(HVI)-Induced Pitting Damage Using Active Guided Ultrasonic Waves: from Linear to Nonlinear

materials

Article Characterizing Hypervelocity Impact (HVI)-Induced Pitting Damage Using Active Guided Ultrasonic Waves: From Linear to Nonlinear

Menglong Liu 1,2, Kai Wang 2, Cliff J. Lissenden 3, Qiang Wang 4, Qingming Zhang 5, Renrong Long 5, Zhongqing Su 2,* and Fangsen Cui 1

1 Institute of High Performance Computing, A*STAR, Singapore 138632, Singapore; [email protected] (M.L.); [email protected] (F.C.) 2 Department of Mechanical Engineering, The Hong Kong Polytechnic University, Kowloon, Hong Kong, China; [email protected] (M.L.); [email protected] (K.W.) 3 Department of Engineering Science and Mechanics, The Pennsylvania State University, University Park, State College, PA 16802, USA; [email protected] 4 School of Automation, Nanjing University of Posts and Telecommunications, Nanjing 210023, China; [email protected] 5 State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing 100081, China; [email protected] (Q.Z.); [email protected] (R.L.) * Correspondence: [email protected]; Tel.: +852-2766-7818

Academic Editor: Dirk Lehmhus Received: 12 April 2017; Accepted: 9 May 2017; Published: 18 May 2017

Abstract: Hypervelocity impact (HVI), ubiquitous in low Earth orbit with an impacting velocity in excess of 1 km/s, poses an immense threat to the safety of orbiting spacecraft. Upon penetration of the outer shielding layer of a typical two-layer shielding system, the shattered projectile, together with the jetted materials of the outer shielding material, subsequently impinge the inner shielding layer, to which pitting damage is introduced. The pitting damage includes numerous craters and cracks disorderedly scattered over a wide region. Targeting the quantitative evaluation of this sort of damage (multitudinous damage within a singular inspection region), a characterization strategy, associating linear with nonlinear features of guided ultrasonic waves, is developed. Linear-wise, changes in the signal features in the time domain (e.g., time-of-ﬂight and energy dissipation) are extracted, for detecting gross damage whose characteristic dimensions are comparable to the wavelength of the probing wave; nonlinear-wise, changes in the signal features in the frequency domain (e.g., second harmonic generation), which are proven to be more sensitive than their linear counterparts to small-scale damage, are explored to characterize HVI-induced pitting damage scattered in the inner layer. A numerical simulation, supplemented with experimental validation, quantitatively reveals the accumulation of nonlinearity of the guided waves when the waves traverse the pitting damage, based on which linear and nonlinear damage indices are proposed. A path-based rapid imaging algorithm, in conjunction with the use of the developed linear and nonlinear indices, is developed, whereby the HVI-induced pitting damage is characterized in images in terms of the probability of occurrence.

Keywords: hypervelocity impact; ultrasonic guided waves; nonlinear; structural health monitoring; space structures

1. Introduction Numerous man-made spacecraft (e.g., satellites, space stations, and shuttles) co-exist with innumerable meteoroids and orbital debris (MOD) particles in low Earth orbit. MOD particles travel at such high speeds (on the order of the ﬁrst cosmic velocity) that even a small particle, upon impacting a

Materials 2017, 10, 547; doi:10.3390/ma10050547 www.mdpi.com/journal/materials Materials 2017, 10, 547 2 of 20 spacecraft, can severely jeopardize the integrity of the spacecraft [1,2]. This sort of impact is commonly referred to as a “hypervelocity impact” (HVI), a scenario typically involving an impacting velocity in excess of 1 km/s. HVI is ubiquitous in space, as evidenced by numerous impact lesions seen on spacecraft after every flight mission. Addressing such significance, HVI has received a great deal of attention since the 1950s when humans began extensive exploration into outer space [3]. According to National Aeronautics and Space Administration (NASA), more than 20,000 pieces of MOD particles that are larger than 10 cm are known to exist in low Earth and geosynchronous orbits, along with over 500,000 particles sized 1–10 cm and tens of millions smaller than 1 cm [4]. Nowadays NASA and the U.S. Department of Defense have been cooperating to establish a Space Surveillance Network, aimed at tracking MOD particles bigger than 5 cm, via which the conjunction assessments and collision avoidance maneuvers can be implemented to counter objects included in the network and prevent HVI. However, such a network is unwieldy to monitor those MOD particles that are smaller than 5 cm, and the impact from any of them, though small in size, can functionally compromise the spacecraft performance, with possible catastrophic consequences. Representatively, in 1996, a French satellite was damaged by MOD particles originating from a French rocket that had exploded a decade earlier [5]. In 2007, China carried out an anti-satellite test and launched a missile to purposefully destroy a decommissioned weather satellite for removing the satellite from the orbit [5]. However, the 3000 new pieces of MOD particles consequently produced in that test have posed vast HVI risk to other spacecraft, arousing a great deal of concern from the public. In 2009 a satellite in the Iridium Communications Constellation System was struck by the debris generated from an exploded satellite which was abandoned by the then Soviet Union, and this satellite failed immediately [5]. Notably, this HVI event created approximately 2000 new pieces of trackable debris, threating other spacecraft in orbit. More recently, the European-built Cupola added to the International Space Station in 2010 was suspected to be impacted by a paint flake or a small metal fragment, leaving a 7 mm-diameter circular chip [6]. To minimize the HVI-associated hazard to orbiting spacecraft, a diversity of shielding mechanisms has been developed and installed on space assets, especially for those with a requirement of a long service period or with habitation of astronauts. Taking a typical two-layer Whipple Shield [7] as an example, after the perforation of the outer shielding layer by an MOD particle in HVI, the shattered MOD particles, together with parts of the jetted materials of the outer shielding layer, form a debris cloud that further impacts the inner structure behind the outer shielding layer, leading to numerous craters or direct penetration of the inner structure. It is therefore that a timely perception of an HVI event and consequent evaluation of HVI-induced damage in space assets, including impact central location and the maximum depth of the formed craters, are of vital importance and necessity. Effective and accurate evaluation can be conducive to implementation of remedial and repair actions. Nevertheless, being significantly distinct from ordinary types of damage such as an orifice or a dent engendered in a low-velocity impact case, HVI-induced damage in the inner shielding layer presents unique and complex features: hundreds of small craters and cracks disorderedly clustered over a wide area, entailing novel detection methods. Amongst various candidate detection approaches, the guided-ultrasonic-wave (GUW)-based method has gained prominence [8–11]. Central to the increased use of GUW is its superior capacity of probing a large area promptly with only a few transducers, consuming low energy, accessing hidden components omni-directionally, as well as responding to various types of damage possession with high sensitivity [12–14]. Without a necessity to terminate the normal service of an inspected system and with the potential to be manipulated in an automatic manner, the GUW-based method has proven to be cost-effective in striking a compromise among resolution, detectability, practicality, and cost, corroborating the philosophy of in situ structural health monitoring (SHM). Most GUW-based detection and monitoring methods have been developed in a linear regime, termed linear ultrasonics, in which responding GUW signals (upon probing GUW’s interaction with damage) are examined and the signal features are extracted at the same frequency as that of the probing Materials 2017, 10, 547 3 of 20

GUW. Linear ultrasonics techniques, however, according to the Huygens principle, are only able to evaluate damage with a size comparable with the probing wavelength [15], thus showing limitations when used for the detection of the unique HVI-induced pitting damage. The probing GUW may surpass most of the small-dimension individual damage without any evident linear features to be evidenced in the perceived signals. On top of that, individual pitting craters or cracks with a relatively large size in a pitted region may possess a dimension comparable to the wavelength of the probing GUW, and each of them acts as an obstacle to scatter the probing GUW, leading to high complexity of captured signals, because of the mixing and overlapping of mutually-interfering waves scattered by the large number of craters and cracks in the pitted region. In contrast, nonlinear ultrasonics, in which the responding GUW signals are explored in the frequencies other than the excitation frequency, has found their superb niche for SHM towards undersized damage [16,17]. The premise of the use of nonlinear ultrasonics is that damage, even when small in dimension, can introduce or augment the nonlinearity in the medium that can modulate probing GUW and generate nonlinear features into GUW signals in the frequency domain. These nonlinear GUW features have been proven to have higher sensitivity to the microstructure and micro-damage than the conventional linear GUW features [18]. Deng [19,20] is among those who first pointed out the condition of the accumulative nonlinear second harmonic wave along the propagation distance. De Lima [21] further analyzed the accumulation and oscillation of second harmonic, sum-, and difference-frequency components of the Lamb wave based on the normal mode expansion. Both concluded that, unlike the longitudinal wave in a bulk medium and the surface acoustic wave in a surface medium with unconditionally accumulative second harmonics, there are only a few accumulative mode pairs available under deliberately chosen frequencies and modes. The reason is that two conditions: (1) synchronism, or phase matching (the same phase velocity) and (2) non-zero power flux, are essential to guarantee an accumulative nonlinear guided wave. Lissenden [22], Liu [23], and Chillara [24] analyzed all the possible mode pairs (including shear horizontal wave) of accumulative second or even third harmonics with a displacement gradient-based formulation, to point out all the possible wave mode pairs to be used. Up to this point, the characterization of HVI-induced pitting damage has yet to be well investigated. By analyzing how the microstructure and undersized damage influence the nonlinear stress-strain relationship, the correspondingly generated nonlinear ultrasonics can trace back to those nonlinear features, which are reflected by the material microstructure and undersized damage barely detectable with linear ultrasonics. Thus, the detection philosophy using nonlinear GUW techniques has ushered a novel avenue of using nonlinear ultrasonics to evaluate damage such as HVI-induced pitting damage in the shielding structures. Addressing the imminent need of evaluation of HVI-induced damage by MOD particles and the lack of relevant research on the use of nonlinear GUWs for characterizing HVI-induced damage, this study is dedicated to evaluating HVI-induced pitting damage using linear/nonlinear features of GUWs. The commercial software Abaqus®/Explicit, together with a self-written vectorized user defined material subroutine (VUMAT) nonlinear material subroutine, is used to simulate the linear/nonlinear features of wave propagation in real three dimensional (3-D) structures. Compared to the predominant application of COMSOL® to simplified two dimensional (2-D) scenarios by other researchers [25], the newly adopted approach is far more efficient in calculation and is thus expanded to a real 3-D case. Upon a correct understanding of the linear/nonlinear features of GUWs, both linear/nonlinear damage indices (DIs) are accordingly extracted to quantitatively image the HVI-induced damage using a path-based probability imaging algorithm. This paper is organized as follows. Section2 briefly introduces the 2-D analytical solution of the nonlinear GUWs and the incremental form of the nonlinear stress-strain relation to be embedded into the VUMAT user subroutine. Section3 concerns both 2-D and 3-D numerical modeling based on Abaqus®/Explicit and VUMAT to quantitatively analyze the accumulation effect of nonlinear second harmonic waves in cases including phase matching and phase mis-matching. The HVI-induced Materials 2017, 10, 547 4 of 20 damage is presented, and the related linear/nonlinear features of GUWs are extracted as DIs in Section4. Finally, damage characterization is implemented in Section5 based on the DIs using the reconstruction algorithm for probabilistic inspection of damage (RAPID), followed with the concluding remarks in Section6.

