Elastic Waves in Composite Media and Structures: With Applications to Ultrasonic Nondestructive Evaluation 9781420053395, 1420053396, 978-1-4200-5338-8

Table of contents :
Content: Introduction. Waves in Composite Media. Guided Waves in Laminated Plates. Guided Waves in Laminated Cylinders. Coupled Fields. Scattering of Waves in Laminated Plates and Cylinders.

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Elastic Waves in Composite Media and Structures With Applications to Ultrasonic Nondestructive Evaluation

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Mechanical Engineering Series Frank Kreith & Roop Mahajan - Series Editors

Published Titles Computer Techniques in Vibration Clarence W. de Silva Distributed Generation: The Power Paradigm for the New Millennium Anne-Marie Borbely & Jan F. Kreider Elastic Waves in Composite Media and Structures: With Applications to Ultrasonic Nondestructive Evaluation Subhendu K. Datta and Arvind H. Shah Elastoplasticity Theor y Vlado A. Lubarda Energy Audit of Building Systems: An Engineering Approach Moncef Krarti Energy Converstion D. Yogi Goswami and Frank Kreith Energy Management and Conser vation Handbook Frank Kreith and D. Yogi Goswami Finite Element Method Using MATLAB, 2nd Edition Young W. Kwon & Hyochoong Bang Fluid Power Circuits and Controls: Fundamentals and Applications John S. Cundiff Fundamentals of Environmental Discharge Modeling Lorin R. Davis Handbook of Energy Efficiency and Renewable Energy Frank Kreith and D. Yogi Goswami Heat Transfer in Single and Multiphase Systems Greg F. Naterer Introductor y Finite Element Method Chandrakant S. Desai & Tribikram Kundu Intelligent Transportation Systems: New Principles and Architectures Sumit Ghosh & Tony Lee Machine Elements: Life and Design Boris M. Klebanov, David M. Barlam, and Frederic E. Nystrom Mathematical & Physical Modeling of Materials Processing Operations Olusegun Johnson Ilegbusi, Manabu Iguchi & Walter E. Wahnsiedler Mechanics of Composite Materials Autar K. Kaw Mechanics of Fatigue Vladimir V. Bolotin Mechanism Design: Enumeration of Kinematic Structures According to Function Lung-Wen Tsai Mechatronic Systems: Devices, Design, Control, Operation and Monitoring Clarence W. de Silva MEMS: Applications Mohamed Gad-el-Hak © 2009 by Taylor & Francis Group, LLC 53388_C000.indd 2

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MEMS: Design and Fabrication Mohamed Gad-el-Hak The MEMS Handbook, Second Edition Mohamed Gad-el-Hak MEMS: Introduction and Fundamentals Mohamed Gad-el-Hak Multiphase Flow Handbook Clayton T. Crowe Nanotechnology: Understanding Small Systems Ben Rogers, Sumita Pennathur, and Jesse Adams Optomechatronics: Fusion of Optical and Mechatronic Engineering Hyungsuck Cho Practical Inverse Analysis in Engineering David M. Trujillo & Henry R. Busby Pressure Vessels: Design and Practice Somnath Chattopadhyay Principles of Solid Mechanics Rowland Richards, Jr. Thermodynamics for Engineers Kau-Fui Wong Vibration Damping, Control, and Design Clarence W. de Silva Vibration and Shock Handbook Clarence W. de Silva Viscoelastic Solids Roderic S. Lakes

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Elastic Waves in Composite Media and Structures With Applications to Ultrasonic Nondestructive Evaluation

Subhendu K. Datta Arvind H. Shah

Boca Raton London New York

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CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2009 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-13: 978-1-4200-5338-8 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Datta, S. K. (Subhendu K.) Elastic waves in composite media and structures : with applications to ultrasonic nondestructive evaluation / Subhendu K. Datta and Arvind H. Shah. p. cm. -- (Mechanical engineering series) Includes bibliographical references and index. ISBN 978-1-4200-5338-8 (alk. paper) 1. Fibrous composites--Testing. 2. Ultrasonic testing. 3. Elastic waves. I. Shah, Arvind H. II. Title. III. Series. TA418.9.C6D2886 2009 620.1’187--dc22

2008040950

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Dedication Dedicated to our wives Bishakha and Ranjan, and to our children Kinshuk, Ketki, and Seema.

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Table of Contents

Preface...................................................................................................................... xi

1 

Introduction

2 

Fundamentals of Elastic Waves in Anisotropic Media

1.1 1.2 2.1

2.2 2.3

Historical Background...........................................................................................1 Scope of the Book...................................................................................................8

Waves in Homogeneous Elastic Media: Formulation of Field Equations...............................................................................................................11 Plane Waves in a Homogeneous Anisotropic Medium................................. 20 Numerical Results and Discussion....................................................................33

3 

Periodic Layered Media

4 

Guided Waves in Fiber-Reinforced Composite Plates

3.1 3.2 3.3 3.4

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11

Introduction..........................................................................................................37 Description of the Problem................................................................................ 38 Numerical Results................................................................................................51 Remarks................................................................................................................. 56

Introduction......................................................................................................... 64 Governing Equations.......................................................................................... 64 Numerical Results................................................................................................76 Application to Materials Characterization...................................................... 85 Thin Layers........................................................................................................... 88 Guided Waves in Plates with Thin Coating and Interface Layers................ 98 Transient Response due to a Concentrated Source of Excitation...............114 Laminated Plate with Interface Layers........................................................... 142 Remarks............................................................................................................... 146 Laser-Generated Thermoelastic Waves...........................................................147 Results for Thermoelastic Dispersion and Laser-Generated Waves.......... 157 ix

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5 

Wave Propagation in Composite Cylinders

6 

Scattering of Guided Waves in Plates and Cylinders

5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8

Introduction........................................................................................................165 Governing Equations........................................................................................ 166 Analytical Solution for Transversely Isotropic Composite Cylinder........ 168 Stiffness Method I...............................................................................................174 Stiffness Method II.............................................................................................179 Numerical Results—Circular Cylinder..........................................................181 Guided Waves in a Cylinder of Arbitrary Cross Section............................ 186 Numerical Results—Cylinder with Rectangular and Trapezoidal Cross Sections.................................................................................................... 195 5.9 Harmonic Response of a Composite Circular Cylinder due to a Point Force.......................................................................................................... 205 5.10 Forced Motion of Finite-Width Plate..............................................................213 6.1 6.2 6.3

Introduction....................................................................................................... 223 Scattering in a Plate........................................................................................... 224 Scattering in a Pipe............................................................................................ 257

Appendix A: Computer Programs

A.1 Programs............................................................................................................. 283 A.2 Executing Programs.......................................................................................... 286

References ......................................................................................................... 289

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Preface

This book is an outgrowth of over 30 years of research and teaching done by our groups at the Department of Mechanical Engineering at the University of Colorado at Boulder and the Department of Civil Engineering at the University of Manitoba at Winnipeg. The book deals with the fundamentals of waves in fiber-reinforced laminated plates and shells. Composite materials have mechanical and other physical properties that are often superior to traditional monolithic metallic or polymeric materials for use in civil, mechanical, and aerospace structures. They are also generally lighter and more economical to use. They can be tailored to the needs of particular structural applications. For these reasons, composite materials have found widespread use in many structural applications. These materials have complex microstructures and are, in general, anisotropic and inhomogeneous. Thus, they present considerable challenges for characterization of their mechanical properties and prediction of dynamic response. Understanding of elastic wave propagation characteristics in such materials and structures is essential for prediction and interpretation of their dynamic response, and for ultrasonic nondestructive evaluation of mechanical properties of, and defects in, these structures. The goal of this book is to present analytical and numerical techniques that have been found to be effective in solving a wide class of problems involving wave propagation and scattering by defects in anisotropic layered plates and shells. It contains a systematic treatment of elastic waves in unbounded and bounded layered media composed of fiber-reinforced materials using analytical and numerical tools. In addition, extensive numerical results and key executable computer programs are included in the accompanying CD. While an introductory knowledge of elastic waves is desirable when reading this book, essential concepts and equations governing elastic waves in unbounded and bounded anisotropic media are discussed in sufficient detail so that the readers should be able to follow and use the material presented. They will get a good understanding of the characteristics of ultrasonic wave propagation in composite structures. Also, the book will help practitioners simulate and interpret measured dynamic data. The goal here is to provide the reader with theoretical tools to perform tasks such as ultrasonic nondestructive material characterization, nondestructive testing, and impact response of aircraft components, pipelines, coatings, interfaces, and other layered structures. It is believed xi

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that the book will be found useful by the beginning graduate students, experienced researchers, and practitioners as a source for rigorous mathematical models of dispersive wave propagation in laminated structures. In addition, the reader will be able to use the computer programs to solve many of the problems that arise in practical applications. Elastic waves propagating in composite media and structures are significantly influenced by the elastic properties and geometry of the reinforcing phase (fiber, particle, coating, or interface/interphase), layers or laminae, defects (cracks, pores, cavities), and overall geometry. Thus, they provide an effective means of nondestructive characterization of these properties. For this purpose, it is necessary to understand clearly the salient features of dispersive guided modes and how they are modified by defects, inhomogeneities, and boundaries. These features are carefully examined in this book. Topics covered include: waves in unbounded periodically layered media and modeling of effective quasistatic elastic properties; guided waves in laminated plates and shells; Green’s functions and transient response of plates and shells; thermal effects on guided waves in plates; scattering of waves by cracks, delaminations, and joints; and reflection from the edges of plates and cylinders. Treatments of these topics presented here are believed to be sufficiently complete, and extensive references are provided so that the book can be used by students, researchers, and practitioners to solve problems or to use it as a reference. The authors have benefited immensely from their interactions over the years with many graduate students and professional colleagues. In particular, they would like to acknowledge the help they have received from their former and present graduate students Drs. O. M. Mukdadi, H. Al-Qahtani, T. H. Ju, S. W. Liu, R. L. Bratton, W. M. Karunasena, N. Rattanawangcharoen, H. Bai, J. Zhu, W. Zhuang, and Mr. J. K. T. Yeo. They are especially indebted to Dr. M. J. Frye for his untiring work in drawing all of the figures appearing in this book, to Mr. Faisal Shibley for making the computer programs interactive, to Dr. H. Bai for his unhesitating help with preparing the programs, and to Mr. D. Stoyko for help in the preparation of the manuals and the programs. The authors would like to express special thanks to the Natural Science and Engineering Research Council of Canada (NSERC) and the Manitoba Hydro for their continued support of much of the work that appears in this book. Support from the National Science Foundation, the Office of the Basic Energy Sciences (DOE), and the Office of Naval Research is also gratefully acknowledged.

