Generation Mechanism of Nonlinear Rayleigh Surface Waves ... - MDPI

materials Article

Generation Mechanism of Nonlinear Rayleigh Surface Waves for Randomly Distributed Surface Micro-Cracks Xiangyan Ding 1 , Feilong Li 1 , Youxuan Zhao 1, * and Mingxi Deng 1 1

2 3

*

ID

, Yongmei Xu 1 , Ning Hu 1,2, *

ID

, Peng Cao 3

College of Aerospace Engineering, Chongqing University, Chongqing 400044, China; [email protected] (X.D.); [email protected] (F.L.); [email protected] (Y.X.); [email protected] (M.D.) Key Laboratory of Optoelectronic Technology and Systems of the Education Ministry of China, Chongqing University, Chongqing 400044, China Department of Hydraulic Engineering, Tsinghua University, Beijing 100084, China; [email protected] Correspondence: [email protected] (Y.Z.); [email protected] (N.H.)

Received: 11 April 2018; Accepted: 19 April 2018; Published: 23 April 2018

Abstract: This paper investigates the propagation of Rayleigh surface waves in structures with randomly distributed surface micro-cracks using numerical simulations. The results revealed a significant ultrasonic nonlinear effect caused by the surface micro-cracks, which is mainly represented by a second harmonic with even more distinct third/quadruple harmonics. Based on statistical analysis from the numerous results of random micro-crack models, it is clearly found that the acoustic nonlinear parameter increases linearly with micro-crack density, the proportion of surface cracks, the size of micro-crack zone, and the excitation frequency. This study theoretically reveals that nonlinear Rayleigh surface waves are feasible for use in quantitatively identifying the physical characteristics of surface micro-cracks in structures. Keywords: nonlinearity; Rayleigh surface waves; micro-cracks; numerical simulation

1. Introduction Stress corrosion micro-cracks, on or near a material surface, can susceptibly occur in metallic structures due to fatigue loads and corrosive environments [1,2]. Thus, it is of crucial importance to develop various non-destructive testing methods to detect and evaluate surface cracks for ensuring the integrity and safety of metallic structures. Unfortunately, it is well-known that linear ultrasonic methods based non-destructive testing technologies are not capable of detecting closed micro-cracks [3,4]. On the other hand, many studies have demonstrated that nonlinear ultrasonic-based non-destructive testing technologies are effective for evaluating early material degradation and micro-crack initiation by analyzing the nonlinear effects of ultrasonic waves [5–12]. The energy of Rayleigh surface waves is generally concentrated in the region near the surface within a depth of about one wavelength, and therefore, it is convenient and essential to utilize them to detect and evaluate surface cracks. In recent years, great efforts have been devoted to the detection of early material degradation using the higher harmonics of Rayleigh surface waves [13–21]. For instance, Achenbach et al. [13] have conducted an analytical far-field solution for the cumulative third harmonic surface wave, propagating on a half-space of isotropic incompressible cubically nonlinear material, in a relatively simple and systematic manner. Herrmann et al. [14] have reported a linear increase of the amplitude of a second harmonic surface wave using propagation distance by an experimental investigation. Materials 2018, 11, 644; doi:10.3390/ma11040644

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It is supposed that when elastic waves reach a micro-crack, the crack can be closed and opened under compression and tension [22–24], and the compressive part of Rayleigh waves can penetrate through the crack while the tensile part cannot. The higher harmonics are generated due to the apparent local stiffness changes at the crack region under tension and compression. However, compared with works on early material degradation, research on the detection of micro-cracks using higher harmonics of Rayleigh surface waves is still rarely reported [6]. Oberhardt et al. [6] have modelled randomly distributed surface-breaking micro-cracks and investigated their relationship to second harmonic generation in Rayleigh surface waves with micro-crack density. In their study [6], a hyperelastic constitutive law was derived and implemented to simplify the micro-crack model. Rjelka et al. [25] have determined the relationship between material properties (third-order elastic constants) and inhomogeneous distributions of micro-cracks. They then investigated the acoustic nonlinearity parameter for surface acoustic waves based on second harmonic generation. In this work, and different from Oberhardt et al. [6] and Rjelka et al. [25], the simplified micro-crack model is not used. We employ a true contact model for modelling the micro-cracks. Besides considering the normal contact stiffness, the Coulomb friction law for the tangential direction is adopted in the contact model, and the linear elastic law for material is employed. We focus on the mechanism of second harmonic generation and propagation of Rayleigh surface waves in metallic structures with randomly distributed surface micro-cracks. In addition, the quantitative relationships between the acoustic nonlinear parameter and the micro-cracks are also studied. This work will provide a theoretical basis for detection and evaluation techniques based on nonlinear Rayleigh surface waves. 2. Acoustic Nonlinearity Parameter for Rayleigh Waves To date, a substantial number of experiments have demonstrated that the nonlinear ultrasonic detection technique, based on the second harmonics of bulk waves, Lamb waves, and Rayleigh surface waves, is effective for detecting changes to the microstructure of metal materials [26]. The amplitude of the second harmonic waves can be quantitatively evaluated using the acoustic nonlinearity parameter. Considered a weak quadratic nonlinearity, the acoustic nonlinearity parameter of Rayleigh surface waves at the surface uy = uy (y = 0) can be expressed as [14,18,26]: β=

uy (2ω ) i8p 2κ 2R ( 1 − ) κ 2R + s2 u2y (ω ) x κ 2P κ R

(1)

where p2 = κ 2R − κ 2P , s2 = κ 2R − κS2 , κ R , κ P , and κS are the wavenumbers for the Rayleigh, longitudinal, and shear waves, respectively. However, the contact acoustic nonlinearity caused by cracks is different from classical acoustic nonlinearity. Therefore, a different acoustic nonlinearity parameter β0 = A2 /A1 is used in this study [22–24], where A1 and A2 are, respectively, the amplitudes of the fundamental and second harmonic waves. This parameter is related to the crack density, the frequency of fundamental waves, the propagation distance, and the friction coefficient. 3. Numerical Simulation Since it is a challenge to directly obtain the nonlinear response when a Rayleigh wave passes through micro-cracks by an analytic method, numerical simulations are employed here. A two-dimensional finite element method (FEM) model of a structure with randomly distributed micro-cracks is built using the commercial FEM software ABAQUS (Version 6.14) and Python (Version 2.7). The problem of Rayleigh surface waves propagating through a structure with surface micro-cracks is described in Figure 1. The micro-cracks are at or near the surface of the middle region of the specimen, which is named as a micro-crack zone with length L (Note that the range of L in this paper is from 10 mm to 30 mm). In order to excite Rayleigh surface waves, longitudinal waves are applied using a dynamic displacement excitation on the flank of the wedge. Rayleigh waves are then excited due to