2. Theory and Rationale: Nonlinear Guided Waves The nonlinear elastic mechanics of a solid medium can represent the 3-D stress–strain relation with a second-order approximation as follows [26] σij = Cijkl + 1/2Mijklmnεmn εkl, (1) where σij is the stress tensor, εmn and εkl are the strain tensors; Cijkl is the second-order elastic (SOE) tensor; and Mijklmn is a tensor associated with the material and geometric nonlinearities. If only the ﬁrst term in the parenthesis, Cijkl, is retained, Equation (1) will represent the linear elastic 3-D Hooke’s Law. The tensor Mijklmn in Equation (1) can be expressed in terms of the notation by Landau and Lifshitz [27] as follows

Mijklmn = Cijklmn + Cijlnδkm + Cjnklδim + Cjlmnδik, (2) where 1 Cijklmn = ( Aδ Ijlmn + δil Ijkmn + δjk Iilmn + δjl Iikmn) 2 ik (3) + 2B δij Iklmn + δkl Imnij + δmn Iijkl + 2Cδijδklδmn,

In Equations (2) and (3), Cijklmn is the introduced third-order elastic (TOE) tensor due to material nonlinearity, while geometric nonlinearity is manifested in the last three terms of Equation (2). δkm and those in similar forms with different subscripts are the Kronecker deltas; Ijlmn and those in similar forms are the fourth-order identity tensors. As displayed in Equation (3), the material nonlinearity-related term, Cijklmn, is directly related to three TOE constants A, B and C, which are the inherent properties of the material, to be measured experimentally [28]. To start with, the inner shielding layer can be simpliﬁed as a 2-D plane strain waveguide for wave propagation, in which four stress components σ11, σ22, σ33, and σ12 and three strain components ε11, ε22, and ε12 are reserved. In this way, the 2-D stress-strain relations for the nonlinear material can be easily deduced from the 3-D stress-strain relation in Equation (1). If indices 1 and 2 in the Cartesian coordinate are projected to y and z as the thickness and propagation direction (see Figure1), the amplitude of the second harmonic can be expressed as [21],

∗ i(ka+k )z ik z Am(z) = Am(z)e b − Am(0)e n , (4)

where i f vol + f sur f n n ∗ Am(z) = ∗ , kn 6= ka + kb, (5) 4Pmn[kn − (ka + kb)] f vol + f sur f n n ∗ Am(z) = z, kn = ka + kb, (6) 4Pmn where κa and κb denote the wave numbers of two fundamental modes. In all the cases studied in the paper, κa = κb, which gives the second harmonic (whose wave number is κm) twice the frequency as the fundamental mode. κn is the wave number of the mode identiﬁed by the index n that is not orthogonal to the mode with wave number κm. Pmn is the complex power ﬂux in the direction nz of the mth propagating mode in the expansion of the secondary solution. It is calculated by the integral of the particle velocity multiplied by the stress tensor corresponding to the second harmonics over the Materials 2017, 10, 547 5 of 19

∗ =0, ≠, (7) Thus a propagating mode m is orthogonal to all modes except itself, which makes m = n. In Equations (5) and (6), and denote the power flux through the surface and through the volume, respectively, due to nonlinear terms of the primary wave. Both variables contain the square of the fundamental mode displacement as shown in [21]. Thus a relative acoustic nonlinearity parameter (RANP) is defined as 2 = , (8) 1 whereMaterials 22017 ,and10, 547 1 denote the amplitude of the second harmonic and fundamental mode,5 of 20 respectively.

In the case of ≠ +, A2 remains bounded and oscillates with a spatial periodicity, often entire waveguide cross section. The orthogonality relation for the propagating and evanescent modes called the dispersion length , expressed as is given as [29] 2 ∗ Pmn == 0, km 6= k,n ,(9) (7) −2 Thus a propagating mode m is orthogonal to all modes except itself, which makes m = n. In Equations (5) where surdenotes f vol or ( = in all the studied cases), the wave number of the fundamental and (6), fn and f denote the power ﬂux through the surface and through the volume, respectively, mode. n due to nonlinear terms of the primary wave. Both variables contain the square of the fundamental In the case of = + , the condition of phase matching is satisfied between the mode displacement as shown in [21]. Thus a relative acoustic nonlinearity parameter (RANP) β0 is fundamental mode and specific mode in the expansion of the secondary solution. If the power flux is deﬁned as not zero ( + ≠0), A2 grows linearly alongA2 the propagation direction, which is called β0 = , (8) ‘internal resonance’. In conclusion, the internal resonanceA12 (linear growth of the second harmonic) requireswhere A two2 and conditions,A1 denote (1) the phase amplitude matching of the and second (2) non-zero harmonic power and flux fundamental ( + mode, ≠0 respectively.).

Figure 1. Stress-free 2-D plate structure. Figure 1. Stress-free 2-D plate structure. Numerical simulation offers an alternative and a supplement to the analytical solution. An incrementalIn the caseform of ofκ then 6= nonlinearκa + κb, A stress-strain2 remains bounded relation is and deduced oscillates in this with section. a spatial In real periodicity, applications, often ancalled excitation the dispersion signal has length a finiteLn, expressedtime duration. as Thus, the signal has a frequency spectrum of finite bandwidth instead of a single frequency that is assumed in the analytical analysis. In particular, the 2π aforementioned analytical analysis residesLn =on the 2-D ,plane strain assumption, while in real (9) − experiments, waves propagate in 3-D structures, inkn which2k f the wave diffraction will exert an influence on the amplitude of waves. where κ denotes κ or κ (κ = κ in all the studied cases), the wave number of the fundamental mode. The fmodeling aplatformb a Abaqusb ®/Explicit with the user defined VUMAT subroutine is adopted In the case of κ = κ + κ , the condition of phase matching is satisfied between the fundamental to realize the numericaln moa deling.b Compared to Abaqus®/Standard and COMSOL® that utilize an mode and specific mode in the expansion of the secondary solution. If the power flux is not implicit algorithm, Abaqus®/Explicit is suitable for the transient wave propagation problem, with a sur f vol controlledzero ( fn time+ fn increment6= 0), A2 to grows ensure linearly the solving along accura the propagationcy and stability. direction, As the which simulation is called is a ‘internalprocess ofresonance’. transient dynamics In conclusion, in nature, the internal the Forward resonance Euler (linear algorithm growth (explicit of the integration) second harmonic) is used requires to convert two sur f vol theconditions, constitutive (1) phase rate matchingequation, anddefined (2) non-zero according power to Equation flux ( fn (10),+ fn to 6=an0). incremental equation. AbaqusNumerical®/Explicit simulationprovides a user offers subroutine an alternative block called and aVUMAT, supplement which to allows the analytical users to program solution. theAn specified incremental stress-strain form of therelation nonlinear into an stress-strain incremental relation form. A is two-state deduced architecture in this section. is adopted In real whereapplications, in each aniteration excitation variables signal including has a finite the time strain duration. increment Thus, and the stress signal increment has a frequency have their spectrum initial of valuesfinite bandwidthin the old state instead and ofupdated a single values frequency in the that new is state. assumed The inupdated the analytical stress in analysis. Equation In (1) particular, can be expressedthe aforementioned as analytical analysis resides on the 2-D plane strain assumption, while in real experiments, waves propagate in 3-D structures, in which the wave diffraction will exert an influence 1 on the amplitude of waves. +∆ =( + ( +∆ ))( +∆ ), (10) 2 The modeling platform Abaqus®/Explicit with the user defined VUMAT subroutine is adopted to realize the numerical modeling. Compared to Abaqus®/Standard and COMSOL® that utilize an implicit algorithm, Abaqus®/Explicit is suitable for the transient wave propagation problem, with a controlled time increment to ensure the solving accuracy and stability. As the simulation is a process of transient dynamics in nature, the Forward Euler algorithm (explicit integration) is used to convert the constitutive rate equation, defined according to Equation (10), to an incremental equation. Abaqus®/Explicit provides a user subroutine block called VUMAT, which allows users to program the specified stress-strain relation into an incremental form. A two-state architecture is adopted where in each iteration variables including the strain increment and stress increment have their initial values Materials 2017, 10, 547 6 of 20 in the old state and updated values in the new state. The updated stress in Equation (1) can be expressed as 1 σ + ∆σ = C + M (ε + ∆ε ) (ε + ∆ε ), (10) ij ij ijkl 2 ijklmn mn mn kl kl where ∆ε and ∆σ are the strain and stress increments, respectively. Expanding the right side of Equation (10), the stress increment can be expressed as

1 1 1 ∆σij = Cijkl + 2 Mijklmnεmn + 2 Mijklmn∆εmn (εkl + ∆εkl) − Cijkl + 2 Mijklmnεmn εkl 1 = Cijkl + 2 Mijklmn∆εmn ∆εkl + Mijklmnεmn (11) 1 = Cijkl + 2 Mijklmn(∆εmn + 2εmn) ∆εkl ,

Transforming Equation (11) into the VUMAT written in Fortran 77, the material and geometric nonlinearities are embedded into the constitutive relation. Based on the explicit integration, these variables are all vectorized to facilitate the computing process. In order to keep the simulation stability and accuracy, the time increment is controlled automatically by the global stable increment estimator in Abaqus®/Explicit. Moreover, considering that the higher-order harmonic components are relatively small, double precision and excitation of considerably large amplitudes are speciﬁed in simulation to ensure computation accuracy and excite ﬁnite-amplitude waves.

3. Numerical Simulation of Nonlinear Guided Waves Both 2-D and 3-D numerical models with nonlinear stress-strain relations are built, featuring the frequently investigated S0-S0 and S1-S2 modes, to gain an insight into the accumulative nature of RANP.

3.1. 2-D Scenario

3.1.1. Problem Description

A plate with a thickness of 1 mm is modeled for the mode pair S0-S0, to which Plexiglas angle beam assemblies (see Table1 for material parameters) are bonded in order to realistically reflect the mechanism of how an ultrasonic wave is excited into the plate structure (Figure2). To model the linear/nonlinear features of wave propagation, the material parameters including density, Young’s modulus, Poisson’s ratio, and TOE constants listed in Table1[ 21] are defined as the input variables of VUMAT. A uniform displacement of twenty-cycle Hanning-window modulated sinusoidal toneburst at a central frequency of 400 kHz with the corresponding mode pair S0-S0 shown in Figure3a, is introduced at the inclined plane of the wedge. This S0-S0 case of phase mis-matching (case 1 in Table2) is deliberately set to quantitatively analyze the dispersion length. The incident angle of 30◦ is set according to Snell’s law [30]. Through a fixed coupling between the angle beam assemblies and the plate, the ultrasonic longitudinal vibration is converted to in-plane wave propagation. A fine mesh with the element length of 0.2 mm, 1/30 of the wavelength of the second harmonic S0 mode, is assigned to the whole model. As illustrated in Figure2, measurement points are defined from d = 0 to 0.87 m at an interval of 10 mm for the acquisition of the in-plane displacement. To model the mode pair S1-S2 with phase matching (case 2 in Table2), a plate with a thickness of 3 mm is used. Based on the calculation using Snell’s law, the incident angle is adjusted to be 25◦ to ensure the in-plane propagation of the refracted S1 wave from the wedge into the plate. A uniform displacement of twenty-cycle Hanning-window modulated sinusoidal toneburst at a central frequency of 1.178 MHz (with corresponding mode pair S1-S2 of 3.534 MHz mm–7.068 MHz mm displayed in Figure3b), is introduced at the inclined plane of the wedge. A fine mesh with the element length of 0.05 mm, 1/50 of the wavelength of the second harmonic S2 mode, is assigned to the whole model. Materials 2017, 10, 547 7 of 20 Materials 2017, 10, 547 7 of 19 Materials 2017, 10, 547 7 of 19

Figure 2. Schematic of 2-D plane strain simulation model for the mode pairs S0-S0 and S1-S2. Figure 2. Figure 2. Schematic of 2-D plane strain simulation model for the mode pairspairs SS00-S0 andand S S11-S-S22. .

