Foundations of Acoustic Methods Used in Non-Destructive Inspection of Laminated Materials

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(1)Foundations of Acoustic Methods Used in Non-Destructive Inspection of Laminated Materials.

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(3) Foundations of Acoustic Methods Used in Non-Destructive Inspection of Laminated Materials. PROEFSCHRIFT. ter verkrijging van de graad van doctor aan de Technische Universiteit Delft, op gezag van de Rector Magnificus prof. dr. ir. J.T. Fokkema, in het openbaar te verdedigen ten overstaan van een commissie, door het College voor Promoties aangewezen, op dinsdag 13 januari 2004 om 16:00 uur. door. Alexei Vasilievich KONONOV. Master of Science (physics of acoustic and hydrodynamic wave processes), The Nizhny Novgorod State University geboren te Kirov, Russia.

(4) Dit proefschrift is goedgekeurd door de promotoren: Prof. dr. ir. R. de Borst. Samenstelling promotiecommissie: Rector Magnificus, Prof. dr. ir. R. de Borst, Prof. dr. ir. D.J. Rixen, Prof. ir. L.B. Vogelesang, Prof. dr. ir. C.P.A. Wapenaar, Prof. dr. ir. D.H. van Campen, Dr. Sc. A.V. Metrikine,. Voorzitter Technische Technische Technische Technische Technische Technische. Universiteit Universiteit Universiteit Universiteit Universiteit Universiteit. Delft, promotor Delft Delft Delft Eindhoven Delft,. Published and distributed by: DUP Science DUP Science is an imprint of Delft University Press P.O. Box 98 NL-2600 MG Delft The Netherlands Telephone: +31 15 2785678 Telefax: +31 15 2785706 E-mail: Info@Library.TUDelft.NL ISBN 90-407-2466-0 Keywords: non-destructive inspection, planar laminate, laser ultrasonics, moving load, moving laser beam, transition radiation, diffraction radiation c 2003 by A.V. Kononov Copyright Cover design by A.V. Kononov All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any other information storage and retrieval system, without written permission from the publisher: Delft University Press. c 2003 by A.V. Kononov Printed in The Netherlands Copyright .

(5) Contents 1 Introduction 2 Conventional methods : Basic concepts 2.1 Model and approach . . . . . . . . . . . 2.2 Transfer matrix . . . . . . . . . . . . . . 2.3 Case study . . . . . . . . . . . . . . . . . 2.4 Sources . . . . . . . . . . . . . . . . . . . 2.5 Notes to the methods . . . . . . . . . . .. 1 . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. 9 9 12 22 30 35. 3 Laser ultrasonics 3.1 Introduction . . . . . . . . . . . . . . . . . 3.2 Thermoelastic waves . . . . . . . . . . . . 3.3 Directivity of laser generated ultrasound in 3.4 Preface to the moving sources . . . . . . . 3.5 Vavilov-Cherenkov radiation . . . . . . . . 3.6 Transition radiation . . . . . . . . . . . . . 3.7 Diffraction radiation . . . . . . . . . . . . 3.8 Radiation in non-uniform motion . . . . .. . . . . . . . . solids . . . . . . . . . . . . . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. 47 47 51 57 68 69 73 81 88. 4 Moving photothermal sources 4.1 Problem statement . . . . . . . . . . . . 4.2 Solution of the thermo-elastic problem . 4.3 Periodic sequence of laser pulses . . . . . 4.4 Uniformly moving photo-thermal source 4.5 Oscillatory motion . . . . . . . . . . . . 4.6 Motion in a circle . . . . . . . . . . . . . 4.7 Concluding remarks . . . . . . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. 101 102 108 113 118 121 129 133. . . . . .. . . . . . . .. . . . . .. . . . . .. . . . . . . .. . . . . .. . . . . . . .. . . . . . . .. 5 Introduction to the anisotropic lamina properties 137 5.1 Elastic constants . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 5.2 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . 141.

(6) ii. CONTENTS. 5.3 5.4 5.5. Stiffness matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Dispersion properties . . . . . . . . . . . . . . . . . . . . . . . . . 151 Photo-thermal excitation . . . . . . . . . . . . . . . . . . . . . . . 155. 6 Conclusions A. 159. 163 A.1 Plane waves superposition . . . . . . . . . . . . . . . . . . . . . . 163. B B.1 B.2 B.3 B.4 B.5 B.6. Inverse Fourier transform . . . . . . . . . . . . . . . Kramer-Kronig relations . . . . . . . . . . . . . . . Fundamental solution . . . . . . . . . . . . . . . . . Auxiliary problem . . . . . . . . . . . . . . . . . . . Mindlin plate: fundamental solution . . . . . . . . . Mindlin plate: derivation of the energy flux density. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. 165 165 166 167 168 171 173. C.1 C.2 C.3 C.4 C.5. Transfer matrix :: 3D Material constants . Path integral . . . . Potentials . . . . . . 2D integral . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. 175 175 177 177 180 180. C. D. case . . . . . . . . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. 183 D.1 Stiffness matrix for isotropic medium . . . . . . . . . . . . . . . . 183 D.2 Anisotropic material constants . . . . . . . . . . . . . . . . . . . . 184. Bibliography. 185.

(7) Chapter 1 Introduction Ultrasonic inspection has become one of the most popular nondestructive testing (NDT) technique because of its versatility and easy operation. It can detect internal cracks and inclusion type defects in homogeneous or layered materials, often without much difficulty. Layered materials, which are also called laminated materials, have become widely used in the aerospace industry, naval engineering and many other industries, and thus have attracted considerable interest of researchers in the last two decades. Here we will give a concise overview of the research work conducted in the area of nondestructive testing of laminated materials, where emphasis will be placed on those NDT techniques and theoretical aspects that are closely related to experimental techniques. Some analytical and experimental research in this direction has been done at Delft University of Technology. This work we shall consider in more detail at the end of the overview. At an abstract level, ultrasonic inspection of materials can be formulated as the following problem: when a solid specimen is irradiated by an acoustic source with known properties, an incident wave field interacts with the specimen, which results in reflected and transmitted wave fields. Both fields possess some information about the structure and the integrity of the specimen. If the internal structure and properties of the specimen are known then we are dealing with a forward problem of determination of expressions for the reflected and transmitted fields. However, in ultrasonic inspection one has to deal with the inverse problem of determining acoustic parameters of the specimen using the measured reflected or transmitted (or both) wave fields. From a mathematical point of view such a problem is related to the class of ill-posed problems and its analysis is usually rather complex. When a solution to the inverse problem has been obtained, it has to be compared and verified with the solution to the forward problem. The forward problem for layered materials is known as wave radiation and propagation in layered media. This topic has been studied thoroughly (although it is still developing), which has resulted in a large number of publications and.