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using a dynamic displacement excitation on the flank of the wedge. Rayleigh waves are then excited due the Rayleigh critical angle θ. These propagate along Rayleigh the positive direction of x, and usingto a dynamic displacement excitation on waves the flank of the wedge. waves are then excited the interact Rayleigh critical angle θ. These waves propagate along the positive direction ofsecond x, and harmonic interact with with the micro-cracks when they reach the micro-crack zone. As a result, the due to the Rayleigh critical angle θ. These waves propagate along the positive direction of x, and waves are generated.they Because the size of micro-cracks is much smaller than the wavelength of are theinteract micro-cracks reach thethey micro-crack zone. As a result, the second harmonic waves with thewhen micro-cracks when reach the micro-crack zone. As a result, the second harmonic Rayleigh surface waves, most fundamental waves pass through the micro-crack zone and are generated. Because the size of micro-cracks much smaller wavelength of Rayleigh surface waves are generated. Because the size of ismicro-cracks is than muchthe smaller than the wavelength of collected atfundamental three signal detection positions (named as transmitted waves), which are 0 mm, 40 mm, waves, most waves pass through the micro-crack zone and are collected at three Rayleigh surface waves, most fundamental waves pass through the micro-crack zone and signal are and 80positions mm away(named from theasright boundary of the micro-crack respectively, as shown in Figure detection transmitted waves), which are 0zone, mm, waves), 40 mm, which and 80 mm away collected at three signal detection positions (named as transmitted are 0 mm, 40from mm,the 1. right of the micro-crack zone, respectively, as shown in Figure 1. andboundary 80 mm away from the right boundary of the micro-crack zone, respectively, as shown in Figure 1.

y

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Plastic wedge Excitation Plastic wedge signal Excitation signal

Signal detection position Signal detection 40mm 40mm position 40mm 40mm

θ

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Al

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Mircocrack

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Al

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Figure 1. Schematic of Rayleigh surface waves propagating through a region with micro-cracks.

L Figure 1. Schematic of Rayleigh surface waves propagating through a region with micro-cracks. Figure 1. Schematic of Rayleigh waves propagating through a region with micro-cracks. To investigate the influence ofsurface micro-cracks on the acoustic nonlinearity parameter, N microTo investigate the influence of micro-cracks the acoustic nonlinearity parameter, N Note micro-cracks cracks are randomly distributed at or near theon surface of the middle part with length L. that Tomicro-cracks investigate the influence of micro-cracks on the acoustic nonlinearity parameter, N microare when randomly distributed at or near the surface of the middle part with length L. Note that when are near the surface, the distances from the surface to the centers of micro-cracks cracks are randomly distributed at or near the surface of the middle part with length L. Note that are less than wavelength of the parameter w% centers is defined as the ratio ofare theless micro-cracks arethe near the surface, the Rayleigh distanceswave. from The the surface to the of micro-cracks when are near the surface, the distances fromw% theof surface to the centers number of surface of micro-cracks divided by theparameter total number micro-cracks, which represents the of than the micro-cracks wavelength the Rayleigh wave. The is defined as the ratioof ofmicro-cracks the number are less thanofthe wavelength ofthe the Rayleigh wave. parameter defined as the ratio of on the proportion micro-cracks Thus, 𝑤𝑤% = 100% meansw% thatisall the micro-cracks are surface micro-cracks dividedon by thesurface. total number ofThe micro-cracks, which represents the proportion of surface micro-cracks divided by the total number the the surface. Additionally, all the cracks have the same length 2amicro-cracks, (themicro-cracks range of awhich is from 25 µm 75 of number micro-cracks on the surface. Thus, w% = 100% means thatofof all the arerepresents on theto surface. proportion of micro-cracks on the surface. Thus, 𝑤𝑤% = 100% means that all the micro-cracks are µm in this study). Therefore, the crack density can be defined as c = Na/L, where c is a dimensionless Additionally, all the cracks have the same length of 2a (the range of a is from 25 µm to 75 µm inon this the surface. Additionally, all theare cracks have the same length ofrandom 2a (the range of a is from 25 µm to 75 parameter. In this study, there two variables that affect the distribution of micro-cracks, study). Therefore, the crack density can be defined as c = Na/L, where c is a dimensionless parameter. µm this study). Therefore, theorientation crack density can bewhich defined c = Na/L, by where c is a dimensionless i.e.,inthe center position and the of cracks, areasdescribed the probability density In this study, there are two variables that affect the random distribution of micro-cracks, i.e., the center parameter. In this study, there are two variables that the random distribution of micro-cracks, function (PDF) with a uniform random variable in theaffect simulations. position and the orientation of cracks, which are described by the probability density function (PDF) i.e., the position and the orientation of cracks, which are for described by the and probability density A center linear elastic constitutive law of aluminum (AL-6061-T6) the specimen plastics for the with a uniform random variable in the simulations. wedge are considered in the FEM analysis, and in thethe material properties used in the simulations are function (PDF) with a uniform random variable simulations. AA linear elastic law aluminum (AL-6061-T6) for3,the the specimen and plastics 3, constitutive 𝜌𝜌Al = 2704 kg/m 𝐸𝐸constitutive = 0.33, 𝜌𝜌Plastic = 1180 kg/m 𝐸𝐸Plastic = 4 Gpa, and 𝑣𝑣Plastic = the linear elastic law𝑣𝑣of of (AL-6061-T6) for specimen and plastics forfor the Al = 68.9 GPa, Al aluminum wedge are considered in the FEM analysis, and the material properties used in the simulations 0.33. Thus, the Rayleigh critical θ = and 53.28° be calculated by used the material properties are of are wedge are considered in the FEMangle analysis, thecan material properties in the simulations 3 3 ρAl𝜌𝜌aluminum = 2704 kg/m ρPlastic = 1180 kg/m and vPlastic = 0.33. 3E and ,plastics. , Al 𝐸𝐸Al= =68.9 68.9GPa, GPa,vAl 𝑣𝑣Al= 0.33, = 0.33, 𝜌𝜌Plastic = 1180 kg/m, 3E , Plastic 𝐸𝐸Plastic= =4 Gpa, 4 Gpa, and 𝑣𝑣Plastic = Al = 2704 kg/m ◦ can be calculated by the material properties of aluminum Thus, the Rayleigh critical angle θ = 53.28 The size of the FEM model is 240 × 6 mm. Since at least 20 elements are required in the 0.33. Thus, the Rayleigh critical angle θ = 53.28° can be calculated by the material properties of and plastics. and wavelength of plastics. the highest excitation frequency waves (2000 kHz in this study), from the aspect of aluminum numerical accuracy, leastmodel 6iselements crack are employed certainty, thewavelength element The sizesize of the model 240is×240 6 for mm. at least 20least elements are required in the The of FEM the at FEM × each 6 Since mm. Since at 20with elements areand required in the sizehighest for the of non-crack zone is set towaves Lfrequency max = 0.08 mm. The FEM model isfrom constructed using four-of of wavelength the excitation frequency (2000 kHz in thiskHz study), the aspect ofthe numerical the highest excitation waves (2000 in this study), from the aspect node plane strain (CPE4R) elements, as shown in Figure 2. numerical accuracy, at leastfor 6 elements for are each crack are with employed with and certainty, and thesize element accuracy, at least 6 elements each crack employed certainty, the element for the

size for the non-crack Lmax =The 0.08FEM mm.model The FEM model is constructed the fournon-crack zone is set to zone Lmax is= set 0.08tomm. is constructed using the using four-node plane node(CPE4R) plane strain (CPE4R) elements, as shown strain elements, as shown in Figure 2. in Figure 2.