(a) (b) (a) (b) Figure 3. Phase velocities for the mode pair (a) S0-S0 0.4–0.8 MHz mm and (b) S1-S2 at 3.534–7.068 MHz Figure 3. Phase velocities for the mode pair (a) S0-S0 0.4–0.8 MHz mm and (b) S1-S2 at 3.534–7.068 MHz Figuremm. 3. Phase velocities for the mode pair (a)S0-S0 0.4–0.8 MHz mm and (b)S1-S2 at 3.534–7.068 MHz mm. mm. Table 1.1. Material parameters for thethe plateplate structure.structure. Table 1. Material parameters for the plate structure. 3 Part Elastic Modulus (GPa) Poisson’s Ratio Density (kg/m3 ) Part Elastic Modulus (GPa) Poisson’s Ratio Density (kg/m ) 3 PlexiglasPart wedge Elastic Modulus4.12 (GPa) Poisson’s0.39 Ratio Density1180 (kg/m ) Plexiglas wedge 4.12 0.39 1180 Plexiglas wedge 4.1268.9 0.390.33 11802780 Aluminum plate A68.9 (GPa)68.9 B0.33 (GPa) C 27802780 (GPa) AluminumAluminum plate plate A A(GPa)−320(GPa) B (GPa)−200 CC(GPa) (GPa)−190 −320−320 −−200200 −−190190 Table 2. Parameters for the mode pairs. TableTable 2. 2. ParametersParameters for for the the mode mode pairs. pairs. Case Plate Thickness Excitation Frequency Thickness Mode Phase Velocity Case Plate Thickness Excitation Frequency Thickness Mode Phase Velocity No. (mm)Plate ThicknessFrequencyExcitation (kHz) ProductFrequency (MHz Thickness mm) Pair Phase(m/s) Velocity No.Case No. (mm) Frequency (kHz) Product (MHz mm) ModePair Pair (m/s) 1 1 (mm) Frequency400 (kHz) Product0.4–0.8 (MHz mm) S0-S0 5261–5221(m/s) 1 1 400 0.4–0.8 S0-S0 5261–5221 2 1 3 1 1178400 3.534–7.0680.4–0.8 SS0-S1-S0 2 5261–5221 6241–6241 3 3.534–7.068 1 2 2 2 3 11781178 3.534–7.068 S1-S-S2 6241–6241 3.1.2. Results and Discussion 3.1.2.3.1.2. Results Results and and Discussion Discussion The simulation with the excitation of the central frequency of 400 kHz is performed first. Short TimeTheThe Fourier simulation simulation Transform with with (STFT)the the excitation excitationis applied of the ofto the thecentral extracted central frequency frequency in-plane of 400 displacement, of kHz 400 is kHz performed isto performedobtain first. the Shortwave first. TimeShortenergy Fourier Time packets Fourier Transform corresponding Transform (STFT) (STFT) isto applied the is fundamental applied to the toextracted the and extracted in-planesecond in-plane harmonicdisplacement, displacement, modes. to obtain The to maximum obtainthe wave the energywavevalues energy ofpackets the first packets corresponding wave corresponding packets areto theextracted to thefundamental fundamental to indicate and the and second amplitude second harmonic harmonic of both modes. modes. modes. TheAs The plane maximum maximum waves values of the first wave packets are extracted to indicate the amplitude of both modes. As plane waves valuespropagate of the without first wave wave packets diffraction, are extracted the fundamental to indicate S0 the mode amplitude (Figure of4a) both remains modes. largely As plane unchanged waves propagate without wave diffraction, the fundamental S0 mode (Figure 4a) remains largely unchanged propagateexcept at the without near-field wave area. diffraction, It is noticed the fundamental that for the S0 Smode0-S0 mode, (Figure the4a) accumulation remains largely of unchangedthe second except at the near-field area. It is noticed that for the S0-S0 mode, the accumulation of the second exceptharmonic at theS0 mode near-field lasts up area. to a It distance is noticed around that for400 themm, S 0making-S0 mode, a dispersion the accumulation length around of the 750 second mm harmonic S0 mode lasts up to a distance around 400 mm, making a dispersion length around 750 mm harmonic(corresponding S0 mode to the lasts valley up to in a distanceFigure 4b,c). around An 400anal mm,ytical making solution a dispersionof the dispersion length length around based 750 mm on (corresponding to the valley in Figure 4b,c). An analytical solution of the dispersion length based on (correspondingEquation (9) is togiven. the valley In the in current Figure4 b,c).case, An analytical=2⁄ solution, and of the= dispersion⁄ , which length gives based the Equationdispersion (9) length is given. as In the current case, =2⁄ , and =⁄ , which gives the dispersion length as 1 = = 0.8584 m., 11 1 (12) = 2( − )= 0.8584 m., 1 1 (12) 2( − )

Materials 2017, 10, 547 8 of 20

on Equation (9) is given. In the current case, κn = 2w/cp−2w, and κa = w/cp−w, which gives the dispersion length as 1 L = = 0.8584 m., (12) n 1 1 Materials 2017, 10, 547 2 fa − 8 of 19 cp−2w cp−w

The obtainedobtained analyticalanalytical result result corresponds corresponds well well to the to numericalthe numerical result (circaresult 750(circa mm), 750 which mm), validates which validatesthe accuracy the accuracy of the built of numericalthe built numerical model. The model. rationale The rationale of the variation of the variation trend is thattrend within is that 400 within mm, 400the mm, condition the condition of phase of matching phase matching can be satisﬁed can be sa approximatelytisfied approximately to reach to the reach accumulative the accumulative second secondharmonic harmonic S0 mode. S0 However,mode. However, after that after distance, that distance, the second the harmonicsecond harmonic mode will mode gradually will gradually decrease decreasein its amplitude in its amplitude to a valley to asa valley the phase as the mis-matching phase mis-matching dominates. dominates. Then the Th seconden the harmonic second harmonic S0 mode Sre-accumulates0 mode re-accumulates since the since phase the matching phase matching is satisﬁed is satisfied again. again.

(a) (b) (c)

FigureFigure 4.4. MagnitudesMagnitudes of of the the S0 -SS00-Smode0 mode pair (0.4pair MHz (0.4 mm–0.8MHz mm–0.8 MHz mm) MHz extracted mm) extracted through Short-time through Short-timeFourier Transform Fourier (STFT) Transform in the 2-D(STFT) simulation in the model:2-D simulation (a) fundamental model: mode (a) fundamental A1; (b) second mode harmonic A1; (modeb) second A2; andharmonic (c) A2/(A1) mode2 A2;. and (c) A2/(A1)2.

The mode pair S1-S2 is also investigated following the same analysis procedure as the previous The mode pair S1-S2 is also investigated following the same analysis procedure as the previous S0-S0 mode. A slight decrease of the S1 mode along the propagation distance is displayed in Figure 5a, S0-S0 mode. A slight decrease of the S1 mode along the propagation distance is displayed in Figure5a, which can be attributed to the severe dispersion of phase velocity for the S1 mode (see Figure 3b). which can be attributed to the severe dispersion of phase velocity for the S1 mode (see Figure3b). As shown in Figure 5b, the second harmonic S2 mode shows a monotonic increase of amplitude over As shown in Figure5b, the second harmonic S 2 mode shows a monotonic increase of amplitude over thethe propagation propagation distance distance up up to to 400 400 mm. Taking Taking the the amplitude decrease decrease of of the fundamental mode mode (Figure(Figure 55a)a) into account, account, the the calculated calculated RANP RANP indica indicatestes a monotonic a monotonic and andalmost almost linear linear increase increase up to ~500up to mm, ~500 which mm, does which not does corroborate not corroborate the analytical the analyticalanalysis of analysis infinite accumulation of inﬁnite accumulation of the second of harmonic S2 mode according to Equation (6). The inconsistency, which is frequently observed in other the second harmonic S2 mode according to Equation (6). The inconsistency, which is frequently studiesobserved as inreviewed other studies in [31], as may reviewed arise infrom [31 ],severa may arisel aspects: from (1) several the selected aspects: central (1) the frequency selected central may deviate from the frequency for precise phase matching of the S1-S2 mode; (2) the severe dispersion of frequency may deviate from the frequency for precise phase matching of the S1-S2 mode; (2) the phase velocity for the S1 mode may prevent the accumulation of the S2 mode to a long distance; severe dispersion of phase velocity for the S1 mode may prevent the accumulation of the S2 mode to a (3)long the distance; ability of (3) the the current ability commercial of the current Finite commercial Element code Finite to accurately Element codecapture to accuratelythe faint displacement. capture the Nevertheless, a qualitative conclusion that the S2 mode accumulates along the propagation direction faint displacement. Nevertheless, a qualitative conclusion that the S2 mode accumulates along the is achieved. In addition, among all these excited modes, S1 and S2 propagate with the fastest group propagation direction is achieved. In addition, among all these excited modes, S1 and S2 propagate velocitywith the fastest[32], which group makes velocity them [32], whichsuperior makes to other them mode superior pairs to other when mode dealing pairs with when material dealing characterizationwith material characterization or damage detection or damage [32,33]. detection [32,33].

(a) (b) (c)

Figure 5. Magnitudes of the S1-S2 mode pair (3.534–7.068 MHz mm) extracted through STFT in the 2-D simulation model (a) fundamental mode A1; (b) second harmonic mode A2, and (c) A2/(A1)2.

Materials 2017, 10, 547 8 of 19

The obtained analytical result corresponds well to the numerical result (circa 750 mm), which validates the accuracy of the built numerical model. The rationale of the variation trend is that within 400 mm, the condition of phase matching can be satisfied approximately to reach the accumulative second harmonic S0 mode. However, after that distance, the second harmonic mode will gradually decrease in its amplitude to a valley as the phase mis-matching dominates. Then the second harmonic S0 mode re-accumulates since the phase matching is satisfied again.

(a) (b) (c)

Figure 4. Magnitudes of the S0-S0 mode pair (0.4 MHz mm–0.8 MHz mm) extracted through Short-time Fourier Transform (STFT) in the 2-D simulation model: (a) fundamental mode A1; (b) second harmonic mode A2; and (c) A2/(A1)2.

The mode pair S1-S2 is also investigated following the same analysis procedure as the previous S0-S0 mode. A slight decrease of the S1 mode along the propagation distance is displayed in Figure 5a, which can be attributed to the severe dispersion of phase velocity for the S1 mode (see Figure 3b). As shown in Figure 5b, the second harmonic S2 mode shows a monotonic increase of amplitude over the propagation distance up to 400 mm. Taking the amplitude decrease of the fundamental mode (Figure 5a) into account, the calculated RANP indicates a monotonic and almost linear increase up to ~500 mm, which does not corroborate the analytical analysis of infinite accumulation of the second harmonic S2 mode according to Equation (6). The inconsistency, which is frequently observed in other studies as reviewed in [31], may arise from several aspects: (1) the selected central frequency may deviate from the frequency for precise phase matching of the S1-S2 mode; (2) the severe dispersion of phase velocity for the S1 mode may prevent the accumulation of the S2 mode to a long distance; (3) the ability of the current commercial Finite Element code to accurately capture the faint displacement. Nevertheless, a qualitative conclusion that the S2 mode accumulates along the propagation direction is achieved. In addition, among all these excited modes, S1 and S2 propagate with the fastest group velocityMaterials 2017[32],, 10 ,which 547 makes them superior to other mode pairs when dealing with material9 of 20 characterization or damage detection [32,33].

