transmission). Thus, the transmission amplitude is expected to reduce with the increase in distance between the crack faces, if the latter is accompanied by a reducing number and/or surface area of the wave transmitting contacts between crack surfaces.

We qualitatively illustrate the abovementioned tendency in Figure 6, where the points of the horizontal axis correspond to possible intermediate crack states between the completely closed and completely open cracks, whereas those of the vertical axis correspond to the values of wave transmission coefficients with or without mode conversion. The orange monotonic curve, which is presented as a straight line only for simplicity, illustrates the expected dependence of the transmission coefficient of Rayleigh wave, “tR,” on the state of the crack. The dependence of the coefficient which characterizes the mode conversion of the incident Rayleigh wave into “transmitted” skimming longitudinal acoustic wave, “tL-R” on the state of the crack, is expected to be very different. Furthermore, this coefficient should be $0 %$ not only in the completely open state of the crack (because of the absence of any transmission in the absence of the contact between opposite surfaces of the crack) but also in the completely closed state of the crack (because of the absence of any mode conversion process in a homogeneous medium). Thus, the dependence of the “tL-R”“transmission” coefficient on the crack state should be non-monotonous, as illustrated qualitatively by the simplest curve with a single local maximum in Figure 6. The scheme in Figure 6 with an additional hypothesis that the initial state of the crack (before any photothermal loading) is located between the completely closed state and the state providing the maximum “transmission” coefficient for the mode-converted wave is sufficient for a qualitative interpretation of all the experimental data in Figure 4(a).

In Figure 6, the hypothetical initial state is represented by the dashed vertical line (0). We assume that the thermoelastic loading of the crack caused by laser-induced heating reduces the separation of the surfaces of the imperfectly/incompletely closed crack; thus, heating modifies the state of the crack from its initial state to the states that are approaching the completely closed state, as indicated by the red arrow. With the increase in heating laser power, the crack state progressively approaches the closed state of the crack. The dashed lines (1) and (3) indicate two crack states, when the heating laser is irradiating the crack, and state (3) is achieved at a higher heating laser power than state (1). Thus, in accordance with Figure 6 for the assumed initial state, the transmission coefficient for the Rayleigh wave without mode conversion increases continuously, whereas that of the Rayleigh wave via its mode conversion in the longitudinal wave decreases continuously with the increase in heating power. This explains the experimental amplitudes of the signals “tR(heat)” and “tL-R(heat)” in Figure 4(a).

As mentioned earlier, some asperities are irreversibly modified (broken) during the closure of the crack. Consequently, when the heating laser is turned off, and the crack is reopened, the crack does not return to its original state, but attains a different state that is close to the completely open state. Thus, cooling, which follows heating, displaces the state of the crack (Figure 6) from its initial state toward the completely open state (in the direction indicated by the green arrow). As the heating laser power increases continuously, at the end of the cooling phases, which follow the interruption of heating, the crack states progressively approach the completely open state of the crack. In Figure 6, the states realized at the end of the three cooling states are represented by the dashed lines (2), (4), and (5). State (4) is achieved after the subcycle is realized with a higher heating laser power than state (2). State (5) is achieved after the subcycle is realized with a higher heating laser power than state (4). Thus, consistent with Figure 6, for the assumed initial state, the transmission coefficient for the Rayleigh wave without mode conversion decreases continuously, whereas that of the Rayleigh wave via its mode conversion in the longitudinal wave exhibits a non-monotonic behavior with increasing heating power. The latter coefficient initially increases and then decreases. This explains the experimental results for the amplitudes of the signals “tR(cool)” and “tL-R(cool)” in Figure 4(a), particularly the non-monotonic dependence of “tL-R(cool).”