(8) 2. 1. Introduction. several monographs. In particular, the cases where the layers are isotropic and homogeneous, and thus the wave field can be separated into the longitudinal and transverse parts, have been well studied, see for example [1–3]. In contrast, in the case of anisotropic layers, which occurs in fiber-reinforced laminates, such separation cannot be carried out and, consequently, the general equations of motion can not be reduced to classical wave equations. This substantially complicates the analysis, because of the large number of normal wave modes. A few methods have been developed that are able to solve this. These methods fall into three main categories: methods based on the classic plate theories [4–6], numerical methods [7–11] and exact methods [12–14]. The first two categories can be considered as approximate approaches, whereas the major advantage of the exact methods is the ability to compute responses at high frequencies. The various experimental techniques used in NDT can be classified into several main categories. The first category encompasses the traditional ultrasonic methods such as A-scan, B-scan and C-scan. In particular, individual pulseechoes are referred to as an A-scan, a juxtaposed collection of A-scans is known as a B-scan, and the pulse-echoes measurements that are performed over two dimensions are called a C-scan [15, 16]. These methods are well known and are characterized by a good resolution and reliability. However, these methods tend to be time-consuming for large specimens and require a liquid couplant between a transducer and a specimen, which may be rather inconvenient in certain cases. This is especially true for the C-scan method, which will be discussed at the end of this section. The second category consists of techniques related to the enhancing of the traditional (or conventional) ultrasonic technique. In this category the following techniques are worth mentioning: the new and fast developing ultrasonic imaging techniques such as adaptive focusing (also known as self-focusing) technique [17– 19] and ultrasonic synthetic aperture focusing technique [20, 21]. The general theory behind these methods has been developed in radio radar-location about forty years ago. During the last two decades these methods, applied to solids, have been studied intensively and the rapid progress in computer technology has made them feasible. In general, the adaptive focusing method can be described as follows. An ordinary ultrasonic focused transducer is rather inflexible in searching for small cracks because its focal point and radiation direction (i.e. directivity) are fixed. However, in ultrasonic NDT, it is often necessary to focus ultrasound in order to detect small cracks in solid media. This can be accomplished by using a selffocusing array of ultrasonic transducers. Each transducer in the array is controlled by a computer, so that the phase of the transducer can be adjusted in order to maintain the directivity of the total array in a desired direction, as illustrated in Figure 1.1..

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(17) . Sample. T. Figure 1.1: Adaptive focusing method. The array of transducers is shown by the column of dashed rectangles. The time delay is proportional to the length of the bold line to the left of each transducer.. The system focuses in an iterative manner. An incident acoustic wave, radiated by one transducer from the array, is scattered by a target T . The scattered wave is sampled, and the phase shift for each transducer is calculated. Then, the total array emits a combined wave field, which consists of an elementary field radiated by each transducer according to its particular time delay. This field is supposed to focus (or to have substantially higher intensity) at some area near the target. This focusing procedure is repeated a few times in order to obtain the optimal focusing. By repeating this procedure, the focal point can be steered through the whole specimen without mechanical motion of the transducer array. This technique is quite effective in the cases where the position of the flaw is known approximately and it is necessary to gather some specific information about it. Another efficient method is the ultrasonic synthetic aperture imaging method. Synthetic aperture focusing refers to a procedure in which the focal properties of a large aperture of a focused transducer (lens or antenna), are synthesized from a series of measurements made using a small-aperture transducer which has been scanned over a large area. The processing required to focus the data is known as synthetic aperture focusing. This technique can be implemented in ultrasonic inspection using a single conventional piezoelectric element (transducer) with a wide ultrasonic beam. The transducer is mechanically scanned over a specimen to form a synthetic aperture as depicted in Figure 1.2. At each transducer position, an ultrasonic impulse is transmitted into the target. Then the transducer is switched to receiving mode and wavefronts reflected by a flaw inside the ultrasonic beam are recorded coherently to form so-called hologram data, i.e. the amplitude and the phase of the signal are stored..

(18) 4. 1. Introduction x Transducer. Scanning line Specimen. α. z. Figure 1.2: Ultrasonic synthetic aperture focusing technique. Next, at the data processing stage, a spatial fast Fourier transform (FFT) algorithm is applied to the stored data in conjunction with special coherent summation of signals for different depths resulting in numerical focusing. With inverse FFT, this gives a high resolution (x, z) image. In fact, the numerical algorithms and the methodology used in synthetic aperture focusing can be very useful in some specific cases when the resolution of the measured data needs to be improved. The third category of NDT experimental techniques consists of methods which are non-traditional in the sense that the physical principles they are based on do not rely explicitly on the phenomenon of ultrasound propagation. In this category, we would like to mention a successful and promising application of Lamb waves in NDT [22, 23]. Lamb waves may be defined as the elastic perturbations propagating in solid plates with free boundaries. Lamb was the first who derived a mathematical formula for plate waves in 1917 [24]. His study showed that a plate could transmit an infinite number and kind of waves (symmetric and anti-symmetric modes). Lamb waves are very attractive for the quick inspection of large structures, since these waves (just like the guided elastic waves) can propagate along a plate-like structure parallel to the boundary surfaces over distances of several meters , depending on the material and geometry of the structure. Moreover, Lamb waves produce stresses throughout the plate thickness and consequently, the entire thickness of the plate is disturbed (explored). This makes it possible to find defects originating from either surface, and also to detect internal defects such as delamination in laminates. If a receiving transducer is positioned at a remote point on the structure, the signal received contains information about the integrity of the line between the transmitting and receiving transducers. Alternatively, echoes returning to the transmitting transducer may be monitored. The waves can be excited using several methods: such as a standard two-transducer scheme, vibrating steel pins or the photo-acoustic method [25] as is illustrated in Figure 1.3..

(19) 5 R. T. R T y x . y x . (a). (b). Figure 1.3: Different approaches for the excitation of Lamb waves in solids. (a) Transmitter (T) and receiver (R) are placed in defocus position, (b) transmitter and receiver are bonded to steel buffer pins.. The practical advantage of Lamb wave imaging (or L-scan) is that in an experimental set it is not necessary to use water as a couplant. Another advantage is related to the fact that in the traditional ultrasonic methods acoustic waves of a high frequency range (1-10MHz) are employed, which are strongly attenuated in the epoxy resins used in the laminates. Since Lamb waves are usually generated in a relatively low frequency band (below 1MHz) L-scan is suitable, as already mentioned, for long range scanning. The application of lasers and related optical techniques in NDT deserve special attention. The considerable advantages of using lasers rather than more conventional methods for the generation and detection of ultrasound are increasingly being emphasized, both in non-destructive evaluation and in the study of material properties [26, 27]. The phenomenon of the excitation of acoustic waves by laser is based on the thermo-expansion property of solids. There are two major advantages of laser ultrasonic techniques over the conventional piezoelectric transducer techniques: 1) the laser methods are non-contact (no surface couplants need be applied) and 2) they are remote, i.e. they may be used in hostile environments, for example, for acoustic molten metal depth sensing [28]. While on the contrary, conventional transducers require contact and coupling fluid, thus introducing transmitting errors and drastically reducing processing speed. The primary disadvantage of the laser techniques is that acoustic signals generated by pulsed laser sources in the thermoelastic regime have relatively low amplitudes. However, this is problematic only for some low-sensitive optical detection systems and can be overcome by using the use of the Fabry-Perot interferometer [29] or by using an array of laser sources [30]. So far, we have not mentioned other techniques used for the material evaluation and testing. This is motivated by our main goal: to describe those techniques which in our opinion can be especially promising in the area of NDT for laminates. Additionally, we would like to note that the methods discussed above must be specially adjusted in order to be suitable for the testing of laminates..