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1 Introduction 1.1 Historical Background ...............................................................1 Elastic Waves in Layered Media with Isotropic Layers • Waves in Fiber-Reinforced Media • Waves in Anisotropic Layered Media • Guided Waves in Laminated Plates and Circular Cylindrical Shells • Cylinders of Arbitrary Cross Section • Particle-Reinforced Composites 1.2 Scope of the Book ........................................................................8

1.1 Historical Background 1.1.1 Elastic Waves in Layered Media with Isotropic Layers Wave propagation in elastic composite media has been studied extensively since the 1950s. Early investigations dealt with plane-layered composite media and were motivated by seismic exploration. Postma (1955), White and Angona (1955), and Rytov (1956) derived expressions for effective wave speeds of plane waves propagating in a periodically laminated medium. They assumed that wavelengths of such waves are much longer than the thicknesses of the layers. In this limit, the wave speeds are independent of frequency, and the layered medium can be modeled as a homogeneous transversely isotropic medium with the symmetry axis parallel to the direction of layering. Expressions for the effective static elastic moduli of the anisotropic medium were derived in these papers. Carcione et al. (1991) carried out one-dimensional and two-dimensional numerical simulations of longitudinal and shear wave propagation in periodically bilayered elastic media, and they showed that, for sufficiently long wavelengths, dispersion was negligible and that the effective homogeneous transversely isotropic approximation was valid. When the wavelengths became comparable with the layer thicknesses, strong dispersion (frequency-dependent wave speeds) was observed. This would be expected because, at short wavelengths, refraction and reflection of waves at the interfaces of the layers give rise to dispersive behavior. Various approximate theories that account for dispersion of elastic waves in a periodically laminated medium were proposed in the 1960s and 1970s. Among these were the 1

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effective stiffness theory proposed by Sun et al. (1968b) and Achenbach et al. (1968) (see also Achenbach and Sun 1972), the mixture theory used by Bedford and Stern (1971, 1972) and McNiven and Mengi (1979), and the interacting continuum theory (Hegemier 1972; Hegemier and Bache 1974; and Nayfeh and Gurtman 1974). In addition to the approximate theories mentioned above, exact solutions for harmonic wave propagation in a laminated medium have been presented by Sun et al. (1968 a) and Lee and Yang (1973) for antiplane strain problems, and by Sve (1971) for the plane strain case. Later, Delph et al. (1978, 1979, 1980) presented exact solutions for harmonic wave propagation using Floquet’s theory. In all of these papers mentioned here, each layer material was considered to be linearly elastic, homogeneous, and isotropic. During the last half century, there has been a continued interest in the dynamic behavior of layered anisotropic medium due to the increasing use of advanced composite materials in aerospace, naval, and civil structures. Such structural components are typically made of fibers of high strength and stiffness-reinforcing plastics, metals, or ceramics. The morphology of these materials makes their mechanical response much more complicated than the (usually) isotropic homogeneous matrix materials. In general, these are anisotropic and inhomogeneous. Furthermore, the mechanical properties are strongly dependent upon the properties of the interfaces between the constituent phases. Composite structural components are usually made up of a stack of layers (plies or laminae), i.e., they are laminated. This adds another layer of complexity to their dynamic behavior.

1.1.2 Waves in Fiber-Reinforced Media To model ultrasonic wave propagation in a composite laminate, it is necessary to model the dynamic properties of a lamina that is made of a matrix reinforced by fibers. Wave propagation in a homogeneous elastic medium reinforced by aligned continuous fibers has been investigated by Achenbach (1976) and Hlavacek (1975) when the fiber distribution is periodic. Other studies on periodic distribution of fibers include Hegemier et al. (1973), Nemat-Nasser and Yamada (1981a), and Nayfeh (1995). Wave propagation in the presence of a random distribution of aligned fibers has been studied by Bose and Mal (1973, 1974), Datta (1975), Datta and Ledbetter (1983), Datta et al. (1984), Willis (1983), Varadan et al. (1978, 1986), and Beltzer and Brauner (1985), among others. When the wavelength of the propagating plane waves is much longer than the fiber diameter, effective wave speeds are found to lead to the static effective elastic constants. Static effective elastic properties of aligned, continuous, fiber-reinforced composites have been studied in great detail since the early 1960s. Among the pioneering works are those by Hashin and Rosen (1964), Hill (1964), Hashin (1979), and Christensen and Lo (1979). References to many other works on static effective thermoelastic properties can be found in the NASA report by Hashin (1972) and in the monograph by Christensen (1991). There are now well-established theories for the modeling of effective anisotropic thermoelastic properties of aligned fiber-reinforced composite media. These anisotropic properties are used to then model the dynamic behavior of laminated media.

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Introduction

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1.1.3 Waves in Anisotropic Layered Media Wave propagation in an infinitely layered medium when each layer (lamina) is anisotropic has been studied by Hegemier (1972), Hegemier et al. (1973), and Hegemier and Nayfeh (1973) using mixed spatial and asymptotic expansions as well as a mixture theory. Exact dispersion relations for propagating waves in periodic anisotropic layered media have been studied by Nemat-Nasser and Yamada (1981b), Yamada and Nemat-Nasser (1981), Shah and Datta (1982), Braga and Herrmann (1992), Nayfeh (1995), and Datta (2000), who have used the concept of Floquet waves. Approximate solutions to the Floquet waves using variational principles have also been presented in Nemat-Nasser (1972) and Nemat-Nasser and Minagawa (1975) (see also Minagawa and Nemat-Nasser 1977).

1.1.4 Guided Waves in Laminated Plates and Circular Cylindrical Shells Dispersive behavior of guided waves in laminated plates and shells has been studied extensively in the last 20 years or so by many investigators. Dispersive modal propagation behavior is strongly influenced by the anisotropic properties of each lamina and the stacking sequence used. Thus, this can be exploited to determine material properties of each lamina. Early studies of the propagation of free guided waves (Lamb waves) in an anisotropic plate were reported by Ekstein (1945), Newman and Mindlin (1957), and Kaul and Mindlin (1962a, 1962b). Later, Solie and Auld (1973) gave a detailed description of guided waves in cubic plates, and Li and Thompson (1990) studied the dispersion characteristics of orthotropic plates. Nayfeh and Chimenti (1989) developed the equations for a generally anisotropic plate. Extensive numerical results presented in these papers showed many characteristic features that arise due to anisotropy and that can be exploited to determine the anisotropic properties of the material composition of the plate. Guided waves in plates composed of uniaxial or multidirectional fiber-reinforced laminates show very complex behavior because of the complicated reflection and refraction phenomena arising at the interfaces between the anisotropic laminae (layers). Various schemes have been developed for the theoretical studies of this problem. One of these is the method of partial waves that was developed by Rayleigh (1885, 1889) and was used by Lamb (1917) to study guided waves. Many problems of guided wave propagation in free and fluid-loaded plates and layered semi-infinite spaces have been solved by this method and by its extension using the transfer-matrix approach (Thomson 1950; Haskell 1953). Mention may be made of the works by Bogy and Gracewski (1983), Nayfeh and Chimenti (1988, 1989), Mal (1988a, 1988b), Chimenti and Nayfeh (1990a, 1990b), Nayfeh and Chimenti (1991), Nayfeh (1991), Karunasena et al. (1991b, 1991c), and Pan and Datta (1999). Nayfeh (1995) gives a good exposition of this method and its applications. Comparison of theoretical and experimental results has led to efficient techniques for the inverse characterization of individual lamina properties. Although this method has been widely used since the early works of Thomson and Haskell, it has been found to have precision problems at high frequencies and for thick layers. There have been several different modifications proposed by many investigators to overcome these instability problems. Among these, mention may be made of the

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delta matrix approach (Dunkin 1965; Kundu and Mal 1985; Castaings and Hosten 1994; Zhu et al. 1995a); global matrix method (Knopoff 1964; Schmidt and Jensen 1985; Mal 1988a; Mal 1988b; Mukdadi et al. 2001); reflectivity method (Kennett and Kerry 1979; Kennett 1983; Fryer and Frazer 1984); and the stiffness matrix method (Kausel and Roesset 1981; Wang and Rajapakse 1994; Wang and Rokhlin 2001, 2002a). Comprehensive reviews of guided waves in composite plates and their use in material characterization have been published by Chimenti (1997) and Datta (2000) (see also Rokhlin and Wang 2002). In all the matrix methods mentioned above, finding the roots of the transcendental dispersion equations is quite cumbersome and time consuming, especially when the number of layers is large. Also, addition of each layer involves a new equation and a new search. Additional complications arise when it is necessary to obtain not only the roots corresponding to the propagating modes, but also those corresponding to the evanescent (and nonpropagating) modes. The latter are needed to study scattering by defects in the plate or reflection of waves from the edges. They are also needed to express the Green’s function for the plate as a modal sum. For reasons of numerical efficacy and general applicability, an alternative procedure was developed by Datta et al. (1988c), Karunasena et al. (1991a, 1991b, 1991c), and Karunasena (1992) to model dispersion of guided waves in single- as well as many-layered plates. This was a stiffness method that was originally proposed by Dong and Nelson (1972). In this approach, each lamina was divided into several sublayers. The variation of the displacement through the thickness of each sublayer was approximated by polynomials in a thickness variable with coefficients chosen such that the displacement (or displacement and traction) continuity was maintained at the interfaces between sublayers. Then, using Hamilton’s principle, the dispersion equation was obtained as a standard algebraic eigenvalue problem. Eigenvalues of this equation yielded the wave numbers (real and complex) corresponding to different frequencies for the guided modes. Corresponding eigenvectors were the displacements (or, displacements and tractions) at the nodes. Discussion of other numerical methods can be found in Liu and Xi (2002). In many applications, laminated composite plates are composed of periodic layers, where the layering is the repetition of unit cells, each cell being made up of uniaxial fiber-reinforced plies oriented in different in-plane directions. Lamb wave propagation in such a periodically laminated plate can be analyzed in terms of Floquet modes in an infinite medium having the same periodic structure. Dispersion of Lamb waves in such a periodic laminated plate was studied by Shull et al. (1994), and they found interesting features of mode clustering and gaps in the dispersion behavior. It was suggested that these unusual features were related to the Floquet wave pass and no-pass zones. In this paper, the stiffness method described above was used to obtain the dispersion curves. Safaeinili and Chimenti (1995) and Safaeinili et al. (1995) used the Floquet wave analysis to simplify significantly the solution to the dispersion equation for a multilayered plate. Wang and Rokhlin (2002b, 2002c) used the Floquet wave method to derive the homogenized properties of a multilayered cross-ply composite plate and for the determination of the single-ply properties of a multidirectional composite. Homogenized effective properties of a multilayered cross-ply composite plate were derived earlier by Karunasena et al. (1991b) and Datta et al. (1992) using the stiffness method.