(a) (b) (c)

Figure 5. Magnitudes of the S1-S2 mode pair (3.534–7.068 MHz mm) extracted through STFT in the MaterialsFigure 2017,5. 10Magnitudes, 547 of the S1-S2 mode pair (3.534–7.068 MHz mm) extracted through STFT in the 2-D9 of 19 2-D simulation model (a) fundamental mode A1; (b) second harmonic mode A2, and (c) A2/(A1)2. simulation model (a) fundamental mode A1; (b) second harmonic mode A2, and (c) A2/(A1)2. 3.2. 3-D Scenario 3.2. 3-D Scenario All the above models are built in the assumption of 2-D plane strain, which, compared to the actualAll case the of above wave modelspropagation are built in the in the3-D assumptionplate structures, of 2-D neglects plane strain,the infl which,uence of compared diffraction. to theTo quantitativelyactual case of study wave the propagation influence inof diffraction, the 3-D plate a 3-D structures, model with neglects only the mode inﬂuence pair of S0-S diffraction.0 is built. TheTo quantitatively 3-D S1-S2 model study is not the built inﬂuence here, ofconsidering diffraction, more a 3-D calculation model with effort only is therequired mode for pair higher S0-S0 frequency.is built. The 3-D S1-S2 model is not built here, considering more calculation effort is required for higher frequency. 3.2.1. Problem Description 3.2.1. Problem Description In the 3-D model, the plate thickness of 1 mm and the excitation central frequency of 400 kHz are setIn (see the 3-DFigure model, 6), the the same plate as thickness that in the of 12-D mm S and0-S0 model. the excitation Enormous central calculation frequency resources of 400 kHz are consumedare set (see to Figure model6), wave the same propagation as that in in the a 3-D 2-D stru S 0-Scture.0 model. Two operations Enormous calculationto reduce the resources model size are areconsumed accordingly to model performed. wave propagation Firstly, a symmetric in a 3-D structure. boundary Two condition operations is applied to reduce to thereduce model the sizemodel are toaccordingly half of its performed.original size. Firstly, Secondly, a symmetric the beam boundary divergence condition angle is of applied the fundamental to reduce the mode, model which to half is simulatedof its original to be size. less Secondly, than 30°, theenables beam a divergencefurther reduction angle of ofthe the fundamentalcalculation domain mode, whichthrough is simulatedmodeling ◦ ofto an be isosceles less than triangle 30 , enables with a a base further angle reduction of 30° in of the the in-plane calculation dimension. domain A through mesh size modeling of 0.25 mm of anis ◦ assignedisosceles triangleto the whole with a basemodel, angle with of 30approximatelyin the in-plane 26 dimension. nodes assigned A mesh in size one of 0.25wavelength mm is assigned of the concernedto the whole second model, harmonic with approximately wave mode 26 S nodes0, resulting assigned in ina onecalculation wavelength model of thecomprising concerned over second 10 millionharmonic elements. wave mode S0, resulting in a calculation model comprising over 10 million elements.

Figure 6. Schematic of the 3-D simulation model for the mode pair S0-S0. Figure 6. Schematic of the 3-D simulation model for the mode pair S0-S0. 3.2.2. Results and Discussion 3.2.2. Results and Discussion The variation trend of magnitudes along the propagation distance (Figure 7) in the 3-D case is The variation trend of magnitudes along the propagation distance (Figure7) in the 3-D case significantly different from that in the 2-D case (Figure 4). A drastic decrease in the magnitude of the is signiﬁcantly different from that in the 2-D case (Figure4). A drastic decrease in the magnitude fundamental mode occurs due to wave diffraction (Figure 7a). The magnitude of the second harmonic S2 mode only accumulates up to a propagation distance of 100 mm (Figure 7b), beyond which the diffraction dominates and suppresses the further accumulation of the second harmonic wave. However, RANP continues to accumulate up to 300 mm. Although the correction of diffraction based on the analytical 3-D nonlinear Raleigh wave is available [34], the analytical solution of the nonlinear Lamb wave taking the diffraction into account is yet to be developed. RANP deduced from the 2-D case (see Equation (8)) was used in most studies [32,33,35,36], which is also applied in this paper, while some works [37] tried to correct RANP for guided Lamb waves. Using RANP calculated from Equation (8), a qualitative agreement of RANP in the numerical 3-D case (Figure 7c) corresponds well with the 2-D case (Figure 4c).

Materials 2017, 10, 547 10 of 20 of the fundamental mode occurs due to wave diffraction (Figure7a). The magnitude of the second harmonic S2 mode only accumulates up to a propagation distance of 100 mm (Figure7b), beyond which the diffraction dominates and suppresses the further accumulation of the second harmonic wave. However, RANP continues to accumulate up to 300 mm. Although the correction of diffraction based on the analytical 3-D nonlinear Raleigh wave is available [34], the analytical solution of the nonlinear Lamb wave taking the diffraction into account is yet to be developed. RANP deduced from the 2-D case (see Equation (8)) was used in most studies [32,33,35,36], which is also applied in this paper, while some works [37] tried to correct RANP for guided Lamb waves. Using RANP calculated from Equation (8), a qualitative agreement of RANP in the numerical 3-D case (Figure7c) corresponds well with the 2-D case (Figure4c). Materials 2017, 10, 547 10 of 19

(a) (b) (c)

Figure 7. Magnitudes of the S0-S0 mode pair (0.4 MHz mm–0.8 MHz mm) extracted through STFT in Figure 7. Magnitudes of the S0-S0 mode pair (0.4 MHz mm–0.8 MHz mm) extracted through STFT in thethe 3-D 3-D simulation simulation model model ( (aa)) fundamental fundamental mode mode A1; A1; ( (bb)) second second harmonic harmonic mode mode A2; A2; and and ( c) A2/(A1) 22. .

4.4. Experimental Experimental Validation Validation UponUpon the the establishment establishment of of the the analytical analytical and and numerical numerical foundation foundation of linear/nonlinear GUWs GUWs withwith nonlinear nonlinear stress-strain stress-strain relation relation in in the the last last two two sections, sections, the the DIDIss based based on on linear/nonlinear linear/nonlinear GUWs GUWs areare extracted extracted to to characterize characterize HVI- HVI-inducedinduced pitting pitting damage damage in in the the inner inner shielding shielding layer. layer. This This kind kind of of damagedamage poses poses a a threat threat to to the the struct structuralural integrity integrity and and leads leads to to further further potential potential injury injury to to the the equipment equipment andand crew crew inside. inside.

4.1.4.1. HVI HVI Experiment Experiment TheThe HVI HVI facilities facilities at atthe the State State Key Key Laboratory Laboratory of Explosion of Explosion Science Science and Technology, and Technology, China,China, were usedwere for used the HVI for the experiments, HVI experiments, of which the of whichcore eq theuipment core equipmentis a two-stage is light a two-stage gas gun. light Through gas gun.the propulsionThrough the of propulsiongas of extremely of gas ofhigh extremely pressure, high a pressure,projectile acan projectile be accelerated can be accelerated to impinge to a impinge target structure,a target structure, at a desired at avelocity desired up velocity to 10 km/s–the up to 10 impact km/s–the velocity impact in velocitya typical inHVI a typical event in HVI the event low Earthin the orbit low between Earth orbit a MOD between particle a MOD and spacecraft. particle and A two-layer spacecraft. shielding A two-layer assembly shielding is immobilized assembly atis the immobilized testing chamber at the of testing the two-stage chamber light of thegas two-stagegun, which light launches gas gun, a projectile which launches(aluminum a projectile2024-T4, ~6(aluminum km/s, ø4.5 2024-T4,mm) into ~6 the km/s, thinner ø4.5 outer mm) layer into (aluminum the thinner 2024-T4, outer layer 1 mm (aluminum in thickness, 2024-T4, 300 mm 1 mm× 300 in mmthickness, in the 300in-plane mm ×dimension),300 mm in to the produce in-plane a dimension),controlled HVI to produce (see Figure a controlled 8a). Upon HVI penetrating (see Figure the8a). outerUpon layer penetrating (see Figure the 8b) outer using layer the (see vast Figure instant8b) kine usingtic energy, the vast the instant shattered kinetic projectile, energy, together the shattered with someprojectile, jetted together portion of with the some outer jetted layer, portion further ofimpa thects outer the layer,thicker further inner impactslayer (3 mm the thickerin the thickness). inner layer As(3 mmshown in thein Figure thickness). 8c, multitudinous As shown in Figure small-scale8c, multitudinous pitting damage small-scale is scattered pitting over damage a large is area, scattered the centerover aof large which area, is filled the center with ofcontinuous which is ﬁlledcraters, with while continuous the surrounding craters, while area is the filled surrounding with separated area is craters.ﬁlled with separated craters.

(a) (b) (c)

Figure 8. (a) Schematic of the Hypervelocity Impact (HVI) experiment; (b) photograph of the outer layer being penetrated; and (c) photograph of the inner layer with multitudinous small craters.

4.1.1. Experimental Setups Figure 9 shows the schematic and photograph of the experiment. The RITEC advanced measurement system RAM-5000 SNAP (RITEC, Warwick, US) is used to generate a Hanning-window modulated ultrasonic voltage excitation around 1000 Vp-p. The piezoelectric transducer converts the

Materials 2017, 10, 547 10 of 19

(a) (b) (c)

Figure 7. Magnitudes of the S0-S0 mode pair (0.4 MHz mm–0.8 MHz mm) extracted through STFT in the 3-D simulation model (a) fundamental mode A1; (b) second harmonic mode A2; and (c) A2/(A1)2.

4. Experimental Validation Upon the establishment of the analytical and numerical foundation of linear/nonlinear GUWs with nonlinear stress-strain relation in the last two sections, the DIs based on linear/nonlinear GUWs are extracted to characterize HVI-induced pitting damage in the inner shielding layer. This kind of damage poses a threat to the structural integrity and leads to further potential injury to the equipment and crew inside.

4.1. HVI Experiment The HVI facilities at the State Key Laboratory of Explosion Science and Technology, China, were used for the HVI experiments, of which the core equipment is a two-stage light gas gun. Through the propulsion of gas of extremely high pressure, a projectile can be accelerated to impinge a target structure, at a desired velocity up to 10 km/s–the impact velocity in a typical HVI event in the low Earth orbit between a MOD particle and spacecraft. A two-layer shielding assembly is immobilized at the testing chamber of the two-stage light gas gun, which launches a projectile (aluminum 2024-T4, ~6 km/s, ø4.5 mm) into the thinner outer layer (aluminum 2024-T4, 1 mm in thickness, 300 mm × 300 mm in the in-plane dimension), to produce a controlled HVI (see Figure 8a). Upon penetrating the outer layer (see Figure 8b) using the vast instant kinetic energy, the shattered projectile, together with some jetted portion of the outer layer, further impacts the thicker inner layer (3 mm in the thickness). As shown in Figure 8c, multitudinous small-scale pitting damage is scattered over a large area, the Materialscenter2017 ,of10 ,which 547 is filled with continuous craters, while the surrounding area is filled with separated11 of 20 craters.