(20) 6. 1. Introduction. As has already been noted work has been done before at Delft University of Technology by Coenen and Vos [31, 32]. Both dissertations are concerned with the problem of nondestructive evaluation of laminated materials. The first author has described the C-scan technique. For this technique, two ultrasonic transducers are typically used. The first transducer generates an incident acoustic wave field, while the second transducer, placed at the other side of the specimen, records the transmitted acoustic field. After the wave field has been recorded, the transducers are moved to a next position. In this way the transducers will scan the whole surface. This principle is illustrated in Figure 1.4. y x . (a). (b). Figure 1.4: (a) Traditional C-scan: The transducers move simultaneously in two dimensions parallel to the specimen; (b) C-scan image; regions with a high attenuation level are dark colored.. For large specimens this process can be time consuming. In order to reduce the virtually total reflection of ultrasound because of the large difference in acoustic impedances of the air and the solid specimen, a coupling media or couplant, such as water is introduced. This implies that either the experimental set has to be submerged in water or a water jet has to be introduced between transducers and the specimen. At the next stage, an attenuation level of the transmitted field is estimated and used further as a characteristic for qualifying the condition of the specimen. The attenuation level is defined as the logarithm of the ratio of the maximum absolute value of the input signal - generated by the first transducer and the maximum absolute value of the signal received by the second transducer. A high attenuation level indicates the presence of damage, or high porosity, or void inclusion. Therefore, regions with a high attenuation level are considered potentially dangerous. The specimen is rejected if the number of weak spots exceeds the critical number. In the output of the method, which follows data processing routine, one can obtain some 2D-image or, rather, a ”shadow” of the imperfections inside the specimen, as is illustrated in Figure 1.4(b). The method is fully developed and known to be reasonably effective. However, a C-scan as a conventional scanning technique has its shortcomings. First, the technique generally does not give much information about the geomet-.

(21) 7. rical and physical parameters of the imperfectness such as its geometry, depth, crack or void inclusion etc. Second, it is not effective in detecting cracks that are normal to the plate surface, because the ultrasonic signal is not reflected by the crack when the signal propagation direction is parallel to the crack surface. Third, this technique can be rather inconvenient for the testing of curved specimens, for which it is quite difficult to maintain simultaneous motion of the transducers. Some of these limitations can possibly be resolved using a so-called advanced C-scan processing method, which has been described by Vos [32]. This method is based on an analysis of the reflected wave field. Thus, a transducer and a receiver now have to be placed at the same side of the specimen, which can be convenient for the cases when a specimen can not be accessed from two opposite sides, see Figure 1.5. 0.8. 0.3. 0.6. 0.2 0.4. 0.1 0.2. y x . (a). 0.5. 1. t 1.5. 2. 2.5. 0.5. 3. 1. 1.5. 2. 2.5. 3. 0. 0 –0.2. –0.1. –0.4. –0.2 –0.6. –0.3 –0.8. –1. –0.4. (b). (c). Figure 1.5: (a) Advanced C-scan method: (b) incident signal, (c) reflected signal. The data processing used in the advanced C-scan differs substantially from the data processing used in a conventional C-scan. In the C-scan method only one number is preserved - the maximum absolute value that occurs in a timewindow in contrast to the advanced method where the complete time trace of the reflected signal is used in the data processing. It is clear that the full time trace of the signal possesses much more detailed information about the region under observation. Usually, the time-dependence of a reflected signal has quite a complex form. This is the result of the multiple reflections of the incident field on the number of layers, superposed with the noise and losses in the system. In order to simulate these processes, a mathematical model of acoustic field propagation in layered media has been developed in [32]. This gave an opportunity to simulate the phenomenon approximately and also a possibility to solve the inverse problem, i.e. to obtain material parameters by studying the parameters of the reflected field..

(22) 8. 1. Introduction. Nevertheless, the model developed in [32] has a number of shortcomings that do not allow the application of this model in the general case. In particular, the following limitations occur: (1) a specimen which originally consists of a number of solid layers is modeled as a structure in which each layer behaves as an ideal fluid or an isotropic acoustic medium; (2) due to the first model limitation only the case of normal wave incidence is described; (3) for the same reason only plane wave incidence is analyzed, which implicitly imposes the assumption that the aperture of a transducer is infinitely large; (4) due to normal incidence the technique fails to detect cracks that are vertical to the surface. The model described by Vos can be considered as a starting point for the present study. In Chapter 2 we consider conventional ultrasonic methods applied for a planar laminated structure immersed in a fluid from a general point view. In Chapter 3 we extend our study onto laser ultrasonics methods. In particular, in the first part of the chapter we consider phenomena of elastic wave excitation due to laser irradiation. Then, we focus on the directivity of the laser generated ultrasound and various methods of its control. In the second part we study the elastic wave generation due to moving sources form a general point of view in the context of the radiation directivity and spectrum. Chapter 4 is devoted to the theoretical study of ultrasound generation in the laminated solids by a moving laser source. The following cases are investigated: a) rectilinear motion of the laser beam; b) oscillatory motion of the beam; c) saw-tooth motion; d) uniform circular motion of the beam. Finally, in Chapter 5 we introduce more realistic laminates with anisotropic elastic properties. Such laminates consist of one or many fiber-reinforced lamina that are bonded together in order to achieve better structural properties and performance over conventional materials..

(23) Chapter 2 Conventional methods : Basic concepts. In this chapter the interaction between an acoustic wave and a solid laminate immersed in water is studied. The focus will be on the most basic and, at the same time, the conceptual problem of the plane wave propagation. More specific, the problem of reflection and transmission of a plane sound wave through a planar laminate which consists of an arbitrary number of parallel solid layers. This problem can be called a ”classical” problem of conventional NDE techniques, i.e. inspection of laminates, which are immersed in fluid, in reflection or transmission mode. The laminate response is described in a frame of the transfer matrix approach which is commonly used for such problems. The method has been enhanced in a way that it becomes applicable to a wide range of frequencies. The derived model is studied numerically with respect to the frequency spectrum and the angle of incidence. Further, the acoustic fields due to distributed acoustic sources are investigated. In the last section, the methods that applied to the problems of wave propagation in layered media are illustrated and discussed. In particular, recursive methods are outlined.. 2.1. Model and approach. Consider the model which is depicted in Figure 2.1. The model consists of two semi-infinite fluid media, denoted by 1 and n + 1, and n − 1 solid elastic layers (laminate), denoted by 2, 3, . . . , n..

(24) 10. 2. Conventional methods : Basic concepts θ n+1. z zn. n+1. hn. z n−1. n. h n−1. z n−2. n−1. h n−2. z n−3. n−2. z2 h2. 2 1. z1. x θ1. Figure 2.1: Model and reference system. Each nth layer in the laminate is characterized by its own set of parameters in which: pρn is the density, λn , µn are the Lam´e constants, p hn is the thickness, an = (λn + 2µn )/ρn , bn = µn /ρn are the velocities of the compressional and shear waves. For further convenience, the total thickness of the entire system of layers is denoted by H, and the origin of the coordinates system is placed at the lower boundary of the first layer. A sound or acoustic wave can be defined as an oscillatory motion of small amplitude in a compressible fluid. The viscosity of the fluid should, in general, be taken into account. This is because the effect of viscosity becomes noticeable at high frequency excitations, especially in the case of interface problems due the strong absorption that occurs when a sound wave is reflected from a solid wall as was shown in [33]. Moreover, using similar wave-reflection techniques the viscosities of some fluids can be measured [34]. A sound wave, at each point of the fluid, causes alternating compression and rarefaction. The fluid velocity v f and the pressure p in a sound wave vary in time and space according the linearized Navier-Stokes equations %. η ∂vf = −∇p + η∆vf + ξ + ∇(∇vf ), ∂t 3 ∂% ∂p ∂% = −%∇vf , = c2 , ∂t ∂t ∂t. (2.1). where η and ξ are the dynamic and volume coefficients of the viscosity, % is the fluid density, and c is the adiabatic velocity of sound [35]. From the statement, it follows that we neglect the local variations of the temperature and the thermodiffusion processes. By means of the following decomposition of the vector v f ,.