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5

Guided elastic waves in composite circular cylinders have many similarities with those in composite plates and have been studied analytically as well as numerically. Propagation of elastic waves in hollow circular cylinders has been the subject of extensive investigations in the past. Gazis (1959a, 1959b) presented exact three-dimensional solutions to the problem of waves in hollow isotropic cylinders. Extensive numerical results were presented in Gazis (1959b) for the flexural wave modes (see also Armenakas et al. 1969). Three-dimensional wave propagation in hollow cylinders was also studied by Greenspon (1959). McNiven et al. (1966a, 1966b) presented exact and approximate solutions for axisymmetric waves in hollow cylinders. Guided waves in a composite circular cylindrical shell were studied by Armenakas (1967, 1971). An early study of longitudinal waves in a solid cylinder having transversely isotropic properties was published by Chree (1890), who derived solutions to the governing equations using a power-series method. Morse (1954) obtained the solutions in terms of Bessel functions for axisymmetric guided waves in solid, transversely isotropic cylinders. Einspruch and Truell (1959) also derived the dispersion equations for this case and for the torsional waves. Wave propagation in a hollow, transversely isotropic cylinder was studied by Mirsky (1965a, 1965b) using a displacement potential representation due to Buchwald (1961). Eliot and Mott (1968) studied solid and hollow cylinders having transversely isotropic symmetry. McNiven and Mengi (1971) analyzed in detail axisymmetric modes in transversely isotropic rods (including a uniaxial fiber-reinforced composite rod). Recently, Berliner and Solecki (1996a, 1996b) and Honarvar et al. (2007) have studied, respectively, dispersion of guided waves in fluid-loaded hollow and solid transversely isotropic cylinders. For a cylindrically orthotropic symmetry, Mirsky (1964a, 1964b) obtained the solution to the axisymmetric motion of a cylindrical shell using an approximate theory. A similar approximate theory was also used by Mengi and McNiven (1971) to study axially symmetric waves in transversely isotropic rods. For axisymmetric waves in orthotropic cylinders, solutions were obtained by Mirsky (1964b), who used the Frobenius method. Chou and Achenbach (1972) and Armenakas and Reitz (1973) studied the flexural motion of orthotropic cylinders, also using the Frobenius method. Dispersion of guided waves in composite rods (having a solid core bonded to an outside shell, both isotropic) was studied by McNiven et al. (1963), Whittier and Jones (1967), Armenakas (1970), and Lai et al. (1971). Clad rods or wires have been investigated for use as acoustic delay lines and fiber acoustic waveguides. There have been several investigations dealing with guided waves in isotropic clad rods (fibers). A survey of the early literature was given by Thurston (1978). More recently, fiber acoustic waveguides having isotropic properties have been investigated by Safaai-Jazi et al. (1986) and Jen et al. (1986) under the assumption of weak guidance and by Dai et al. (1992a, 1992b) for transversely isotropic materials. The equation governing dispersion of guided waves in a clad cylinder is complicated, even when both materials are isotropic. They are much more complicated if the materials are anisotropic, as in the case of fiber-wound tubes. As mentioned above, for isotropic homogeneous cylinders, extensive analytical, numerical, and experimental research on vibration of guided waves has been reported since the early work of Pochhammer (1876). Early studies of dispersion of waves in elastic

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circular cylinders include Onoe et al. (1962), Pao and Mindlin (1960), and Mindlin and McNiven (1960). Detailed accounts can be found in Meeker and Meitzler (1964), Achenbach (1973), Miklowitz (1978), Graff (1991), and Rose (1999). In the past, there were very few detailed studies of waves in composite cylinders of general anisotropy. Even though, for the transversely isotropic materials, it is possible to obtain analytical solutions, the dispersion equations for nonaxisymmetric motion are rather complicated, and solutions to these equations representing not only propagating waves, but also evanescent waves, are best accomplished by some efficient numerical schemes. Nelson et al. (1971) and Huang and Dong (1984) developed a stiffness method to study propagation of guided waves in laminated anisotropic cylinders with arbitrary lamina layup. This is a generalization of the stiffness method used for laminated plates (see the foregoing discussion). In this, the cylinder is discretized into coaxial cylinders, and radial variation is accounted for by an appropriate polynomial interpolation function in each subcylinder (sublayer). As in the case of laminated plates, quadratic and cubic interpolation functions have been used, and both show excellent agreement with exact solutions when available. In regard to experimental work, the resonance method was used by Zemanek (1972) to verify theoretical predictions of dispersion of waves in a cylinder. Curtis (1982) reviewed wave-propagation techniques available for the determination of material properties of cylinders. Oblique insonification of circular cylinders immersed in fluid has been used to generate and measure guided waves in isotropic cylinders (Flax et al. 1980; Maze et al. 1985; Molinero and de Billy 1988; Li and Ueda 1989). Acoustic scattering by a transversely isotropic cylinder has been studied recently by Honarvar and Sinclair (1995) and Ahmad and Rahman (2000). Guided waves in a transversely isotropic cylinder in a fluid medium has been investigated by Nagy (1995) (see also Berliner and Solecki 1996a, 1996b; Ahmad 2001). Guided waves in a solid cylinder having a transversely isotropic core with an interface layer lying between the core and the outer layer were studied by Xu and Datta (1991). Nayfeh and Nagy (1996) analyzed axisymmetric waves in multilayered transversely isotropic cylinders. It may be noted that similar problems for isotropic cylinders were studied by Huang et al. (1995). Huang et al. (1996) investigated both theoretically and experimentally scattering by multilayered isotropic cylinders in fiber-reinforced composite media. Niklasson and Datta (1998) reported wave scattering and propagation in a transversely isotropic medium containing a transversely isotropic cylinder. Xu and Datta (1991) used a hybrid method that combined finite-element representation of the core and exact eigenfunction expansion for the isotropic outer cylinder. They also used the exact solution to obtain dispersion curves for axisymmetric motion of a transversely isotropic cylinder. Since the analytical formulation of wave propagation in a laminated cylinder with arbitrarily oriented fiber layups in the laminae is intractable, several approximate schemes have been proposed. The most common ones are the various shell theories in which the constitutive relations for the radially inhomogeneous cylinders are replaced by integral forms to reduce the problem to that of homogeneous equivalent cylinders (Tsai and Roy 1971; Sun and Whitney 1974). Other references to approximate shell theories can be found in the work by Barbero et al. (1990). The stiffness method used by Nelson et al. (1971) and Huang and Dong (1984) was generalized by Kohl

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Introduction

7

et al. (1992a, 1992b), Rattanawangcharoen et al. (1992, 1994), and Rattanawangcharoen (1993) to study dispersive guided waves in laminated cylinders.

1.1.5 Cylinders of Arbitrary Cross Section As evidenced from the above review, the vast majority of the studies have been concerned with cylinders of circular cross sections. However, there have been a few studies that have dealt with cylinders of noncircular cross sections. In most of these studies, the material has been taken to be homogeneous and isotropic. For particular ratios of width to depth of a rectangular bar, Mindlin and Fox (1960) found a discrete set of points on the branches of the frequency spectra. Later, general solutions were presented by Morse (1948, 1950), Kynch (1957), Nigro (1966), Fraser (1969), Aalami (1973), Nagaya (1981), SeGi et al. (1994), and Taweel et al. (2000). Recently, attention has been given to composite and anisotropic materials that have applications to aerospace structures and quantum wires. Volovoi et al. (1998) considered beams of fiber-reinforced composite materials, and Nishiguchi et al. (1997), Mukdadi et al. (2002a, 2002b, 2005), and Mukdadi and Datta (2003) considered anisotropic and layered plates having rectangular as well as more general cross sections.

1.1.6 Particle-Reinforced Composites The above brief historical review of waves in composite media has been confined to laminated and fiber-reinforced composites. There have also been significant research studies on wave propagation in particle-reinforced media. Early studies of wave propagation in a two-phase medium were motivated by the need to model seismic wave velocities in rocks permeated by fluids. These include wave propagation in a fluid medium with a suspension of spherical particles by Ament (1953) and elastic wave propagation in a medium with spherical inclusions by Yamakawa (1962) and Mal and Knopoff (1967). These were limited to low concentrations of particles and long wavelengths compared with the dimensions of the scatterers. For arbitrary concentrations, long-wavelength propagation of elastic waves in a medium containing spherical inclusions has been studied by Waterman and Truell (1961), Mal and Bose (1974), and Berryman (1980a). In this limit, one obtains effective static elastic properties of an elastic medium containing spherical inclusions. In recent years, there have appeared several publications that contain various approximate theories for dispersion and attenuation of effective plane waves propagating in an elastic medium containing spherical inclusions. Among these are Beltzer et al. (1983), Beltzer and Brauner (1986, 1987), Sabina and Willis (1988), Shindo et al. (1995), Kim et al. (1995), Yang (2003), and Aggelis et al. (2004). Compared with the large volume of work dealing with wave propagation in a medium containing spherical inclusions that has accumulated over the last 40 years, there are very few studies that have been concerned with the effect of inclusion shape on the effective dynamic properties of particle-reinforced composites. Kuster and Toksöz (1974) and Berryman (1986) presented long-wavelength results when the inclusions were randomly oriented spheroids. Ledbetter and Datta (1986) considered both aligned and randomly oriented ellipsoids (see also Datta 1977). Recently, Sabina et al. (1993) and Smyshlayaev et al. (1993) have considered effective wave speeds and attenuation in a medium containing

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Elastic Waves in Composite Media and Structures

aligned and randomly oriented spheroids. There are very few experimental investigations of wave propagation in a particle-reinforced composite. Some of these are: Kinra et al. (1980, 1982), Kinra and Anand (1982), Ledbetter and Datta (1986, 2000), Datta and Ledbetter (1986a), and Ledbetter et al. (1984, 1995). In the above, we have mentioned major works dealing with waves in composite materials having elastic constituents. However, anelastic effects have a large influence on wave speeds and attenuation. Wave propagation in a composite with viscoelastic constituents can be obtained essentially along the same lines if the elastic properties are treated as complex functions of frequency, and there have been many works in the literature. In this book, we will restrict our attention mostly to elastic composites, with one exception. This is the case of thermoelastic wave propagation in an anisotropic plate.