These irreversible modifications of the crack caused by laser-induced thermoelastic stresses during the first photothermal loading cycle are further illustrated in Figure 7. The initial state of the crack is illustrated in Figure 7(a). When the crack is heated at a relatively low laser power, the thermoelastic loading of the crack caused by laser-induced heating reduces the separation between the surfaces of the imperfectly/incompletely closed crack, as illustrated in Figure 7(b). This process corresponds to the transition of the crack state from the initial state to state (1), as shown in Figure 6. The irreversible modification (breaking) of asperities during this crack closure can reduce the number/surface area of contacts between the opposite surfaces of the crack, when the crack is reopened after interruption of heating, as illustrated schematically in Figure 7(c). This process corresponds to the transition of the crack state from state (1) to state (2). Figure 7(d) schematically illustrates a scenario, where the crack is heated at a higher heating laser power than the previous subcycle, causing more complete closure of the crack than previously. This results in (a) more severe destruction of the asperities on the crack faces, that is, in Figure 6 state (3) closer to the completely closed state than state (2) is attained and (b) the crack with smaller number/surface of contacting asperities, when the heating is removed and the subcycle is complete by cooling, that is, state (4) which is closer in Figure 6 to the completely open state than state (2). With the continuous increase in heating laser power, the damage of asperities in the crack faces accumulates with the continuous decrease in crack width, when the heating laser is on, as illustrated in Figure 7(f). After the crack is reopened when the heating laser is off, smaller number of contact points remains, as illustrated in Figure 7(g); thus, the crack state moves closer to the completely open state than the earlier attained state (4), that is, to state (5) in Figure 6.

It must be noted that this non-monotonic evolution of the signal “tL-R(cool)” during the first photothermal loading cycle is observed repeatedly. Figure 8 shows the signal “tL-R(cool)” detected at various locations along the crack. Except at $h ~ 1 mm$ , the non-monotonic signal evolution can be observed at all experimental locations up to $h ~ 10 mm$ . The monotonic dependence of the “tL-R(cool)” signal amplitude on the heating power at the position $h ~ 1 mm$ indicates that the maximum power applied for heating in this experimental position is still insufficient to attain the crack state corresponding to the maximum of the “transmission” coefficient with mode conversion (see Figure 6). Generally, the non-monotonic characteristic of the dependence of the “transmission” coefficient with mode conversion on the crack state makes the measurements of the mode-converted waves more sensitive to the crack state than the measurements of the Rayleigh wave transmitted without mode conversion. For example, during the measurement at the lowest heating power, the tendency of the amplitude of any of the signals “tL-R(cool)” and “tL-R(heat)” to either decrease or increase with the increase in heating power provides an opportunity to position the initial state of the crack on the qualitative scheme presented in Figure 6.

In addition to the amplitude dependence on heating power, some other data exist in Figure 4(a) that are worth discussing. As explained above, both “tR” and “tL-R” are expected to become zero, when the crack is completely open. However, in Figure 4(a), none of the amplitudes “tR(cool)” and “tL-R(cool)” become zero. This indicates that, at the current experimental location, when the crack is reopened after laser heating at maximum power, contacting asperities, transmitting Rayleigh wave, and generating mode-converted wave still exist. The completely open state of the crack is not attained in these experiments. Similarly, a non-zero signal “tL-R(heat)” is measured at the maximum heating laser power (Figure 4(a)), indicating that the crack is not completely closed at this heating laser power. However, a complete crack closure revealed by zero mode-conversion can be found at other experimental locations in this study, as well as in our earlier work.^66,67

Physical nature of the non-monotonic variation of acoustic signals with the increase in photothermal loading in the case of reversible elastic dynamics of the crack

Here, we explain the dependencies of acoustic signal amplitude on the continuously increasing heating power during the second and third photothermal loading cycles presented in Figure 4(b) and (c), respectively. As discussed above, the physical phenomena occurring during the second and third photothermal loading cycles mostly include reversible elastic modifications of the crack. In Figure 4(b) and (c), quasi-reversibility is observed in highly weak (compared with Figure 4(a)) dependence of amplitudes of both “tR(cool)” and “tL-R(cool)” on the heating power applied during each subcycle of heating/cooling, indicating the successive return of the crack to the same quasi-equilibrium state. In addition, quasi-reversibility is observed in the quantitative similarity in Figure 4(b) and (c) of the dependence of the amplitudes of both “tR(heat)” and “tL-R(heat)” on the heating power. The dependences of the latter signals on increasing the heating power provide replicas (to scale) of the dependences of the transmission coefficient of the Rayleigh wave and the mode-conversion coefficient on the separation distance between the opposite crack surfaces. It is expected that increasing the heating power during the subcycles of heating/cooling progressively reduces the separation between the crack surfaces, which is attainable at the end of heating phases, thereby progressively establishing contacts between increasing number of asperities with smaller heights. Furthermore, surface asperities of different sizes are observed experimentally by AFM imaging. A magnified AFM image, which was shot within the observation area depicted in Figure 5(b), is presented in Figure 9. Two large asperities are represented by “A,” and three smaller visible asperities are represented by “B.”