(25) 2.1 Model and approach. 11. using the scalar φf and the vector Af potentials, vf = ∇φf + ∇ × Af ,. ∇Af = 0,. the system of Eqs. (2.1) is rewritten in the form ∆φf −. ∂∇φf ∂Af 1 ∂ 2 φf + = 0, = ν∆Af , 2 2 c ∂t ∂t ∂t ∂p with = −%c2 ∆φf , ∂t. (2.2). where ∆ ≡ ∇2 , = (ξ + 4η/3)/(%c2 ), ν = η/%. As can be seen from Eqs. (2.2) in a viscous fluid two types of motion exist: longitudinal and transverse. The transverse motion is strongly damped and can be neglected in some cases. The stress-velocity relations in a viscous fluid are ! f f f ∂v ∂v ∂v 2 ∂v f f i l σik = −p δik + η + k − δik (2.3) + ξδik l ∂xk ∂xi 3 ∂xl ∂xl In elastic medium, which is assumed to be isotropic and homogeneous with the mass density ρ, the particle velocity v(r, t) at any point is described by ρ. ∂2v = µ∆v + (λ + µ)∇(∇v) ∂t2. An arbitrary solution of this equation can be presented in the form v = ∇φ + ∇ × Ψ,. (2.4). where the scalar potential φ(x, y, z, t) and the vector potential Ψ(x, y, z, t) have to satisfy the following wave equations ∆φ −. 1 ∂2φ = 0, a2 ∂t2. ∆Ψ −. 1 ∂2Ψ = 0, b2 ∂t2. (2.5). where a , b are the velocities of the compressional and shear waves respectively, and Ψ is normally chosen to satisfy the constraint condition [36] ∇Ψ = 0. (2.6). At the fluid-solid interface the components of the stress tensor and the velocity vector must be continuous across the boundary. This implies that at the fluidsolid interface on the upper and the lower surface of laminate: z = zn+1 , z = z1 the following relations hold v = ∇φf ,. f σik = σik. (2.7).

(26) 12. 2. Conventional methods : Basic concepts. In the non-viscous fluid, the normal components of the stress tensor the velocity vector must be continuous, but the tangential stresses have to vanish. Thus, the above condition reduces to the requirement that the tangential components must be zero at the boundary ∂φf , ∂z τxz = 0,. (2.8). vz = σzz = −p,. with p = −%. τyz = 0. ∂φf ∂t. Inside the laminate the stress tensor and particle velocity have to be continuous across the boundaries of the layers, so that the following relations have to be satisfied at each layer-to-layer interface v(j) = v(j+1) , j = 2, . . . , n − 1,. (j) (j+1) σzz = σzz ,. (j) (j+1) τxz = τxz ,. (2.9). (j) (j+1) τyz = τyz. At this point, the formal mathematical statement of the problem has been completed. As can be seen in Figure 2.1, the direction of the axes x, y are chosen so that the normal to the front of the incident wave lies in the (x, z) plane (the plane of incidence), and the z-axis is normal to the boundaries of the layers directed toward the media n + 1. In this case (i.e. plane wave excitation), the problem is two-dimensional.. 2.2. Transfer matrix. Let a harmonic plane wave of frequency ω be incident on the upper surface of the laminate at the angle of incidence θn+1 , see Figure 2.1. In general, two waves exist in each medium as a result of multiple reflections at the boundaries of the layers, with the exception of medium 1. One of these waves propagates in the direction of positive z-axis, and the other in the direction of negative z-axis. In medium 1 the only wave exists, which has passed through the entire system of layers and propagates in the negative z direction. Because of specially chosen plane of the incidence (xz-plane), all quantities depend only on the coordinates x and z. Therefore, the potential Ψ can be chosen such that only its y-component differs from zero Ψ = (0 , ψ(x, z) , 0). ⇒. ∇Ψ ≡ 0.

(27) 2.2 Transfer matrix. 13. by doing so, the constraint condition is automatically satisfied. Then, according to Eq. (2.4) vector v has the components vx =. ∂φ ∂ψ − , ∂x ∂z. vy = 0,. vz =. ∂φ ∂ψ + , ∂z ∂x. (2.10). The stress tensor can be expressed in terms of the particle velocity because of the fact that all values are varying harmonically with time, hence. σzz. iv , u ∼ exp(−iωt) ⇒ u = ω iλ ∂vx ∂vz iµ ∂vx ∂vz 2iµ ∂vz = + , τxz = + + ω ∂x ∂z ω ∂z ω ∂z ∂x. According to Eqs. (2.10) and using Eqs. (2.5), the components of the stress tensor can be written as follows ∂2ψ ∂2φ iµ 2 −κ φ − 2 2 + 2 , (2.11) σzz = ω ∂x ∂z∂x ∂2ψ iµ ∂ 2 φ 2 +κ ψ+2 2 τxz = ω ∂z∂x ∂x where κ = ω/b. Let us consider the nth layer with thickness dn . As a result of reflections from the boundaries of the layer, a system of longitudinal and transverse waves exists, which are propagating in the direction of the positive and negative z-axis. The potentials of the longitudinal and the transverse waves in the layer can be chosen of the form (zn ≥ z ≥ zn−1 ) φ = [C + eiαn (z−zn−1 ) + C − eiαn (zn −z) ] ei(κn x−ωt) , ψ = [B + eiβn (z−zn−1 ) + B − eiβn (zn −z) ] e. (2.12). 0 x−ωt) i(κn. where αn = kn cos θn ,. κn = kn sin θn ,. kn = ω/a(n). βn = κn cos γn ,. κn0 = κn sin γn ,. κn = ω/b(n). Let us note that, such a form of general wave solutions inside a layer is important for eliminating the numerical overflow of the exponential terms in the global matrix method (which will be discussed later). However, such a form does not alter the resulting transfer matrix as follows from the uniqueness of the matrix. In accordance with the Snell’s law κn sin γn = kn sin θn = kn+1 sin θn+1 = . . . ⇒ κn = κn0 ..

(28) 14. 2. Conventional methods : Basic concepts. Thus, κ determines the phase velocity along any boundary and the factor exp(i(κn x− ωt)) will be omitted in the intermediate calculations. Eqs. (2.10), and (2.11) are elaborated at the upper boundary of the nth layer, i.e. at z = zn , which results in the following matrix form  (n)      C− vx  (n)  B−   vz      (2.13)  (n)  = Ah  C+  σzz  (n) B+ τxz. where the matrix Ah is defined as follows   −iβn eiBn iκn iβn iκn eiAn  −iαn iκn eiBn  iκn iαn eiAn  Ah =   −iD1 iD2 βn −iD1 eiAn −iD2 βn eiBn  iD1 eiBn iD2 αn iD1 −iD1 αn eiAn. and the following notation has been used zn − zn−1 = hn , αn hn = An , βn hn = Bn and D1 = µ(κ2n − 2κn2 )/ω, D2 = 2µn κn /ω. An analogous matrix can be elaborated at lower side of layer z = zn−1 . Since the components of stress tensor and both components of the velocity have to be continuous at the boundary (n−1) (n−1) z = zn−1 , the same expressions hold for the quantities vz , . . . , τzz at the upper boundary of the (n − 1) layer, which yields  (n−1)      C− vx  (n−1)  B −   vz   (2.14)  (n−1)  = A0   C+  σzz  (n−1) B+ τxz. where A0 is given by .  iκn −iβn iβn eiBn iκn eiAn  −iαn eiAn iαn iκn  iκn eiBn  A0 =   −iD1 eiAn iD2 βn eiBn −iD1 −iD2 βn  iD1 iD2 αn eiAn iD1 eiBn −iD2 αn. The solution of the system (2.14) with respect to C ± , B ± is (index n is omitted)    i(κ2 −2κ 2 ) −iκ iω −iωκ  − (n−1) vx κ2 p 1 2ακ2 p1 2µκ2 p1 2µακ2 p1 C  2 −2κ 2 ) −iωκ −iω  iκ B −   −i(κ   (n−1)  − 2 2 2  κ p2 2µβκ p2 2µκ2 p2   vz  +  =  2βκ p2   −i(κ2 −2κ 2 )  C   −iκ iω iωκ   (n−1)  σzz 2 2 2 2   κ 2ακ 2µκ 2µακ (n−1) B+ i(κ2 −2κ 2 ) −iκ iωκ −iω τxz 2βκ2. κ2. 2µβκ2. 2µκ2.

