Nonlinear Assessment of Material and Interface Imperfections Based on Non-Collinear Shear Wave Mixing

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2 Nonlinear Assessment of Material and Interface Imperfections Based on Non-Collinear Shear Wave Mixing A thesis submitted to the Division of Research and Advanced Studies of the University of Cincinnati in partial fulfillment of the requirements for the degree of Master of Science in the Department of Aerospace Engineering and Engineering Mechanics of the College of Engineering 016 By Ziyin Zhang B.S., Nanjing University, 01 Committee Chair: Professor Peter B. Nagy

3 ABSTRACT Non-collinear shear wave mixing has been reported to exhibit potential for assessing the additional nonlinearity caused by both bulk material degradation and interface imperfections between solids. When this technique is used to assess the material bulk nonlinearity, certain resonance condition has to be satisfied while when it is used to assess the nonlinearity of an imperfect plane interface between two half-solids, the resonance condition can be relaxed. However, even when misaligned from the bulk resonance conditions, this technique is still somewhat sensitive to the intrinsic bulk nonlinearity and the relatively strong bulk nonlinearity of the surrounding material adversely reduces the nonlinear contrast produced by interface imperfections. In an effort to increase the nonlinear contrast produced by the imperfect interfaces, this research focused on conducting a combined analytical and computational investigation to identify the optimal inspection conditions for ultrasonic characterization of imperfect interfaces based on non-collinear shear wave mixing. The mechanism of nonlinear shear wave mixing technique was further investigated to identify the main factors which affect the strength of the detected bulk nonlinearity of the material. Then two analytical models for nonlinear imperfect interfaces were developed by using a finite nonlinear interfacial stiffness representation of an imperfect interface of vanishing thickness and a thin nonlinear interphase layer in a quasi-static approximation, respectively. Both analytical models were numerically verified by comparison to COMSOL finite element simulations. The results show that the imperfect interface generates the same amount of nonlinearity in both reflection and transmission fields and detecting the nonlinear reflection completely eliminates the adverse influence of bulk nonlinearity and therefore increases the nonlinear contrast generated by the imperfect interface. The feasibility of ii

4 detecting the nonlinear reflection from the imperfect interface has also been confirmed to be possible by the experiment on Ti-6Al-4V diffusion bonded specimens using enhanced noncollinear ultrasonic wave mixing system implemented with a novel four-way polarity flipping technique. iii

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6 ACKNOWLEDGMENTS1 I would like to first express the most sincere appreciation to Professor Peter B. Nagy for his tremendous help, guidance, patience and generosity throughout this research. His pursuit for perfection in science and his deepest love for research have greatly inspired me to finish this thesis. The past three years of working with Professor Peter B. Nagy has already become the most beneficial experience in my life. I would also like to show my appreciation to the thesis committee members, Professor Francesco Simonetti and Dr. Waled Hassan for their time, advice and patience during the time I finished this thesis. I would also like to give my special thanks to Curtis W. Fox for all the technical support that he provided to me throughout the years. I am also thankful for his friendship and all his bad jokes. To my dearest parents Bofeng Zhang and Xun Ping, I would like to express my love and gratitude for their everlasting support and love for me during all these years. Their advice and comfort always inspire me to keep pursuing my dreams. Finally, the most importantly, I would like to thank my dear husband Ningxi Yu. Without his love, care and support for me, I would have never been able to finish this thesis. This work was supported by the FAA under Contract No. DTFACT-13-C This work was supported by the Federal Aviation Administration under Contract No. DTFACT-13-C v

7 CONTENTS Abstract... ii Acknowledgments...v Contents...1 List of Tables and Figures...4 List of Often Used Symbols...8 Chapter I Introduction 1.1 Overview Literature Review Theory of Non-Collinear Ultrasonic Wave Mixing Intrinsic Material Nonlinearity Imperfect Interface Nonlinearity Organization of the Rest of the Thesis...6 Chapter II Material Bulk Nonlinearity Assessment Based on Non-Collinear Shear Wave Mixing.1 Analytical Model Material Mixing Efficiency Function Geometric Mixing Efficiency Function...9. Numerical Validation Validation of the Material Mixing Efficiency Function

8 .. Chapter III Validation of the Geometric Mixing Efficiency Function...44 Imperfect Interface Nonlinear Assessment Based on Non-Collinear Shear Wave Mixing 3.1 Analytical Model Bulk Nonlinearity Mixing at a Perfectly Reflecting Interface Nonlinear Finite Interfacial Stiffness Model Thin Nonlinear Interphase Layer Model Numerical Validation Validation of Bulk Mixing at a Perfectly Reflecting Interface Validation of Nonlinear Finite Interfacial Stiffness Model Validation of Thin Nonlinear Interphase Layer Model Interface Versus Bulk Nonlinearity...76 Chapter IV Experimental Nonlinear Assessment of Diffusion Bonded Ti-6Al-4V Specimens Based on Non-Collinear Shear Wave Mixing 4.1 Experimental System Four-Way Polarity Flipping Technique Experimental Results and Data Analysis...86 Chapter V Conclusions and Future Work 5.1 Conclusions Future Work...96

9 Bibliography...98 Appendices A. Derivation of Scattered Nonlinear Longitudinal Wave Amplitude B. Interfacial Compliance of a Thin Nonlinear Interphase Layer C. Thin Nonlinear Interphase Layer versus Bulk Nonlinearity

10 LIST OF TABLES AND FIGURES CHAPTER I Figure 1.1 Schematic illustration of shear wave mixing Figure 1. Normalized linear reflection coefficient measured in Ti-6Al-4V specimens with diffusion bonds as the imperfect interface by Escobar-Ruiz et al. [5]. Higher percentage of the cross-boundary grain growth (CBGG) qualifies better interface bonding quality....4 Figure 1.3 Schematic illustration of normalized linear and nonlinear reflection responses versus different qualities of an imperfect interface...5 CHAPTER II Figure.1 Schematic illustration of the D ray approximation Figure. Interacting area of two shear waves from numerical simulation....3 Figure.3 Illustration of the envelope function e(x) and the corresponding three triangular pulse functions calculated with D = 4 mm, θ1 = 69.0 and θ = Figure.4 Schematic illustration of the D computational model...40 Figure.5 Example of the received longitudinal pulse....4 Figure.6 Relationship between the Murnaghan coefficient m and the amplitude of the mixed longitudinal wave Figure.7 Comparison between analytical approximation and numerical results on geometrical mixing efficiency function η of Ti-6Al-4V

11 Figure.8 Comparison between analytical approximation and experimental results on geometrical mixing efficiency function η of IN Table.1 List of material properties used in the numerical validation of bulk nonlinearity. 39 CHAPTER III Figure 3.1 Coordinate system used to study shear wave mixing at an imperfect interface. Only the two obliquely incident shear waves and the longitudinal reflected and transmitted waves produced by nonlinear mixing are shown Figure 3. Normalized reflection coefficient versus angle of incidence for various interfacial compliance ratios for Ti-6Al-4V...57 Figure 3.3 Interaction of two shear waves at a perfectly delaminated interface at (a) t =.95 µs, (b) t = 3.11 µs, (c) t = 3.43 µs and (d) t = 3.46 µs Figure 3.4 Numerical validation of the predicted loss in the amplitude of the mixed longitudinal wave caused by a perfectly reflecting interface...65 Figure 3.5 Schematic illustration of the imperfect interface model used in the D numerical simulations Figure 3.6 Examples of longitudinal (a) transmitted and reflected signals simulated by the finite interfacial stiffness model and (b) the resulting combined and filtered signal Figure 3.7 Comparison between analytical approximation and numerical results for the nonlinear reflection from an imperfect interface of finite interfacial stiffness with (a) Kn / Kn3 = 1 and (b) Kn / Kn3 = 10 (see Table 3.1 for details)...7 5

12 Figure 3.8 Comparison between analytical approximation and numerical results for the nonlinear reflection from a thin interphase layer (see Table 3. for details) Figure 3.9 Comparison between bulk and interface mixing of non-collinear shear waves in a Ti-6Al-4V specimen (see Table 3.3 for details) Figure 3.10 Examples of (a) transmitted and (b) reflected longitudinal pulses generated with and without nonlinear interface imperfections at θ1 = 67.0º (see Table 3.3 for details) Table 3.1 Material parameters used in the nonlinear finite stiffness model for Figure 3. and Figure Table 3. Material parameters used in the thin nonlinear interphase layer model for Figure Table 3.3 Material parameters used in the thin nonlinear interphase layer model for Figure 3.9 and Figure CHAPTER IV Figure 4.1 Simplified block diagram of the upgraded pulser/receiver system Figure 4. Schematic illustrations of the nonlinear inspection system in one-sided (a) transmission and (b) reflection modes of operation Figure 4.3 Nonlinear signatures obtained from the first batch of diffusion bonded Ti-6Al-4V specimens in transmission mode of operation by (a) Escobar-Ruiz et al. with frequency filtering [5] and (b) current study with four-way polarity flipping and linear amplitude averaging

13 Figure 4.4 Nonlinear transmission amplitude images of the second batch of diffusion bonded Ti-6Al-4V specimens (1 1 ) in interface configuration. The listed percentage value is the measured crushing strain for each specimen Figure 4.5 Nonlinear reflection (a) amplitude and (b) phase images of the second batch of diffusion bonded Ti-6Al-4V specimens ( ) in interface configuration. The percentage value is the measured crushing strain for each specimen Figure 4.6 Normalized amplitudes of the nonlinear reflections acquired from the second batch of specimens after data processing with (a) incoherent averaging and (b) coherent averaging Table 4.1 Alignment parameters for bulk and interface configurations. S denotes the separation distance between the footprints of the two incident waves on the top surfaces of the specimens

14 LIST OF OFTEN USED SYMBOLS ω angular frequency k wave vector φ interaction angle between two incident shear waves cd longitudinal wave velocity cs shear wave velocity λ first Lamé constant μ second Lamé constant a frequency ratio c shear-to-longitudinal wave velocity ratio U1, U amplitudes of the shear waves U3 amplitude of the nonlinear longitudinal wave M(φ) material mixing efficiency function V interaction volume r distance from the point of observation to the center of the mixing volume A, B, C third-order elastic constants K bulk modulus of compression ρ material density u particle displacement t time Cijkl linear stiffness tensor Cijklmn nonlinear stiffness tensor β acoustic nonlinearity parameter 8

15 A1 amplitude of the fundamental wave A amplitude of the second harmonic wave l, m, n Murnaghan coefficients c(x) harmonic carrier function Λ periodicity of the interference pattern f1, f frequencies of the two shear waves e(x) envelope function D width of the transducer θ1, θ incident angles of the two shear waves Tri(x) triangular pulse function A peak of the triangular pulse B half-width of the triangular pulse δ(x) Dirac delta function λd longitudinal wave length η normalized geometrical mixing efficiency function T1, T, T3 ultrasonic transducer p propagation distance N1, N cycle numbers of the two shear waves σxx normal stress in the x direction σyy normal stress in the y direction σxy tangential stress Δux normal interface opening displacement Δuy tangential interface opening displacement 9

16 Kn normal interfacial stiffness coefficient Kt tangential interfacial stiffness coefficient Sn normal interfacial compliance coefficient St tangential interfacial compliance coefficient Rss linear shear wave reflection coefficient θs incident and reflected shear wave angle θd reflected longitudinal wave angle α phase angle τ magnitude of pure shear stress Zs shear wave acoustic impedance T3 nonlinear longitudinal wave transmission coefficient R3 nonlinear longitudinal wave reflection coefficient d thickness of the nonlinear interphase layer g1, g input signals driving the two shear transducers h received signal S1 linear coefficient in the Taylor series of the system s overall transfer function S quadratic nonlinear coefficient in the Taylor series of the system s overall transfer function ɛ plane strain wij infinitesimal rotation tensor ξ D interaction factor 10