(a)

(a) Figure 2. Cont.

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(b) Figure 2. 2. FEM FEM model model of of the zone: (a) (b) finite finite element element mesh mesh Figure the micro-crack micro-crack zone: (a) distribution distribution of of micro-cracks; micro-cracks; (b) (the red red line line represents representsone one crack crackin inthe themicro-crack micro-crackzone). zone). (the

A micro-crack is modelled as two contact surfaces, which can be separated from, but cannot A micro-crack is modelled as two contact surfaces, which can be separated from, but cannot interpenetrate into, each other. In addition, the Coulomb friction law is employed to model the interpenetrate into, each other. In addition, the Coulomb friction law is employed to model the relative relative sliding of the surfaces, and the friction coefficient is µ. Thus, the actual traction applied to the sliding of the surfaces, and the friction coefficient is µ. Thus, the actual traction applied to the crack crack faces is given by [8]: faces is given by [8]: )]𝐻𝐻[𝜏𝜏 (2) 𝜇𝜇𝜎𝜎00H𝐻𝐻(−𝜎𝜎 + µσ 𝜇𝜇𝜎𝜎00H 𝐻𝐻(−𝜎𝜎 q𝐪𝐪==σ𝜎𝜎00n𝐧𝐧++[τ[𝜏𝜏00++µσ H [τ00 + (2) (−σ00)] (−σ00)]𝐬𝐬 )]s

andτ0𝜏𝜏are, respectively, normal and shear stresses surfaces of the crack, is where σ𝜎𝜎00and 0 are, where respectively, thethe normal and shear stresses on on thethe surfaces of the crack, H is𝐻𝐻the the Heaviside step function, n is a unit vector that is perpendicular to the crack face, and s is a unit Heaviside step function, n is a unit vector that is perpendicular to the crack face, and s is a unit vector vector in theofplane of theface. crack face. in the plane the crack The dynamic displacement excitation applied applied to the flank flank of of the the wedge wedge can can be be expressed expressed as as The dynamic displacement excitation to the 22 −4− 4 ⁄ 𝑢𝑢(𝑥𝑥, 𝑡𝑡) = 𝐴𝐴 sin(2𝜋𝜋𝜋𝜋𝜋𝜋) × sin(𝜋𝜋𝜋𝜋𝜋𝜋 10 ) , where 𝐴𝐴 is the amplitude of excitation signal (1 × 10 mm in u( x, t) = A00 sin(2π f t) × sin(π f t/10) , where A00 is the amplitude of excitation signal (1 × 10 mm this study [8,27,28]) and f isf the frequency of of thethe excitation signal (ranging from 800~2000 kHz). In in this study [8,27,28]) and is the frequency excitation signal (ranging from 800~2000 kHz). addition, the frequency needs to be low enough to satisfy the low frequency assumption, leading to In addition, the frequency needs to be low enough to satisfy the low frequency assumption, leading to λ/a ≈ 20, where λ is the wavelength and a is the half-crack length [24]. Furthermore, the non-reflecting λ/a ≈ 20, where λ is the wavelength and a is the half-crack length [24]. Furthermore, the non-reflecting or absorbing absorbing boundary boundary conditions conditions at at the the bottom bottom and and right right edges edges of of the the FEM FEM model model are are used used to to or maximally eliminate the influence of boundary reflection. maximally eliminate the influence of boundary reflection. In this this study, study, aaABAQUS/Explicit ABAQUS/Explicit solver, on the the central central different different method, method, is is employed employed to to In solver, based based on simulate the propagation of Rayleigh surface waves in the time domain, which is conditionally stable. simulate the propagation of Rayleigh surface waves in the time domain, which is conditionally stable. The smallest smallest time time increment increment for for the the current currentproblem problemthat thatleads leadsto toaastable stablesolution solutionisis∆t ∆𝑡𝑡= =1 1××10 10−−99 s. The s. However,considering consideringthethe higher harmonic waves generated by the interactions between the However, higher harmonic waves generated by the interactions between the excitation −10 s, to be on the safe excitation signals and micro-cracks, the time increment is set 2 ×s,10 signals and micro-cracks, the time increment is set to be ∆t to = 2be×∆𝑡𝑡 10=−10 to be on the safe side. side. In addition, a double precision operation is to used to reduce the accumulative error here. In addition, a double precision operation is used reduce the accumulative error here. Moreover, an an additional additional consideration consideration is is the the randomness randomness of of the the crack crack distribution. distribution. To To capture capture Moreover, the stochastic nature of the random distribution, multiple FEM models are constructed with different the stochastic nature of the random distribution, multiple FEM models are constructed with different realizations of of the the random random distribution. distribution. Figure Figure 33 shows shows the the average average acoustic acoustic nonlinearity nonlinearity parameter parameter realizations versus the number of FEM models (U1 and U2 are the displacements in the x and y direction, versus the number of FEM models (U1 and U2 are the displacements in the x and y direction, respectively), where c is the crack density, f is the excitation frequency, µ is the friction coefficient of respectively), where c is the crack density, f is the excitation frequency, µ is the friction coefficient of the the micro-crack surfaces, andis 𝑤𝑤% is the proportion ofcracks. surfaceThe cracks. Thenonlinearity acoustic nonlinearity micro-crack surfaces, and w% the proportion of surface acoustic parameter parameter is more stable when the number of FEM models is larger than 50. Therefore, the is more stable when the number of FEM models is larger than 50. Therefore, the results shownresults in the shown in the following section over are averaged 100 FEM models. following section are averaged 100 FEM over models.


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0.10

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0.00

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0.00

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Average Number 40 60 Average Number

80 80

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Figure 3. Acoustic nonlinearity parameter versus number of FEM models (c = 0.1, f = 1000 kHz, µ = 0.1,Figure 𝑤𝑤% =3.60%). Acoustic nonlinearity parameter versus number FEM models = 0.1, = 1000 kHz,µ µ Figure 3. Acoustic nonlinearity parameter versus number of of FEM models (c (c = 0.1, f =f 1000 kHz, = =0.1, 0.1, 𝑤𝑤% = 60%). w% = 60%).