(29) 2.2 Transfer matrix. 15. where p1 = exp(iAn ), and p2 = exp(iBn ). Substitution of this result into Eq. (2.13) results in the local transfer matrix Sn = Ah A−1 0 ,    (n)    (n−1) v vx x s11 s12 s13 s14  (n−1)   (n)   vz   vz  s21 s22 s23 s24   (2.15)  (n−1)   (n)  =   s31 s32 s33 s34 σzz  σzz  (n−1) (n) s41 s42 s43 s44 τxz τxz. where the elements of matrix S are given by the expressions. s11 = s44 = 2 sin2 γn cos An + cos 2γn cos Bn , s12 = i (tan θn cos 2γn sin An − sin 2γn sin Bn ) , s13 = sin θn (cos Bn − cos An )/cn ρn , s14 = −2i(tan θn sin γn sin An + cos γn sin Bn )/bn ρn , s21 = i(2bn sin γn cos θn sin An /cn − tan γn cos 2γn sin Bn ), s22 = s33 = cos 2γn cos An + 2 sin2 γn cos Bn , s23 = −i(cos θn sin An + tan γn sin θn sin Bn )/cn ρn , s24 = 2 sin γn (cos Bn − cos An )/bn ρn , s31 = 2bn ρn sin γn cos 2γn (cos Bn − cos An ), s32 = −iρn (cn cos2 2γn sin An /cos θn + 4bn sin2 γn cos γn sin Bn ), s34 = 2i(tan θn cos 2γn sin An − sin 2γn sin Bn ),. s41 = −i ρn bn 2 (2cos θn sin2 γn sin An /cn + cos2 2γn sin Bn /(2bn cos γn )), s42 = bn ρn sin γn cos 2γn (cos Bn − cos An ), s43 = i (bn sin γn cos θn sin An /cn − tan γn cos 2γn sin Bn /2), with sin θn = κn /kn = (an /c) sin θn+1 , sin γn = κn /κn = (bn /c) sin θn+1 , The local transfer matrix, as it can be noticed from the definition, maps the state vector through the layer. Accordingly, for the ”zero”-layer (hn = 0) it reduces to the identity matrix: S(h = 0) = I. Moreover by letting hn → ∞ in Ah and A0 together with an assumption that the medium possesses some dissipation, one can obtain the transfer matrices S1 and Sn+1 for elastic half-spaces that relate the state vector at the half-space surface to the unknown wave amplitudes in the half-space, as follows  (1)   (n+1)        C−   0 vx vx  (1)   (n+1)  0 B−    vz      vz    (2.16)  (1)  = S1   ,  (n+1)  = Sn+1   C+  , 0 σzz  σzz  (1) (n+1) B+ 0 τxz τxz.

(30) 16. 2. Conventional methods : Basic concepts. where . iκ1 iβ1  −iα1 iκ 1 S1 =   −iD1 iD2 β1 iD2 α1 iD1 (1). (1). (1). (1). 0 0 0 0.    0 0 iκn+1 −iβn+1 0  0 iαn+1 iκn+1    Sn+1 = 0 0 0 0 0 −iD1 −iD2 βn+1  0 0 0 −iD2 αn+1 iD1. If vx , vz , σzz , τxz , are given at the boundary of separation between media 1 and 2, then by successive application of Eqs. (2.15) in ”ascending” order, one can obtain their values at the boundary between layers n and n + 1, for arbitrary n. Thereupon, the result can be written symbolically in the form    (n)        vx(1) vx  (1)   (n)   vz         vz  (2.17)  (n)  = Sn . . . Sk . . . S2  (1)  σzz  σzz  (1) (n) τxz τxz. where Sk denotes the transition matrix for the kth layer. Multiplication of n − 1 transition matrixes in Eq. (2.17) gives the resultant matrix T that may also be called as global transfer matrix  (n)     (1)  vx vx t11 t12 t13 t14  (n)    (1)  vz   vz  t21 t22 t23 t24   (2.18)  (n)  =   (1)  t31 t32 t33 t34  σzz σzz   (n) (1) t41 t42 t43 t44 τxz τxz. This procedure is similar to a cascaded network. Each layer can be viewed as an four-port device with four acoustic ports, and the multilayer is a cascade of layers that behaves as an equivalent single layer with matrix T, which relates input and output across the entire multilayer. We now study some properties of the matrix T, which can be rewritten in the following form (b) (u) A B v v = (2.19) C D S(u) S(b). where v(u) and v(b) are the velocity vectors at the upper and the bottom surfaces of the laminate, S(u) and S(b) are the mechanical stresses at the surfaces and the matrix T is subdivided into four (2 × 2) sub-matrices A, B, C and D. Each of these matrices has a physical meaning. For instance, matrix A relates transmitted wave velocity amplitudes to the incident wave amplitudes when the surface is free.

(31) 2.2 Transfer matrix. 17. (laminate in vacuum S(u,b) = 0), at the same time matrix C reflects the dispersion properties of the laminate. Since the global matrix T is composed of the local transfer matrices Sn of the individual layers, any property of Sn is also a property of T. The first property is: det Sn = 1 ⇒ det T = 1. For its proof we note that Ah = A0 E with E = diag[eiAn , eiBn , e−iAn , e−iBn ] Further, −1 det Sn = det Ah det A−1 0 = det A0 det E det A0 = 1 det T = det Sn . . . det S2 = 1. →. The second property is: the eigenvalues of the local transfer matrix Sn are [ eiAn , eiBn , e−iAn , e−iBn ] The proof follows from the form of the matrix Sn = A0 E (A0 )−1 . This property has a clear physical meaning: an excitation that passes though a layer acquires a certain phase shift. We return now to the problem and consider the bounding media 1 and n + 1 (1) (n) to be non-viscous fluids. Hence, the tangential stresses τxz and τxz must vanish, so that, from Eqs. (2.18) it can be derived that (1) (1) 0 = t41 vx(1) + t42 vz(1) + t43 σzz + t44 (τxz = 0) ⇒. (2.20). (1) vz(n) = t21 vx(1) + t22 vz(1) + t23 σzz. (2.21). (1) t41 vx(1) + t42 vz(1) + t43 σzz = 0,. and additionally. (n) (1) σzz = t31 vx(1) + t32 vz(1) + t33 σzz (1). (1). (1). Using Eq. (2.20) vx can be expressed in terms of vz and σzz . Upon substitution in Eqs. (2.21) the following system results (1) vz(n) = g11 vx(1) + g12 σzz (n) (1) σzz = g21 vx(1) + g22 σzz ,. where g11 = t22 − t21 t42 /t41 , g21 = t32 − t31 t42 /t41 ,. g12 = t23 − t21 t43 /t41 , g22 = t33 − t31 t43 /t41 ,. (2.22).