17 CHAPTER I INTRODUCTION 1.1 OVERVIEW Nonlinear shear wave mixing has been reported to exhibit great potential in detecting the nonlinearity change caused by material degradation and imperfect interfaces. When this technique is used to assess bulk nonlinearity of the specimen, certain resonance condition, which is determined by the frequency ratio of the two shear waves and the shear-to-longitudinal wave velocity ratio of the specimen being inspected, has to be satisfied. A third longitudinal wave will be generated with the frequency and wave vector equal to the sum of the frequencies and wave vectors of the two primary shear waves, respectively, which is usually referred to as the bulk resonant condition. By analyzing the amplitude of the longitudinal wave, the nonlinearity of the specimen can be assessed. Compared to conventional second harmonic generation, shear wave mixing not only has the advantages of frequency, spatial and mode separation, but it also eliminates the spurious nonlinear component generated by the experimental system. Therefore only the nonlinearity generated in the specimen is detected by this technique. When shear wave mixing is used to assess the nonlinearity of an imperfect plane interface between two half-spaces, the resonance condition can be relaxed to suppress longitudinal wave generation by bulk nonlinearity in the surrounding material and thereby assures that the measured longitudinal wave is primarily caused by nonlinear mixing of two shear waves at the interface to be inspected. However, even when misaligned from the bulk resonance condition, this technique is still somewhat sensitive to the intrinsic bulk nonlinearity of the surrounding material, therefore the transmitted longitudinal wave will be proportional to a combination of 11

18 material and interface nonlinearities. As a result, quantitative characterization of an imperfect interface with high bonding quality is difficult based on this technique alone. In an effort to further reduce the adverse influence of bulk nonlinearity in the host material itself, a combined analytical and computational investigation was conducted to identify the optimal inspection conditions for ultrasonic characterization of imperfect interfaces based on non-collinear shear wave mixing. In this thesis, the bulk nonlinearity assessment of the material using shear wave mixing was further investigated both analytically and numerically. The results showed that the strength of the nonlinear bulk signature is affected by the material mixing efficiency which is determined by one of the three Murnaghan coefficients of the material itself and the geometrical mixing efficiency which is determined by the interaction angle of the two primary shear waves. In the study of shear wave mixing on an imperfect interface, two analytical models, namely the nonlinear interfacial stiffness model and the thin nonlinear interphase layer model, were developed. These two models were numerically verified by comparison to COMSOL finite element simulations. The study shows that in some cases of imperfect interfaces, misaligning from the bulk resonance condition will suppress both the bulk and interface nonlinearity and therefore it does not contribute to the increase of the nonlinear contrast of the imperfect interface. However, the combined analytical and numerical studies of imperfect interface showed that the interface will generate the same amount of additional nonlinearity in both the reflection and transmission fields and detecting the reflected nonlinear signature will completely eliminate the adverse bulk nonlinearity generated in the surrounding material. This finding is also supported by experiments conducted on high-quality diffusion bonded Ti-6Al-4V specimens. The reflected nonlinear signature generated by the diffusion bond was detected with the implementation of a 1

19 novel four-way polarity flipping technique and different bond qualities could be distinguished by analyzing the nonlinear reflection signal. 1. LITERATURE REVIEW 1..1 Theory of Non-Collinear Ultrasonic Wave Mixing The theory of nonlinear ultrasonic wave mixing has been studied throughout the years. In 1963 Jones and Kobett first investigated the scattering of two intersecting plane elastic waves in a homogeneous isotropic medium based on nonlinear elasticity theory [1]. Using perturbation approximation approach, they discussed the resonance conditions for five different nonlinear ultrasonic wave mixing cases. Later on in 1964, Taylor and Rollins not only further clarified the resonance conditions for five different interaction cases, but they also derived the displacement amplitude of the nonlinear scattered wave for each case []. In the same year Rollins et al. validated this theory experimentally by observing the generation of a third longitudinal wave due to the mixing of two shear waves under the required resonance condition [3]. They concluded that in certain non-collinear interaction cases, the amplitude of the resulting scattered wave can be used to evaluate the third-order elastic constants of the material. This finding shed light on applying this theory to experimentally assessing the nonlinearity of the material. Both the studies done by Jones and Kobett and Taylor and Rollins were based on plane wave approach. To better accommodate this theory to experiment, Childress and Hambrick further investigated the nonlinear interaction based on wave-packet approach with limited pulse length [4]. Their results were essentially in accordance with previous work. However, as most of the primary and fundamental theories of nonlinear ultrasonic wave mixing were developed in the early 60 s, the information is scattered and previous publications 13

20 contain typos. Recently, Korneev et al. rederived the equations of the scattering coefficients for all possible nonlinear interactions and put them in a more generalized analytical form for nondestructive evaluation of material nonlinearity [5]. In this thesis, the equations in the paper by Jones and Kobett [1] will be used to explain the detailed analytical theory of non-collinear shear wave mixing which is also the main technique in this thesis research. When two vertically polarized shear waves interact in a homogeneous isotropic medium to generate a third longitudinal wave with the frequency and wave vector equal to the sum of the frequencies and wave vectors of the two shear waves, certain resonance condition must be satisfied. ω3 = ω1 + ω, (1.1) k= 3 k1 + k, (1.) where ωi and ki denote the angular frequencies and wave vectors, respectively, of the noncollinear shear waves (i = 1,) and the mixed longitudinal wave (i = 3). The schematic illustration of this type of nonlinear shear wave mixing is shown in Figure 1.1. Eq. (1.) can be rewritten in the form of the interaction angle φ of the two shear waves as follows ω ω ω ωω ( 3 ) = ( 1 ) + ( ) + 1 cos ϕ, cd cs cs cs (1.3) where cd and cs denote the longitudinal and shear wave velocities of the medium respectively as follows cd = λ + µ and cs = ρ µ. ρ (1.4) 14

21 shear k γ φ longitudinal k3 shear k1 Figure 1.1 Schematic illustration of shear wave mixing. 15

22 Here λ and μ denote the first and second Lamé constants and ρ denotes the density of the ω / ω1 and velocity ratio material. By using Eq. (1.1) and introducing the frequency ratio a = c = cs / cd, Eq. (1.3) can be rewritten as cos ϕ= c (1 c )(1 + a ). a (1.5) Eq. (1.5) is the resonance condition for the interaction case of two vertically polarized shear waves. From Eq. (1.5), we can see that the condition 1 cos ϕ 1 must be satisfied otherwise the nonlinear interaction will not exist. For a certain material, the velocity ratio is determined by its Lamé constants and therefore it is a fixed parameter. As a result, the frequency ratio of the two primary shear waves will have a limited range. By applying the above condition to Eq. (1.5), the selection limit of the frequency ratio is as follows 1 c 1+ c a. 1+ c 1 c (1.6) In Figure 1.1, γ denotes the angle by which the propagation direction of the generated longitudinal wave deviates from the propagation direction of the first primary shear wave. With this notation, the incident angle θ1 of the first shear wave has the same value as γ tan θ1 = a sin φ. 1 + a cos φ (1.7a) Then, the angle of the second incident shear wave θ = φ θ1 can be determined as tan θ = sin φ. a + cos φ (1.7b) 16

23 The amplitude of the generated third longitudinal wave is also predicted in the literature. The derivation procedure is quite complicated and tedious and therefore only the main steps will be presented in Appendix A for reference. The simplified amplitude equation is as follows U U M (ϕ)v ω13a (1 + a ), U3 = 1 8πrrcs cd (1.8) where U1 and U are the amplitudes of the two shear waves, U3 is the amplitude of the mixed longitudinal wave, M is the material mixing efficiency function, V denotes the interaction volume and r is the distance between the observation point and the center of the mixing volume. The material mixing efficiency M is a function of the interaction angle φ between the two shear waves and it can be written as = M 1 cs cd 7 1 [( B + K + A + µ) cos ϕ ( B + µ + A)], 3 (1.9) where A and B are two of the three third-order elastic constants introduced by Landau and Lifshitz [6], and K is the bulk modulus of compression. The material mixing function is of great importance as it directly indicates how the third-order elastic constants affect the amplitude of the generated nonlinear longitudinal wave. Eq. (1.9) will be discussed in further detail in Chapter Intrinsic Material Nonlinearity The lattice anharmonicity is a common characteristic of the crystalline solids. The lattice anharmonicity will cause stress-strain relationship of the solid to deviate from Hooke s Law and therefore becomes the source of intrinsic material nonlinearity. Several seminal papers in 1960 s investigated this phenomenon and separately defined different but related nonlinearity 17

24 parameters to describe the nonlinear elasticity in bulk solids [7-9]. Breazeale et al. [10-11] were the first to report that due to this intrinsic nonlinearity, when a sinusoidal acoustic wave propagates through the solid, it will get distorted and generate second harmonic. Later on Thompson et al. [1] also reported that the intrinsic nonlinearity of the material will generate even higher harmonics and they successfully measured the amplitudes of the higher harmonics in fused silica. Yost and Breazeale [13] defined pure mode acoustic nonlinearity parameters in single crystals to describe the intrinsic material nonlinearity based on the second harmonic generation measurement. In their paper, the acoustic nonlinearity parameter was defined as the linear combination of second-order and third-order elastic constants of the material, which can be determined by measuring the amplitudes of the fundamental and second harmonic waves and the propagation distance. Cantrell [14] later on proposed a more generalized definition of the acoustic nonlinearity parameter for arbitrary propagation modes in single crystals. In 015, Matlack et al. reviewed the second harmonic generation technique for material nonlinear characterization [15]. In this thesis, the simplified definition of acoustic nonlinearity parameter based the papers by Cantrell et al. [14] and Matlack et al. [15] will be used to describe the intrinsic material nonlinearity. Assuming that there is a longitudinal plane wave propagating through an isotropic medium with quadratic nonlinearity, the equation of motion simplified to one-dimension is as follows ρ u t =( E1 + E u u, ) x x (1.10) where ρ is the material density, u is the particle displacement and E1 and E can be further expressed as 18

25 E1 = C11, (1.11a) = E 3 C11 + C111, (1.11b) where C11 and C111 are the second- and third-order Brugger elastic constants written in Voigt notation. Combining Eqs. (1.10) and (1.11) yields ρ u t = [C11 + (3 C11 + C111 ) u u. ] x x (1.1) Eq. (1.1) can be further written as u u u, = cd [1 β ] x x t where cd = (1.13) C11 ρ is the longitudinal wave velocity. β is the sought acoustic nonlinearity parameter and in the intact metals β is purely caused by lattice anharmonicity of the material. Here, β can be expressed as β = ( 3 C11 + C111 ). C11 (1.14) The solution to Eq. (1.13) can be written as = u A1 sin(kx ωt ) + A cos(kx ωt ) +... (1.15) where A1 is the amplitude of the fundamental wave and A is the amplitude of the second harmonic wave. A can be further expressed as follows A = βa1 k x. 8 (1.16) 19