4. Result and Discussion 4. Result and Discussion 4. Result and Discussion When Rayleigh surface waves pass through micro-cracks, they can make the crack faces open When Rayleigh surface waves compression, pass through micro-cracks, they can make the crack faces open under tension and closed generates acoustic nonlinearity When Rayleigh surface under waves pass throughwhich micro-cracks, they can make the crack[24,29]. faces under tension and closed under compression, which generates acoustic nonlinearity [24,29]. Meanwhile, because the wavelength λ (1.4–3.6 mm) of Rayleigh surface waves is much larger than open under tension and closed under compression, which generates acoustic nonlinearity [24,29]. Meanwhile, because the to wavelength λmost (1.4–3.6 mm)waves of Rayleigh surface wavesthe is much larger than the crack length (50 µm 150 µm), of the can pass through micro-crack zone. Meanwhile, because the wavelength λ (1.4–3.6 mm) of Rayleigh surface waves is much larger than the the crack length (50 µm to 150 µm), most of the waves can pass throughnonlinear the micro-crack zone. Furthermore, investigate quantitative between the acoustic parameter and crack length (50toµm to 150 µm), most of therelationships waves can pass through micro-crack zone. Furthermore, Furthermore, to investigate quantitative relationships between the the acoustic nonlinear parameter and the micro-cracks, some representative characteristics or parameters, i.e., the crack density, the to investigate quantitative between the acoustic nonlinear parameter thedensity, micro-cracks, the micro-cracks, somerelationships representative characteristics or parameters, i.e., the and crack the propagation distance in the micro-crack zone, the proportion of surface cracks (i.e., 𝑤𝑤 %), the some representative characteristics or parameters, i.e.,proportion the crack density, the propagation distance propagation distance in the micro-crack zone, the of surface cracks (i.e., 𝑤𝑤 %), the in frequency of the fundamental wave,ofand the friction coefficient offrequency the micro-crack surfaces, are the micro-crack zone, the proportion surface cracks (i.e., w%), the of thesurfaces, fundamental frequency of the fundamental wave, and the friction coefficient of the micro-crack are considered here. wave, and the friction coefficient of the micro-crack surfaces, are considered here. considered here. Figure 44 shows shows the displacement displacement contour of of Rayleigh surface surface waves waves while while propagating propagating in in a Figure Figure 4 showsthe the displacementcontour contour of Rayleigh Rayleigh surface waves while propagating in a a structure with surface micro-cracks at 1000 kHz. We can see clearly that the wavelength of Rayleigh structure with surface micro-cracks clearlythat thatthe thewavelength wavelengthofofRayleigh Rayleigh structure with surface micro-cracksatat1000 1000kHz. kHz. We We can can see see clearly surface waves Moreover, although thewaves waves interact with the surface wavesis muchlarger largerthan thanthe the crack length. although the interact with the surface waves isismuch much larger than thecrack cracklength. length.Moreover, Moreover, although the waves interact with micro-crack surfaces ininthe zone, no change observed the Only micro-crack surfaces the micro-crack zone, no obvious obvious change isisobserved ininthe waveforms. Only the micro-crack surfaces in micro-crack the micro-crack zone, no obvious change is observed inwaveforms. the waveforms. a very weak scattering phenomenon occurred around the micro-cracks. Meanwhile, the 800 times a very weak scattering phenomenon occurred around thethe micro-cracks. Meanwhile, thethe 800800 times Only a very weak scattering phenomenon occurred around micro-cracks. Meanwhile, times zoomed deformed shapes clearly demonstrate the openingand andclosing closingstates statesof ofthe the micro-cracks. zoomed deformed shapes clearlydemonstrate demonstratethe theopening opening and micro-cracks. zoomed deformed shapes clearly closing states of the micro-cracks. Opening Opening

Closing Closing

(a) (a) Figure 4. Cont.

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(b) (b)

Figure 4. 4. Propagation Propagation of of surface surface waves waves in in aa structure structure with withsurface surfacemicro-cracks micro-cracks(c(c== 0.1, 0.1, ff = = 1000 Figure 1000 kHz, kHz, Figure 4. Propagation of surface waves in a structure with surface micro-cracks (c = 0.1, f = 1000shapes kHz, µ = 0.1, 𝑤𝑤% = 60%). (a) Displacement contour in the x direction (800 times zoomed deformed µ = 0.1, w% = 60%). (a) Displacement contour in the x direction (800 times zoomed deformed shapes in µ =dashed 0.1, 𝑤𝑤%box); = 60%). (a) Displacement contourininthe they xdirection. direction (800 times zoomed deformed shapes in the Displacement contour the dashed box); (b)(b) Displacement contour in the y direction. in the dashed box); (b) Displacement contour in the y direction.