1.2 Scope of the Book In Chapter 2, we present the fundamental equations and their solutions for plane wave propagation in an infinite homogeneous anisotropic elastic material. Because composite materials are in most cases anisotropic, it is necessary to present this background material for the convenience of the readers and users of this book as well as for establishing notations that are used throughout this book. Chapter 3 deals with wave propagation in a periodically laminated infinite medium. Here, Floquet theory together with a stiffness method is presented. The stiffness method used is quite versatile and is applicable to multiple laminae in a cell having general anisotropic properties. Results are presented showing the anisotropic and dispersive characteristics of plane waves in such media. Wherever possible, results are compared with experiments and with predictions obtained by other modeling techniques. In Chapter 4, we discuss harmonic and transient guided (Lamb) wave propagation in a free-free multilayered fiber-reinforced laminated plate. The stiffness method discussed in Chapter 3 has been used to obtain frequency–wave-number (dispersion) relations for propagating and evanescent modes. A systematic investigation of the effects of different fiber layups and increasing number of laminae on the dispersion characteristics reveals features that are useful for experimental determination of lamina properties. For this study, each lamina has been treated as a transversely isotropic medium with effective properties that are determined by a wave-scattering theory (Datta and Ledbetter 1983; Datta et al. 1984). In addition to the stiffness method, this chapter also includes an exact analytical treatment of the guided-wave problem. The exact treatment leads to refinements of the modal frequencies and mode shapes that are needed to study transient wave propagation and scattering by defects. This chapter also includes some results of comparison between model predictions and experimental observations for a particular case. In addition, the dynamic response of a composite plate to applied external forces and to laser thermal excitation is analyzed. Both time-harmonic and transient waves are treated analytically as well as numerically. The stiffness method that has been discussed thoroughly in this chapter for the isothermal case is generalized to treat the coupled problem of thermoelastic waves. In this case, the thermoelasticity theory, which includes a relaxation time in the heat-conduction equation, has been used. This makes the thermal transport equation hyperbolic. Because of thermal-diffusion effects, the

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Introduction

9

frequency–wave-number relation is now complex, and the thermal waves are found to have high attenuation. However, the primarily elastic waves suffer small attenuation. Guided wave propagation in composite cylinders and cylindrical shells is the subject of Chapter 5. Here again, as in the case of plates, exact solutions are presented for circular cylinders composed of transversely isotropic materials with the symmetry axis parallel to the axis of the cylinder. Also, stiffness methods (similar to that used for plates), in which the cylinder is subdivided into a number of coaxial cylinders and radial interpolation functions, are used to approximate radial variations of displacements (and tractions) to obtain the dispersion relation showing the frequency–wave-number behavior of wave propagation along the axis. Cylinders of noncircular cross sections composed of homogeneous and layered anisotropic materials are also considered in this chapter. In the stiffness method that is employed here, the finite element discretization is used to approximate the displacement (and traction) over the cross section. Whereas shape functions involve one spatial (radial) variable for circular cylinders, they involve two spatial variables in the plane of noncircular cross sections. Scattering of guided waves by cracks in composite plates and cylinders is treated in Chapter 6. A hybrid method that combines the finite element discretization of a finite region containing the cracks or joints and wave function expansion of the fields in the exterior regions is used. Continuity of displacements and tractions at the (artificial) boundaries between the finite region and the exterior semi-infinite regions is enforced to obtain the coefficients of the modal sum as well as the nodal displacements in the finite region. The second is a combined boundary integral and finite element method in which the defects are again enclosed in a finite region, which is modeled by finite elements. The wave field exterior to the finite region is represented by a boundary integral using the Green’s functions for the composite plate. Both methods are shown to give convergent results. Comparison with some available experimental results shows good agreement between the model results and observations. In addition, a boundary integral method combined with a multidomain decomposition is presented for the analysis of scattering in a composite plate.

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2 Fundamentals of Elastic Waves in Anisotropic Media 2.1 Waves in Homogeneous Elastic Media: Formulation of Field Equations ......................................................................11 Deformation • Balance Laws • Constitutive Equations • Coupled Equations of Linear Dynamic Thermoelasticity 2.2 Plane Waves in a Homogeneous Anisotropic Medium ...... 20 Slowness Surface • Energy Transport • Group Velocity • Special Cases • Transformation of Coordinates 2.3 Numerical Results and Discussion .........................................33

2.1 Waves in Homogeneous Elastic Media: Formulation of Field Equations In this section, we will be concerned with the field equations governing thermoelastic time-dependent deformation of homogeneous elastic media. A brief derivation of the linearized equations of motion is presented first. The readers are referred to many excellent textbooks on this subject, e.g., Sokolnikoff (1956), Mal and Singh (1991), and many others.

2.1.1 Deformation An elastic medium has a natural undeformed state in the absence of any external or internal mechanical or other (thermal, electromechanical) sources of disturbance acting on it. We will choose this as the reference state of the body in which a particle in the body is located at a point x. Here, x is the position vector of the particle and can be written as

x = x1 e 1 + x 2 e 2 + x 3 e 3

(2.1.1)

where x1, x2, x3 are the components of the vector x referred to a fixed Cartesian coordinates system and e1, e2, e3 are the unit vectors along the 1-, 2-, 3-axes, respectively 11

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12

Elastic Waves in Composite Media and Structures x3

e3

x2

e2 e1 x1

FIGURE 2.1  Reference coordinates in the undeformed state.

(Fig. 2.1). Using the summation convention, equation (2.1.1) can be written as x = xi e i , i = 1, 2, 3





(2.1.2)

When a source of disturbance acts on the body to change the reference state to a deformed state, the deformation is measured by the displacements of the particles in the body. Let X denote the new position of the particle that was at x in the reference state initially. The displacement u of the particle is given by u =X-x



(2.1.3)

The new position of the particle, X, is a function of x and time t (observation time). In general, u can be large and the deformation will be nonlinear. Here, we will restrict ourselves to small deformations such that the spatial gradients of u are much smaller than unity. Let x + dx be the position of a particle in a neighborhood of the particle at x. In the deformed position, the position of this particle may be denoted by X + dX. The length of the line element dx is

ds = dxi dxi



dS = dXi dXi



(2.1.4)

In the deformed state, its length is

(2.1.5)

Thus, the change in length of the line element dx is dS 2 - ds 2 = dXi dXi - dxi dxi



 ∂X ∂X  k - δ ij  dxi dx j =  k  ∂xi ∂x j 

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Fundamentals of Elastic Waves in Anisotropic Media

This relation can be written as

dS 2 - ds 2 = 2Eij dxi dx j

(2.1.6)



where Eij is the Lagrangian or the Green strain tensor and is given by Eij =

 1  ∂X k ∂X k - δ ij   2  ∂xi ∂x j 

(2.1.7)

Note that equation (2.1.6) can be written in the alternative form dS 2 - ds 2 = dXi dXi -

∂x k ∂x k dX dX ∂Xi ∂X j i j

= 2eij dXi dX j



(2.1.8)

where the Eulerian or Alamansi strain tensor eij is eij =

∂x k ∂x k  1  δ ij  2 ∂Xi ∂X j 

(2.1.9)

Using equation (2.1.3) in (2.1.7) gives

Eij =

1  ∂ui ∂u j ∂uk ∂uk  + +   2  ∂x j ∂xi ∂xi ∂x j 

(2.1.10)

As mentioned above, the displacement gradients will be assumed to be small. So, keeping only the first-order terms in equation (2.1.10), we obtain 1  ∂u ∂u j  Eij ≈  i +  2  ∂ x j ∂ xi 



(2.1.11)

Similarly, using equation (2.1.3) in (2.1.9) and keeping only first-order terms, there results ∂u j  1  ∂u eij ≈  i +  2  ∂X j ∂Xi 



(2.1.12)



It is easily shown that, to the first order of approximation, the linearized strain tensor is



1  ∂u ∂u j  εij ≈ eij ≈ Eij =  i +  2  ∂x j ∂xi 

(2.1.13)

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Elastic Waves in Composite Media and Structures

Thus, for small deformations in which the displacement gradients are small, the infinitesimal strain tensor is the same in both Eulerian and Lagrangian descriptions and may be written as eij ( x , t ) =

1  ∂ui ∂u j  +   2  ∂x j ∂xi 

(2.1.14)

In the rest of this book, we will use the following equation to express the linearized strain tensor as 1 eij = (ui , j + u j ,i ) 2



(2.1.15)

where a comma denotes derivative with respect to a coordinate xi (i = 1, 2, 3). It would be convenient to introduce the infinitesimal rotation tensor as 1 ω ij = (ui , j - u j ,i ) 2



(2.1.16)

The velocity of a point X is v=



∂X ∂t

(2.1.17)



and the acceleration is a=

∂v Dv ( X , t ) ∂ v ( x ,t ) ∂ v ∂ X i ∂v + = = + vi ∂t ∂X i ∂t ∂t Dt ∂X i



(2.1.18)

Now, keeping only the first-order terms, we obtain from equations (2.1.17) and (2.1.18)



v=

∂u = u� ∂t

(2.1.19)

a=

∂v = v� ∂t

(2.1.20)

and





where the superimposed dot (.) denotes a derivative with respect to time.

2.1.2 Balance Laws In this section, we develop the equations governing conservation of mass, linear and angular momenta, and energy. First, let us consider the principle of conservation of mass. For this purpose, let us denote the density of the material at (a) X at time t in the

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Fundamentals of Elastic Waves in Anisotropic Media

deformed state by r(X, t) and (b) x and initial time t0 at the reference state by r 0(x, t0). Let dV be a volume element containing the point X, with dV0 as its initial value. Then, by the principle of conservation of mass, we have rdV = r 0 dV0



(2.1.21a)

Because dV = JdV0, where J is the Jacobian matrix of transformation from x to X, equation (2.1.21a) gives rJ = r 0



(2.1.21b)

Differentiating equation (2.1.21b) with respect to t keeping x fixed, we find D( ρ J ) =0 Dt

or

ρ



DJ Dρ +J =0 Dt Dt

(2.1.22)

Using the property DJ/Dt = Jvk,k, one obtains from equation (2.1.22) the equation of continuity Dρ + ρv k , k = 0 Dt



(2.1.23)

Note that, correct to the first order, we have 1/J ≈ 1 - uk,k

So, equation (2.1.21) gives

r = r 0 (1 - uk,k)



(2.1.24)

Next we will consider the principle of balance of linear momentum. Here, we will be concerned with mechanical forces acting on the interior of the body B as well as on its boundary, ∂B, causing it to deform. Let V be a part of the solid enclosed by a surface S in the deformed state at time t. There are forces acting on S caused by the action of the material outside S on that within V. Consider a plane element dS of the surface S. Let n be a unit normal to dS pointing outward. It is postulated that the elementary force of action on dS by the material outside S on that in the interior is tdS and that the elementary moment of action on dS is zero. The force t is called the traction. It depends on n and the position Y of dS. The force t has the properties: (a) t(n, Y) = −t(−n, Y) and (b) t(n, Y) = s ijniej. The tensor s is the Cauchy stress tensor. Now, the principle of balance of linear momentum can be stated in the form of the following equation: d