26 Therefore, the acoustic nonlinearity parameter β can be written as β= 8A k A1 x. (1.17) It should be noted that both Eqs. (1.14) and (1.17) are important. Eq. (1.14) clearly shows the physical cause of the intrinsic material nonlinearity of the crystalline solids is lattice anharmonicity and the acoustic nonlinearity parameter β can be used to evaluate the intrinsic material nonlinearity. In the meantime, Eq. (1.17) provides a practical way to measure the acoustic nonlinearity parameter β in an experiment. Such measurements will be robust and simple by using contact piezoelectric transducers. By fixing a transmitting and a receiving transducer on the opposite sides of the specimen being inspected and sending the driving signal with a single frequency component to the transmitting transducer, the transmitted signal through the specimen can be detected by the receiving transducer. The acoustic nonlinearity parameter can then be determined according to Eq. (1.17). However, to evaluate the intrinsic material nonlinearity, the measurement of the acoustic nonlinearity parameter β has to be performed on intact specimen as material degradation such as plastic deformation and fatigue can greatly affect the acoustic nonlinearity parameter β. Hikata et al. reported that dislocation plasticity makes contribution to the material nonlinearity [16-17]. Cantrell and Yost also concluded that both plastic deformation and cyclic fatigue will contribute to the material nonlinearity in addition to its intrinsic nonlinearity caused by lattice anharmonicity [18]. Therefore in those cases where the specimens have been plastically deformed or fatigued, the total acoustic nonlinearity parameter β will be the combination of the following three independent parts: βlattice is the lattice nonlinearity parameter which is also the intrinsic material nonlinearity, βmp is the dislocation monopole nonlinearity parameter which is 0

27 caused by plastic deformation, and βdp is the dislocation dipole nonlinearity parameter which is caused by cyclic fatigue. The second harmonic generation technique measures only the total acoustic nonlinearity parameter β and it is quite effective and widely used to detect the levels of material degradation [19-]. However, to measure the intrinsic material nonlinearity, intact specimens are used to assure that the acoustic nonlinearity parameter is purely caused by the lattice anharmonicity. In this thesis, the investigation is focused on nonlinear assessment using non-collinear shear wave mixing. Similar to the second harmonic generation, non-collinear shear wave mixing is also sensitive to material nonlinearity. However, in contrast to the second harmonic generation which detects the acoustic nonlinearity parameter β, non-collinear shear wave mixing only detects part of the acoustic nonlinearity parameter β which will be explained in Chapter Imperfect Interface Nonlinearity Subtle imperfections at the interface between bonded solids that might remain hidden from conventional nondestructive evaluation methods can greatly reduce the service life of fracture-critical components. Ohsumi et al. showed that less than 5-10 % lack of bond in diffusion bonded Ti-6Al-4V engine components can result in dramatic reduction in some of the dynamic strength-related properties, especially in crack resistance and impact damage tolerance [3]. Partially bonded imperfect interfaces behave much like tight kissing bonds. While a poorly bonded interface can be detected by conventional linear ultrasonic inspection methods, a better bonded interface behaves more like a tight kissing bond which exhibits very low linear ultrasonic contrast, i.e., low reflection and high transmission [4]. When assessing the imperfect interface between dissimilar solids, linear ultrasonic inspection has an inherently limited 1

28 detection threshold due to the acoustic impedance mismatch between the two materials and weak imperfections might remain hidden below this threshold. Even when detecting the imperfect interface between similar metals, strong incoherent scattering from the microstructure of the host material can hide weak interface imperfections. As a result, the sensitivity of linear ultrasonic inspection methods quickly drops as bond quality increases and it becomes impossible to distinguish various bond qualities exhibiting significantly different strength parameters. This phenomenon shown in Figure 1. was reported by Escobar-Ruiz et al. [5] in the investigation of imperfect interfaces in Ti-6Al-4V diffusion bonded specimens with different bond qualities. Previous studies have shown that an unbonded interface cannot support tension and therefore opens up during the tensile phase of acoustic vibration. Periodic opening and closing of the interface distorts the impinging harmonic acoustic wave and thus becomes another source of nonlinearity. It has been recognized for many years that the nonlinear acoustic response of an imperfect interface is likely to change more significantly than its linear response as the bond quality of the interface starts to drop as shown in Figure 1.3. This specific characteristic of a partially closed imperfect interface provides a unique opportunity for assessing imperfect interfaces by detecting the acoustic nonlinearity they generate during ultrasonic inspection [630]. Some of the interface characterization techniques that exploit this fact are quasi-nonlinear in the sense that they use what is essentially linear ultrasonic inspection and measure the modulation that appears in the linear response of the interface under static or very low-frequency dynamic loading of the bond [31-35]. In contrast, truly nonlinear inspection methods rely on high-frequency self-modulation, i.e., harmonics generation by a large-amplitude ultrasonic wave, or cross-modulation, i.e., mixing between two waves at least one of which produces highamplitude vibration [36-46]. In principle, there is no fundamental difference between these

29 quasi-linear and nonlinear inspection methods. They all exploit the breakdown of the law of superposition in the presence of nonlinearity and together they cover a wide range of applications in ultrasonic interface characterization. In a recent paper, An et al. developed a nonlinear spring model and reported that this model is capable of predicting second harmonic generation by obliquely incident longitudinal and horizontally polarized shear (SH) waves at an imperfect interface [47]. Demcenko et al. studied a related phenomenon, namely non-collinear mixing of a shear wave with a longitudinal wave at an imperfect interface [48]. Based on these two models, in this thesis a combined analytical and numerical study will be performed using nonlinear shear wave mixing on an imperfect interface which is normal to the mixed longitudinal wave. When non-collinear shear wave mixing is used to assess the nonlinearity of an imperfect plane interface between two half-spaces, the bulk resonance condition Eq. (1.5) discussed in Chapter 1..1 can be relaxed to suppress longitudinal wave generation by bulk nonlinearity in the surrounding material. However, even when misaligned from the bulk resonance condition, this technique is still somewhat sensitive to the intrinsic bulk nonlinearity of the surrounding material. To further reduce the adverse influence of bulk nonlinearity in the host material itself, in this thesis a combined analytical and computational investigation was conducted to identify the optimal inspection conditions for ultrasonic characterization of imperfect interfaces based on non-collinear shear wave mixing. 3

30 Linear Reflection Coefficient [db] linear detection threshold Figure Cross Boundary Grain Growth [%] 100 Normalized linear reflection coefficient measured in Ti-6Al-4V specimens with diffusion bonds as the imperfect interface by Escobar-Ruiz et al. [5]. Higher percentage of the cross-boundary grain growth (CBGG) qualifies better interface bonding quality. 4

31 Normalized Reflection linear nonlinear Interface Quality dissimilar similar, noisy similar Figure 1.3 Schematic illustration of normalized linear and nonlinear reflection responses versus different qualities of an imperfect interface. 5

32 1.3 ORGNAZATION OF THE REST OF THE THESIS The previous section tried to provide sufficient background knowledge on the theory of the non-collinear shear wave mixing technique and the origins of material bulk nonlinearity and imperfect interface nonlinearity. Chapter I also described the main problem encountered when using shear wave mixing to assess the imperfect interface. In the case of nonlinear interface characterization using non-collinear shear wave mixing, both bulk nonlinearity of the host material and the additional nonlinearity generated by the imperfect interface will contribute to the transmitted nonlinear signature. In order to increase the nonlinear contrast produced by interface imperfections, Chapter II focuses on further investigating the factors that affect the detection of material bulk nonlinearity based on non-collinear shear wave mixing. We proposed a simple analytical approximation of this type of mixing to study how misaligning from the bulk resonance condition will affect the detected strength of the bulk nonlinear signature. The analytical results will then be validated by finite element simulations using COMSOL. Based on the results from Chapter II, Chapter III investigates the additional nonlinearity generated by an imperfect interface. In Chapter III, two analytical models for the imperfect interface are developed and validated separately by finite element methods. By using these two analytical models, we will explain the reason why misaligning from bulk resonance conditions is not able to sufficiently increase the detection contrast of an imperfect interface. In Chapter III, a better option for detecting the excess nonlinearity by the imperfect interface is proposed based on the study of the two analytical models. In Chapter IV, experiments are performed on Ti-6Al-4V diffusion bonded specimens in order to validate the findings from Chapter III. A new design of the experiment system will also be described. Finally, Chapter V summarizes the work done in this thesis including conclusions and proposed future work. 6

33 CHAPTER II MATERIAL BULK NONLINEARITY ASSESSMENT BASED ON NON-COLLINEAR SHEAR WAVE MIXING.1 ANALYTICAL MODEL.1.1 Material Mixing Efficiency Function In the paper by Taylor and Rollins, where they discussed the fundamental theory of non- collinear ultrasonic wave mixing, five possible interaction cases were investigated and the amplitude of the generated third nonlinear wave was derived for all five interaction cases []. Their first interaction case was the mixing of two vertically polarized shear waves to generate a third longitudinal wave and in this case the amplitude of the generated longitudinal wave was given as previously quoted in Eq. (1.8) U1U M (ϕ)v ω13a (1 + a ). U3 = 8πrrcs cd (.1) Here M is the material mixing efficiency function and it has been expressed in Eq. (1.9). Eq. (1.9) is repeated here as = M [( B + K + A + µ) cos ϕ ( B + µ + A)]. 3 cs cd (.) In Eq. (.), the material mixing efficiency function is expressed using the third-order elastic constants A and B introduced by Landau and Lifshitz [6]. M can also be expressed using 7

34 Murnaghan coefficients n, m, and l and the relationship between the Murnaghan coefficients and third-order elastic constants are as follows A= n, (.3a) 1 B= m n, (.3b) 1 C =l m + n. (.3c) Replacing the third-order elastic constants in Eq. (.) with Murnaghan coefficients using Eq. (.3) yields = M 1 cs cd 7 [(m + K + m) cos ϕ (m + m)], 3 (.4) where K = λ + µ / 3. From Eq. (.4) we can see that the material mixing efficiency M is linearly dependent on only one of the Murnaghan coefficients m [49]. This unique feature of nonlinear shear wave mixing makes it different from second harmonic generation. We already know that the second harmonic generation technique is sensitive to the acoustic nonlinearity parameter β which is previously defined in Eq. (1.14). Here, Eq. (1.14) is repeated as C β = ( ), C11 (.5) where the third-order elastic constant C111 can be expressed using Murnaghan coefficients as C111 = (l + m) and second-order elastic constant C11 can be expressed using Lamé constants as 8

35 C11 = λ + µ. Therefore the acoustic nonlinearity parameter β can be defined using Murnaghan coefficients as β = (3 + l + m ). l + m (.6) From Eqs. (.4) and (.6) we can see that second harmonic generation is sensitive to the changes in both l and m while nonlinear shear wave mixing is only sensitive to the change in m. In other words, nonlinear shear wave mixing measures only part of the acoustic nonlinearity parameter β. This characteristic of nonlinear shear wave mixing will reduce its sensitivity in some cases compared to second harmonic generation, where the change in the first Murnaghan coefficient l is most significant and the change in m is barely noticeable. However, the non-collinear shear wave mixing technique still has its advantage over conventional second harmonic generation as it can be easily adapted to a scanning measurement system which can generate nonlinear c-scan images directly showing the defects of the specimen being inspected..1. Geometric Mixing Efficiency Function From Eqs. (.1) and (.4), we can see that the amplitude of the mixed longitudinal wave not only depends on one of the Murnaghan coefficient m, but it is also affected by the interaction angle φ between the two shear waves. In the paper by Taylor and Rollins [], the interacting waves were assumed to be infinite plane waves thus they needed to constrain their interaction to a certain volume V to determine the amplitude of the mixed signal. In our combined analytical and numerical investigation, we intend to capture the main features of our experiments conducted with transducers producing finite-diameter beams, therefore in this study the interaction volume is determined by the intersection volume of the ultrasonic beams [49]. 9