(a) (b) (c)

Figure 8. (a) Schematic of the Hypervelocity Impact (HVI) experiment; (b) photograph of the outer Figure 8. (a) Schematic of the Hypervelocity Impact (HVI) experiment; (b) photograph of the outer layer being penetrated; and (c) photograph of the inner layer with multitudinous small craters. layer being penetrated; and (c) photograph of the inner layer with multitudinous small craters. 4.1.1. Experimental Setups 4.1.1. Experimental Setups Figure 9 shows the schematic and photograph of the experiment. The RITEC advanced MaterialsFiguremeasurement 2017,9 10 shows, 547 system the RAM-5000 schematic SNAP and (RITEC, photograph Warwick, of US) the is experiment. used to generate The a Hanning-window RITEC advanced11 of 19 measurementmodulated system ultrasonic RAM-5000 voltage excita SNAPtion (RITEC, around Warwick, 1000 Vp-p. US) The is piezoelectric used to generate transducer a Hanning-window converts the electricalmodulated signal ultrasonic to the voltage longitudinal excitation ultrasonic around vi 1000bration, Vp-p. which The piezoelectricis then transmitted transducer into converts a Plexiglas the wedgeelectrical with signal ultragel to the as longitudinal the couplant. ultrasonic Based vibration,on Snell’s which law, isthe then vibration transmitted refracts into and a Plexiglas finally wedgepropagates with in ultragel the in-plane as the couplant. direction Based of the on plat Snell’se structure law, the through vibration a further refracts coupling and ﬁnally between propagates the anglein the in-planebeam assemblies direction and of the the plate plate. structure The actuator through is a fixed, further and coupling the receiver between moves the angle along beam the propagationassemblies and direction. the plate. To The suppress actuator the is measurement ﬁxed, and the uncertainty, receiver moves an average along the of propagation256 signals is direction. applied Toto suppressthe acquired the measurement response voltage uncertainty, signal. anFour average repeated of 256 experiments signals is applied are performed to the acquired to verify response the voltagecredibility signal. of the Four measured repeated data. experiments are performed to verify the credibility of the measured data.

(a)

(b)

Figure 9. (a) Schematic of the experimental setup for the S0-S0 and S1-S2 mode pairs; and (b) photographic Figure 9. (a) Schematic of the experimental setup for the S0-S0 and S1-S2 mode pairs; and illustration of the experiment. (b) photographic illustration of the experiment.

Experiments on both mode pairs S0-S0 and S1-S2 are performed. Regarding the mode pair S0-S0 0.4 MHz mm–0.8 MHz mm for experiment validation, a 20-cycle Hanning-window modulated sinusoidal toneburst at a central frequency of 400 KHz is adopted as the excitation signal. Two piston piezoelectric transducers (KRAUTKRAMER®, ø25.4 mm) with respective central frequencies of 0.5 MHz and 1 MHz (as actuator and receiver, respectively) are coupled with an aluminum plate of 1 mm thickness. Water is applied as the couplant to ensure a consistent coupling as the location of the receiver changes. The second test is for the mode pair S1-S2 3.42–6.84 MHz mm. A 20-cycle Hanning- window modulated sinusoidal toneburst at a central frequency of 1.14 MHz is adopted as the excitation signal. Two piston piezoelectric transducers (KRAUTKRAMER®, ø25.4 mm) with respective central frequency of 1 MHz and 2.25 MHz (as actuator and receiver, respectively) are coupled with an aluminum plate of 3 mm thickness. Honey is applied as the couplant to ensure a transfer of in-plane vibration from the angle beam assemblies to the aluminum plate structure.

4.1.2. Results and Discussion

Figure 10 displays the typical waveforms for the S0-S0 mode pair. Taking the raw time-domain signal acquired after a propagation distance of 200 mm (Figure 10a) as an example, the S0 mode is observed to occupy the dominant energy. The extracted magnitudes of the fundamental and second

Materials 2017, 10, 547 12 of 20

Experiments on both mode pairs S0-S0 and S1-S2 are performed. Regarding the mode pair S0-S0 0.4 MHz mm–0.8 MHz mm for experiment validation, a 20-cycle Hanning-window modulated sinusoidal toneburst at a central frequency of 400 KHz is adopted as the excitation signal. Two piston piezoelectric transducers (KRAUTKRAMER®, ø25.4 mm) with respective central frequencies of 0.5 MHz and 1 MHz (as actuator and receiver, respectively) are coupled with an aluminum plate of 1 mm thickness. Water is applied as the couplant to ensure a consistent coupling as the location of the receiver changes. The second test is for the mode pair S1-S2 3.42–6.84 MHz mm. A 20-cycle Hanning-window modulated sinusoidal toneburst at a central frequency of 1.14 MHz is adopted as the excitation signal. Two piston piezoelectric transducers (KRAUTKRAMER®, ø25.4 mm) with respective central frequency of 1 MHz and 2.25 MHz (as actuator and receiver, respectively) are coupled with an aluminum plate of 3 mm thickness. Honey is applied as the couplant to ensure a transfer of in-plane vibration from the angle beam assemblies to the aluminum plate structure.

4.1.2. Results and Discussion

Figure 10 displays the typical waveforms for the S0-S0 mode pair. Taking the raw time-domain Materialssignal 2017 acquired, 10, 547 after a propagation distance of 200 mm (Figure 10a) as an example, the S0 mode12 of 19 is observed to occupy the dominant energy. The extracted magnitudes of the fundamental and second harmonicharmonic modes modes (Figure 1010b,c)b,c) through through STFT STFT show show that that (1) the(1) secondthe second harmonic harmonic mode mode is much is weakermuch weakercompared compared to the fundamentalto the fundamental mode; mode; and (2) and the (2 equal) the arrivalequal arrival time of time the fundamentalof the fundamental and second and secondharmonic harmonic modes, themodes, indicator the indicator of quasi-matching of quasi- propagationmatching propagation velocities, guaranteesvelocities, theguarantees accumulation the accumulationof the second of harmonic the second mode. harmonic mode.

(a) (b) (c)

Figure 10. Signals of the mode pair S0-S0 at a propagation distance of 200 mm in the experiment Figure 10. Signals of the mode pair S0-S0 at a propagation distance of 200 mm in the experiment (second harmonic mode × 20 times in amplitude) (a) raw acquired signals; (b) separated signals of S0-S0; (second harmonic mode × 20 times in amplitude) (a) raw acquired signals; (b) separated signals of and (c) separated energy packets of S0-S0. S0-S0; and (c) separated energy packets of S0-S0. The magnitude of the extracted wave energy packet is used as the index to quantitatively calculateThe RANP. magnitude Figure of the11 shows extracted the wavemagnitudes energy of packet the extracted is used as wave the index packet to along quantitatively the propagation calculate RANP. Figure 11 shows the magnitudes of the extracted wave packet along the propagation distance distance for the S0-S0 mode pair. The average value and error bar calculated by the standard deviation arefor obtained the S0-S0 modeaccording pair. to The four average repeated value experime and errornts. bar Attributed calculated to by the the wave standard diffraction, deviation the are fundamentalobtained according mode to(Figure four repeated 11a) decreases experiments. almost Attributed monotonically. to the wave The diffraction, second harmonic the fundamental mode (Figuremode (Figure11b) decreases 11a) decreases slowly almost within monotonically. a distance of The 0.16 second m by harmonic the superposed mode (Figure influence 11b) of decreases wave diffractionslowly within and nonlinear a distance accumula of 0.16 mtion. by theRANP superposed shows an inﬂuence approximately of wave linear diffraction increase and (Figure nonlinear 11c), whichaccumulation. corroborates RANP both shows the analytical an approximately and numerical linear analyses increase in (Figure Sections 11c), 2.1, which 3.1, and corroborates 3.2. both the analytical and numerical analyses in Sections2, 3.1 and 3.2.

(a) (b) (c)

Figure 11. Magnitudes of the S0-S0 mode pair (0.4 MHz mm–0.8 MHz mm) extracted through STFT in four repeated experiments (a) fundamental mode A1; (b) second harmonic mode A2; and (c) A2/(A1)2.

Figure 12 displays typical waveforms for the S1-S2 mode pair. The fundamental wave (Figure 12a) comprises multiple wave modes according to the dispersion curve in Figure 3b, in which the S1 mode (Figure 12b) arrives first, together with the S2 mode in the second harmonic component. Taking the signal acquired after a propagation distance of 200 mm (Figure 12) as an example, STFT analysis is performed to extract the wave packet of the fundamental S1 mode and second harmonic S2 mode (Figure 12c). S1 and S2 arrive the earliest among the multiple components of the wave modes, which facilitates the quantitative analysis on the accumulation effect of the S2 mode. The magnitude of the extracted wave energy packet (Figure 13) is used as the index to quantitatively calculate RANP. Owing to wave diffraction, the fundamental mode decreases almost monotonically (Figure 13a). The second harmonic mode (Figure 13b) accumulates in the range 0.1–0.14 m owing to the phase matching, then decreases slowly due to the influence of wave

Materials 2017, 10, 547 12 of 19 harmonic modes (Figure 10b,c) through STFT show that (1) the second harmonic mode is much weaker compared to the fundamental mode; and (2) the equal arrival time of the fundamental and second harmonic modes, the indicator of quasi-matching propagation velocities, guarantees the accumulation of the second harmonic mode.

(a) (b) (c)

Figure 10. Signals of the mode pair S0-S0 at a propagation distance of 200 mm in the experiment (second harmonic mode × 20 times in amplitude) (a) raw acquired signals; (b) separated signals of S0-S0;

and (c) separated energy packets of S0-S0.

The magnitude of the extracted wave energy packet is used as the index to quantitatively calculate RANP. Figure 11 shows the magnitudes of the extracted wave packet along the propagation distance for the S0-S0 mode pair. The average value and error bar calculated by the standard deviation are obtained according to four repeated experiments. Attributed to the wave diffraction, the fundamental mode (Figure 11a) decreases almost monotonically. The second harmonic mode (Figure 11b) decreases slowly within a distance of 0.16 m by the superposed influence of wave diffractionMaterials 2017 and, 10, 547nonlinear accumulation. RANP shows an approximately linear increase (Figure13 11c), of 20 which corroborates both the analytical and numerical analyses in Sections 2.1, 3.1, and 3.2.

(a) (b) (c)

FigureFigure 11. Magnitudes ofof the the S 0S-S00-Smode0 mode pair pair (0.4 (0.4 MHz MHz mm–0.8 mm–0.8 MHz MHz mm) extractedmm) extracted through through STFT in STFTfour repeatedin four repeated experiments experiments (a) fundamental (a) fundamental mode A1; (modeb) second A1; harmonic(b) second mode harmonic A2; and mode (c) A2/(A1) A2; and2 . (c) A2/(A1)2.