(32) 18. 2. Conventional methods : Basic concepts. The expression for the combined acoustic potential of the incident and the reflected acoustic wave in a fluid can be written in the form (the factor exp(iκn+1 x) is omitted): φf(n+1) = Ai e−iαn+1 (z−H) + Ar eiαn+1 (z−H) ,. (2.23). αn+1 = kn+1 cos θn+1 = (ω/cn+1 ) cos θi , where θi is the angle of incidence. On the other side of the laminate a transmitted sound wave exists, which is given by φf1 = At e−i α1 z. with α1 = k1 cos θ1 = (ω/c1 ) cos θt ,. (2.24). and the angles θi and θt are connected by the relation kn+1 sin θi = k1 sin θr where θt is the angle of transmission. From these expressions the normal particle velocity and the normal component of the stress tensor of the laminate can be found using the continuity conditions (2.8) z=0: z=H:. (1) vz(1) = −iα1 At , σzz = −i%1 ω At. vz(n). = −iαn+1 (Ai − Ar ),. (n) σzz. (2.25). = −i%n+1 ω(Ai + Ar ). Substitution of Eq. (2.25) in Eq. (2.22) results in the equations for the reflection and the transmission coefficients R and T : g21 + g22 Z1 − (g11 + g12 Z1 )Zn+1 Ar = , Ai g21 + g22 Z1 + (g11 + g12 Z1 )Zn+1 At 2Z1 T = = , Ai g21 + g22 Z1 + (g11 + g12 Z1 )Zn+1. R=. (2.26). where Z1 = %1 c1 / cos θi , Zn+1 = %n+1 cn+1 / cos θt are the impedances of the fluid on both sides of the layered structure. In the viscous fluid the components of the velocity can, similar to Eqs. (2.10), be written as vxf =. ∂φf ∂ψ f − , ∂x ∂z. vyf = 0,. vzf =. ∂φf ∂ψ f + , ∂z ∂x. (2.27). iω f ψ =0 ν. (2.28). where the potentials are solutions of the equations ∆φf +. ω2 φf = 0, c2 (1 − iω). ∆ψ f +.

(33) 2.2 Transfer matrix. 19. Using these expressions together with Eqs. (2.3), one arrives at the following expressions for the stress ∂ 2ψf ∂ 2 φf iωφf f −2 2 +2 σzz = η − , ν ∂x ∂x∂z 2 f iωψ f ∂ 2ψf ∂ φ f + +2 τxz = η 2 ∂z∂x ν ∂x2 where the term related to the volume viscosity (ξ-term) was neglected. Using the formal analogy between the equations of motion and the exerted stresses (Eqs. (2.28) and Eqs. (2.11)) one can introduce the viscous fluid layer with the transfer matrix obtained by formal substitution √ √ a ⇒ c 1 − iω, b ⇒ −iων, with = 4η/(3%c2 ), ν = η/% Such a very thin fluid layer may model to some extent delamination between layers or void inclusions. An incident wave, which is assumed to be purely longitudinal, is given by. αn+1 = kn+1 cos θi ,. φfi = Ai e−i αn+1 (z−H)+iκn+1 x , (2.29) p with kn+1 = ω/cn+1 1 − iωn+1 , =(k(n+1) ) > 0. So, the total acoustic potential of the incident and the reflected acoustic waves in the fluid region located above of the laminate are (2.30) φf(n+1) = Ai e−iαn+1 (z−H) + Ar eiαn+1 (z−H) eiκn+1 x , 0. 0. f = Br eiαn+1 (z−H)+iκn+1 x , ψ(n+1). p 0 0 where αn+1 = ζ cos θi , κn+1 = ζ sin θi , and ζ = iω/νn+1 . At the same time, in the region located under the laminate the transmitted potentials are defined as φf(1) = At e−iα1 (z−H)+iκ1 x , 0. (2.31). 0. f = Bt e−iα1 (z−H)+iκ1 x , ψ(1). Using these expressions for the potentials one can obtain the following expressions at the lower surfaces of the laminate z=0:. vx(1) = i(κ1 At + α10 Bt ),. vz(1) = −i(α1 At − κ10 Bt ), (1) σzz = η1 (−P1 At + 2α10 κ10 Bt ) , (1) τxz = η1 (2α1 κ1 At + P1 Bt ) ,. (2.32).

(34) 20. 2. Conventional methods : Basic concepts. and at the upper surface of the laminate the following relations hold 0 vx(n) = i(κn+1 (Ai + Ar ) − αn+1 Br ),. z=H:. (2.33). 0 = −i(αn+1 (Ai − Ar ) − κn+1 Br ), 0 0 −ηn+1 Pn+1 (Ai + Ar ) + ηαn+1 κn+1 Br ,. vz(n) (n) σzz =. (n) τxz = ηn+1 (2αn+1 κn+1 (Ai − Ar ) + Pn+1 Br ) ,. 02 where P1,n+1 = iω/ν1,n+1 − 2κ1,n+1 . Next, substitution of Eqs. (2.32) and (2.33) into Eqs. (2.18) with subsequent rearrangement of the terms leads to the system of algebraic equations with respect to the unknown coefficients: R = Ar /Ai - the reflection coefficient of the longitudinal wave, Rt = Br /Ai - the coefficient of transformation of the longitudinal wave into the transverse oscillations, T = At /Ai - the transmission coefficient for longitudinal and Tt = Bt /Ai - the coefficient of transmission of the longitudinal wave into and transverse oscillations.    iκn+1 T     −iαn+1  T S1  + Sn+1   Tt  =   R   −ηn+1 Pn+1  Rt 2ηn+1 αn+1 κn+1    . . . . (2.34). where T is the laminate transition matrix and the matrixes S1 and Sn+1 of the viscous fluid half-spaces are defined as follows, see (2.16), subscripts 1 and n + 1 being omitted, . iκ iα0  iα iκ 0 S1 =   −ηP 2ηα0 κ10 2ηακ ηP. 0 0 0 0.  0 0 , 0 0. Sn+1. . 0 0 = 0 0.  0 −iκ iα0 0 −iα −iκ 0   0 ηP 2ηα0 κ 0  0 2ηακ −ηP. At high frequencies the dissipation of energy in the laminate also has to be taken into account. Consequently, the model must be enhanced by introducing an attenuation or internal friction in the solid layers. There are three well known models of linear viscoelasticity: Maxwell, Voigt and a combination of the first two models - standard linear solid model [37]. By Lord Kelvin it was experimentally shown that first two models are inadequate to the physical reality. Nevertheless, the Voigt model is widely used by researchers because of its simplicity. The model can be enhanced using the following formal operations. According to a the Voigt model [37] the Lam´e constants are replaced by the differential operators e λ → λ + λ∂/∂t,. µ→µ+µ e∂/∂t,. (2.35).