∫ ρfdV + ∫ tdS = dt ∫ ρ vdV = ∫ ρa dV V

S

V

V

(2.1.25)

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Elastic Waves in Composite Media and Structures

Here f is the body force per unit mass. In the absence of any distributed moments in V, the principle of balance of moment of momentum can be stated as d

∫ X ∧ ρfdV + ∫ Y ∧ tdS = dt ∫ X ∧ ρ vdV = ∫ ρX ∧ adV



V

S

V

V

(2.1.26)

In the component form, equations (2.1.25) and (2.1.26) can be written as

∫ ρ f dV + ∫ σ n dS = ∫ ρa dV (i, j = 1, 2, 3) i



ji

V

∫ε

ijk

j

(2.1.27)

i

S

V

∫



∫

ρ X j f k dV + εijkY jσ lk nl dS = εijk ρ X j ak dV (i, j, k = 1, 2, 3)

V

S

(2.1.28)

V

Using the divergence theorem, the surface integrals appearing in equations (2.1.27) and (2.1.28) can be written as

∫ σ n dS = ∫ σ ji



∫ε

ijk

j

S

V

Y jσ lk nl dS +

∫ε

S

ji , j

ijk

dV

( X jσ lk ),l dV =

V

∫ε

ijk

(σ lkδ jl + X jσ lk ,l )dV

V

Thus, equations (2.1.27) and (2.1.28) can be rewritten respectively in the forms

∫ (ρ f + σ i



ji , j

- ρai )dV = 0

(2.1.29)

V

and

∫ε

ijk

[ X j ( ρ f k + σ lk ,l - ρak ) + σ jk ]dV = 0

(2.1.30)

V

Since equations (2.1.29) and (2.1.30) must hold for any arbitrary volume V of the body in the deformed state, it follows that the integrands must vanish everywhere within the body. Hence, we obtain Cauchy’s equations of motion from equation (2.1.29) as

ρ fi + σ ji , j = ρai



(2.1.31)

Using equation (2.1.31), it is found from equation (2.1.30) that

εijks jk = 0

This result implies that

s jk = s kj

(2.1.32)

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Fundamentals of Elastic Waves in Anisotropic Media

i.e., the stress tensor is symmetric. Note that for small deformation, the density r in equation (2.1.31) can be taken as the initial density, and ai = üi. Finally, we will give a brief outline of the principle of conservation of energy and derive the equations of thermoelasticity. We will consider thermal energy as well as mechanical energy. According to the first law of thermodynamics, we have



d dt

1



∫ ρ  2 v v + U  dV = ∫ ρ f v dV + ∫ t v dS + ∫ ρhdV - ∫ q n dS i i

i i

V

V

i i

(2.1.33)

i i

S

V

S



Here U is the internal energy and h is the rate of heat generation within V, both with respect to unit mass, and q is the heat-flux vector representing the rate of transfer of heat across S. The left-hand side of equation (2.1.33) represents the rate of change of kinetic and internal energies of the material occupying V. The right-hand side is the sum of the rate of work done by the body forces acting on V and the traction acting on S, and the rate of change of heat energy in V. Using the divergence theorem, the surface integrals appearing in equation (2.1.33) can be converted into volume integrals over V. After some obvious algebraic manipulation, equation (2.1.33) takes the form

∫ [v (ρv� -ρ f - σ i



i

i

ij , j

) + ρ(U� - h) - σ ij vi , j + qi ,i ]dV = 0

(2.1.34)

V

Using the equation of motion (2.1.31), it is found from equation (2.1.34) that

ρ(U� - h) - σ ij vi , j + qi ,i = 0



(2.1.35)



Now, we will apply the second law of thermodynamics. We assume that s is the entropy per unit mass at (x,t). Then, the Clausius–Duhem inequality can be stated as

ρh



∫ ρs�dV ≥ ∫ T dV - ∫ V

V

S

qi ni dS T

(2.1.36)

Here T is the absolute temperature. Applying the divergence theorem to the surface integral in the above equation, we find q  ρh ρs� +  i  ≥0  T  ,i T



(2.1.37)

Using equation (2.1.35), the h appearing in equation (2.1.37) can be eliminated, and equation (2.1.37) can be rewritten as



 qT  ρ(s� - U� /T ) +  σ ij vi , j - i ,i  / T ≥ 0 T  



(2.1.38)

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Elastic Waves in Composite Media and Structures

2.1.3 Constitutive Equations The Helmholtz free energy is defined by the equation F = ρ(U - Ts )



(2.1.39)



For an elastic material, F is a single-valued function of the strain tensor, eij, the absolute temperature T, and its gradient, T,i. The energy equation (2.1.35) can be expressed in the form  d(Ts )  F� + ρ  - h - σ ij vi , j + qi ,i = 0  dt 



(2.1.40)



The inequality in equation (2.1.38) can be written as (σ ij vi , j - qiT,i /T ) - ( F� + ρsT� ) ≥ 0



(2.1.41)



∂F ∂F � ∂F � e� + T+ T , the inequality in equation (2.1.41) takes Furthermore, since F� = ∂eij ij ∂T ∂T,i ,i the form   ∂F  ∂F  � ∂F � qiT,i T≥ 0 T  σ ij  vi , j -  ρ s + T ∂eij  ∂T  ∂T,i ,i  



(2.1.42)

Note that the dependent variables s ij, s, and F are determined by the values of the independent variables eij, T, and T,i. On the other hand, their rates given by vi,j, T� , and T�,i can be chosen arbitrarily so that the inequality will be violated for every possible choice of the independent variables. Thus, for the inequality to hold for all choices of the independent � and T� must vanish. We then have the equations: variables, the coefficients of vi,j, T, ,i

σ ij =



s=-



0=

∂F ∂eij

1 ∂F ρ ∂T ∂F ∂T,i

(2.1.43)



(2.1.44)



(2.1.45)



The inequality in equation (2.1.42) then becomes qiT,i ≤ 0



(2.1.46)

Using equations (2.1.43)–(2.1.45) in the energy equation (2.1.40), we obtain

ρ(s�T0 - h) + qi ,i = 0



(2.1.47)

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19

In writing equation (2.1.47), we have replaced T by the initial temperature T0 because s, h, and q are small. In the linearized approximation, F can be expanded in the form, keeping terms up to the second order F=

1 1 θ2 cijkl eij ekl - βijθ eij - ρc 2 2 T0



(2.1.48)

where q = T – T0. Here cijkl is a constant fourth-order tensor (the elasticity tensor), βij is a constant second-rank symmetric tensor (the thermoelasticity tensor), and c is the specific heat at constant deformation. It is assumed that q is much smaller than T0. Using equation (2.1.48) in (2.1.43) and (2.1.44), we obtain the constitutive relations for linear thermoelasticity as

σ ij = cijkl ekl - βijθ



ρs = βij eij +



ρc θ T0

(2.1.49)



(2.1.50)



Using equation (2.1.50), the energy equation (2.1.47) can then be written as

qi ,i = - βijT0 vi , j - ρcθ� + ρh



(2.1.51)

Because s ij and eij are symmetric, it follows that the elasticity tensor, cijkl, has the following symmetry properties

cijkl = c jikl = cijlk = cklij



(2.1.52)

In the usual coupled theory of thermoelasticity, qi is assumed to be a linear function of q,i, in the form

qi = - kijθ , j

(2.1.53a)



where kij is the thermal conductivity tensor. The condition given in equation (2.1.46) implies that kij is positive-definite symmetric. Substitution of this in equation (2.1.51) leads to the parabolic equation for heat conduction, giving the speed of travel for the thermal disturbance to be infinite. This is not physically realistic, and several theories have been proposed to remove this paradox. Among these are Lord–Shulman (LS), Green–Lindsay (GL), and other theories. The reader is referred to the paper by Chandrasekharaiah (1986) for a thorough review of the literature. The simplest of these is the LS theory, in which a relaxation time τ0 (time lag needed to establish steady-state heat conduction in an element of volume when a temperature gradient is suddenly applied on the volume) is introduced in equation (2.1.53a) (Lord and Shulman 1967). In this case, the modified form of equation (2.1.53a) is

 ∂  1 + τ 0 ∂t  qi = - kijθ j



(2.1.53b)

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Elastic Waves in Composite Media and Structures

2.1.4 Coupled Equations of Linear Dynamic Thermoelasticity Substitution of equation (2.1.53b) in equation (2.1.51) yields the modified equation of heat transport as



 ∂ kijθ ,ij =  1 + τ 0  [ρcθ� + T0 βij u�i , j - ρh] ∂t  

(2.1.54)

Since kij is positive definite, this equation is hyperbolic, provided that t 0 is positive. Substitution of the expression for s ij given by equation (2.1.49) in the equation of motion (2.1.31) yields

cijkl uk ,lj = ρu��i + βijθ , j - ρ fi



(2.1.55)

Equations (2.1.54) and (2.1.55) constitute the coupled equations for linear thermoelastodynamics of a homogeneous anisotropic elastic solid including a thermal relaxation time. These form a system of four coupled equations of the hyperbolic type. The solutions to this system—satisfying appropriate initial and boundary conditions for the body B occupying the volume V and bounded by the surface S—describe the thermoelastodynamic state of B. Under conditions of isothermal or adiabatic deformation of a homogeneous anisotropic elastic medium, one needs to consider equation (2.1.55) after omitting the term on the right-hand side involving the temperature gradient. Then, we obtain the displacement equations of elasticity as

cijkl uk ,lj = ρu��i - ρ fi



(2.1.56)

The stress, strain, and temperature relation in equation (2.1.49) becomes

σ ij = cijkl ekl

(2.1.57)



In the following section, we will consider solutions of equation (2.1.56) in the form of plane waves when the body forces are absent, i.e., f i = 0.