Figure 12 displays typical waveforms for the S1-S2 mode pair. The fundamental wave (Figure 12a) Figure 12 displays typical waveforms for the S1-S2 mode pair. The fundamental wave (Figure 12a) comprises multiple wave modes according to the dispersion curve in Figure3b, in which the S 1 mode comprises multiple wave modes according to the dispersion curve in Figure 3b, in which the S1 mode (Figure 12b) arrives ﬁrst, together with the S2 mode in the second harmonic component. Taking the (Figuresignal acquired12b) arrives after first, a propagation together with distance the S2 ofmode 200 in mm the (Figure second 12 harmonic) as an example, component. STFT Taking analysis the is signalMaterials acquired 2017, 10, 547 after a propagation distance of 200 mm (Figure 12) as an example, STFT analysis13 of 19is performed to extract the wave packet of the fundamental S1 mode and second harmonic S2 mode performed to extract the wave packet of the fundamental S1 mode and second harmonic S2 mode diffraction.(Figure 12c). The S1 andcalculated S2 arrive RANP the earliest shows among an approximately the multiple components linear increase of the (Figure wave modes, 13c), which which (Figure 12c). S1 and S2 arrive the earliest among the multiple components of the wave modes, which corroboratesfacilitates the both quantitative the analytical analysis and on numerical the accumulation analyses effectin Sections of the 2.1 S2 andmode. 3.1. facilitates the quantitative analysis on the accumulation effect of the S2 mode. The magnitude of the extracted wave energy packet (Figure 13) is used as the index to quantitatively calculate RANP. Owing to wave diffraction, the fundamental mode decreases almost monotonically (Figure 13a). The second harmonic mode (Figure 13b) accumulates in the range 0.1–0.14 m owing to the phase matching, then decreases slowly due to the influence of wave

(a) (b)(c)

1 2 FigureFigure 12. 12. SignalsSignals of of the the mode mode pair pair S S1-S-S2 3.423.42 MHz MHz mm–6.84 mm–6.84 MHz MHz mm mm at at a a propagation propagation distance distance of of 200200 mm mm in in the the experiment experiment (second (second harmon harmonicic mode mode × ×2020 times times in inamplitude). amplitude). (a) (Separateda) Separated signals signals of

theof thefundamental fundamental and and second second harmonic harmonic modes; modes; (b ()b zoom-in) zoom-in signals signals of of the the S S11 andand S S22 modes;modes; and and ((cc)) separated separated energy energy packets packets of of the the S S11 andand S S22 modes.modes.

The magnitude of the extracted wave energy packet (Figure 13) is used as the index to quantitatively calculate RANP. Owing to wave diffraction, the fundamental mode decreases almost monotonically (Figure 13a). The second harmonic mode (Figure 13b) accumulates in the range 0.1–0.14 m owing to the phase matching, then decreases slowly due to the inﬂuence of wave diffraction. The calculated RANP shows an approximately linear increase (Figure 13c), which corroborates both the analytical and numerical analyses in Sections2 and 3.1.

(a) (b)(c)

Figure 13. Magnitudes of the S1-S2 mode pair (3.42 MHz mm–6.84 MHz mm) extracted through STFT in four repeated experiments. (a) Fundamental mode A1; (b) second harmonic mode A2; and (c) A2/(A1)2.

4.2. Development of Linear/Nonlinear Damage Indices Based on the above analysis in Sections 3 and 4.1, both the linear and nonlinear features can be extracted to develop DIs for the purpose of characterization of HVI-induced pitting damage in the inner shielding layer. A typical GUW-based damage characterization of HVI-induced damage in the inner shielding layer using linear/nonlinear features, as shown in Figure 14, comprises five basic steps, i.e., excitation, wave propagation, sensing, signal processing, and diagnosis. In the excitation step, through a dedicated control of the excitation signal to satisfy both requirements of phase matching and non- zero power flux, the second harmonic wave S2 from the case of the mode pair S1-S2 is accumulated and strengthened for the extraction of linear/nonlinear DIs to characterize the HVI-induced pitting damage. The rationale behind the deliberate selection of the S1-S2 is that usually the probing wave with a higher frequency and higher mode shows a higher sensitivity to damage. Compared to the S1-S2 mode in the MHz range, the second harmonic in the S0-S0 mode can only be accumulated with the excitation frequency below 200 kHz (plate thickness of 3 mm), whose low frequency and large time span will cumber the sensitivity and effectiveness of damage characterization. In the experiment, twenty scanning paths are defined (Figure 15b,c), where the actuator and receiver move in parallel 20 mm at a time (Figure 15a), covering a monitoring area of 160 mm × 200 mm. The excitation of fifteen-cycle Hanning-window modulated sinusoidal toneburst at a central frequency of 1.14 MHz is applied. The phase matching between the mode pair S1-S2 3.42–6.84 MHz mm (Figure 13c) leads to the accumulative second harmonic wave S2, which is utilized for damage characterization.

Materials 2017, 10, 547 13 of 19

diffraction. The calculated RANP shows an approximately linear increase (Figure 13c), which corroborates both the analytical and numerical analyses in Sections 2.1 and 3.1.

(a) (b)(c)

Figure 12. Signals of the mode pair S1-S2 3.42 MHz mm–6.84 MHz mm at a propagation distance of 200 mm in the experiment (second harmonic mode × 20 times in amplitude). (a) Separated signals of

Materialsthe2017 fundamental, 10, 547 and second harmonic modes; (b) zoom-in signals of the S1 and S2 modes; and14 of 20

(a) (b)(c)

1 2 Figure 13. 13. MagnitudesMagnitudes of of the the S -S S1-S mode2 mode pair pair (3.42 (3.42 MHz MHz mm–6.84 mm–6.84 MHz MHz mm) mm)extracted extracted through through STFT inSTFT four in repeated four repeated experiments. experiments. (a) Fundamental (a) Fundamental mode A1; mode(b) second A1; (harmonicb) second mode harmonic A2; and mode (c) A2/(A1) A2; and2. (c) A2/(A1)2. 4.2. Development of Linear/Nonlinear Damage Indices 4.2. DevelopmentBased on the of above Linear/Nonlinear analysis in Damage Sections Indices 3 and 4.1, both the linear and nonlinear features can be extractedBased to on develop the above DIs analysisfor the purpose in Sections of 3characterization and 4.1, both the of linearHVI-induced and nonlinear pitting features damage can in the be innerextracted shielding to develop layer. DIs for the purpose of characterization of HVI-induced pitting damage in the innerA shielding typical GUW-based layer. damage characterization of HVI-induced damage in the inner shielding layerA using typical linear/nonlinear GUW-based damagefeatures, characterization as shown in Figu ofre HVI-induced14, comprises damagefive basic in steps, the inner i.e., excitation, shielding wavelayer usingpropagation, linear/nonlinear sensing, features,signal processing, as shown in an Figured diagnosis. 14, comprises In the ﬁve excitation basic steps, step, i.e., through excitation, a wavededicated propagation, control of sensing, the excitation signal processing, signal to andsatisfy diagnosis. both requirem In the excitationents of phase step, throughmatching a dedicatedand non- zerocontrol power of the flux, excitation the second signal harmonic to satisfy bothwave requirements S2 from the case of phase of the matching mode pair and S non-zero1-S2 is accumulated power ﬂux, andthe secondstrengthened harmonic for wave the extraction S2 from the of case linear/nonlinear of the mode pair DIs S to1-S characterize2 is accumulated the HVI-induced and strengthened pitting for damage.the extraction The rationale of linear/nonlinear behind theDI deliberates to characterize selection the of HVI-induced the S1-S2 is that pitting usually damage. the probing The rationale wave withbehind a higher the deliberate frequency selection and higher of the mode S1-S2 isshows that usuallya higher the sensitivity probing waveto damage. with a Compared higher frequency to the 1 2 0 0 Sand-S highermode in mode the MHz shows range, a higher the second sensitivity harmonic to damage. in the ComparedS -S mode tocan the only S1-S be2 modeaccumulated in the MHzwith therange, excitation the second frequency harmonic below in the 200 S 0kHz-S0 mode (plate can thickness only be of accumulated 3 mm), whose with low the frequency excitation frequencyand large timebelow span 200 will kHz cumber (plate thickness the sensitivity of 3 mm), and effectiveness whose low frequency of damage and characterization. large time span will cumber the sensitivityIn the andexperiment, effectiveness twenty of damage scanning characterization. paths are defined (Figure 15b,c), where the actuator and Materials 2017, 10, 547 14 of 19 receiver move in parallel 20 mm at a time (Figure 15a), covering a monitoring area of 160 mm × 200 mm. The excitation of fifteen-cycle Hanning-window modulated sinusoidal toneburst at a central frequency of 1.14 MHz is applied. The phase matching between the mode pair S1-S2 3.42–6.84 MHz mm (Figure 13c) leads to the accumulative second harmonic wave S2, which is utilized for damage characterization.

Figure 14. Schematic of linear/nonlinear guided wave-based damage characterization. Figure 14. Schematic of linear/nonlinear guided wave-based damage characterization.

In the experiment, twenty scanning paths are deﬁned (Figure 15b,c), where the actuator and receiver move in parallel 20 mm at a time (Figure 15a), covering a monitoring area of 160 mm × 200 mm. The excitation of ﬁfteen-cycle Hanning-window modulated sinusoidal toneburst at a central frequency of 1.14 MHz is applied. The phase matching between the mode pair S1-S2 3.42–6.84 MHz mm (Figure 13c) leads to the accumulative second harmonic wave S2, which is utilized for damage characterization.

(a) (b)(c)

Figure 15. (a) Photograph of damage characterization through twenty scanning paths; (b) nine vertical paths; and (c) eleven horizontal paths.

In the case of phase matching, S1-S2 3.42–6.84 MHz mm, the magnitude of the signal packet is extracted (as illustrated in Figure 14), which indicates the influence of the HVI-induced damage on the linear (fundamental S1 mode) and nonlinear (second harmonic S2 mode) GUWs (Figure 16). The fundamental probing wave (S1) is sensitive to the HVI-induced pitting damage, resulting in a significant energy scattering and reflection upon encountering the damage (a magnitude decrease of 20 dB in Figure 16a). In addition, as the second harmonic wave S2 accumulates along the propagation distance, the interaction between S2 and damage is manifested via the magnitude of A2 (see Figure 16b), whose decrease of 25 dB indicates a higher sensitivity of the second harmonic mode (S2) to damage compared with the fundamental mode (S1). The RANP is calculated, (see Figure 16c) highlighting the path with damage. All the three parameters, A1, A2, and RANP, which indicate the existence of HVI- induced pitting damage via different mechanisms, are used as DIs.

(a) (b) (c)

Figure 16. Magnitudes of the mode pair S1-S2 (3.42–6.84 MHz mm) extracted through STFT on the twenty paths in the experiment in (a) fundamental mode A1; (b) second harmonic mode A2; and (c) A2/(A1)2.

Materials 2017, 10, 547 14 of 19 Materials 2017, 10, 547 14 of 19

Materials 2017, 10, 547 15 of 20 Figure 14. Schematic of linear/nonlinear guided wave-based damage characterization. Figure 14. Schematic of linear/nonlinear guided wave-based damage characterization.

(a) (b)(c) (a) (b)(c) FigureFigure 15. 15. a(a) Photograph of damage characterization through twenty scanning paths;b (b) nine vertical Figure 15. (a() )Photograph Photograph of of damage damage characterizat characterizationion through through twenty twenty scanning scanning paths; paths; (b ( ) )nine nine vertical vertical paths;paths; and and (c ()c eleven) eleven horizontal horizontal paths. paths. paths; and (c) eleven horizontal paths.