(35) 2.2 Transfer matrix. 21. e µ in which λ, e are the viscous parameters. For harmonical processes, using Eqs. (2.5), this implies that the compressional and shear wave velocities a, b used in the transition matrix Eq. (2.15) have to be replaced according to following definitions a → a(ω) = a. p. 1 − i δa ω,. b → b(ω) = b. p. 1 − i δb ω,. (2.36). e + 2e where δa = (λ µ)/(λ + 2µ), δb = µ e/µ. We mote that the Voigt model is acceptable only at relatively low frequencies. In the case of a standard linear solid the following substitution is valid a → a(ω) = a. r. i ω τa 1− , 1 + i ω τa. b → b(ω) = b. r. 1−. i ω τb , 1 + i ω τb. (2.37). Before running the simulations the following must be outlined. The elements of the local transfer matrix given by Eqs. (2.15) are prone to numerical instability because of the exponential growth of the trigonometric functions (sin cos)(A n Bn ) for complex values of the arguments. A solution to this problem can be found by noting that each element of the matrix can be multiplied by some number without affecting the result, e.g. the scattering coefficients given by Eqs. (2.34). Thus, each element has been divided by cos An , resulting in a new equivalent transfer matrix s11 = s44 = 2 sin2 γn + cos 2γn F1 , s12 = i (tan θn cos 2γn tan An − sin 2γn F2 ) , s13 = sin θn (F1 − 1)/cn ρn , s14 = −2i(tan θn sin γn tan An + cos γn F2 )/bn ρn , s21 = i(2bn sin γn cos θn tan An /cn − tan γn cos 2γn F2 ), s22 = s33 = cos 2γn + 2 sin2 γn F1 , s23 = −i(cos θn tan An + tan γn sin θn F2 )/cn ρn , s24 = 2 sin γn (F1 − 1)/bn ρn , s31 = 2bn ρn sin γn cos 2γn (F1 − 1), s32 = −iρn (cn cos2 2γn tan An /cos θn + 4bn sin2 γn cos γn F2 ), s34 = 2i(tan θn cos 2γn tan An − sin 2γn F2 ),. (2.38). s41 = −iρn bn 2 (2cos θn sin2 γn tan An /cn + cos2 2γn F1 /2bn cos γn ), s42 = bn ρn sin γn cos 2γn (F1 − 1), s43 = i (bn sin γn cos θn tan An /cn − tan γn cos 2γn F2 /2), where the following functions have to be coded in the numerically effective way.

(36) 22. 2. Conventional methods : Basic concepts. (=(αn , βn )) > 0 1 + ei2Bn / 1 + ei2An F2 = sin Bn /cos An = i eihn (αn −βn ) 1 − ei2Bn / 1 + ei2An . F1 = cos Bn /cos An = eihn (αn −βn ). . Moreover, the transmission coefficient T has to be adjusted with respect to the new transfer matrix (the reflection coefficient R remains intact): T = C n . . . C 1 T0. with Ci = 1/cos An = 2 eiAn / 1 + ei2An ,. (2.39). where T0 is computed using Eqs. (2.26) or (2.34). The elements of the matrix do not grow when the arguments become complex valued. As for the implementation method, the so-called Kahan’s summation algorithm [38] can be used in the matrix multiplication routines in order to reduce the roundoff error. With these enhancements, the transfer matrix becomes numerically stable and applicable to a wide range of frequencies.. 2.3. Case study. In this section, the model developed in the previous section is studied numerically in order to describe quantitatively and qualitatively the response of a layered structure (laminate) with respect to the acoustic plane wave excitation with the different parameters. The structure of the laminate is taken similar to the structure of Glare, a material which consists of alternating layers of aluminum and so-called prepreg layers (usually, fiber-reinforced resin or epoxy) as illustrated in Figure 2.2.. Figure 2.2: Layered structures of Glare material used for simulations (1,3,5 layers). Aluminum layers are colored dark, prepreg layers are colored light according to the acoustic impedance of these materials.. The physical properties of the laminate materials and of the coupling material (water) together with the ”defect” material (air) are shown in Table 2.1..

(37) 2.3 Case study. 23. Material. kg Density ( m 3). Long. vel.( m s). Shear vel. m s. Layer thick. (m.). water aluminum prepreg air. 1000. 2790. 1400. 1.3. 1480. 6380. 2730. 330.. 0. 3130. 1300. 0.. ∞ 4. × 10−4 3. × 10−4 1. × 10−7. Table 2.1: Material properties In Figures 2.3 the intensity reflection R = |R|2 and intensity transmission T = |T |2 coefficients of an acoustic plane wave are presented for the case of a single aluminum plate submerged in water. 1. 0.8. 0.8. 0.6. 0.6. R,T. R,T. 1. 0.4. 0.4. 0.2. 0.2. 0. 0. 5e+07. 1e+08. 2e+08. 1.5e+08. 0. 0. 0.0005. 0.001. 0.0015. 0.002. 0.0025. 0.003. λ. ω. (a). (b). Figure 2.3: (a) Intensity reflection (bold line) and transmission (dashed line) coefficients of an aluminum plate for 0.0 angle of incidence, 0.4mm thick, as a function of frequency; and (b) as a function of longitudinal wave length[m] in aluminum.. The graphs have a clear physical explanation. Since the case of normal incidence is considered shear waves are not excited in the plate. The energy is almost completely transmitted through the plate for the frequencies which are below the approximate range of (<1MHz). On the contrary, for the frequencies which are above this range (>1MHz) most part of the energy is reflected by the plate except for the frequencies that are close to the minima. The minima are given, as follows from the analysis of Eqs. (2.26), by ∗ ωm =m. πa d. ⇒. λa =. 2d , m. m = 1, 2 . . .. (2.40). where λa is a wavelength of the longitudinal wave in aluminum. This relation corresponds to the fact that the layer of the half-wave thickness has no effect on.

(38) 24. 2. Conventional methods : Basic concepts. the incident wave. This is indicated in the Figure 2.3(b), where the first minimum of the reflection coefficient and, consequently, the maximum of the transmission coefficient are located at λa = 0.8mm. Further, the reflection coefficient will be plotted only. The transition coefficient T can be found using an approximate relation between coefficients: R + T ≈ 1. When the incidence is not normal, the picture becomes somewhat more complicated, as is shown in Figure 2.4. 0.8. 0.8. 0.6. 0.6. R. 1. R. 1. 0.4. 0.4. 0.2. 0.2. 0. 0. 2e+07. 4e+07. 6e+07. 8e+07. ω. 1e+08. 0. 0. 2e+07. 4e+07. 6e+07. 8e+07. 1e+08. ω. (a). (b). Figure 2.4: Reflection coefficient for (a) 0.0 degrees angle of incidence (dashed line) and 5.0 degrees angle of incidence (bold line); (b) 10.0 - dashed line and 20.0 - bold line.. As is seen in the Figure 2.4 new minima have appeared. These minima are related to the shear wave excitation in the plate. When increasing the angle of incidence, and in particular, if the angle of incidence becomes larger than the angle of total internal reflection for longitudinal waves in the aluminum: θa∗ = arcsin(c/a) ≈ 13.42o , the picture become similar to Figure 2.3(a), see Figure 2.4(b). Now, all the minima in the frequency range are due to the shear wave excitation, and their locations are given by ∗ ωm =m. πb d cos γ with. 2d cos γ λb = , m = 1, 2 . . . m q cos γ = 1 − (b/c)2 sin2 θi , ⇒. (2.41). where λb is a wavelength of the shear wave in the aluminum. When the structure consists of many layers, the physical explanation of the results becomes more complicated, especially in the case of non-normal incidence, see Figure 2.5..

(39) 2.3 Case study. 25 1. 0.8. 0.8. 0.6. 0.6. R. R. 1. 0.4. 0.4. 0.2. 0.2. 0. 0. 2e+07. 4e+07. 8e+07. 6e+07. 0. 1e+08. 0. 2e+07. 4e+07. 8e+07. 6e+07. ω. 1e+08. ω. (a). (b). Figure 2.5: Reflection coefficient: (a) of a 3-layer structure (dashed line) and a 5layer structure (bold line) for 0.0 degrees angle of incidence ; (b) a 5-layer structure 10.0 degrees angle of incidence.. An increase of the number of layers leads, as confirmed by the plots, to an increase of the number of minima. The locations of these minima are close to the minima which were originated in the more simpler structure consisting of 1 or 3 layers. Further analysis shows that with increasing the number of the layers these sets of the minima will be transformed into quasi-continuous frequency bands at which the structure is acoustically transparent. 1. 1 1.0 MHz 0.2 MHz. 1.0 MHz 0.2 MHz. 0.8. 0.8. 0.6. R. R. 0.6. 0.4. 0.4. 0.2. 0. 0.2. 0. 20. 40. 60. 80. 0. 0. Angle of incidence. (a). 20. 40. 60. 80. Angle of incidence. (b). Figure 2.6: Reflection coefficient versus angle of incidence for different frequencies: (a) of an aluminum plate, (b) of a 3-layer structure.. Another interesting phenomenon can be observed from the study of the dependence of the reflection coefficient on the angle of plane wave incidence, see.