2.2 Plane Waves in a Homogeneous Anisotropic Medium In this section, we will derive the equations and the characteristics of their solutions for plane waves that are governed by equation (2.1.56) when the body force f is absent. For this purpose, it will be assumed that the time dependence of u(x,t) is simple harmonic of the form e−iwt. Thus, u(x,t) is written as

u(x ,t ) = U(x )e -iωt

(2.2.1)

where w is the circular frequency and i = √−1. Then, equation (2.1.56) takes the form

cijklU k ,lj = - ρω 2U i



(2.2.2)

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We will seek a solution of equation (2.2.2) of the form U i = Ai e



ikn j x j



(2.2.3)

where k is the wave number, n is the unit vector representing the direction of propagation of the plane wave, and A is a constant vector amplitude of the wave. Substitution of equation (2.2.3) in (2.2.2) gives rv2 Ai = cijklnlnj Ak



(2.2.4)

where v = w/k is the phase velocity of the plane harmonic wave. Equation (2.2.4) is the Christoffel equation for the determination of v for a wave of phase k propagating in the direction n and its amplitude A. Equation (2.2.4) represents three linear homogeneous equations in the three unknown components of A. For a nontrivial solution (A ≠ 0) of equation (2.2.4), rv2 must be an eigenvalue and A is a corresponding eigenvector of equation (2.2.4). Note that A defines the direction of the particle velocity at x. The characteristic equation governing the eigenvalues and the eigenvectors is

Ω(v , n1 , n2 , n3 ) = cijkl n j nl - ρv 2δ ik = 0



(2.2.5)

Here, |…| denotes a 3 × 3 determinant and d ik is the Kronecker delta. Equation (2.2.5) is cubic in v2. Thus, there are three possible values of v2 for each n. Since the Christoffel acoustic tensor, Γik = cijklnjnl, has the property that it is symmetric (Γik = Γki) and positive definite, it follows that the eigenvalues rv2 are positive. Thus, the phase velocities ±v are real (∵ r > 0), with +v signifying a wave moving outward (away from the origin) and –v moving inward. The expression v(n) for all possible directions n traces out a surface of three sheets, called the “velocity surface.”

2.2.1 Slowness Surface Equation (2.2.5) can be written in an alternative form in terms of 1/v as



1 Γ - ρδ ij = 0 v 2 ij

(2.2.6)



Solution of this equation for v−1 as a function of n, v−1(n) traces out a surface having three sheets called the “slowness surface.” Once v2 has been solved from equation (2.2.5), the particle displacement vector A is found from the matrix equation

[Γ ik - ρv 2δ ik ][ Ak ] = 0



(2.2.7)

where the first matrix in the above equation is 3 × 3 and the second is 3 × 1. Since this equation is homogeneous, A can be found except for a constant multiplying factor. Now, it is easily shown that if vM and vN are two different eigenvalues, then the corresponding eigenvectors A M and A N are orthogonal to each other.

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2.2.2 Energy Transport For the complex exponential form of the displacement given by equations (2.2.1) and (2.2.3), the stress tensor s ij takes the form

σ ij = ikl Ak cijkl e i ( kl xl -ωt )



(2.2.8)



The strain tensor eij is given by i eij = (ki A j + k j Ai )e i ( kl xl -ωt ) 2



(2.2.9)

Here we have written kl = knl. Now, the flux of energy across a small element of surface due to the propagating plane wave can be calculated in the following manner. We have, from equation (2.1.56), the time-averaged power flow per unit area as P=-



1 σ n v* 2 ij j i

(2.2.10)

where * denotes a complex conjugate. Using equations (2.2.1) and (2.2.8) in the above equation, we obtain 1 P = ρω 2vAk Ak 2



(2.2.11)

The average power flow vector (Poynting vector) (Auld 1990) is given by P=



1 1 ω2 〈-vi*σ ij 〉 = c AA ne 2 2 v ijkl i k l j

(2.2.12)

Thus, we have the relation

The energy velocity is defined as



Ve =

n.P = P P P = 1 〈E 〉 2 ρω 2 Ak Ak

(2.2.13)

(2.2.14)

where 〈E〉 is the average total energy density. It is seen that

n.Ve = v

(2.2.15)

2.2.3 Group Velocity The group velocity of the plane wave is given by Auld (1990)



Vg = ∇ kω = e i

∂ω ∂v = ei ∂ki ∂ni

(2.2.16)

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23

where we have used the relations v = w/k and k = kn. Now, using equation (2.2.7), we get

ρv 2 Ai Ai = cikjl nk nl Ai A j





(2.2.17)

Differentiating both sides of this equation with respect to nm, we obtain ∂v 1 cimjl nl Ai A j = ∂nm ρv Ai Ai



(2.2.18)

Thus, it is found that Vg =



1 cimjl nl Ai A j em ρv Ai Ai

(2.2.19)

Comparison of this equation with (2.2.14) shows the important identity

Vg = Ve

(2.2.20)

Now, equation (2.2.15) can be written as dv = (Vg )i dni





(2.2.21)

Also, the slowness vector is given by s=



n v

(2.2.22)

Taking the differential of both sides, we get vds + dvs = dn



(2.2.23)

Taking the “dot” product of this with Vg gives, using equations (2.2.21) and (2.2.22),

Vg.ds = 0

(2.2.24)

This shows that the group velocity vector (or the energy velocity vector) is normal to the slowness surface.

2.2.4 Special Cases In this section, we will consider simplifications that occur when the material possesses some symmetry properties. For convenience of subsequent discussion the four-suffix elements of the stiffness tensor cijkl will be represented by two-suffix elements of a 6 × 6 matrix. This is done by the following identification scheme: 11 → 1, 22 → 2, 33 → 3

23,32 → 4, 31,13 → 5, 12,21 → 6

(2.2.25)

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The same contracted notation will be used to write the six components of the stress in the form of a 6 × 1 matrix. Also, we will use the engineering shear strain components defined as

γ 12 = 2e12 , γ 23 = 2e23 , γ 31 = 2e31

(2.2.26)



The stress strain relation in equation (2.1.57) can then be written in the matrix form c11 c12 c13 σ 1      c22 c23 σ 2   σ  c33  3 =   symm σ    4  σ 5     σ 6  

c14 c15 c24 c25 c34 c35 c44 c45 c55

c16  e1    c26  e2    c36  e3  c46  γ 4    c56  γ 5    c66  γ 6 

(2.2.27)

This equation can be written in the matrix notation as [σ ] = [c][e]



(2.2.28)

where [s] is a 6 × 1 matrix, [c] is 6 × 6, and [e] is the second matrix (6 × 1) on the right. The elements of the Christoffel acoustic matrix [Γ] can be written in the contracted notation as Γ11 = c11n12 + c66n22 + c55n32

+ 2c16n1n2 + 2c15n1n3 + 2c56n2n3 Γ12 = c16n12 + c26n22 + c45n32



+ (c12 + c66 )n1n2 + (c14 + c56 )n1n3 + (c46 + c25 )n2n3 Γ13 = c15n12 + c46n22 + c35n32



+ (c14 + c56 )n1n2 + (c13 + c55 )n1n3 + (c36 + c45 )n2n3

(2.2.29)

Γ 22 = c66n12 + c22n22 + c44n32

+ 2c26n1n2 + 2c46n1n3 + 2c24 n2n3 Γ 23 = c56n12 + c24n22 + c34n32



+ (c46 + c25 )n1n2 + (c36 + c45 )n1n3 + (c23 + c44 )n2n3 Γ 33 = c55n12 + c44n22 + c33n32



+ 2c45n1n2 + 2c35n1n3 + 2c34n2n3

The Christoffel acoustic matrix [Γ] simplifies considerably if the material has a high level of symmetry. In the following, we will develop the equations for some special cases of symmetry.

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Fundamentals of Elastic Waves in Anisotropic Media

2.2.4.1 Isotropic Elastic Material If the material is isotropic, then there are no preferred directions in the material, and the elastic stiffness constants are the same independent of the choice of the coordinate axes in which the stress and strain tensor components are expressed. It can be shown that the stiffness tensor cijkl has the form cijkl = λδ ijδ kl + µ(δ ikδ jl + δ ilδ jk )



(2.2.30)



where l and m are known as the Lamé’s constants. Thus, the stress strain relation in equation (2.1.57) takes the form

σ ij = λ ekkδ ij + 2µeij



(2.2.31)



The Lamé’s constants l, m have the properties 0 < 3l + 2m < ∞; 0 < m < ∞



(2.2.32)

In the matrix notation, the stiffness matrix [c] takes the form  λ + 2µ    c  =   symmetric   



λ λ + 2µ

λ λ λ + 2µ

0 0 0 µ

0 0 0 0 µ

0  0 0  0 0  µ 

(2.2.33)



The Christoffel matrix elements are now given by Γ11 = (λ + µ )n12 + µ Γ 22 = (λ + µ )n22 + µ

(2.2.34)

Γ 33 = (λ + µ )n32 + µ Γ ij = (λ + µ )ni n j , i ≠ j





Then, equation (2.2.4) becomes

(rv2 − m)Ai = (l + m)ninkAk

(2.2.35)

This equation implies that either (a) Ai = Ani, i.e., A is parallel to n or (b) Aknk = 0, i.e., A is perpendicular to n. In case (a), we get rv2 = (l + 2m), and in case (b) we find rv2 = m. The wave is pure longitudinal in case (a) and it is pure shear in case (b). The three sheets

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of the velocity surface are spherical in this case. The two sheets corresponding to the shear waves are coincident. The average power flow vector is obtained by using equation (2.2.30) in (2.2.12) P=



1 ω2 [(λ + µ )Al nl A + µ A2 n] 2 v

Now, for the longitudinal wave, A = An and v = vl = gives 1 P = ρω 2 vl A 2 n 2

(2.2.36)

λ + 2µ . Then, equation (2.2.36) ρ (2.2.37)

Using equation (2.2.14), we find that the energy velocity Ve is given by Ve = vln



On the other hand, for the shear wave, Alnl = 0 and v = vs = P=



(2.2.38) µ . So, P is given by ρ

1 ρω 2 v s A2 n 2

(2.2.39)

Ve = vsn

(2.2.40)

The energy velocity is then found to be

Thus, in an isotropic homogeneous medium, the energy (or group) velocity associated with a plane wave is coincident with its phase velocity. 2.2.4.2 Transversely Isotropic Material A major emphasis of this book is on wave propagation in an infinite or finite elastic medium having a laminated structure, where each lamina may be homogeneous isotropic or is a composite made up of a homogeneous matrix reinforced by aligned fibers. In the former case, the laminated medium can be characterized as a homogeneous anisotropic medium (in the long-wavelength limit) having an axis of elastic symmetry that is perpendicular to the plane of each lamina. In the latter instance, each lamina may be treated as homogeneous anisotropic having the axis of symmetry parallel to the fiber axes. Such an anisotropic medium is called transversely isotropic. Planes perpendicular to the axis of symmetry (unique axis) are isotropic. If the x1-axis is taken parallel to the unique axis, then the stiffness matrix [c] takes the form



c11 c12 c12   c22 c23  c22 c  =   symmetric    

0 0 0   0 0 0  0 0 0   c44 0 0   c66 0  c66 

(2.2.41)