36 Although the ultrasonic beams produced by finite-diameter transducers are inherently divergent, in our analytical approximation we relied on ray approximation of collimated beams as it is illustrated in Figure.1. In the numerical simulation, the interaction area is shown in Figure. for more details. In the linear field, the red and blue spots shown in Figure. indicate the positive and negative peaks of the ultrasonic wave. However in the quadratic nonlinear field, there is no difference between positive and negative peaks (red and blue) and both can be considered as the peak of the nonlinear longitudinal wave generated by mixing. In another word, the interference pattern shown in Figure. will produce a harmonic carrier function c( x= ) cos (π x / Λ ), (.7) where x is the spatial coordinate in the direction of the mixed longitudinal wave generation shown in Figure.1 and.. Λ is the periodicity of the interference pattern produced by the two shear waves which is also the interval between each horizontal lines formed by red and blue dots shown in Figure.. Therefore, Λ can be expressed by using the frequencies and the incident angles of the two shear waves as follows Λ= cs. f1 cos θ1 + f cos θ (.8) The effective mixing area is a hexagon determined by the diameters of the transmitting and receiving transducers and the incident angles of the two mixing shear waves. In our D analysis, the strength of the mixed nonlinear longitudinal wave depends on the length of each horizontal line formed by red and blue dots in Figure.. As those horizontal lines propagate, the generated longitudinal wave will get accumulated. 30

37 y D θ x D θ1 Λ D Figure.1 Schematic illustration of the D ray approximation. 31

38 y x Figure. Interacting area of two shear waves from numerical simulation. 3

39 In order to calculate the amplitude of the final accumulated longitudinal wave, the envelope function e(x) of the width of the effective mixing area needs to be calculated. The approximated effective mixing area can be sectioned into five different parts as shown in Figure.. In the first part where the two shear waves start to interact with each other, the increase of the width of the effective mixing area is the fastest. For this part, the maximum length in the x direction is denoted as b1 and it is calculated as follows b1 = D sin(θ1 + θ ) + sin θ1 sin θ, tan θ1 sin(θ1 + θ ) (.9) where θ1 and θ are the incident angles of the two primary shear waves and D denotes the diameter of the transducer, which is also the beam width of ultrasonic waves in this ray approximation. The increasing rate of the width of the mixing area in the first part can be easily determined as s1= tan θ1 + tan θ. Followed by the first part, in the second part of the interaction area the width of the effective mixing area also increases but at a slower rate which can be calculated as s= tan θ. The maximum length of this part in the x direction is denoted as b and it is calculated as follows b D sin(θ1 + θ ) sin θ1 + sin θ sin(θ1 + θ ) + sin θ1 sin θ [ ]. tan θ sin(θ1 + θ ) tan θ1 sin(θ1 + θ ) (.10) In the third part of the interaction area, the width of the mixing area remains the same and it equals the width of the transducer. The width of the mixing area starts to decrease in the fourth part. Due to the fact that the hexagon mixing area is symmetrical, the decreasing rate of the width of the fourth part is the same as the increasing rate s in the second part. The maximum length of the fourth part in x direction is also the same as b. Similarly, the width of the mixing 33

40 area in the fifth part decreases at the same rate s1 as it increases in the first part and the maximum length of the fifth part is also the same as b1. To calculate the maximum length of the third part of the mixing area in the x direction, the total length of the effective mixing area in the x direction is calculated as b=d cos θ1 + cos θ. sin(θ1 + θ ) (.11) Then the length of the third part of the mixing area can be calculated as b3 = b b1 b. (.1) If the origin of the coordinates is selected to be the center of the effective mixing area, then the envelope function e(x) of the width of the effective mixing area can be plotted as Figure.3. For future calculation convenience, the envelope function e(x) can also be considered as the sum of three different triangular pulses shown in Figure.3 as follows e( x) = Tri1 ( x) + Tri ( x) + Tri3 ( x). (.13) These triangular pulses can be written as A (1 x / Bi ), x Bi Trii ( x) = i, 0, otherwise (.14) where i = 1,,3. From Figure.3, Ai and Bi (i = 1,,3) can be easily calculated from simple geometrical considerations as 34

41 10 Tri1 ( x) 9 Tri ( x) 8 Tri3 ( x) 7 e ( x) y [mm] b1 b1-4 b b -5-6 Figure.3 b x [mm] Illustration of the envelope function e(x) and the corresponding three triangular pulse functions calculated with D = 4 mm, θ1 = 69.0 and θ =

42 = A1 b D 1 1 θ ) + tan(θ1 += ( ), cos θ1 cos θ (.15a) b = A b1 tan θ1 + tan θ A1, (.15b) A3 = (b1 + b b ) tan θ, (.15c) and B1 = b /, (.16a) = B b / b1, (.16b) B3= b / b1 b. (.16c) Then, the geometrical approximation of the interference pattern is the product of the carrier and envelope functions f ( x ) = c ( x )e( x ). (.17) The Fourier transform of the interference pattern, F(k), is the convolution of the spatial Fourier transforms of the carrier and envelope functions, C(k) and E(k), respectively. F= (k ) F{ f (= x)} F{c( x) e(= x)} C (k ) E (k ), (.18) where k denotes the spatial frequency. The Fourier transform of the harmonic carrier function is C (k ) = δ( k + π π ) + δ( k ), Λ Λ (.19) where δ is the Dirac delta function and a constant factor that depends on the particular definition of Fourier transform. Therefore, the Fourier transform of the interference pattern can be obtained as 36

43 F (= k) k E (k C ( k) E (k k)d= π π ) + E (k + ). Λ Λ (.0) Assuming that the carrier function changes much faster than the envelope function, the spatial frequency spectrum of the envelope function is sufficiently band limited so that the second term in Eq. (.0) can be neglected. The main purpose of this analysis is to find the amplitude Fd of the longitudinal wave generated by mixing between the two shear waves and based on Eq. (.0), Fd can be written as Fd = F ( π π π ) E( ), λd λd Λ (.1) where λd = cd / (f1 + f) which denotes the wavelength of the longitudinal mixed wave. The amplitude of the longitudinal wave Fd can be normalized to its maximum value Fd0 = E(0) reached at Λ = λd to get the final normalized geometrical mixing efficiency E( η= π π ) λd Λ. E (0) (.) It has been mentioned earlier that the envelope function e(x) can be written as the sum of three triangular pulses, therefore the Fourier transform of the envelope function E(k) can be written as 3 3 Bk = E (k ) = Ei (k ) Ai Bi sinc ( i ), i i (.3) where Ai and Bi are the peaks and half-widths of the triangular pulses which are determined in Eqs. (.15) and (.16). Finally, the normalized geometrical mixing efficiency function η is derived as 37

44 3 1 1 Ai Bi sinc πbi ( ) λd Λ. η= i 3 Ai Bi (.4) i. NUMERICAL VALIDATION..1 Validation of the Material Mixing Efficiency Function To validate the material mixing efficiency function M, we used COMSOL Multiphysics finite element software to perform the numerical simulation. Figure.4 shows a D illustration of the computation model used in this study [49]. T1 and T represent the two shear wave transmitters and T3 represents the longitudinal receiver. In this numerical simulation, all three transducers have the same width of 3 mm. The distances between the centers of the transducers and the center (O) of the interaction zone are all chosen to be p = 5 mm so that the two shear waves arrive to it at the same time. Except for the apertures of the two shear transmitters, the material model is covered all around with absorbing boundaries to prevent spurious interference caused by reflected shear waves. The top and bottom boundaries are of longitudinal wave absorption types to avoid the need for reflection corrections of the measured displacement amplitudes. To validate the material mixing efficiency function M, we need to validate that the amplitude of the mixed longitudinal wave is linearly dependent on only one of the Murnaghan coefficient, m. In this simulation, aluminum was used to validate the relationship between m and the amplitude of the mixed longitudinal wave. The relevant material properties of aluminum are listed in Table.1. 38

45 Table.1 List of material properties used in the numerical validation of bulk nonlinearity. material ρ [kg/m3] λ [GPa] μ [GPa] l [GPa] m [GPa] n [GPa] aluminum Ti-6Al-4V The two frequencies of the shear waves were selected as f1 = 4.5 MHz and f = 5.5 MHz. The cycle numbers of the Hanning windowed tone burst of the two shear waves were N1 = 6 and N = 7 respectively. Based on the Lamé constants and the density of the material listed in Table.1, both longitudinal and shear wave velocities can be easily calculated using Eq. (1.4). The frequency ratio a and velocity ratio c can then be determined and, according to Eqs. (1.7), the two incident shear wave angles were calculated as θ1 = 69.30º and θ = 50.38º. The amplitude of the two transmitting shear waves were chosen to be U1 = U = 100 nm. It has to be noted that in a real experiment, a transducer can never generate ultrasonic wave with such high amplitude. However, the nonlinear effect is rather weak compared to the linear effect and in order to accurately measure the nonlinear response, the amplitudes of the driving signals were set to high values in the numerical simulation. 39

46 B T1 C shear (ω1) θ θ1 shear (ω) T d d D O A longitudinal (ω3) d F Figure.4 T3 E Schematic illustration of the D computational model. 40

47 Figure.5 shows an example of the received longitudinal pulse. In order to prevent diffracted components of the primary shear waves from interfering with the much weaker nonlinear signals, all simulations were run twice by flipping the polarities of the excitation signals and averaging the two received signals [0, 50]. This simple technique very efficiently suppresses the fundamental harmonics that would otherwise completely overshadow the sought mixed signals of nonlinear origin. As all the transducers are separated from the center of the interaction area by p = 5 mm and the shear and longitudinal wave velocities of aluminum are cs = 3.10 mm/µs and cd = 6.18 mm/µs, the propagation time for shear wave was ts = p/cs 1.6 µs and the propagation time for longitudinal wave was td = p/cs 0.81 µs. The nonlinear signal shown in Figure.5 arrived at ts + td =.4 µs as expected. The frequency of the mixed longitudinal wave is equal to the sum of the frequencies of the two shear waves, i.e., f3 = 10 MHz. The amplitude of the received longitudinal pulse was measured. We then ran a parametric study by changing the Murnaghan coefficient and monitored the amplitude of the mixed longitudinal wave. The simulation results showed that when only l or n was changed, the longitudinal wave amplitude remained constant. However, when m was changed, the amplitude changed proportionally as shown in Figure.6. In Figure.6, when m equals zero the amplitude of the mixed longitudinal wave is not zero. In this case, the detected nonlinear longitudinal wave is not caused by the intrinsic material nonlinearity but by the geometrical nonlinearity. We can also predict this phenomenon by analyzing Eq. (.4). In Eq. (.4) when m is set to be zero the material mixing function M is also not zero. Therefore, theoretically a weak longitudinal wave can still be generated. The results shown in Figure.6 are in perfect agreement with the theoretical prediction that the amplitude of the mixed longitudinal wave is linearly dependent on one of the Murnaghan coefficient m. 41

48 6 Amplitude [nm] Time [µs] Figure.5 Example of the received longitudinal pulse. 4

49 Amplitude [nm] Figure Absolute Value of m [GPa] 1500 Relationship between the Murnaghan coefficient m and the amplitude of the mixed longitudinal wave. 43