Figures 5 and 6 show wave signals collected at the signal detection position (0 mm away from Figures 5 and 6 show wave signalscollected collectedatatthe thesignal signal detection detection position Figures 5 and 6 show wave signals position(0 (0mm mmaway awayfrom from the right boundary of the micro-crack zone) in the x and y directions, respectively, wherein the wave right boundary micro-crackzone) zone)ininthe thexxand and yy directions, directions, respectively, the the right boundary of of thethe micro-crack respectively,wherein whereinthe thewave wave signals for the non-cracked case are marked by the solid line, and the wave signals for the cracked signals non-cracked marked by the solid wave signals cracked signals for for the the non-cracked casecase are are marked by the solid line,line, andand the the wave signals for for the the cracked case case are marked by the dashed line. We can see that the energies of the wave signals in the x and y case are marked by the dashed line. We can see that the energies of the wave signals in the x and y are marked by the dashed line. We can see that the energies of the wave signals in the x and y directions directions areare almost identical. Moreover, the signals for the the non-cracked non-crackedcase caseand andthe the almost identical. thetime-domain time-domain signals for are directions almost identical. Moreover, theMoreover, time-domain signals for the non-cracked case and the cracked case cracked case in in thethe x and y directions are very similar, as in Figures Figures5a 5aand and6a, 6a,respectively. respectively. cracked x and directions similar, as shown shown in in the x andcase y directions are yvery similar,are asvery shown in Figures 5a and 6a, respectively. More obvious More obvious differences can be observed from the frequency analysis of the wave signals (Figures More obvious differences be observed from the frequency analysis the wave signals5b (Figures differences can be observedcan from the frequency analysis of the waveofsignals (Figures and 6b). 5b and 6b). We find that the wave signals for the non-cracked case marked by the solid line only 5b and 6b). We find that the wave signals for the non-cracked case marked by the solid line only We find that the wave signals for the non-cracked case marked by the solid line only contain the contain thethe 1000 kHz wave thatthe thesecond secondharmonic harmonic contain 1000 kHz wavesignals signals(the (thefundamental fundamental waves), waves), indicating indicating that 1000 kHz wave signals (the fundamental waves), indicating that the second harmonic waves cannot be waves cannot be be generated forfor the wave signals signalsfor forthe thecracked crackedcase, case, waves cannot generated thenon-cracked non-crackedcase. case.However, However, the wave generated for the non-cracked case. However, the wave signals for the cracked case, marked by the marked by by thethe dashed line, contain butalso alsothe the2000 2000kHz kHzwave wave marked dashed line, containnot notonly onlythe the1000 1000kHz kHz wave wave signals, signals, but dashed line, contain not only the 1000 kHz wave signals, but also the 2000 kHz wave signals (second signals (second harmonicwaves) waves)and andeven evendistinct distinct 3000/4000 3000/4000 kHz wave signals (second harmonic wave signals signals(third/quadruple (third/quadruple harmonic waves) andimplying even distinct 3000/4000 wave signals (third/quadruple harmonic waves), harmonic waves), thatthe the existencekHz micro-cracks harmonic waves), implying that existence ofofmicro-cracks is the critical critical factor factorfor forgenerating generatingthe the implying that the existence of micro-cracks is the critical factor for generating the second harmonic second harmonic waves. Meanwhile, the third and quadruple harmonic waves for the case of second harmonic waves. Meanwhile, the third and quadruple harmonic waves for the case of waves. Meanwhile, the third and quadruple theofcase of Rayleigh surface waves Rayleigh surface waves more remarkableharmonic comparedwaves to the the for cases Rayleigh surface waves areare more remarkable compared to cases of bulk bulk[24] [24]and andLamb Lamb[8] [8]waves. waves. are Here, more from remarkable compared to the cases of bulk [24] and Lamb [8] waves. Here, from numerical numerical simulations,we wefind findthat, that,when when using using Rayleigh Rayleigh surface Here, from numerical simulations, surfacewaves, waves,micro-cracks micro-cracks simulations, we find that, when using Rayleigh surface waves, micro-cracks could generate the second could generate the second harmonic waves. Therefore, this study provides a foundation could generate the second harmonic waves. Therefore, this study provides a foundationfor forsome some harmonic waves. Therefore,for this study provides foundation for possible new techniques for possible new techniques identifying surfaceamicro-cracks micro-cracks in aasome structure possible new techniques for identifying surface in structure using usingRayleigh Rayleighsurface surface identifying waves. surface micro-cracks in a structure using Rayleigh surface waves. waves. 6.0x10-5 6.0x10-5

No Cracks No Cracks Crack Density=0.1 Crack Density=0.1

Amplitude(mm) Amplitude(mm)

4.0x10-5 4.0x10-5

2.0x10-5 2.0x10-5 0.0 0.0 -2.0x10-5

-2.0x10-5 -4.0x10-5 -4.0x10-5 50 50

52 52

54 56 54 Time(us) 56 Time(us) (a)

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Figure 5. Cont.

58 58

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1.5x10 1.5x10-5 1.2x10-5 1.2x10-5 9.0x10-6 9.0x10-6 6.0x10-6 6.0x10-6 3.0x10-6 3.0x10-6 0.0 0 0.0 0

1

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Figure 5. Wave signals in the x direction (a) time(b) domain; (b) frequency domain (c = 0.1, f = 1000 kHz, Figure 5. Wave signals in the x direction (a) time domain; (b) frequency domain (c = 0.1, f = 1000 kHz, µ = 0.1, 𝑤𝑤% = 60%). 5. Wave signals in the x direction (a) time domain; (b) frequency domain (c = 0.1, f = 1000 kHz, µFigure = 0.1, w% = 60%). µ = 0.1, 𝑤𝑤% = 60%).

8.0x10-5

No Cracks Crack Density=0.1

-5-5 6.0x10 8.0x10


-5-5


4.0x10 6.0x10

-5-5 2.0x10 4.0x10 -5 2.0x100.0 -5 -2.0x10 0.0

-5 -4.0x10 -5 -2.0x10 -6.0x10-5 -4.0x10-5 50 -6.0x10-5

50

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Time(us) 52

54 (a)

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60


(a)

-5 -5 2.0x10 2.5x10



-5 -5 1.5x10 2.0x10

1.0x10-5 1.5x10-5 5.0x10-6 1.0x10-5

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Figure 6. Wave signals in the y direction (a)Frequency(MHz) time domain; (b) frequency domain (c = 0.1, f = 1000 kHz, µ = 0.1, 𝑤𝑤% = 60%).

(b)

Next,6.the wave signals collected at 0(a) mm, 40domain; mm, and mm, away from (c the right Figure Wave signals in the y direction time (b)80 frequency domain = 0.1, f =boundary 1000 kHz, of Figure 6. Wave signals in the y direction (a) time domain; (b) frequency domain (c = 0.1, f = 1000 kHz, theµ micro-crack zone, are shown in Figure 7 (in the y direction). We can find that the waveforms of = 0.1, 𝑤𝑤% = 60%). µ = 0.1, w% = 60%). Rayleigh surface waves collected at different locations are the same, indicating that the second

Next, the wave signals collected at 0 mm, 40 mm, and 80 mm, away from the right boundary of the micro-crack zone, are shown in Figure 7 (in the y direction). We can find that the waveforms of Rayleigh surface waves collected at different locations are the same, indicating that the second

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Next, the wave signals collected at 0 mm, 40 mm, and 80 mm, away from the right boundary of the Materials 2018, zone, 11, x FOR REVIEW 8 of 14 micro-crack arePEER shown in Figure 7 (in the y direction). We can find that the waveforms of Rayleigh surface waves collected at different locations are the same, indicating that the second harmonics from harmonics theare received signal are independent from theposition. signal detection And thecan same the receivedfrom signal independent from the signal detection And theposition. same tendency be tendency canxbe found inIfthe x direction. If there exists damping, of thethe amplitudes of the fundamental found in the direction. there exists damping, the amplitudes fundamental waves and the waves harmonics and the second harmonics could be reduced, but nevertheless, the acoustic cannot nonlinear second could be reduced, but nevertheless, the acoustic nonlinear phenomenon be phenomenon cannot be affected significantly. affected significantly.