(40) 26. 2. Conventional methods : Basic concepts. Figure 2.6. From Figures 2.6, 2.7 it can be noted that for some angles of incidence which are greater than the angle of total internal reflection of the shear waves in aluminum: θb∗ = arcsin(c/b) ≈ 28.22o , the energy reflection coefficient has a deep minimum even in the low frequency range. Subsequent plots show that the location of these minima is structure dependent: the minimum shifts to smaller angles with an increase of the number of layers (i.e. the total thickness).. 1. 1 1.0 MHz 0.2 MHz. 1.0 MHz 0.2 MHz. 0.8. 0.6. 0.6. R. R. 0.8. 0.4. 0.4. 0.2. 0.2. 0. 0. 20. 40. 60. 80. 0. 0. Angle of incidence. (a). 20. 40. 60. 80. Angle of incidence. (b). Figure 2.7: Reflection coefficient versus angle of incidence for different frequencies: (a) of a 5-layer structure, (b) of a 7-layer structure.. Furthermore, it can be concluded that the plots corresponding to the acoustic wave excitation with the low frequency (0.2MHz) are the most representative, since they are qualitatively similar to each other, and only differ with respect to the coordinate of the minimum. These minima correspond to the resonance excitation of the waves in the laminate which are similar to the plate Lamb waves. In this case the phase velocity of the incident wave along the plate, vph = c/ sin θi , is close to the phase velocity of the waves propagating along the plate. As can be understood from the Figures, for an increasing the number of layers the velocity of the Lamb waves is increasing. The sensitivity of the velocity of the L-waves to the change in the structure parameters, as illustrated in Figure 2.8, is repeatedly used in NDI..

(41) 2.3 Case study. 27. 1. 0.2. 0.40 mm 0.41 mm. 0.40 mm 0.41 mm. 0.8. R. R. 0.6 0.1. 0.4. 0.2. 0. 0. 20. 40. 0 54. 80. 60. 55. 56. Angle of incidence. 57. 58. 59. 60. Angle of incidence. (a). (b). Figure 2.8: Sensitivity of the reflection coefficient to variations of the aluminum plate thickness at 1.0 MHz. 1. 1 25 MHz 50 MHz. 25 MHz 50 MHz. 0.6. 0.6. R. 0.8. R. 0.8. 0.4. 0.4. 0.2. 0.2. 0. 0. 10. 20. 30. Angle of incidence. (a). 40. 50. 0. 0. 10. 20. 30. 40. 50. Angle of incidence. (b). Figure 2.9: Reflection coefficient for a high frequency range: (a) of an aluminum plate, (b) of a 5-layer structure. In a high frequency range, see Figure 2.9, the reflection coefficient varies substantially for the incidence angles that belong to the interval at which the upper aluminum layer is acoustically transparent, namely [0o . . . θb∗ ≈ 30o ]. Beyond this interval the waves are damped completely in the upper layer. Therefore, the dynamics is mostly determined by the properties of this layer. As can be noticed from the Figure 2.9, the angle coordinate of the first minimum from the left approaches the angle of total internal reflection of shear waves in aluminum θb∗ . This fact can be used for the experimental determination of the shear wave velocity of some specimen..

(42) 28. 2. Conventional methods : Basic concepts. Next, we study the effect of delamination (or void inclusion) between layers on the reflected field. We consider the following configuration: (1) a laminate consists of three layers - aluminum-prepreg-aluminum (a-p-a) and (2) a laminate with defect - aluminum-void-prepreg-aluminum (a-v-p-a), see Figure 2.10 1. 1. a-p-a a-v-p-a. a-p-a a-v-p-a 0.8. 0.6. 0.6. R. R. 0.8. 0.4. 0.4. 0.2. 0.2. 0. 0. 20. 40. 60. 0. 80. 0. 20. 40. 60. 80. Angle of incidence. Angle of incidence. (a). (b). Figure 2.10: Reflection coefficient versus angle of incidence at 0.5 MHz for a laminate with inclusion of void layer of thickness : (a) 0.001 mm, (b) 0.0001 mm.. In Figure 2.10(a) the thickness of the void layer is such that almost all energy is reflected back due to sharp difference in the layers acoustic impedances. Such a high contrast in the reflected fields intensities is the key principle of the void detection used in the C-scan technique. Moreover, at higher frequencies the reflected field is sensitive to the depth of the defect as confirmed in Figure 2.11. 1. 1. a-p-a a-v-p-a a-p-v-a. 0.8. a-v-p-a a-p-v-a. 0.9. 0.8. R. R. 0.6. 0.7. 0.4. 0.6. 0.2. 0. 0.5. 0. 20. 40. Angle of incidence. (a). 60. 0.4. 0. 20. 40. 60. 80. Angle of incidence. (b). Figure 2.11: Reflection coefficient versus angle of incidence for laminate with inclusion of void layer of thickness 0.001 mm: (a) 1.5 MHz, (b) 4.0 MHz..

(43) 2.3 Case study. 29. As shown in the Figure 2.11(a), the low frequency field is not sensitive to the depth of the defect, since the wave-length is comparable with the total thickness of the laminate. In contrast, see the Figure 2.11(a), the acoustic field of higher frequency already ”feels” the inner structure of the specimen. Such an effect can be considered as the basis for an NDI technique that is capable to detect the location of a defect. In the previous cases it was assumed that the fluid depth under the laminate is infinite. In practical situations the underlying fluid layer has a finite depth. Moreover, the fluid at the top side also has finite depth. However, the reflections from the upper fluid boundary can easily be elaborated by using certain measurement techniques or by special data processing. Therefore, the influence of the finite depth of the underlying fluid will be studied next. Consider a fluid layer of the thickness hf : −hf ≤ z ≤ 0. At the bottom of the fluid, the normal velocity component must be zero, i.e. vz = ∂φf1 /∂z = 0 at z = −hf . A general solution that is satisfying these conditions is φf = At cos [α1 (z + hf )]. According to this, the scattering coefficients yield g21 p1 + g22 Z1 − (g11 p1 + g12 Z1 )Z1 Ar = , R= Ai g21 p1 + g22 Z1 + (g11 p1 + g12 Z1 )Z1 At 2Z1 T = , = Ai cos(α1 hf ) (g21 p1 + g22 Z1 + (g11 p1 + g12 Z1 )Z1 ). (2.42). where p1 = −i tan(α1 hf ). The depth hf can be adjusted in the way that a particular laminate sample has its own ”signature”, see Figure 2.12. 1. 1. 5-layers 7-layers. 0.98. 0.95 0.96. R. R. 0.9 0.94. 0.85 0.92. 0.8. 0.9. 0. 20. 40. 60. Angle of incidence. (a). 80. 0.88. 0. 20. 40. 60. 80. Angle of incidence. (b). Figure 2.12: Reflection coefficient versus angle of incidence for laminate(s) lying over fluid layer for: (a) 5-layers laminate, 1.0 MHz, 0.1 m. fluid depth (b); 0.5 MHz, 0.005 m. fluid depth..