In the case of phase matching, S1-S2 3.42–6.84 MHz mm, the magnitude of the signal packet is InIn the the case case of of phase phase matching, matching, S1 S-S1-S2 3.42–6.842 3.42–6.84 MHz MHz mm, mm, the the magnitude magnitude of ofthe the signal signal packet packet is extracted (as illustrated in Figure 14), which indicates the influence of the HVI-induced damage on extractedis extracted (as (asillustrated illustrated in Figure in Figure 14), 14 which), which indi indicatescates the theinfluence influence of the of theHVI-induced HVI-induced damage damage on the linear (fundamental S1 mode) and nonlinear (second harmonic S2 mode) GUWs (Figure 16). The theon linear the linear (fundamental (fundamental S1 mode) S1 mode) and nonlinear and nonlinear (second (second harmonic harmonic S2 mode) S2 mode) GUWs GUWs (Figure (Figure 16). The 16). fundamental probing wave (S1) is sensitive to the HVI-induced pitting damage, resulting in a fundamentalThe fundamental probing probing wave wave (S1) (Sis 1)sensitive is sensitive to the to the HVI-induced HVI-induced pitting pitting damage, damage, resulting resulting in in a a significant energy scattering and reflection upon encountering the damage (a magnitude decrease of significantsignificant energy energy scattering scattering and and reflection reflection upon upon encountering encountering the the damage damage (a (a magnitude magnitude decrease decrease of of 20 dB in Figure 16a). In addition, as the second harmonic wave S2 accumulates along the propagation 2020 dB dB in in Figure Figure 16a). 16a). In In addition, addition, as as the the second second harmonic harmonic wave wave S S2 2accumulatesaccumulates along along the the propagation propagation distance, the interaction between S2 and damage is manifested via the magnitude of A2 (see Figure 16b), distance,distance, the the interaction interaction between between S S2 2andand damage damage is is manifested manifested via via the the magnitude magnitude of of A2 A2 (see (see Figure Figure 16b), 16b), whose decrease of 25 dB indicates a higher sensitivity of the second harmonic mode (S2) to damage whosewhose decrease decrease of of 25 25 dB dB indicates indicates a a higher higher sensitivity sensitivity of of the the second second harmonic harmonic mode mode (S (S22)) to to damage damage compared with the fundamental mode (S1). The RANP is calculated, (see Figure 16c) highlighting the compared with the fundamental mode (S ). The RANP is calculated, (see Figure 16c) highlighting comparedpath with with damage. the fundamental All the three mode parameters, (S1).1 The A1, RANP A2, andis calculated, RANP, which (see Figureindicate 16c) the highlighting existence of theHVI- paththe pathwith withdamage. damage. All the All three the threeparameters, parameters, A1, A2 A1,, and A2, RANP, and RANP, which which indicate indicate the existence the existence of HVI- of induced pitting damage via different mechanisms, are used as DIs. inducedHVI-induced pitting pitting damage damage via different via different mechanisms, mechanisms, are used are as used DIs. as DIs.

(a) (b) (c) (a) (b) (c) Figure 16. Magnitudes of the mode pair S1-S2 (3.42–6.84 MHz mm) extracted through STFT on the Figure 16. Magnitudes of the mode pair S1-S2 (3.42–6.84 MHz mm) extracted through STFT on the Figuretwenty 16. pathsMagnitudes in the experiment of the mode in pair(a) fundamental S1-S2 (3.42–6.84 mode MHz A1; mm) (b) second extracted harmonic through mode STFT A2; on theand twenty paths in the experiment in (a) fundamental mode A1; (b) second harmonic mode A2; and twenty(c) A2/(A1) paths2. in the experiment in (a) fundamental mode A1; (b) second harmonic mode A2; and 2 ((cc)) A2/(A1) A2/(A1). 2.

5. Damage Characterization of HVI-Induced Pitting Damage Using Damage Indices

Both linear/nonlinear features of GUWs in the mode pair S1-S2 of Section 4.2 are utilized in the condition of phase matching, in conjunction with a path-based imaging algorithm, to image the HVI-induced pitting damage.

5.1. Construction of Probabilistic Imaging Using the Damage Index The reconstruction algorithm for the probabilistic inspection of damage (RAPID) [38] is frequently applied in the path-based damage imaging. Inspired by the stimulus of ‘visualizing’ HVI-induced damage, RAPID is performed to project the characterization results to synthetic images. In RAPID, a damage event is manifested according to the probability of its presence, rather than deterministic parameters (damage location, size, shape, etc.). Once an area in the imaging presents a high Materials 2017, 10, 547 16 of 20 probability of damage, conventional Nondestructive Evaluation methods can be adopted for a further determination of the damage parameter. In particular, it is almost impossible to use a conventional parameter to define the severity of HVI-induced pitting damage in a large area. Thus, the RAPID, using a probability to describe the HVI-induced pitting damage, indicates its advantage over conventional damage manifestation. In this way, the distribution probability of HVI-induced pitting damage is depicted in a contour map, allowing an intuitive and rapid diagnosis of the inner shielding layer under HVI. In RAPID, 2-D spatial meshes are firstly built into the inspection area of the inner shielding layer, to which a 2-D pixelated image with each pixel correlated exclusively with a spatial point is mapped. The value of each corresponding image pixel, called the filed value, is able to indicate the presence probability of damage at the spatial point. Thus, an HVI-induced zone with pitting damage can be graphically defined by highlighting those pixels with remarkably high field values as the central region of pitting damage and also noticing some other pixels with relatively small field values as the adjacent region of pitting damage, allowing users to obtain a quantitative and intuitive perception about the HVI-induced pitting damage. Using the setup of damage characterization in Section 4.2, twenty scanning paths are defined to cover the inspection area. Given a sensing path Pm (m = 1, 2, 3, ... , 20) with the transducer and receiver at (yi,zi) and (yj,zj), respectively, the field value at pixel (y, z) sensed by this sensing path is expressed as ς − R (y, z) ξ(y, z) = DI m , (13) m m ς − 1 where DI is the defined damage index in the designated path, ς is the scaling parameter controlling the size of the influence area, and Rm is a weight reflected by a distance relationship, which reads,

 q q 2 2 (y−y )2+(z−z )2+ (y−y ) +(z−z )  i i j j <  q 2 2 when ς, Rm(y, z) = (yi−yj) +(zi−zj) , (14)  ς when > ς,

Through Equation (13), the field value at each pixel of the image corresponding to the designated path is obtained, representing the probability distribution of damage, as illustrated in Figure 17a when the reciprocal of A1 on path five is used as the DI. Generally speaking, if the inspection area is free of damage, all the influenced pixels in the designated sensing path, for which Rm(y, z) < ς according to Equation (14), show a very low field value owing to the associated DI close to zero. On the contrary, if the inspection area is damaged, the field values in the influenced pixels comprise various elliptical loci focused on the two pixels corresponding to the transducer and receiver. The field value gradually decreases as the distance increases for the concerned pixel to the two foci. Once the pixel is outside of the controlled area, Rm(y, z) = ς and accordingly ξ(y, z) = 0, the field value is always zero, irrespective of the value of DI. In accordance with Equation (13), any individual sensing path Pm draws a source image, describing the probability as to the HVI-induced damage in the inner shielding layer regarding the single sensing path Pm. Nevertheless, damage may occur distant from most of the paths and can only be sensed by certain paths, and the measuring noise and uncertainty are other concerns if only a sparse path network is introduced. To address both issues, an image fusion scheme is introduced. By amalgamating the available source images of all the sensing paths at the pixel level, an ultimate superposed image is constructed, which renders a collective consensus as to the HVI-induced damage from the entire scanning network. The image fusion scheme, according to the twenty scanning paths in Figure 15, is defined as 20 ξ(y, z)sum = ∑ ξ(y, z)j, (15) j=1 Materials 2017, 10, 547 16 of 19

In accordance with Equation (13), any individual sensing path Pm draws a source image, describing the probability as to the HVI-induced damage in the inner shielding layer regarding the single sensing path Pm. Nevertheless, damage may occur distant from most of the paths and can only be sensed by certain paths, and the measuring noise and uncertainty are other concerns if only a sparse path network is introduced. To address both issues, an image fusion scheme is introduced. By amalgamating the available source images of all the sensing paths at the pixel level, an ultimate superposed image is constructed, which renders a collective consensus as to the HVI-induced damage from the entire scanning network. The image fusion scheme, according to the twenty scanning paths in Figure 15, is defined as

(, ) =(,), (15) Materials 2017, 10, 547 17 of 20 where (,) is the field value at pixel (y, z) in the ultimate superposed image. This process of imagewhere ξfusion(y, z)sum improvesis the ﬁeld the value signal-to-noise at pixel (y, zratio,) in the and ultimate the damage-incurred superposed image. singularity This process stands of image out comparedfusion improves to the theindividual signal-to-noise source ratio,images. and Thus the damage-incurred, the identified HVI-induced singularity stands damage out zone compared can be to highlightedthe individual in the source image images. pixels Thus, quantitatively the identiﬁed and intuitively. HVI-induced damage zone can be highlighted in the image pixels quantitatively and intuitively. 5.2. Imaging: Results and Discussion 5.2. Imaging: Results and Discussion The imaging results of HVI-induced pitting damage, using RAPID based on the linear/nonlinear featuresThe of imaging GUWs, results are presented of HVI-induced and discuss pittinged. damage, Damage using imaging RAPID basedusing onRAPID the linear/nonlinear based on the linear/nonlinearfeatures of GUWs, features are presentedin the case andof phase discussed. matching Damage between imaging the S1-S2 usingmode RAPIDis performed. based on is theset aslinear/nonlinear 1.05. As the nonlinear features feature—the in the case of second phase matching harmonic between wave—accumulates the S1-S2 mode along is performed. the propagationς is set distance,as 1.05. As damage the nonlinear can manifest feature—the through second the harmonicmagnitude wave—accumulates decrease of both A1 along and the A2. propagation Thus, the reciprocaldistance, damage values of can A1 manifest and A2 throughare taken the as magnitude the DIs for decreasethe linear of and both nonlinear A1 and A2. features, Thus, therespectively, reciprocal withvalues both of A1 imaging and A2 results are taken indicating as the DI thes for damage the linear occurrence and nonlinear (Figure features, 17b,c). respectively, The region withwithboth the highestimaging pixel results value indicating (the orange the damage area) corresponds occurrence to (Figure the central 17b,c). area The of region the HVI-induced with the highest damage. pixel Consideringvalue (the orange the form area) of corresponds HVI-induced to thedamage central that area the of central the HVI-induced area suffers damage. more severely Considering than the the surroundingform of HVI-induced area, the damagedamage thatimaging the central well presents area suffers the moreHVI-induced severely damage than the to surrounding guide a further area, repairthe damage or replacement. imaging well Finally, presents RANP the is HVI-induced also used as damagethe DI for to damage guide a furtherimaging repair (Figure or replacement.17d), which alsoFinally, highlights RANP isthe also damage used as location. the DI for In damage addition, imaging one advantage (Figure 17 d),of whichRAPID also over highlights other algorithms the damage is thatlocation. all the In DI addition,s for imaging one advantage are obtained of RAPID in the overcurrent other detected algorithms status is without that all the DIneeds for to imagingacquire anyare obtainedprior knowledge, in the current which detected improves status the applicability without the needof the to extracted acquire any DIs priorfor the knowledge, characterization which ofimproves HVI-induced the applicability pitting damage. of the extracted DIs for the characterization of HVI-induced pitting damage.

Materials 2017, 10, 547 17 of 19 (a) (b)

Figure 17. Damage imaging with reconstruction algorithm for the probabilistic inspection of damage Figure 17. Damage imaging with reconstruction algorithm for the probabilistic inspection of damage (RAPID) in the case of phase matching using various damage indices (DIs). (a) reciprocal of A1 on (RAPID) in the case of phase matching using various damage indices (DIs). (a) reciprocal of A1 on path path five; (b) reciprocal of A1 on all paths; (c) reciprocal of A2 on all paths; and (d) A2/(A1)2 on all ﬁve; (b) reciprocal of A1 on all paths; (c) reciprocal of A2 on all paths; and (d) A2/(A1)2 on all paths. paths.