8.0x10-5

Transmitted wave position 0mm Transmitted wave position 40mm Transmitted wave position 80mm

-5

6.0x10

Amplitude(mm)

4.0x10-5 2.0x10-5 0.0

-2.0x10-5 -4.0x10-5 -6.0x10-5 50

55

60

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80

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90

(a) -5

2.5x10

Transmitted wave position 0mm Transmitted wave position 40mm Transmitted wave position 80mm

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2.0x10-5 1.5x10-5 1.0x10-5

5.0x10-6 0.0

0

1

2

3

4

5

6

Frequency(MHz) (b) Figure 7. 7. Waveforms at at different locations (a) (a) time domain; (b) Figure Waveformsofofwave wavesignals signalsininthe they direction y direction different locations time domain; frequency domain (c =(c0.1, f = 1000 kHz, µ = 0.1, 𝑤𝑤%w% = 60%). (b) frequency domain = 0.1, f = 1000 kHz, µ = 0.1, = 60%).

Furthermore, we investigate the relationship between the acoustic nonlinearity parameter and Furthermore, we investigate the relationship between the acoustic nonlinearity parameter and the proportion of surface micro-cracks, as well as the crack density and propagation distance in the the proportion of surface micro-cracks, as well as the crack density and propagation distance in the micro-crack zone. The acoustic nonlinearity parameter increases almost linearly with the proportion micro-crack zone. The acoustic nonlinearity parameter increases almost linearly with the proportion of of surface micro-cracks as shown in Figure 8. When all the micro-cracks are on the surface (𝑤𝑤% = surface micro-cracks as shown in Figure 8. When all the micro-cracks are on the surface (w% = 100%), 100%), the acoustic nonlinearity parameter reaches the maximum, which is the most serious case in the acoustic nonlinearity parameter reaches the maximum, which is the most serious case in practice. practice. Figure 9 shows the acoustic nonlinearity parameter versus the crack density c. The crack density (c = Na/L) is a dimensionless number, which can be used as a quantitative indicator for acoustic nonlinearity caused by micro-cracks. Because the crack density can be affected by parameters N and a, we consider the following two cases: the first case is to increase N and to keep a the same, and the second is to increase a and to keep N the same. When the excitation frequency and the friction coefficient of the micro-crack surfaces remain unchanged, the acoustic nonlinearity parameter is more

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Acoustic AcousticNonlinearity NonlinearityParameter Parameter

Figure 9 shows the acoustic nonlinearity parameter versus the crack density c. The crack density (c = Na/L) is a dimensionless number, which can be used as a quantitative indicator for acoustic nonlinearity caused by micro-cracks. Because the crack density can be affected by parameters N and a, we consider the following two cases: the first case is to increase N and to keep a the same, and theMaterials second2018, is to a REVIEW and to keep N the same. When the excitation frequency and the friction Materials 2018, 11,increase FORPEER PEER REVIEW 14 11, xxFOR 99ofof14 coefficient of the micro-crack surfaces remain unchanged, the acoustic nonlinearity parameter is more sensitive to crack crack length than crack number, and the acoustic acoustic nonlinearity parameter increases sensitive to length number, and the nonlinearity parameter increases sensitive to crack length thanthan crackcrack number, and the acoustic nonlinearity parameter increases linearly linearly with the increase in crack density. We also find that the acoustic nonlinearity parameters in y linearly with the increase in crack density. We also find that the acoustic nonlinearity parameters in with the increase in crack density. We also find that the acoustic nonlinearity parameters in the x and the x and y directions are consistent. the x and y directions are consistent. directions are consistent. 0.24 0.24

U1 U1 U2 U2

0.20 0.20 0.16 0.16 0.12 0.12 0.08 0.08 0.04 0.04 0.00 0.00

0% 0%

20% 20%

40% 40%

60% 60%

80% 80%

100% 100%

Theproportion proportionof ofsurface surfacecracks cracks The

Acoustic AcousticNonlinearity NonlinearityParameter Parameter

Figure 8. Acoustic nonlinearity parameter versus thethe proportion of surface cracks (c =(c(c 0.1, f0.1, = f1000 kHz, Figure Acoustic nonlinearity parameter versus the proportion ofsurface surface cracks f==1000 1000 Figure 8.8.Acoustic nonlinearity parameter versus proportion of cracks ==0.1, kHz, 0.1). µ =kHz, 0.1).µµ==0.1). U1(IncreasingN) N) U1(Increasing U2(IncreasingN) N) U2(Increasing U1(Increasinga)a) U1(Increasing U2(Increasinga)a) U2(Increasing

0.36 0.36 0.30 0.30 0.24 0.24 0.18 0.18 0.12 0.12 0.06 0.06 0.00 0.00

0.050 0.050

0.075 0.075

0.100 0.100

0.125 0.125

0.150 0.150

Crackdensity density Crack Figure 9. Acoustic nonlinearity parameter versus = 1000 1000 kHz, kHz, µ =0.1, 0.1,𝑤𝑤% w% =100%). 100%). Figure Acoustic nonlinearity parameter versusthe thecrack crackdensity density(f Figure 9.9.Acoustic nonlinearity parameter versus the crack density (f(f==1000 kHz, µµ==0.1, 𝑤𝑤% ==100%).

The lengths ofeach each micro-crack inpractice practice should bedifferent. different. Thus, wealso also consider thetwo two TheThe lengths of each micro-crack in practice should be different. Thus, we we also consider the the two cases lengths of micro-crack in should be Thus, consider cases with the crack length of uniform and Gaussian random distributions. In the two cases, the mean cases with the crack length of uniform and Gaussian random distributions. In the two cases, the mean with the crack length of uniform and Gaussian random distributions. In the two cases, the mean values values arebut thesame, same, butthe thedeviations standarddeviations deviations areFigures changed. Figures 10and and 11acoustic showthe the acoustic are the but standard are changed. 10 11 show acoustic arevalues the same, the standard are changed. 10Figures and 11 show the nonlinearity nonlinearity parameter versus the standard deviation of micro-crack lengths. It is shown that the the nonlinearity parameter versus the standard deviation of micro-crack lengths. is shown that parameter versus the standard deviation of micro-crack lengths. It is shown that theIt acoustic nonlinearity acousticdoes nonlinearity parameter doesnot not changewith with standard deviation, which only relatedto to acoustic nonlinearity parameter does change standard isisonly related parameter not change with standard deviation, which is onlydeviation, related to which the mean value. themean meanvalue. value. the Figure 12 shows the acoustic nonlinearity parameter versus the propagation distance in the Figure 12 12 shows the the acoustic acoustic nonlinearity nonlinearity parameter parameter versus versus the the propagation propagation distance in in the the Figure micro-crack zone.shows The acoustic nonlinearity parameter also increases linearly with distance the propagation micro-crack zone. The acoustic nonlinearity parameter also increases linearly with the propagation micro-crack zone. The acoustic nonlinearity parameter also increases linearly with the propagation distance in the micro-crack zone, leading to its linear accumulation property. distancein inthe themicro-crack micro-crackzone, zone,leading leadingto toits itslinear linearaccumulation accumulationproperty. property. distance