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Here c44 = 12 (c22 - c23 ). In this case, there are five independent elastic constants. For an orthotropic material with the 12-, 23-, and 31-planes as symmetry planes, the Christoffel matrix can be written as Γ11 = c11n12 + c66n22 + c55n32 Γ12 = (c12 + c66 )n1n2 Γ13 = (c13 + c55 )n1n3

(2.2.42)

Γ 22 = c66n12 + c22n22 + c44n32 Γ 23 = (c44 + c23 )n2n3 Γ 33 = c55n12 + c44n22 + c33n32





For transverse isotropy, we have c12 = c13, c22 = c33, and c55 = c66, and c44 is given by the relation given above. In the case of transverse isotropy, since the x1-axis is the axis of symmetry, the x1–x3plane can be chosen as the plane containing the direction of propagation of the plane wave. In that case, n2 = 0. Then, equation (2.2.7) takes the form  Γ11 - ρv 2 0 Γ13   A1      0   A2  = 0 Γ 22 - ρv 2 0     Γ 2 A  ρ v 0 Γ 13 33   3 



(2.2.43)

where

Γ11 = c11n12 + c55n32 , Γ 33 = c55n12 + c33n32 , Γ 22 = c66n12 + c44n32 , and Γ13 = (c13 + c55 )n1n3

It follows from equation (2.2.43) that, of the three sheets of the velocity surface, there is one with displacement polarized perpendicular to the plane of propagation containing the axis of symmetry. This is a pure shear motion with the phase velocity given by v3 =



c66n12 + c44n32 ρ

(2.2.44)

The other two sheets are traced out by the velocities v1 and v2, where v12 and v22 are the roots of the quadratic equation

2 =0 Ω(ω , k1 , k3 ) = ( ρv 2 )2 - ρv 2 (c11n12 + c33n32 + c55 ) + Γ11 Γ 33 - Γ13



(2.2.45)

The velocities of the two waves with displacement polarizations in the plane of x1–x3 are then given by



ρv12 =

1 c n2 + c n2 + c + D   2  11 1 33 3 55

(2.2.46)

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ρv 22 =

where

1 c n2 + c n2 + c - D   2  11 1 33 3 55

(2.2.47)

D = [(c11 - c55 )n12 + (c55 - c33 )n32 ]2 + 4(c13 + c55 )2 n12n32



(2.2.48)

Note that D > 0 and ρ(v12 - v 22 ) = D > 0 . The wave moving with the larger of the two velocities, v1 = v qL , is called quasi-longitudinal, and the slower wave with the velocity, v2 = v qS, is called quasi-shear. They are neither purely longitudinal nor purely shear because the displacements associated with them are not parallel or perpendicular to the velocity direction n. Now, to obtain the group velocities of the three waves, we note that, for the shear wave polarized in the x2 direction, we get from equation (2.2.43) So, it is found that

Ω SH (ω , k1 , k3 ) = (c66 k12 + c44 k32 ) - ρω 2 = 0

(VgSH )1 =

n1c66 ρv 3

(VgSH )3 =

n3c44 ρv 3



(2.2.49)

(2.2.50a)



(2.2.50b)

These equations show that V is not parallel to n. Thus, the group velocity direction deviates from the phase velocity direction. Note that VgSH .n is equal to v3 as per equation (2.2.15). On the other hand, for the wave propagating and polarized in the plane of x1–x3, we obtain the components of the group velocity of the quasi-longitudinal (qL) or quasishear (qS) wave as SH g

(Vg )1 = -

∂Ω /∂k1 ∂Ω /∂ω

n1[c11 {ρv 2 - (c55n12 + c33n32 )} + =

c55 {ρv 2 - (c11n12 + c55n32 )} + (c13 + c55 )2 n32 ] ρv[2ρv 2 - (c11n12 + c33n32 + c55 )]

(Vg )2 = -

(2.2.51a)

∂ Ω/∂k3 ∂ Ω/∂ω

n3[c33 {ρv 2 - (c11n12 + c55n32 )} + =

c55 {ρv 2 - (c55n12 + c33n32 )} + (c13 + c55 )2 n12 ] ρv[2ρv 2 - (c11n12 + c33n32 + c55 )]

(2.2.51b)



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29

� by taking v as v1 or v2, respectively, in equation (2.2.51). It can be shown that Vg given by equations (2.2.51a) and (2.2.51b) satisfies the equation � � Vg .n = v It is found from equations (2.2.42) and (2.2.43) that, when the wave is propagating along the x1-axis, the phase velocities and the polarizations are given by the equations

(c11 - ρv 2 )A1 = 0

(2.2.52a)



(c66 - ρv 2 )A2 = 0

(2.2.52b)



(c55 - ρv 2 )A3 = 0

(2.2.52c)

It is seen that a pure longitudinal wave with velocity vl = √(c11/r) propagates along the x1-axis, and two pure shear waves, one polarized parallel to the x2-axis and the other polarized parallel to the x3-axis, propagate with the velocities v2 = √(c66/r) and v3 = √(c55/r), respectively. Similarly, for propagation along the x3-axis, the equations are

(c55 - ρv 2 )A1 = 0

(2.2.53a)



(c44 - ρv 2 )A2 = 0

(2.2.53b)



(c33 - ρv 2 )A3 = 0

(2.2.53c)

In this case, the longitudinal wave polarized in the x3 direction propagates with velocity v3 = √(c33/r), and the two shear waves with polarizations, respectively, in the x1 and x2 directions propagate with velocities v1 = √(c55/r) and v2 = √(c44/r).

2.2.5 Transformation of Coordinates It is sometimes convenient to choose coordinate axes that are different from the symmetry axes of the material. To write the stress and strain tensors in the new reference frame, let us suppose that the new x′ y′ z′ system is obtained from the xyz system by the orthogonal matrix



axx axy axz    [a] = a yx a yy a yz    azx azy azz 

(2.2.54)

xi′ = aijxj

(2.2.55)

so that we have

Then the stress vector transformation can be written as

sI′ = MIJsJ, I,J = 1,2,3,4,5,6

(2.2.56)

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where [M] is the 6 × 6 transformation matrix 2 2 2   axx 2axy axz 2axz axx 2axx axy axy axz   2 a 2yy a 2yz 2a yy a yz 2a yz a yx 2a yx a yy   a yx   2 2 2 a a a 2 a a 2 a a 2 a a zy zz zy zz zz zx zx zy   zx [M] =  a yx azx a yy azy a yz azz a yy azz + a yz azy a yx azz + a yz azx a yy azx + a yx azy    a a azy axy azz axz axy azz + axz azy axz azx + axx azz axx azy + axy azx  zx xx   axx a yx axy a yy axz a yz axy a yz + axz a yy axz a yx + axx a yz axx a yy + axy a yx   



(2.2.57)

In the same way, the strain vector transformation is given by eI′ = NIJeJ



(2.2.58)

where N is obtained from M by shifting the factor 2 from the upper right-hand corner to the lower left-hand corner. It is noted that [N]−1 = [M]T



(2.2.59)

It then follows that the stiffness matrix [c] transforms as [c′] = [M][c][M]T



(2.2.60)

For future use, we will consider the transformation of the stiffness matrix given by equation (2.2.60) when the coordinates are rotated clockwise by an angle θ about the 3-axis. The coordinate transformation matrix from x1-, x2-, and x3-axes to x1′-, x2′-, and x3′(= x3)-axes is (see Fig. 2.2) cosθ - sinθ 0    [a] =  sinθ cosθ 0     0  0 1  



(2.2.61)

x3(x3´) n

α x2´ –θ

x1´

x1 Fiber direction

FIGURE 2.2  Coordinate transformation.

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The [M] matrix is then given by  cos 2 θ  2  sin θ  0  [M ] =  0  0  1  2 sin 2θ



sin 2 θ cos 2 θ 0 0 0 1 - sin 2θ 2

0 0 1 0 0

0 0 0 cosθ - sinθ

0 0 0 sin θ cosθ

0

0

0

- sin 2θ   sin 2θ  0   0  0    cos 2θ  

(2.2.62)

The transformed stiffness coefficients are then found to be 1  c11′ = c11 cos 4 θ + c22 sin 4 θ +  c12 + c66  sin 2 2θ 2   1  c12′ =  (c11 + c22 ) - c66  sin 2 2θ + c12 (cos 4 θ + sin 4 θ ) 4  c13′ = c13 cos 2 θ + c23 sin 2 θ c16′ =

1 sin 2θ[c11 cos 2 θ - c22 sin 2 θ - (c12 + 2c66 )cos 2θ ] 2

(2.2.63a)

1  c22 ′ = c11 sin 4 θ + c22 cos 4 θ +  c12 + c66  sin 2 2θ 2  c23 ′ = c13 sin 2 θ + c23 cos 2 θ

c26 ′ =

1 sin 2θ[c11 sin 2 θ - c22 cos 2 θ + (c12 + 2c66 )cos 2θ ] 2 c33 ′ = c33



1 c36 ′ = (c13 - c23 )sin 2θ 2

(2.2.63b)

c44 ′ = c44 cos 2 θ + c55 sin 2 θ 1 c45 ′ = (c55 - c44 )sin 2θ 2 c55′ = c44 sin 2 θ + c55 cos 2 θ



1 c66 ′ = (c11 + c22 - 2c12 ) sin 2 2θ + c66 cos 2 2θ 4 c14′ = c15′ = c24 ′ = c25 ′ = c34 ′ = c35 ′ = c46 ′ = c56′ = 0

(2.2.63c)



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It may be noted that the stiffness matrix in this rotated coordinate system has a similar structure as that of a monoclinic material having the 12-plane as the plane of symmetry. Relation (2.2.63) holds for an orthotropic material having the 12- and 13-planes as planes of symmetry. For a plane wave propagating in the plane of x1′–x3′, let a be the angle made by the direction of propagation n with the x3-axis. Then,

n1 = sina cosq, n2 = −sina sinq, n3 = cosa

(2.2.64)

The velocities of the three propagating waves can be obtained from equations (2.2.44)– (2.2.47) in the following manner. Let the angle between the wave propagation direction and the x1-axis be β and the projection of n on the x2–x3-plane be taken as the x3′-axis. Then, the direction cosines of the wave-propagation direction referred to the x1-, x2′-, and x3′-axes are given by

n1 = cos β , n2′′ = 0, n3′′= sin β

(2.2.65)



These can be expressed in terms of a and q as

n1 = sin α cosθ , n2′′ = 0, n3′′= 1 - n12



(2.2.66)

The elements of the Christoffel matrix are given by Γ11 = c11n12 + c55n3′′ 2 Γ12 = 0 Γ13 = (c13 + c55 )n1n3′′ Γ 22 = c n + c44n3′′ 2 66 1

(2.2.67)

2

Γ 23 = 0 Γ 33 = c55n12 + c33n3′′ 2





Equation (2.2.7) then takes the form



 Γ11 - ρv 2  0   Γ 13 

0 Γ 22 - ρv 2 0

A  Γ13   1    0   A2′′ = 0 2 Γ 33 - ρv     A3′′

(2.2.68)

As seen before, the wave with displacement polarized in the x2″ direction is uncoupled from the quasi-longitudinal and quasi-shear waves polarized in the x1–x3″-plane. The velocities of the latter two waves are obtained from equations (2.2.46)–(2.2.48) after replacing n3 by n3″. The polarizations of these waves are given by the eigenvectors of equation (2.2.68). As expected, for a transversely isotropic medium, quasi-longitudinal and quasi-shear waves have displacement polarizations in the plane containing the symmetry axis and the direction of propagation of the wave, whereas the pure shear wave is polarized perpendicular to this plane.