50 .. Validation of the Geometrical Mixing Efficiency Function Theoretical predictions indicated that, besides the material mixing efficiency function M, the geometrical mixing efficiency function η also affects the amplitude of the mixed longitudinal wave. To numerically validate the geometrical mixing efficiency function η, Ti-6Al-4V was used and the related material parameters are also listed in Table.1 [49]. The simulation model is similar to the one used in the validation of material mixing efficiency function, except that due to the changes in the material properties, the two incident angles of the shear waves are changed. The frequencies of the two shear waves were the same as the ones in the validation of the material mixing efficiency function M, f1 = 4.5 MHz and f = 5.5 MHz. The amplitudes of the shear waves were chosen to be U1 = U = 00 nm. As we have mentioned in the Chapter..1, such high amplitude only exists in the numerical simulation in order to increase the weak nonlinear effect and reduce the numerical errors. In the validation of the geometrical mixing efficiency function η, the periodicity pattern Λ is the variable and therefore according to the Eq. (.8), the incident angles of the shear waves will be changed. To make sure that the propagation direction of the mixed longitudinal wave was always normal to the receiving transducer, the two incident angles always satisfied the following relationship ω1 sin θ1 =ω sin θ. (.5) The comparison between the analytical approximation based on Eq. (.4) and the numerical results are shown in Figure.7. The numerical results are in good agreement with the analytical prediction, though there is some deviation between them. This deviation is caused by the fact that in the numerical simulation the ultrasonic beams had certain degrees of divergence which was neglected in the analytical approximation. The analytical approximation of the geometrical 44

51 mixing efficiency function η was also experimentally validated on IN718 specimens and the results are shown in Figure.8. In spite of some systematic deviations due to the fact that the experiment was done under immersion in water, the experimental results are still in fairly good agreement with the analytical predictions. 45

52 Normalized Magnitude analytical approximation numerical First Shear Wave Angle of Incidence, θ 1 [deg] Figure.7 Comparison between analytical approximation and numerical results on the geometrical mixing efficiency function η of Ti-6Al-4V. 46

53 Normalized Magnitude analytical approximation experimental bulk angle First Shear Wave Angle of Incidence, θ 1 [deg] Figure.8 Comparison between analytical approximation and experimental results on the geometrical mixing efficiency function η of IN

54 CHAPTER III IMPERFECT INTERFACE NONLINEARITY ASSESSMENT BASED ON NON-COLLINEAR SHEAR WAVE MIXING 3.1 ANALYTICAL MODEL Bulk Nonlinearity Mixing at a Perfectly Reflecting Interface In a seminal paper on this subject, Baik and Thompson introduced a quasi-static model to study the linear acoustic reflection and transmission from imperfect interfaces [51]. According to their linear finite interfacial stiffness model for imperfect interfaces of negligible thickness, both normal σxx and tangential σxy stresses are required to be continuous at the interface. The interface imperfection exhibits itself through small normal = u x u x (0+ ) u x (0 ) and tangential = u y u y (0+ ) u y (0 ) interface opening displacements that are proportional to the normal and tangential interface tractions [51] σ xx K n σ = xy 0 0 u x K t u y (3.1a) 0 σ xx, S t σ xy (3.1b) or alternatively u x Sn u = y 0 where Kn and Kt are the normal and tangential interfacial stiffness coefficients, respectively, and Sn and St are the corresponding normal and tangential interfacial compliance coefficients. When 48

55 the interface is perfectly bonded, both Sn and St vanish and the non-collinear ultrasonic wave mixing technique measures the bulk nonlinearity of the material only. In contrast, when the interface is perfectly delaminated, both Sn and St are approaching infinity and the incident shear waves undergo perfect reflection. Since the shear wave incident angles θ1 and θ in this case are both above the longitudinal critical angle, there is no mode conversion. Therefore, the interaction volume is separated into two parts. One is formed by the two incident shear waves and the other part is formed by the two reflected shear waves, while the total mixing volume remains the same [49]. The linear shear wave reflection coefficient of the delaminated interface can be written as follows Rss c sin θs sin θd cos θs = Rss eiα. c sin θs sin θd + cos θs (3.) Here, θs denotes the incident and reflected shear wave angle, θd is the reflected longitudinal wave angle and c is the previously introduced shear-to-longitudinal velocity ratio. Since there is no mode conversion, the magnitude of the reflection coefficient is unity but it has a phase angle α. The first component of the mixed longitudinal wave generated by the two incident shear waves hits the perfectly reflecting boundary at normal incidence and then it is reflected back with a phase angle change of 180 at the delaminated interface. The second component of the mixed longitudinal wave is generated by the two reflected shear waves. Due to the reflection, the first shear wave incident at θ1 exhibits a phase angle change of α1 and the second shear wave incident at θ exhibits a phase angle change of α. As a result, when the second component of the mixed longitudinal wave adds up, the phase angle difference between them results in a reduced amplitude U 3 < U 3i + U 3r that can be calculated from 49

56 U= 3 U 3i + U 3r + U 3iU 3r cos α, (3.3) where U3i and U3r are the amplitudes of the two components of the longitudinal wave respectively and Δα denotes the phase angle difference between them. Eq. (3.3) will be numerically validated in Chapter Nonlinear Finite Interfacial Stiffness Model When a subtle interface imperfection cannot be detected by the linear method, it still might be detectable using nonlinear methods. Figure 3.1 shows a schematic illustration of noncollinear shear wave mixing at an imperfect plane interface. To simplify the problem, the media on the two sides of the interface are assumed to be identical. It should be pointed out that nonlinear ultrasonic inspection techniques are inherently far more complex than their conventional linear counterparts, therefore in practice nonlinear techniques are called upon only when simpler linear methods fail. Therefore, we will focus on interface imperfections that remain hidden from linear ultrasonic inspection because they do not produce measurable increase in reflection and decrease in transmission at the highest feasible inspection frequencies. Accordingly, Figure 3.1 shows only the two obliquely incident and perfectly transmitted shear waves and the longitudinal reflected and transmitted waves produced by nonlinear mixing, but no reflected shear waves. Similar to the linear finite interfacial stiffness model developed by Baik and Thompson [51] to study imperfect interfaces, the nonlinear finite interfacial stiffness model also requires that the normal σxx and tangential σxy stresses be continuous at the interface. 50

57 y shear wave (ω ) ϕ longitudinal reflected wave (ω 3) imperfect interface Δux θ θ1 longitudinal transmitted wave (ω 3) x shear wave (ω 1) Figure 3.1 Coordinate system used to study shear wave mixing at an imperfect interface. Only the two obliquely incident shear waves and the longitudinal reflected and transmitted waves produced by nonlinear mixing are shown. 51

58 The nonlinear interface imperfection exhibits itself through normal Δux and tangential Δuy interface opening displacements that are nonlinear functions of σxx and σxy as follows [5] σ xx σ xy K n1 0 0 K n K n3 K t1 K t K t3 u x u y K n4 u x +... K t3 u y u u x y (3.4a) or u x u y S n1 0 0 Sn Sn3 S t1 S t S t3 σ xx σ xy Sn4 σ xx S t3 σ xy σ σ xx xy (3.4b) Here, for the sake of simplicity, the discussion is limited to quadratic nonlinearity. Kni and Kti are linear (i = 1) and nonlinear (i =,3,4) normal and tangential interfacial stiffness coefficients. Similarly, Sni and Sti are linear (i = 1) and nonlinear (i =,3,4) normal and tangential interfacial compliance coefficients. The stiffness and compliance forms given in Eq. (3.4) are equivalent and can be converted into each other by simple inversion as follows K n3 K t4 K 1 1 Sn1 =, Sn = n, Sn3 =, S t1 ==, S t4 3 K n1 K t1 K n1 K n1k t1 K n1k t1 (3.5a) or S S S 1 1 K n1 =, K n = n, K n3 = n3, K t1 ==, K t4 t4. 3 Sn1 S t1 Sn1 Sn1S t1 Sn1S t1 (3.5b) 5

59 In Eqs. (3.1) and (3.4) the linear elements that must vanish for an isotropic interface because of symmetry requirements were replaced with zeros. In addition, assuming that the nonlinear interface exhibits symmetry for tangential deformations of opposite signs, Kn4 = Kt = Kt3 = 0, and consequently Sn4 = St = St3 = 0, will also hold. It should be pointed out that the interfacial spring model is usually presented in the literature as a stiffness relationship. However, the compliance formulation is more straightforward to use in the case under consideration here, therefore we will use the compliance form given in Eq. (3.4b). As it was mentioned before, the investigation will be limited to interface imperfections that cannot be detected using conventional linear reflection or transmission measurements. In such cases the linear interfacial compliance can be assumed to vanish and the normal σxx and tangential σxy stress components at the interface can be approximated by the unperturbed superposition of the shear stresses produced by the two incident waves. Then, the weak perturbation produced by the nonlinear interfacial compliance can be directly calculated from these normal and tangential stress components using Eq. (3.4b). Following the simple rules of stress transformation, pure shear stress of magnitude τ in a coordinate system rotated by an angle θ relative to the xy coordinate system produces the following normal and shear stress components s xx = s yy = τ sin(θ), (3.6a) s xy = τ cos(θ). (3.6b) The two shear waves that mix at the interface as shown in Figure 3.1 are assumed to produce shear stress levels of τ1 and τ that could be expressed as 53

60 t1 = Zs u1, t (3.7a) t = Zs u, t (3.7b) where Zs denotes the shear wave acoustic impedance of the host material and u1 and u are the transverse displacements produced by the first and second incident shear waves, respectively. The spatial and temporal variation of these transverse displacements can be expressed as follows = u1 U1 cos(k1 r ω1t ), (3.8a) = u U cos(k r ωt ), (3.8b) where r denotes the position vector, and U1 and U are the displacement amplitudes of the first and second incident shear waves, respectively. Both phase terms are neglected for the sake of simplicity. Then, the linearly combined stresses at the interface (x = 0) are σ(xxl ) = ω1 Zs U1 sin(k y1 y ω1t ) sin(θ1 ) + ω Zs U sin(k y y ωt ) sin(θ ), (3.9a) σ(xyl ) = ω1 Zs U1 sin(k y1 y ω1t ) cos(θ1 ) ω Zs U sin(k y y ωt ) cos(θ ). (3.9b) Assuming that the tangential wave vector components of the two shear waves cancel each other, then the nonlinear normal interface opening displacement of angular frequency ω3 = ω1 + ω is u x3 = ( ) Sn σ(xxl ) ( ), + Sn3 σ(xyl ) (3.10) where 54

61 ( σ(xxl ) ) ω1 ω Zs U1 U cos[(ω1 + ω )t ]sin(θ1 ) sin(θ ) (3.11a) and ( ) (l ) s xy = ω1 ω Zs U1 U cos[(ω1 + ω ) t ]cos(θ1 ) cos(θ ). (3.11b) Here, only one half of the mixed signal of ω3 = ω1 + ω angular frequency is used for interface characterization while the other half of ω4 = ω1 - ω angular frequency is disregarded. Because of the symmetric normal interface opening displacement in both directions, the reflected and transmitted mixed longitudinal waves will have the same amplitude U3 = u x3. (3.1) The longitudinal wave transmission T3 and reflection R3 coefficients for the mixed nonlinear signal can be defined as follows U T3 = R3 = 3. U1 U (3.13) Combining Eqs. (3.10) through (3.13) yields the sought reflection and transmission coefficients 1 T3 = R3 = ω1 ω Zs ( Sn sin θ1 sin θ Sn3 cos θ1 cos θ ). (3.14) It should be mentioned that Eq. (3.14) is not applicable directly for second harmonic generation at normal incidence, i.e., when θ1 = θ = 0 and ω1 = ω = ω, since it represents only the mixed term of ω3 = ω1 + ω angular frequency without the ω1 and ω terms, therefore it yields only half of the ω3 = ω second harmonic longitudinal signal amplitude when ω1 = ω = ω. 55