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AcousticNonlinearity NonlinearityParameter Parameter Acoustic Nonlinearity Parameter Acoustic

0.36 0.36 0.36

U1 U1 U1 U2 U2 U2

0.30 0.30 0.30 0.24 0.24 0.24 0.18 0.18 0.18 0.12 0.12 0.12 0.06 0.06 0.06

0.00 0.00 0.00 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.00 0.00 0.01 0.01 0.02 0.02 0.03 0.03 0.04 0.04 0.05 0.05 0.06 0.06

Standard deviation Standard Standard deviation deviation


Figure 10. Acoustic nonlinearity parameter versus standard deviation of micro-crack lengths with Figure 10. Figure 10. Acoustic nonlinearity parameter versus standard deviation of micro-crack lengths with Figure 10.Acoustic Acousticnonlinearity nonlinearityparameter parameterversus versusstandard standard deviation deviation of of micro-crack micro-crack lengths lengths with with uniform distribution (c = 0.1, f = 1000 kHz, µ = 0.1, 𝑤𝑤% = 100%). uniform distribution (c = 0.1, f = 1000 kHz, µ = 0.1, w% = 100%). uniform distribution (c = 0.1, f = 1000 kHz, µ = 0.1, 𝑤𝑤% = 100%). uniform distribution (c = 0.1, f = 1000 kHz, µ = 0.1, 𝑤𝑤% = 100%).

0.36 0.36 0.36

U1 U1 U1 U2 U2 U2

0.30 0.30 0.30 0.24 0.24 0.24 0.18 0.18 0.18 0.12 0.12 0.12 0.06 0.06 0.06 0.00 0.00 0.00

0.00 0.00 0.00

0.01 0.01 0.01

0.02 0.02 0.02

Standard deviation Standard Standard deviation deviation

0.03 0.03 0.03


Figure 11. Acoustic nonlinearity parameter versus standard deviation of micro-crack lengths with Figure Acoustic nonlinearity parameter versus standard deviation of micro-crack lengths with Figure 11.11. Figure 11.Acoustic Acousticnonlinearity nonlinearityparameter parameterversus versusstandard standard deviation deviation of of micro-crack micro-crack lengths lengths with Gaussian distribution (c = 0.1, f = 1000 kHz, µ = 0.1, 𝑤𝑤% = 100%). Gaussian distribution 0.1, 1000 kHz, 0.1, w% 𝑤𝑤% 100%). Gaussian distribution Gaussian distribution(c(c (c= ==0.1, 0.1,f ff===1000 1000kHz, kHz, µµ µ=== 0.1, 𝑤𝑤%=== 100%). 100%).

0.30 0.30 0.30

U1 U1 U1 U2 U2 U2

0.25 0.25 0.25 0.20 0.20 0.20 0.15 0.15 0.15 0.10 0.10 0.10 0.05 0.05 0.05

10 10 10

15 15 15

20 20 20

25 25 25

Distance(mm) Distance(mm) Distance(mm)

30 30 30

Figure 12. nonlinearity parameter versus the propagation distance in the micro-crack zone Figure 12.Acoustic Acoustic nonlinearity parameter versus the propagation distance in the micro-crack zone Figure Acousticnonlinearity nonlinearityparameter parameterversus versusthe thepropagation propagationdistance distance in in the the micro-crack micro-crack zone zone Figure 12.f12. Acoustic (c(c==0.1, = 1000 kHz, µ = 0.1, 𝑤𝑤% = 100%). 0.1, f = 1000 kHz, µ = 0.1, 𝑤𝑤% = 100%). (c = 0.1, f = 1000 kHz, µ = 0.1, 𝑤𝑤% = 100%). (c = 0.1, f = 1000 kHz, µ = 0.1, w% = 100%).

Acoustic Nonlinearity Parameter

reasons could be the more pronounced clapping behavior of the cracks than the frictional one, and asymmetry due to the random micro-cracks modelling. Thus, it could be inferred that the clapping behavior of the cracks is more remarkable than the frictional one under the condition of randomly distributed micro-cracks. Figure 14 shows the acoustic nonlinearity parameter versus the excitation Materials 2018,It11, 644be seen that the acoustic nonlinearity parameters of the x and y directions increase 11 of 14 frequency. can linearly with the excitation frequency. The main reason is that the wavelength of the Rayleigh surface wave decreases when the excitation frequency increases, and the wave with a smaller wavelength is In contrast to classic acoustic nonlinearity, the factors affecting acoustic nonlinearity also include more sensitive to the micro-crack. Therefore, the nonlinear interactions between Rayleigh surface excitation frequency and the friction coefficient of micro-crack surfaces. In this study, the Coulomb waves with a smaller wavelength and the micro-cracks became stronger, and the second harmonic law of friction is used to calculate the frictional force between the micro-crack surfaces. When the was more noticeable. crack density and the excitation frequency remain unchanged, the acoustic nonlinearity parameter has The above cases are all based on the assumption that crack angles have a uniform and random almost no relation to the friction coefficient, as shown in Figure 13. It should be noted that the acoustic distribution with a range from 0° to 180° (the angle between the crack face and the x direction). nonlinearity parameter of the x direction is slightly higher than that of the y direction. The reasons However, the angles of micro-cracks [22,23] can affect the acoustic nonlinearity parameter. In this could be the more pronounced clapping behavior of the cracks than the frictional one, and asymmetry case, we consider that all the micro-crack angles are uniform, and the positions of micro-cracks are due to the random micro-cracks modelling. Thus, it could be inferred that the clapping behavior of also of a uniformly random distribution. The acoustic nonlinearity parameter is shown as a function the cracks is more remarkable than the frictional one under the condition of randomly distributed of the angle of micro-cracks in Figure 15. When the angle is 90°, the acoustic nonlinearity parameter micro-cracks. Figure 14 shows the acoustic nonlinearity parameter versus the excitation frequency. is at the maximum, which is essentially caused by the considerable clapping behavior of the cracks It can be seen that the acoustic nonlinearity parameters of the x and y directions increase linearly rather than the frictional one. Meanwhile, we also find that the friction coefficient slightly affects the with the excitation frequency. The main reason is that the wavelength of the Rayleigh surface wave acoustic nonlinearity parameter under the condition of regularly arranged micro-cracks. The reason decreases when the excitation frequency increases, and the wave with a smaller wavelength is more could be the weak energies of Rayleigh waves, which hardly generate the slipping trend between the sensitive to the micro-crack. Therefore, the nonlinear interactions between Rayleigh surface waves surfaces of micro-cracks. Therefore, we can also infer that the clapping behavior is of more with a smaller wavelength and the micro-cracks became stronger, and the second harmonic was importance than the frictional one under the condition of regularly arranged micro-cracks. more noticeable.