Although the “tR(heat)” signals in the second and third photothermal loading cycles constitute intermediate saturation of amplitude growth between ~100 mW and ~300 mW (Figure 4(b) and (c)), their main features of monotonic dependence on the heating power can still be (similar to “tR(heat)” signal in Figure 4(a)) qualitatively associated with the continuously increasing number/surface area of contacts between the opposite crack surfaces with the increase in heating. The new feature measured in the signals after the heating phase of the heating/cooling subcycle in Figure 4(b) and (c) in comparison with those in Figure 4(a) is the non-monotonic dependence of the “tL-R(heat)” signal amplitude on the heating power. In Figure 4(a) the considered dynamics of the mode-converted signal “tL-R(heat)” is monotonic, whereas in Figure 4(b) and (c) it include local minima in the amplitudes of mode-converted waves at ~275 mW. In addition to the non-monotonic dependence of “tL-R(cool)” during the first cycle, the non-monotonic dependence of “tL-R(heat)” in both the second and third cycles is another indication of the physical principle that was qualitatively formulated in the previous subsection: “increase in the number/surface area of contacts between the opposite surfaces of the crack could lead either to increased or decreased efficiency of mode-conversion processes.” Similar to other dependences presented in Figure 4, these non-monotonic features of mode-converted signals are repeatedly observed in different experimental locations along the crack. We expect that these reliable experimental observations could lead to the development of elasticity theory at the nanometric scale in the future, which could provide insights into the interaction between the distributions of 3D asperities and explain the experimental results at a quantitative level. In our opinion, in addition to describing the elastic/inelastic response of two rough surfaces in contact to arbitrary in orientation and polarization acoustical, that is, low amplitude loading,^82–85 such a theory should account for the destruction of the asperities, modification of the shape, and statistical distribution of asperities caused by strong thermoelastic loading. The development of such a theory is an extremely challenging task. To elaborate the numerical methods describing the asperities destruction, the ideas of strong/soft discontinuities^86–89 could be potentially useful.

Conclusion

Experiments on monitoring of the periodic laser-induced contactless local heating of the crack were conducted using the laser ultrasonics technique in the point-source-point-receiver configuration in the ultrasound transmission geometry at various locations along the crack length. Compared with the earlier experiments, measurements of Rayleigh wave transmission across the crack were performed not only at the end of the heating phase of each photo-induced thermoelastic loading subcycle but also after each corresponding cooling phase. The latter measurements provided direct evidence of the continuously accumulating irreversible modifications of the crack until the heating laser power increased continuously with successive heating/cooling subcycles. When this cycling photothermal loading was stopped at a certain maximum heating power and restarted from the lowest heating power, reversible processes were mostly observed. The irreversible modifications of the crack subjected to photothermal loading are attributed to the destruction of nanoscale asperities at the crack surfaces. This hypothesis was corroborated by comparing the AFM images of the crack before and after photothermal loading. Furthermore, the non-monotonic dependence on the heating power of the amplitude of the surface skimming longitudinal acoustic wave excited by the incident Rayleigh wave on the crack was revealed. These observations indicate that the mode-conversion process is more sensitive to the state of the crack than the transmission of the Rayleigh wave without mode conversion, demonstrating a monotonic dependence on heating power. We believe that the findings of this research, which provide potential opportunities to enhance the sensitivity of the laser ultrasonics technique in crack monitoring, will impel the theoretical research on the mode-conversion processes occurring in the interaction of surface acoustic waves with cracks. From the experimental perspective, the prospects are real-time monitoring of the photothermal loading of the crack using laser ultrasonics, that is, not only at the end of heating and cooling phases of the subcycles but also during the subcycles. Such a monitoring technique could provide beneficial information on the immediate modification of the crack state following the destruction of a single surface asperity or a group of asperities.

Footnotes

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (No. 61627802, No. 61975080 and No. 61405093).

ORCID iDs

Chen-Yin Ni

Vitalyi Gusev

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