62 For the purposes of distinguishing the nonlinear signature of the imperfect interface from that of the surrounding host material, it is important to establish how the magnitude of the mixed longitudinal wave changes with the angle of incidence. According to Eq. (3.14), this angular dependence is determined by the interfacial compliance ratio Sn3/Sn. Figure 3.3 illustrates the normalized reflection coefficient versus angle of incidence θ1 for various interfacial compliance ratios. Here, the reflection coefficient for any given value of the Sn3/Sn ratio was normalized to its corresponding value at the bulk resonance angle of incidence θ1r 69.º, i.e., where the background nonlinearity of the host material is at its maximum (see Table 3.1 for the linear material properties of the Ti-6Al-4V host material). For small values of Sn3/Sn < the nonlinear transmission (and reflection) produced by an imperfect interface decreases with increasing angle of incidence θ1. Below the resonant angle θ1r the bulk nonlinear signature increases with the angle of incidence, therefore it can be readily suppressed relative to the nonlinear interface signature by lowering the angle of incidence. However, for large values of Sn3/Sn > 0 the nonlinear transmission produced by an imperfect interface increases with increasing angle of incidence θ1, therefore the bulk nonlinearity can be suppressed in the transmitted longitudinal wave by increasing rather than decreasing θ1 relative to θ1r, which is technically much more difficult. 56

63 Normalized Reflection Coefficient Sn3/Sn First Shear Wave Angle of Incidence, θ1 [deg] Figure 3. Normalized reflection coefficient versus angle of incidence for various interfacial compliance ratios for Ti-6Al-4V. 57

64 3.1.3 Thin Nonlinear Interphase Layer Model The thin interphase layer model of imperfect interfaces assumes that the additional nonlinear compliance of the interface can be accounted for by introducing a homogeneous interphase layer of small thickness d around the imperfect interface. The third-order, i.e., nonlinear, elastic properties of this otherwise perfectly bonded interphase layer are obtained by appropriately modifying the corresponding elastic properties of the host material. In this section we will show that, in order to model nonlinear interface imperfections with a thin homogeneous layer of hyperelastic material exhibiting quadratic nonlinearity, the nonlinear interfacial compliances defined in Eq. (3.4b) must satisfy not only the previously mentioned symmetry condition (Sn4 = 0) but also a second condition that Sn = Sn3. In addition, the interphase layer must be very thin relative to both the longitudinal and shear wavelengths. The hyperelastic material will be characterized by its two Lamé constants, λ and μ, three Murnaghan constants, l, m, and n, and density ρ. As before, in order to limit the solution to imperfect interfaces that are hidden from linear ultrasonic inspection, we will assume that the density and linear Lamé constants of the interphase layer are identical to those of the surrounding host material. Chapter II has shown that out of the three nonlinear Murnaghan coefficients, only the second coefficient m influences longitudinal wave generation by shear wave mixing. Therefore, the other two Murnaghan constants (l and n) of the interphase layer will be also taken to be identical to those of the host material. In this way, the nonlinear interphase layer will be characterized by the product of its thickness d and the additional value Δm of its second Murnaghan coefficient over that of the host material. The finite interfacial stiffness model assumes a vanishing interface thickness but still produces an interface opening displacement discontinuity. The thin interphase layer model 58

65 assumes a thickness that is much smaller than the longitudinal and shear wavelengths in it, but also includes some of the neighboring host material within x = ± d/ distance of the interface on both sides. In general, for a thin, but not vanishingly thin, nonlinear layer the displacement discontinuities between the opposite faces can be expressed in a form that is similar to Eq. (3.4b) but also includes terms of σyy [5] u x u y (layer) 0 s d 11 0 s66 sn st sn3 st3 sn4 st4 s1 0 sn5 st5 sn6 st6 σ xx σ xy σ xx σ xy sn7 σ xx σ xy +..., st7 σ yy σ yy σ yy σ xx σ yy σ xy (3.15) where s11, s1, and s66 are the linear compliance constants of the host material (in Voigt s abbreviated notation). In terms of Lamé constants, these linear compliance coefficients are s11 = λ + μ, 4μ (λ + μ) (3.16a) s1 = λ, 4μ (λ + μ) (3.16b) s66 = 1. μ (3.16c) In the case of pure shear stresses, Eq. (3.15) can be reduced to the simpler form of Eq. (3.4b) by exploiting that σyy = -σxx so that 59

66 u x u y (l ayer) s s d s66 Sn1 0 Sn 0 S 0 t1 sn + sn5 sn6 st + st5 st6 Sn3 0 sn3 st3 σ xx σ xy sn4 sn7 σ xx +... st4 st7 σ xy σ σ xx xy σ xx σ xy (layer) 0 σ xx +... S t4 σ xy σ σ xx xy. (3.17) Again, the elements in the last column of the compliance matrix, i.e., the elements multiplying σxx σxy, are zero by virtue of symmetry. Equation 3.17 indicates one of the two reasons why a thin but finite interphase layer is difficult to model in the interfacial compliance approximation. The interfacial compliance model assumes that the interface region has infinitesimally small thickness and produces and interface opening displacement that depends only on the normal and tangential traction components acting on the opposite surfaces of the interface. A thin interphase layer has a small, but non-vanishing, thickness that causes problems in the adaptation of the interfacial compliance model. First, it makes it necessary to account for the in-plane normal stress σyy that is not a surface traction at all. Second, in addition to strain that is related to stress according to the nonlinear constitutive relationship, the total nonlinear interphase layer opening displacement also includes geometrical nonlinearity caused by rotation that is not related to stress directly as strain is. The role of rotation in the total nonlinear interphase layer opening displacement of a thick interface layer will be discussed in Appendices B and C. 60

67 The total compliance of the interphase layer is then the sum of the compliance of the intact layer formed from the host material and the extra compliance of the imperfect interface inside it (layer) u x = u y u x u y (intact layer) u x + u y (interface). (3.18) Therefore, the non-vanishing effective interfacial compliance coefficients of the imperfect interface can be obtained by subtracting the compliance coefficients of the intact layer of the host material from those of the homogenized interphase layer (interface) Sn = (layer) (intact layer) Sn Sn, (3.19a) (interface) Sn3 = (layer) (intact layer) Sn3 Sn3. (3.19b) For the purposes of the discussion here, the effect of rotation could be neglected in Eq. (3.17) since it cancels out when the interface compliance is calculated. In Appendix B, the main steps of deriving the effective interfacial compliances of a homogeneous isotropic interphase layer of quadratic nonlinearity were summarized. The final result is that the two non-vanishing normal and tangential nonlinear interfacial compliances are identical and they are proportional to the product of the interphase layer thickness d and the additional value Δm of its second Murnaghan coefficient over that of the host material that represents the additional nonlinearity caused by the presence of the imperfect interface [5] mat Sn = mat Sn3 = m d μ (λ + μ) (3.0) 61

68 Here, the superscript mat indicates that only the material nonlinearity of the thin interphase layer is included in the calculation of the equivalent interfacial stiffness. The negative sign on the right side of the above equation means that nonlinear softening of the material expressed by a decrease of its second Murnaghan coefficient increases the nonlinear compliance of the interphase layer therefore corresponds to positive nonlinear interfacial compliance. When the normal and tangential interfacial stiffness coefficients are equal in Eq. (3.14), they can be factored out and, with φ = θ1 + θ denoting the total angle subtended by the two interacting shear waves, Eq. (3.14) can be rewritten as U mat 1 mat T3 = R3 = 3 = ω1 ω Zs Sn cos(φ). U1U (3.1) In Appendix C, the theory of Taylor and Rollins [] for bulk mixing of two non-collinear shear waves due to material nonlinearity yields the following similar result U 3mat = U1 U 1 mat ξ ω1 ω Zs Sn cos( φ), (3.) where ξ is a factor accounting for the differences between the 3D volumetric scattering considered in the theory of Taylor and Rollins and our D plane wave model that does not allow beam divergence. Comparing Eqs. (3.1) and (3.) reveals that, except for the factor ξ, our predictions for the thin interphase layer model are consistent with the bulk mixing theory of Taylor and Rollins. 6

69 3. NUMERICAL VALIDATION 3..1 Validation of Bulk Mixing at a Perfectly Reflecting Interface When we discussed the bulk mixing at a perfectly reflecting interface in Chapter 3.1.1, it has been pointed out that there is an amplitude loss in the mixed longitudinal wave due to the reflection of the incident shear waves from a perfectly delaminated interface and this loss was given in Eq. (3.3) in terms of the phase shift of the otherwise perfectly reflected shear waves. To numerically validate this prediction, the computational model used in the validation of bulk material and geometrical mixing efficiency functions described in Chapter. was cut in half leaving only the top part and the resulting bottom boundary was set as a perfectly reflecting traction-free surface [49]. The receiving transducer was moved from the bottom boundary to the top of the model to measure the longitudinal wave reflection from the free surface representing a perfectly delaminated interface. Ti-6Al-4V was chosen as the material for this simulation and the related material properties can be found in Table.1. As usual, the two shear waves had the frequencies of f1 = 4.5 MHz and f = 5.5 MHz and the amplitudes were 100 nm. Figure 3.3 shows the interaction process of the two shear waves mixing at a perfectly delaminated interface. The results of the numerical validation of the predicted loss in the amplitude of the mixed longitudinal wave caused by a perfectly reflecting interface are shown in Figure 3.4. In Figure 3.4, the position of the interface is considered at the origin when the mixing area of the two incident shear waves hitting the interface is equal to the mixing area of the two reflected shear waves. Figure 3.4 shows that besides some minor difference, the numerical simulation results essentially agree with the previously introduced analytical approximation of Eq. (3.3). 63

70 Figure 3.3 (a) (b) (c) (d) Interaction of two shear waves at a perfectly delaminated interface at (a) t =.95 µs, (b) t = 3.11 µs, (c) t = 3.43 µs and (d) t = 3.46 µs. 64

71 Amplitude Loss Coefficient analytical numerical Interface Position [mm] Figure 3.4 Numerical validation of the predicted loss in the amplitude of the mixed longitudinal wave caused by a perfectly reflecting interface. 65

72 3.. Validation of Nonlinear Finite Interfacial Stiffness Model Figure 3.5 shows a schematic illustration of the imperfect interface model used in the D numerical simulations [5]. The model is similar to the one used for the validation of bulk mixing described in Chapter. except that there is an imperfect interface lying at the center of the interaction zone. To validate the nonlinear finite interfacial stiffness model, the capability of the COMSOL Multiphysics FE simulation software was exploited to explicitly define the normal and tangential interface opening displacements as nonlinear functions of the prevailing surface tractions. The material parameters used for the host material (Ti-6Al-4V) and the interface are listed in Table 3.1. Table 3.1 Material parameters used in the nonlinear finite stiffness model for Figure 3. and Figure 3.7. ρ [kg/m3] λ [GPa] µ [GPa] Kn1 [N/m3] Kn [N/m4] Kn3 [N/m4] Kt1 [N/m3] host material 4, N/A N/A N/A N/A imperfect interface #1 N/A N/A N/A imperfect interface # N/A N/A N/A To simplify the problem, the host material is set to be linear elastic with geometrical nonlinearity only, i.e., hyperelastic without material nonlinearity (l = m = n = 0) so that the detected nonlinear signature is caused solely by the imperfect interface. The model is cut into half horizontally through the center. The bottom boundary of the top volume and the top boundary of the bottom volume are selected as a contact pair to represent the imperfect interface. Since the interface is assumed to be isotropic, Kn4, Kt, Kt3, in Eq. (3.4a) must vanish due to symmetry. In addition, since the generation of the mixed longitudinal wave is not affected by Kt4, for the sake of 66