0.36 U1 U2

0.30 0.24 0.18 0.12 0.06 0.00 0.0

0.2

0.4

0.6

0.8

1.0

Friction coefficient


Figure parameter versus thethe friction coefficient (c = (c 0.1,= f 0.1, = 1000 = Figure 13. 13. Acoustic Acousticnonlinearity nonlinearity parameter versus friction coefficient f =kHz, 1000 𝑤𝑤% kHz, 100%). w%2018, = 100%). Materials 11, x FOR PEER REVIEW 12 of 14

0.42

U1 U2

0.36 0.30 0.24 0.18 0.12 0.06

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Frequency(MHz) Figure 14. 14.Acoustic Acousticnonlinearity nonlinearity parameter versus excitation frequency (c =µ = 0.1, = 0.1, 𝑤𝑤% = Figure parameter versus the the excitation frequency (c = 0.1, 0.1,µw% = 100%). 100%).

arameter

0.40 0.32

U1 (µ =0.1) U2 (µ =0.1) U1 (µ =0.3) U2 (µ =0.3)

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12 of 14

0.42

U1 U2

0.36

12 of 14

0.30 The above cases are all based on the assumption that crack angles have a uniform and random distribution with a range0.24 from 0◦ to 180◦ (the angle between the crack face and the x direction). However, the angles of micro-cracks [22,23] can affect the acoustic nonlinearity parameter. In this case, we consider that all the micro-crack angles are uniform, and the positions of micro-cracks are also of 0.18 a uniformly random distribution. The acoustic nonlinearity parameter is shown as a function of the 0.12 15. When the angle is 90◦ , the acoustic nonlinearity parameter is at angle of micro-cracks in Figure the maximum, which is essentially caused by the considerable clapping behavior of the cracks rather 0.06 than the frictional one. Meanwhile, that the slightly affects the acoustic 0.8we also 1.0 find1.2 1.4 friction 1.6 coefficient 1.8 2.0 nonlinearity parameter under the condition of regularly arranged micro-cracks. The reason could be Frequency(MHz) the weak energies of Rayleigh waves, which hardly generate the slipping trend between the surfaces Figure 14. Acoustic nonlinearity parameter versus the excitation frequency = 0.1, µ = 0.1, 𝑤𝑤% = the of micro-cracks. Therefore, we can also infer that the clapping behavior is of (c more importance than 100%). frictional one under the condition of regularly arranged micro-cracks.


0.40

U1 (µ =0.1) U2 (µ =0.1) U1 (µ =0.3) U2 (µ =0.3) U1 (µ =0.5) U2 (µ =0.5)

0.32 0.24 0.16 0.08 0.00

0

30

60

90

120

150

180

Angle(degrees) Figure the angle of of micro-cracks (c =(c0.1, f = 1000 kHz,kHz, 𝑤𝑤% Figure 15. 15. Acoustic Acousticnonlinearity nonlinearityparameter parameterversus versus the angle micro-cracks = 0.1, f = 1000 = 100%). =w% 100%).

5. 5. Conclusions Conclusions A numerical model model containing containing randomly randomly distributed constructed to to A numerical distributed surface surface micro-cracks micro-cracks is is constructed investigate from investigate the the propagation propagation phenomenon phenomenon of of Rayleigh Rayleigh surface surface waves waves by by statistical statistical analysis analysis from numerous results from random micro-crack models. The following paragraphs outline the main numerous results from random micro-crack models. The following paragraphs outline the main conclusions drawn from from this this study. study. conclusions drawn Firstly, whenRayleigh Rayleigh surface waves are used the fundamental waves, the ultrasonic Firstly, when surface waves are used as the as fundamental waves, the ultrasonic nonlinear nonlinear effect, in terms of second harmonics, can be generated by surface micro-cracks. It found effect, in terms of second harmonics, can be generated by surface micro-cracks. It is found isthat the that third and quadruple harmonics of Rayleigh surfaceare waves more noteworthy thanofthose thirdthe and quadruple harmonics of Rayleigh surface waves moreare noteworthy than those bulk of bulk andwaves. Lamb waves. and Lamb Second, lengths and and angles angles of of surface surface micro-cracks micro-cracks were were Second, the the different different distributions distributions of of the the lengths investigated. investigated. The The results results show show that that the the acoustic acoustic nonlinearity nonlinearity parameter parameter is is only only related related to to the the mean mean value of the lengths of surface micro-cracks, and the acoustic nonlinearity parameter is at maximum value of the lengths of surface micro-cracks, and the acoustic nonlinearity parameter is at maximum ◦. when the the angle angle of of the the surface surface micro-crack micro-crack is is 90 90°. when Finally, wesystematically systematically investigated the relationships between the nonlinear acoustic parameter nonlinear Finally, we investigated the relationships between the acoustic parameter and some key factors, i.e., the micro-crack density, the propagation distance in the cracking and some key factors, i.e., the micro-crack density, the propagation distance in the cracking region, region, the proportion ofcracks, surfacethe cracks, the excitation and the frictionof coefficient of the the proportion of surface excitation frequency,frequency, and the friction coefficient the micro-crack micro-crack surfaces. The results reveal that the acoustic nonlinear parameters in the x-axis and ysurfaces. The results reveal that the acoustic nonlinear parameters in the x-axis and y-axis are axis are basically consistent, which is linearly proportional to the crack density, the proportion of basically consistent, which is linearly proportional to the crack density, the proportion of surface cracks, the propagation distance of surface waves in the cracking region, and the excitation frequency. However, it is not correlated with the friction coefficient of the micro-crack surfaces. Therefore,

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the acoustic nonlinear parameter can be used to effectively characterize the degeneration of material properties caused by surface micro-cracks using the Rayleigh surface wave technique. Furthermore, a proper increase of the fundamental frequency can generate more significantly higher harmonics and improve the accuracy of this technique. These quantitative relationships provide the foundation for developing nonlinear ultrasonic based quantitative non-destructive evaluation techniques for assessing surface micro-cracks. Acknowledgments: This work is supported by the National Science Foundation of China (No. 11604033, No. 11632004, No. 51769028 and No. 11602040), and Key Program for International Science and Technology Cooperation Projects of Ministry of Science and Technology of China (No. 2016YFE0125900), Key Project of Natural Science Foundation of CQ CSTC (No. 0241002432002). Author Contributions: X.D., F.L., Y.X., P.C. and M.D. conceived and designed the simulations; F.L. performed the simulations; X.D. analyzed the data; N.H. contributed analysis tools; Y.Z., N.H. and P.C. revised the paper; Y.Z. and N.H. wrote the paper. Conflicts of Interest: The authors declare no conflict of interest.

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