6. Conclusions The active GUW-based characterization of HVI-engendered damage in the inner shielding layer of spacecraft is pursued. Addressing the understanding of nonlinear second harmonic waves due to various nonlinearity sources, particularly the second harmonic waves, the analytical solution, numerical modeling, and experimental validations are completed and compared, to show their consistency through the analysis on the accumulation of second harmonics under phase matching and phase mis-matching between the fundamental and second harmonic wave modes in both 2/3-D fashions. On the strength of the understanding of all the nonlinear sources, the mode pair S1-S2 with phase matching is used for the characterization of HVI-induced pitting damage in the inner shielding layer. The linear and nonlinear signal features of guided waves are extracted to calibrate the damage. Compared with the linear components, the nonlinear components feature a larger decrease in amplitude when a sensing path passes the damaged area. Thus, those paths across the pitting damage can be spotlighted using the defined linear and nonlinear DIs. Besides, the acquired signals on the paths without crossing damage offer the baseline for the GUW-based damage characterization. In this way, there is no need to record the prior condition, which facilitates the applicability of the currently developed methods. Finally, using DIs extracted from both linear/nonlinear features of GUWs, the RAPID algorithm shows the HVI-induced pitting damage to the inner shielding layer.

Author Contributions: Menglong Liu prepared the paper, and carried out the simulations and part of the experiment and algorithm development. Kai Wang assisted with the design and implementation of the experiment. Cliff J. Lissenden made contributions to the theoretical foundation of the nonlinear guided waves. Qiang Wang developed the measurement system. Qingming Zhang and Renrong Long participated in the HVI experiment. Zhongqing Su proposed the idea of nonlinear GUW-based detection, supervised this research, and modified the whole paper. Fangsen Cui gave critical review.

Conflicts of Interest: The authors declare no conflict of interest.

References

Materials 2017, 10, 547 18 of 20

6. Conclusions The active GUW-based characterization of HVI-engendered damage in the inner shielding layer of spacecraft is pursued. Addressing the understanding of nonlinear second harmonic waves due to various nonlinearity sources, particularly the second harmonic waves, the analytical solution, numerical modeling, and experimental validations are completed and compared, to show their consistency through the analysis on the accumulation of second harmonics under phase matching and phase mis-matching between the fundamental and second harmonic wave modes in both 2/3-D fashions. On the strength of the understanding of all the nonlinear sources, the mode pair S1-S2 with phase matching is used for the characterization of HVI-induced pitting damage in the inner shielding layer. The linear and nonlinear signal features of guided waves are extracted to calibrate the damage. Compared with the linear components, the nonlinear components feature a larger decrease in amplitude when a sensing path passes the damaged area. Thus, those paths across the pitting damage can be spotlighted using the deﬁned linear and nonlinear DIs. Besides, the acquired signals on the paths without crossing damage offer the baseline for the GUW-based damage characterization. In this way, there is no need to record the prior condition, which facilitates the applicability of the currently developed methods. Finally, using DIs extracted from both linear/nonlinear features of GUWs, the RAPID algorithm shows the HVI-induced pitting damage to the inner shielding layer.

Acknowledgments: This project is supported by a Key Project of the National Natural Science Foundation of China (Grant No. 51635008). This project is also supported by the Hong Kong Research Grants Council via General Research Fund (No. 15214414 and No. 15201416). Author Contributions: Menglong Liu prepared the paper, and carried out the simulations and part of the experiment and algorithm development. Kai Wang assisted with the design and implementation of the experiment. Cliff J. Lissenden made contributions to the theoretical foundation of the nonlinear guided waves. Qiang Wang developed the measurement system. Qingming Zhang and Renrong Long participated in the HVI experiment. Zhongqing Su proposed the idea of nonlinear GUW-based detection, supervised this research, and modified the whole paper. Fangsen Cui gave critical review. Conflicts of Interest: The authors declare no conflict of interest.

References

1. IADC W G. Members. Sensor Systems to Detect Impacts on Spacecraft IADC-08-03; IADC: Houston, TX, USA, 2013. 2. Christiansen, E.; Hyde, J.; Bernhard, R. Space shuttle debris and meteoroid impacts. Adv. Space Res. 2004, 34, 1097–1103. [CrossRef] 3. Drolshagen, G. Impact effects from small size meteoroids and space debris. Adv. Space Res. 2008, 41, 1123–1131. [CrossRef] 4. NASA. Space Debris and Human Spacecraft. Available online: http://www.nasa.gov/mission_pages/ station/news/orbital_debris.html (accessed on 13 May 2017). 5. NASA Orbital Debris Program Office. Monthly number of objects in earth orbit by object type. Orbital Debris Q. News 2016, 20, 13. 6. European Space Agency. Impact Chip. Available online: http://www.esa.int/spaceinimages/Images/2016/ 05/Impact_chip (accessed on 13 May 2017). 7. Taylor, E.A.; Herbert, M.K.; Vaughan, B.A.M.; McDonnell, J.A.M. Hypervelocity impact on carbon fibre reinforced plastic/aluminium honeycomb: Comparison with whipple bumper shields. Int. J. Impact Eng. 1999, 23, 883–893. [CrossRef] 8. Su, Z.; Ye, L.; Lu, Y. Guided lamb waves for identification of damage in composite structures: A review. J. Sound Vib. 2006, 295, 753–780. [CrossRef] 9. Qiu, L.; Yuan, S.F.; Zhang, X.Y.; Wang, Y. A time reversal focusing based impact imaging method and its evaluation on complex composite structures. Smart Mater. Struct. 2011, 20, 105014. [CrossRef] 10. Yuan, S.; Ren, Y.; Qiu, L.; Mei, H. A multi-response-based wireless impact monitoring network for aircraft composite structures. IEEE Trans. Ind. Electron. 2016, 63, 7712–7722. [CrossRef] Materials 2017, 10, 547 19 of 20

11. Qiu, L.; Yuan, S.; Liu, P.; Qian, W. Design of an all-digital impact monitoring system for large-scale composite structures. IEEE Trans. Instrum. Meas. 2013, 62, 1990–2002. [CrossRef] 12. Giurgiutiu, V. Tuned lamb wave excitation and detection with piezoelectric wafer active sensors for structural health monitoring. J. Intell. Mater. Syst. Struct. 2005, 16, 291–305. [CrossRef] 13. Li, F.; Su, Z.; Ye, L.; Meng, G. A correlation filtering-based matching pursuit (cf-mp) for damage identification using lamb waves. Smart Mater. Struct. 2006, 15, 1585–1594. [CrossRef] 14. Zhou, C.; Su, Z.; Cheng, L. Quantitative evaluation of orientation-specific damage using elastic waves and probability-based diagnostic imaging. Mech. Syst. Signal Process. 2011, 25, 2135–2156. [CrossRef] 15. Alleyne, D.N.; Cawley, P. The interaction of lamb waves with defects. IEEE Trans. Ultrason. Ferroelectr. 1992, 39, 381–397. [CrossRef][PubMed] 16. Jhang, K.-Y. Nonlinear ultrasonic techniques for nondestructive assessment of micro damage in material: A review. Int. J. Precis. Eng. Manuf. 2009, 10, 123–135. [CrossRef] 17. Su, Z.; Zhou, C.; Hong, M.; Cheng, L.; Wang, Q.; Qing, X. Acousto-ultrasonics-based fatigue damage characterization: Linear versus nonlinear signal features. Mech. Syst. Signal Process. 2014, 45, 225–239. [CrossRef] 18. Hong, M.; Su, Z.; Lu, Y.; Sohn, H.; Qing, X. Locating fatigue damage using temporal signal features of nonlinear lamb waves. Mech. Syst. Signal Process. 2015, 60, 182–197. [CrossRef] 19. Deng, M. Cumulative second-harmonic generation of lamb-mode propagation in a solid plate. J. Appl. Phys. 1999, 85, 3051–3058. [CrossRef] 20. Deng, M. Cumulative second-harmonic generation accompanying nonlinear shear horizontal mode propagation in a solid plate. J. Appl. Phys. 1998, 84, 3500–3505. 21. De Lima, W.J.N.; Hamilton, M.F. Finite-amplitude waves in isotropic elastic plates. J. Sound Vib. 2003, 265, 819–839. [CrossRef] 22. Lissenden, C.J.; Liu, Y.; Choi, G.W.; Yao, X. Effect of localized microstructure evolution on higher harmonic generation of guided waves. J. Nondestruct. Eval. 2014, 33, 178–186. [CrossRef] 23. Liu, Y.; Chillara, V.K.; Lissenden, C.J.; Rose, J.L. Third harmonic shear horizontal and rayleigh lamb waves in weakly nonlinear plates. J. Appl. Phys. 2013, 114, 114908. [CrossRef] 24. Chillara, V.K.; Lissenden, C.J. Interaction of guided wave modes in isotropic weakly nonlinear elastic plates: Higher harmonic generation. J. Appl. Phys. 2012, 111, 124909. [CrossRef] 25. Chillara, V.K.; Lissenden, C.J. Nonlinear guided waves in plates: A numerical perspective. Ultrasonics 2014, 54, 1553–1558. [CrossRef][PubMed] 26. Norris, A.N. Finite-amplitude waves in solids. In Nonlinear Acoustics; Hamilton, M.F., Ed.; Academic Press: San Diego, CA, USA, 1998; pp. 263–277. 27. Landau, L.; Lifshitz, E. Theory of Elasticity, 3rd ed.; Pergamon Press: Oxford, UK, 1986. 28. Thomas, J.F., Jr. Third-order elastic constants of aluminum. Phys. Rev. 1968, 175, 955–962. [CrossRef] 29. Auld, B.A. Acoustic Fields and Waves in Solids; Wiley: New York, NY, USA, 1973. 30. Rose, J.L. Ultrasonic Guided Waves in Solid Media; Cambridge University Press: Cambridge, UK, 2014. 31. Chillara, V.K.; Lissenden, C.J. Review of nonlinear ultrasonic guided wave nondestructive evaluation: Theory, numerics, and experiments. Opt. Eng. 2016, 55, 011002. [CrossRef] 32. Hong, M.; Su, Z.; Wang, Q.; Cheng, L.; Qing, X. Modeling nonlinearities of ultrasonic waves for fatigue damage characterization: Theory, simulation, and experimental validation. Ultrasonics 2014, 54, 770–778. [CrossRef][PubMed] 33. Liu, Y.; Chillara, V.K.; Lissenden, C.J. On selection of primary modes for generation of strong internally resonant second harmonics in plate. J. Sound Vib. 2013, 332, 4517–4528. [CrossRef] 34. Torello, D.; Thiele, S.; Matlack, K.H.; Kim, J.-Y.; Qu, J.; Jacobs, L.J. Diffraction, attenuation, and source corrections for nonlinear rayleigh wave ultrasonic measurements. Ultrasonics 2015, 56, 417–426. [CrossRef] [PubMed] 35. Matlack, K.H.; Kim, J.-Y.; Jacobs, L.J.; Qu, J. Experimental characterization of efficient second harmonic generation of lamb wave modes in a nonlinear elastic isotropic plate. J. Appl. Phys. 2011, 109, 014905. [CrossRef] 36. Zuo, P.; Zhou, Y.; Fan, Z. Numerical and experimental investigation of nonlinear ultrasonic lamb waves at low frequency. Appl. Phys. Lett. 2016, 109, 021902. [CrossRef] Materials 2017, 10, 547 20 of 20

37. Bermes, C.; Kim, J.-Y.; Qu, J.; Jacobs, L.J. Experimental characterization of material nonlinearity using lamb waves. Appl. Phys. Lett. 2007, 90, 021901. [CrossRef] 38. Zhao, X.L.; Gao, H.D.; Zhang, G.F.; Ayhan, B.; Yan, F.; Kwan, C.; Rose, J.L. Active health monitoring of an aircraft wing with embedded piezoelectric sensor/actuator network: I. Defect detection, localization and growth monitoring. Smart Mater. Struct. 2007, 16, 1208–1217. [CrossRef]

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