73 simplicity, it was also set to zero. Therefore, we only have to assign representative values to Kn1, Kn, Kn3, and Kt1. For a perfect linear interface, both normal ux and tangential uy displacement components are continuous through the interface, therefore the linear interfacial stiffness coefficients must approach infinity, which presents an obvious numerical problem in finite element simulations. To better elucidate the nonlinear effect, we only consider cases when the interface is essentially transparent for linear inspection at the given inspection frequency. However, if the linear normal and transverse interfacial stiffness coefficients are not sufficiently high, the imperfect interface will be easily detected by conventional linear reflection and/or transmission measurements. To compromise, the two linear stiffness parameters were set to their empirically determined highest possible value without causing numerical problems during FE simulations. In this way, the finite interfacial stiffness model caused minimal linear perturbation in the incident shear waves, therefore the numerical results could be used to validate the predictions of the above presented analytical approximation that is based on the weak perturbation approximation. For the same reason, it was also necessary to minimize the inherent geometrical nonlinearity of the interface that could be achieved using sufficiently low displacement amplitudes U1 = U = 0. nm for the two primary shear waves. In the following example the frequencies of the two shear waves are f1 = 4.5 MHz and f = 5.5 MHz and the number of cycles in the Hanning windowed tone bursts are N1 = 6 and N = 7, respectively, to assure that the two pulses are of approximately the same length. 67

74 longitudinal reflection shear shear (ω 3 ) incident incident T T (ω ) 1 (ω 1 ) θ 1 R r θ longitudinal transmission (ω 3 ) O imperfect interface R t Figure 3.5 Schematic illustration of the imperfect interface model used in the D numerical simulations. 68

75 Figure 3.6a shows examples of typical nonlinear longitudinal reflected and transmitted pulses. Again, all simulations were run twice by flipping the polarities of the excitation signals and averaging the two received signals to prevent diffracted primary shear waves. Figure 3.6b shows the combined and filtered signal obtained by calculating the asymmetric part of the reflected and transmitted longitudinal pulses shown in Figure 3.6a and then using a high-pass filter to remove the low-frequency spectral content of the result. The transmitting and receiving transducers were all separated from the center of the interaction zone by p = 5 mm propagations distance and the shear and longitudinal velocities were cs = 3.1 mm/µs and cd = 6.4 mm/µs, respectively, therefore the shear propagation time was ts = p/cs 1.6 µs and the longitudinal propagation time was td = p/cd 0.8 µs. The nonlinear signals shown in Figure 3.6 arrive at around ts + td =.4 µs as expected and their center frequencies are 10 MHz which is the sum of the frequencies of the two incident shear waves. These results confirm our prediction that an imperfect interface generates both reflected and transmitted longitudinal waves with the same amplitude but opposite phase. 69

76 (a) reflected transmitted Amplitude [fm] Time [µs] (b) combined and filtered Amplitude [fm] Time [µs] Figure 3.6 Examples of longitudinal (a) transmitted and reflected signals simulated by the finite interfacial stiffness model and (b) the resulting combined and filtered signal. 70

77 Figures 3.7a and 3.7b show the simulation results from two parametric studies with Kn / Kn3 = 1 and Kn / Kn3 = 10, respectively. The analytical predictions were obtained from Eq. (3.14) with the nonlinear interfacial compliances Sn and Sn3 calculated using Eq. (3.5a) from the nonlinear interfacial stiffness constants listed in Table 3.1. According to Figure 3.7, the numerical results obtained from the finite interfacial stiffness model agree with the analytical prediction except for a minor discrepancy caused by the inherent divergence of acoustic waves simulated in the D FE model. In the derivation of the analytical approximation, the nonlinear mixing of two shear plane waves at the imperfect interface is assumed to produce reflected and transmitted longitudinal plane waves with infinite beam width. In contrast, in the numerical simulations the primary shear waves were generated by 3 mm wide transmitters and the resulting longitudinal waves were detected by 3 mm wide receivers. Because of inherent diffraction effects caused by these finite-width beams over their respective propagation length of 5 mm, the numerical results exhibit a slight deviation from the corresponding analytical predictions. In spite of this small discrepancy, within the uncertainties caused by systematic differences between the numerical and analytical models, these simulation results validate the finite nonlinear interfacial stiffness model we developed for non-collinear shear wave mixing at an imperfect interface. It should be noted that in this model, the ratio between Sn and Sn3 can be selected arbitrarily while in the thin interphase layer model to be discussed in the next Chapter 3..3 these two nonlinear interfacial compliance coefficients will be inherently equal. 71

78 (a) Reflection Amplitude [fm] 5 analytical approximation interfacial stiffness (COMSOL) 4 ( Kn / Kn3 = 1 ) First Shear Wave Angle of Incidence, θ1 [deg] (b) Reflection Amplitude [fm] 5 analytical approximation interfacial stiffness (COMSOL) 4 ( Kn / Kn3 = 10 ) First Shear Wave Angle of Incidence, θ1 [deg] Figure 3.7 Comparison between analytical approximation and numerical results for the nonlinear reflection from an imperfect interface of finite interfacial stiffness with (a) Kn / Kn3 = 1 and (b) Kn / Kn3 = 10 (see Table 3.1 for details). 7

79 3..3 Validation of Thin Nonlinear Interphase Layer Model In the following numerical validation of the thin nonlinear interphase layer model, we will use another capability of the COMSOL software that has a built-in option called Murnaghan material to simulate the nonlinear interaction between an acoustic wave and an isotropic hyperelastic solid of quadratic nonlinearity [5]. The geometric model for the simulation is similar to the schematic illustration previously shown in Figure 3.5. In order to realistically represent the relatively high vibration amplitudes achieved under typical experimental conditions, the displacement amplitudes of the two primary shear waves were chosen to be U1 = U = 0 nm in the following simulations. The host material is selected as Ti6Al-4V and for the imperfect interface a thin layer is defined at the center of the model with the same Lamé constants as the host material to prevent linear reflection. The material nonlinearity of both the host material and the interphase layer are defined by their respective Murnaghan coefficients. Since only the second Murnaghan constant m influences the longitudinal wave generation by non-collinear shear wave mixing, the other two Murnaghan constants (l and n) of the interphase layer were taken to be identical to those of the host material. In this way, the nonlinear interphase layer can be characterized by its thickness d and the additional nonlinearity Δm over that of the host material. The presence of bulk nonlinearity in the host material causes only a transmitted longitudinal wave by nonlinear mixing between the two incident shear waves but not a reflected one. Therefore, the reflection coefficient of a nonlinear interphase layer embedded in a nonlinear host material can be directly determined from the simulation. In contrast, the transmission coefficient of the nonlinear interphase layer can be determined only by subtracting from the simulated signal through the interphase layer the reference signal produced by bulk nonlinearity that must be calculated separately by setting Δm = 0. 73

80 Figure 3.8 shows a comparison between our analytical approximation and numerical results for the nonlinear reflection from a thin nonlinear interphase layer. The material parameters used in this simulation are listed in Table 3.. Table 3. Material parameters used in the thin nonlinear interphase layer model for Figure 3.8. d [µm] ρ [kg/m3] λ [GPa] µ [GPa] l [GPa] m [GPa] n [GPa] host material N/A 4, imperfect interface #3 10 4, ,000 0 The thickness of the interphase layer d = 10 µm was selected to be much less than the shortest wavelength encountered in the simulation. According to Figure 3.8, except for the previously seen small systematic discrepancy between the analytical approximation and numerical simulation results, the overall agreement is quite reasonable. 74

81 Reflection Amplitude [fm] analytical approximation thin interphase layer (COMSOL) First Shear Wave Angle of Incidence, θ 1 [deg] Figure 3.8 Comparison between analytical approximation and numerical results for the nonlinear reflection from a thin interphase layer (see Table 3. for details). 75

82 3.3 INTERFACE VERSUS BULK NONLINEARITY One of the main questions we would like to answer based on the developed analytical approximations and computational simulations of nonlinear mixing between two non-collinear shear wave is whether the spurious bulk nonlinearity of the surrounding host material can be suppressed for better interface characterization by optimizing the inspection parameters. Escobar-Ruiz et al. suggested in a recent paper that misaligning the inspection system from the bulk resonance condition produces some improvement in this respect [5]. Nonlinear ultrasonic inspection based on the non-collinear shear wave mixing method is usually conducted with obliquely incident longitudinal transducers in either immersion or contact mode of operation. Unfortunately, lowering the incident angles of the interacting shear waves is limited by the requirement that the longitudinal waves generated by the transmitters hit the surface at incident angles above the first (longitudinal) critical angle in the coupling medium (water or polymer wedge). For example, in Ti-6Al-4V this condition sets a limit of θm = sin-1(cs/cd) 31º on the smaller of the shear wave angles and θ1m = sin-1(a cs/cd) 39º on the larger one. In immersion inspection, the two refraction angles must be chosen above θ1 54º and θ 4º to retain sufficiently high energy transmission into the specimen to be tested [5]. In order to estimate the achievable suppression of bulk nonlinearity in the host material due to beam misalignment, the simple geometrical mixing efficiency function η was proposed in Chapter II. Figure 3.9 compares bulk and interface mixing in a Ti-6Al-4V host containing a thin nonlinear interphase layer. The material parameters used in this simulation are listed in Table 3.3. Like before, the displacement amplitudes of the two primary shear waves were U1 = U = 0 nm. Based on this comparison, misalignment to θ1 57º should substantially reduce the bulk nonlinearity relative to that of a thin interphase layer when Sn = Sn3. 76

83 Unfortunately the improvement is much smaller when Sn << Sn3 because the mixing efficiencies of the imperfect interface and the bulk material exhibit similar angular dependence to each other. In such cases, reflection mode of operation offers a better alternative. Table 3.3 Material parameters used in the thin nonlinear interphase layer model for Figure 3.9 and Figure d [µm] ρ [kg/m3] λ [GPa] µ [GPa] l [GPa] m [GPa] n [GPa] host material N/A 4, imperfect interface #4 10 4, , Figure 3.10 illustrates the (a) transmitted and (b) reflected longitudinal pulses generated with and without a nonlinear interface in a Ti-6Al-4V specimen (see Table 3.3 for details). In order to illustrate the typical dominance of bulk nonlinearity over weaker interface nonlinearity in transmission mode of operation, the incident angles of the two interacting shear waves were θ1 = 67.0º and θ = 48.9º, i.e., quite close to the phase-matching bulk resonance condition of θ1r = 69.º and θr = 49.9º. Figure 3.10a shows that the transmitted mixed signal is fairly strong (U3 8.6 pm) even in the absence of an imperfect interface because of bulk mixing that occurs in the host material and the additional nonlinearity of the imperfect interface adds relatively little (ΔU3 1.8 pm) to the amplitude of the total transmitted nonlinear signal. Weak nonlinearity produced by an imperfect interface can be more easily detected in the reflection field. Figure 3.10b illustrates that, in the absence of linear reflection caused by impedance mismatch at the interface between dissimilar materials and additional spatially incoherent nonlinear backscattering from the inhomogeneous microstructure of the material, bulk mixing produces no 77

84 backward propagating longitudinal wave, therefore in reflection mode of operation the additional nonlinearity of an imperfect interface can be detected without interference from bulk nonlinearity. 78

85 Reflection Amplitude [fm] interface analytical interface simulation bulk analytical bulk simulation First Shear Wave Angle of Incidence, θ 1 [deg] Figure 3.9 Comparison between bulk and interface mixing of non-collinear shear waves in a Ti-6Al-4V specimen (see Table 3.3 for details). 79

86 (a) Transmitted Amplitude [pm] imperfect interface perfect interface Time [µs] (b) Reflected Amplitude [pm] imperfect interface perfect interface Time [µs] Figure 3.10 Examples of (a) transmitted and (b) reflected longitudinal pulses generated with and without nonlinear interface imperfections at θ 1 = 67.0º (see Table 3.3 for details). 80

87 CHAPTER IV EXPERIMENTAL NONLINEAR ASSESSMENT OF DIFFUSION BONDED Ti-6Al-4V SPECIMENS BASED ON NON-COLLINEAR SHEAR WAVE MIXING 4.1 EXPERIMENTAL SYSTEM The combined analytical and numerical study described in the previous chapters has indicated that instead of detecting the transmitted nonlinear signal, detecting the reflected nonlinear signal will completely eliminate the adverse influence of bulk nonlinearity and therefore is the better option to assess the nonlinearity of the imperfect interface. To further validate the findings from the analytical and numerical study, a nonlinear assessment experiment was also conducted on diffusion bonded Ti-6Al-4V specimens with various bond qualities [53]. From previous investigations, it has been recognized that the nonlinear reflection from the interface tends to be very weak. The signal-to-noise ratio (SNR) of the old nonlinear ultrasonic imaging system that was originally developed for bulk nonlinearity measurements is not sufficient enough to detect the nonlinear reflection from hidden interface imperfections. Therefore, a new nonlinear inspection system was designed and built. The simplified block diagram of the enhanced nonlinear ultrasonic imaging system is shown in Figure 4.1. The excitation signal for the primary shear waves are calculated by a single CompuGen 430 four-channel Arbitrary Function Generator (AFG) card that plugs directly into the computer through a PCI bus. The output signals of two RF Amplifiers after filtering by two 5 MHz low pass filters are driving two 5 MHz, 0.5 diameter immersion transducers. The 81

88 nonlinear signal received by a 10 MHz, 0.5 diameter, 5 focal length transducer goes through a customized two-stage signal conditioning amplifier before it is sent to an AD-IPR-110 ADC/Pulser/Receiver PCI card installed in the computer for further analysis. The first stage of the two-stage signal conditioning includes a 40 db low-noise preamplifier and a band pass filter (BPF) with center frequency of 10 MHz and bandwidth of 3 MHz. The second stage consists of another 4 db amplifier and a second BPF with center frequency of 10 MHz and a narrower bandwidth of 0.75 MHz. The upgrade and integration of the hardware not only greatly reduced the electrical noise of the whole system, but also made it possible to implement a novel four-way polarity flipping scheme which can effectively suppress all spurious signals regardless of their frequency content. The theory of four-way polarity flipping technique will be further discussed in the following Chapter 4.. A new LabView software controlling the whole system including pulse generation, scanning control, nonlinear signature analysis and imaging processing was also developed. The SNR of the enhanced nonlinear ultrasonic imaging system has increased by more than 40 db and thereby it can be used to detect the weak nonlinear reflection from the imperfect interface. The schematic illustration of the transducer alignment of the nonlinear imaging system is shown in Figure 4.. In the transmission mode, the back wall of the specimen to be tested is used to detect the nonlinear signature. The reflection mode is used to detect the nonlinear reflection from the interface. The receiving transducer also picks up the reverberations of the transmitted nonlinear signal. Therefore, the system is required to have sufficient enough temporal resolution to distinguish the nonlinear reflection from the reverberations of the nonlinear transmitted signal. The alignment of the transducers in the enhanced system can also be controlled by a computer and the alignment process is automated. 8

89 CompuGen 430 Arbitrary Function Generator (PCI) 1 AD-IPR-110 ADC/Pulser/ Receiver (PCI) 3 Computer BTO4000 AlphaS RF Amplifier BTO4000 AlphaS RF Amplifier customized two-stage signal conditioning 1 GaGe/DynamicSignals LLC Tomco Technologies 3 NDT Automation/Mistras 5MHz/0.5 transducer 3 5MHz/0.5 transducer 3 10MHz/0.5 /3 transducer 3 Figure 4.1 Simplified block diagram of the upgraded pulser/receiver system. 83

90 (a) Transducer 1 (Tx) Transducer 3 (Rx) Transducer (Tx) ω 3 ω 1 ω θ 1i θ i θ 1s θ s bondline specimen requires reflecting backwall x-y scanner (b) Transducer 1 (Tx) Transducer 3 (Rx) Transducer (Tx) ω 3 ω ω 1 θ 1i θ i θ 1s θ s bondline specimen x-y scanner Figure 4. Schematic illustrations of the nonlinear inspection system in one-sided (a) transmission and (b) reflection modes of operation. 84

91 4. FOUR-WAY POLARITY FLIPPING TECHNIQUE It has been mentioned earlier that in order to increase the SNR of the nonlinear ultrasonic imaging system, a novel four-way polarity flipping scheme was implemented. In nonlinear shear wave mixing, in addition to the sought mixed nonlinear signal, strong second harmonic signals of the primary waves will also be generated. For example, when the frequencies of the shear waves are set to be 4.5 MHz and 5.5 MHz, the sought mixed signal will have the summed frequency of 10 MHz while the second harmonic signals will have the frequencies of 9 MHz and 11 MHz. In the old system, external narrow BPF filters were used to suppress those unwanted second harmonic signals. However this frequency separation becomes particularly difficult when the specimen under test has relatively small thickness since the applied tone bursts of the shear waves must be limited in length, which in turn widens the frequency bandwidth. Because of this widening, in the previous example the 10 MHz mixed signal cannot be sufficiently discriminated from the nearby nonlinear spectral components of 9 and 11 MHz frequency using narrow band filtering only. In order to further suppress the unwanted second harmonics, four-way polarity flipping technique was implemented into the system to suppress all spurious signals regardless of their frequency content [53]. If the input signals driving the two shear wave transducers are denoted as g1 and g and the received signal is denoted as h, then the four possible polarity combinations can be expressed as h( ++ )= S1{g1 + g } + S [ g1 + g + g1g ] +..., (4.1a) h( + )= S1{g1 g } + S [ g1 + g g1g ] +..., (4.1b) 85

92 h( ) = S1{ g1 g } + S [ g1 + g + g1g ] +..., (4.1c) h( + ) = S1{ g1 + g } + S [ g1 + g g1g ] +..., (4.1c) where S1 and S are the linear and quadratic coefficients in the Taylor series of the system s overall transfer function. The received signals acquired from these four polarity combinations are averaged as follows h h( ++ ) h( + ) + h( ) h( + ) = S g1g. 4 (4.) The averaged signal is proportional to the sought nonlinear mixed signal g1g and both the fundamental linear components and their spurious nonlinear second harmonic components are completely eliminated. This technique can dramatically increase the SNR and makes it possible to adjust the frequencies of the two primary shear waves without any limitation on the frequency selectivity of the BPF, e.g. the a = f / f1 frequency ratio can be much lower than the usually used a = 1. value or even be set to unity. This technique has been implemented in the pulse generation software by linking four segments of the driving signal containing all possible polarity combinations listed in Eqs. (4.1) into four-pulse sequences that can be averaged after reception in a single operation at very high processing speeds. It has been found that this technique suppressed the spurious nonlinear signals by as much as 30 db, which can be exploited to dramatically increase the sensitivity of the nonlinear imaging system. 4.3 EXPERIMENTAL RESULTS AND DATA ANALYSIS In this example of nonlinear assessment of diffusion bonded Ti-6Al-4V specimens, two batches of specimens made of 0.5 thick flat plates were tested [53]. The bonding conditions 86

93 were adjusted so that the specimens covered a wide range of bond quality. The first batch of specimens were the same ones used in an earlier study reported by Escobar-Ruiz et al. [5], therefore they were first tested to demonstrate the reproducibility of our enhanced nonlinear ultrasonic imaging system in the same transmission mode. The scanning area was selected to be by as in the previous study. Throughout all the experiments performed in the current study, two alignment configurations were used and the related parameters are listed in Table 4.1. Table 4.1 Alignment parameters for bulk and interface configurations. S denotes the separation distance between the footprints of the two incident waves on the top surfaces of the specimens. φ [ ] θ1i [ ] θi [ ] θ1 [ ] θ [ ] S [mm] bulk interface For each specimen, the nonlinear transmission amplitude image was obtained. To quantitatively analyze the detected nonlinear signatures of different bond qualities, for each image the acquired amplitude data for every pixel was linearly averaged and the averaged value was used to represent the strength of the nonlinear signature for that particular specimen. The comparison between the results from [5] by Escobar-Ruiz et al. and those from our current study using the enhanced system is shown in Figure 4.3. Figure 4.3 demonstrates that the results obtained by the enhanced nonlinear imaging system from the first batch of specimens not only reproduce those obtained earlier, but the suppression of bulk nonlinearity when using the interface configuration is slightly improved. 87

94 (a) Normalised Response [db] -1-4 Interface configuration Bulk configuration Cross-Boundary Grain Growth [%] 100 Normalized Magnitude [db] (b) Figure interface configuration bulk configuration Cross-Boundary Grain Growth [%] 100 Nonlinear signatures obtained from the first batch of diffusion bonded Ti-6Al-4V specimens in transmission mode of operation by (a) Escobar-Ruiz et al. with frequency filtering [5] and (b) current study with four-way polarity flipping and linear amplitude averaging. 88

95 Before the second batch of six new diffusion bonded Ti-6Al-4V specimens were tested with the enhanced nonlinear imaging system, their microstructures were examined. The results showed that all six new specimens exhibited apparently perfect diffusion bonds. In addition, the measurement of relative shear wave birefringence showed that the second batch of Ti-6Al-4V specimens had much higher degree of anisotropy than the specimens in the first batch. As the shear wave velocity varied significantly in the orthogonal two directions of the new specimens, it was important that the nonlinear measurement be taken at the same direction for all the specimens. The second batch of Ti-6Al-4V diffusion bonded specimens was first tested using the enhanced nonlinear imaging system in transmission mode of operation with interface alignment. Since these specimens were smaller than those in the first batch, the scanning area was 1 by 1. The acquired amplitude images are shown in Figure 4.4. From Figure 4.4, we can see that there is no perceivable difference between the nonlinearity detected in transmission mode from the six new specimens. It has been demonstrated that the enhanced nonlinear imaging system has the ability to detect nonlinear reflection from imperfect interfaces. Therefore the second batch of Ti-6Al-4V diffusion bonded specimens were also tested using the nonlinear imaging system in reflection mode of operation. Unlike in transmission mode, in nonlinear reflection mode both the amplitude and phase of the nonlinear signal were recorded and these images are shown in Figure 4.5. For the data analysis based on the acquired nonlinear reflection amplitude and phase images, the following incoherent and coherent averaging methods were used U incoh = 1 M N Amn, MN m= 1 n= 1 (4.3a) and 89

96 #T3T4 4. % #T5T6 4.6 % #T7T8 5.4 % #T1T 5.5 % maximum minimum #T11T1 5.5 % #T9T % Figure 4.4 Nonlinear transmission amplitude images of the second batch of diffusion bonded Ti-6Al-4V specimens (1 1 ) in interface configuration. The listed percentage value is the measured crushing strain for each specimen. 90

97 (a) #T3T4 4. % #T5T6 4.6 % #T7T8 5.4 % #T1T 5.5 % maximum minimum #T11T1 5.5 % #T9T % (b) #T3T4 4. % #T5T6 4.6 % #T7T8 5.4 % #T1T 5.5 % #T11T1 5.5 % #T9T % Figure 4.5 Nonlinear reflection (a) amplitude and (b) phase images of the second batch of diffusion bonded Ti-6Al-4V specimens ( ) in interface configuration. The percentage value is the measured crushing strain for each specimen. 91