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Fundamentals of Elastic Waves in Anisotropic Media

2.3 Numerical Results and Discussion As an illustrative example, we now consider a plane wave propagating in a fiberreinforced composite medium. Because continuous graphite-fiber-reinforced polymeric matrix materials are widely used in applications, numerical results will be presented for a particular case of graphite–epoxy-composite medium. It is now well established that an elastic homogeneous matrix reinforced by a random distribution of aligned continuous cylindrical elastic fibers can be modeled as a transversely isotropic homogeneous elastic medium when the wavelength of a propagating wave is much longer than the fiber diameter. The effective properties of such a medium are the static effective properties. For a typical graphite-fiber-reinforced epoxy matrix, these anisotropic elastic constants are as follows, in units of 109 N/m2,

c11 = 160.73, c12 = c13 = 6.44, c22 = c33 = 13.92, c55 = c66 = 7.07, c44 = 3.5

(2.3.1)

Here, the x-axis is taken to be parallel to the fiber axis. The density r = 1578 kg/m3. Thus, the phase velocity of the longitudinal wave propagating in the fiber direction is v1L = 10.1 km/s. On the other hand, phase velocities of shear waves polarized in the 3- or 2directions (S or SH directions) and propagating along the 1-direction (fiber direction) are v1S = v1SH = 2.12 km/s—see equations (2.2.52a,b,c). The phase velocities of the longitudinal (L), S, and SH waves propagating along the 3-direction are v3L = 2.97 km/s, v3S = 2.12 km/s, and v3SH = 1.49 km/s, respectively—see equations (2.2.53a,b,c). Equations (2.2.44), (2.2.46), and (2.2.47) can be used to calculate the phase velocities of SH, qL, and qS waves propagating along the direction making an angle a with the x3-axis in the 13-plane. Figure 2.3 shows the polar plot of normalized values of the phase velocities for different angles of propagation a x = 90° − a. Velocities are normalized with respect to v1S . 90

180

1

270

2

3

4

5

0

qL qS SH

FIGURE 2.3  Polar plot of normalized phase-velocity surfaces versus propagation angle (in degrees).

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Elastic Waves in Composite Media and Structures 90

180

0.5

1.0

1.5

0

qL qS SH 270

FIGURE 2.4  Polar plot of normalized slowness surfaces versus propagation angle (in degrees).

Normalized VgSH-z, -VgQL-z, -VgQS-z

n The slowness s = is shown as a polar plot in Fig. 2.4. The x1 and x3 components of v/v1S the group velocities of the qL, qS, and SH waves are calculated using equations (2.2.50) and (2.2.51) and are plotted in Fig. 2.5. Note the characteristic cusps appearing in the group velocity plot of the qS wave. It is noted that the directions of group-velocity propagation are different than the phase-velocity propagation directions, a. As discussed above, the group-velocity direction is normal to the slowness surface at the point. The skew angle measuring the difference between the angles made by the phase and group 1.5

qL qS SH

1.0

0.5

0.0

0

3 1 2 4 Normalized VgSH-x, -VgQL-x, -VgQS-x

5

FIGURE 2.5  x–z plot of normalized group velocities.

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35

Fundamentals of Elastic Waves in Anisotropic Media 80 qL qS SH

Skew Angle

60

40 20

0

–20

10

20

30

40 50 60 Propagation Angle

70

80

90

FIGURE 2.6  Skew angle versus propagation angle (in degrees).

velocities with the x3-direction is shown in Fig. 2.6. It can be seen that this can be quite large in this particular case of strong anisotropy. A program for calculating the results shown graphically in Figs. 2.3–2.6 is included in the disc accompanying this book. The readers are encouraged to use the data for the material properties for their particular applications in this program to generate the plots for their use. It should be emphasized that this program can be used for orthotropic materials as well for propagation in a plane of symmetry. Furthermore, for a transversely isotropic material, the phase velocity, slowness, and group velocity can be obtained by using equations (2.2.66)–(2.2.68) for propagation in an arbitrary plane.

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3 Periodic Layered Media 3.1 Introduction ...............................................................................37 3.2 Description of the Problem..................................................... 38 Stiffness Method • Effective Modulus Method 3.3 Numerical Results .....................................................................51 Isotropic Laminates • Anisotropic Laminates 3.4 Remarks ..................................................................................... 56

3.1 Introduction In Chapter 2 (Section 2.2), plane waves in a homogeneous anisotropic elastic medium were considered. Equations governing the phase and group velocities were derived. Since the medium is homogeneous, these velocities are independent of frequency, i.e., waves are nondispersive. If the medium is nonhomogeneous, e.g., a composite medium made up of a distribution of fibers or particles of different material properties embedded in a matrix of some other material properties, then an incident plane wave propagating in the medium will be scattered, and constructive and destructive interferences will take place. If the wavelength is much longer than the characteristic dimensions of the fibers or inclusions, then a coherent plane wave will emerge with frequency-dependent phase velocity and with amplitude decaying with distance of travel. Thus, the wave will be dispersive and attenuative. A similar phenomenon takes place when a plane wave propagates through a planelayered medium with layers having different material properties. Here, an incident plane wave will be reflected and refracted (scattered) at the plane interfaces of adjacent layers. Again, if the wavelength is large compared with the thicknesses of the layers, then a plane wave will propagate through the composite medium, having velocity that will be dependent upon frequency and amplitude that will be decaying. If the medium has a periodic layered structure, Floquet wave theory leads to a dispersion equation governing various modes (propagating and evanescent) of propagation of harmonic waves. In the long-wavelength limit, this equation yields three speeds of wave propagation that correspond to those for an effective homogeneous anisotropic medium. The dispersion equation defines a surface in the frequency–wave-number space that has the characteristic feature of exhibiting passing and stopping bands found in wave propagation in periodic 37

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Elastic Waves in Composite Media and Structures

media. A review of significant contributions in this important field has been presented in Chapter 1 (Section 1.1). See also Nayfeh (1995), Lee (1972), Braga and Herrmann (1988, 1992), and Ting and Chadwick (1988). In this chapter, we present a model for studying dispersion of elastic waves in an infinite medium composed of periodic layers of orthotropic material. For the purpose of generality, three-dimensional motion is dealt with first. Results for the special cases of antiplane and plane-strain motion are given later. A stiffness method is used here for its general applicability to the case when each layer has monoclinic symmetry about its bounding planes. An exact treatment of the problem when each layer is transversely isotropic is given at the end of the chapter. In the stiffness method, an interpolation function is assumed to represent the displacement within each layer in terms of a discrete set of generalized coordinates. These generalized coordinates are the displacements and tractions at the bounding interfaces of the layer, thus ensuring continuity of displacements and tractions across the interfaces between adjacent layers. By applying Hamilton’s principle and using Floquet theory, the dispersion equation is obtained as a generalized algebraic eigenvalue problem whose solution yields the frequency–wave-number relation as well as the variation of stresses and displacements. As will be seen in later chapters, the method is adapted well for bounded plates and cylinders.

3.2 Description of the Problem Consider a stack of plane layers having orthotropic symmetry with a common plane of symmetry that is taken as the x1–x2-plane. The layers are assumed to be rigidly bonded to one another. A global Cartesian coordinates system (x1, x2, x3) is chosen so that the x3-axis is normal to the layers. Let superscript (i) identify the variables of interest associated with the ith layer, which is assumed to be orthotropic, having symmetry planes parallel to the global coordinate planes. It will be convenient to use a local Cartesian coordinate system in the ith layer with origin at the midplane of the layer and axes parallel to the global coordinate axes. For the purpose of keeping the algebra simple, it will be assumed that the stack is a two-layer periodically laminated body of unbounded extent. Any two adjacent laminates in the body then comprise a unit cell (see Fig. 3.1). We are concerned with a harmonic wave propagating in an arbitrary direction through such a medium.

3.2.1 Stiffness Method Let the wave-propagation direction make an angle a with the x3-axis and its projection on the x1–x2-plane make an angle −θ with the x1-axis (see Fig. 2.2). A new global coordinate system (x1′, x2′, x3′) is chosen such that the transformation from the unprimed system to the primed system is given by the matrix [a] in equation (2.2.61). Thus, the wave vector lies in the plane of x1′–x3′, and the displacement will be independent of x2′. Note that the displacement will have all of the three components. The two adjacent laminae comprising a typical unit cell Cn have elastic constants given, respectively, by cij(i) and cij(i+1), thicknesses 2h(i) and 2h(i+1), and densities r (i) and

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39

Periodic Layered Media d1 = 2h(1) d2 = 2h(2)

B z y

L

x

FIGURE 3.1  Geometry of the unit cell.

r (i+1). For the ith and (i + 1)th laminae, assuming orthotropic symmetry, the stress relations are given by



σ xx ( j )  c ( j ) c ( j ) c ( j ) 0 0 12 13   11  j ( )  σ yy c ( j ) c ( j ) c ( j ) 0 0 22 233   12  σ ( j )   ( j ) ( j) ( j) 0 0  zz  c13 c23 c33 =   0 c44( j ) 0 0 σ yz ( j )   0    σ zx ( j )   0 0 0 c55( j ) 0    σ ( j )   0 0 0 0 0  xy 

 e xx ( j )    ( j) 0  e yy      0  ezz ( j )    (j = i, i + 1)  0  γ yz ( j )    0  γ zx ( j )    c66( j )  γ xy ( j )    0

(3.2.1)

1 where s ij(j), eij(j), and (γij(j)) are the stress and strain components. In the transformed 2 (primed) coordinates system, the stiffness coefficients cij(j) appearing in equation (3.2.1) will be transformed as given by equation (2.2.60). We will write the transformed stress– strain relation for the jth layer as

σ ij ′ = Dijkl ekl ′



(3.2.2)

where Dijkl = cijkl ′. Note that the stiffness matrix Dijkl has the same form as that for a monoclinic material having the x1 – x2-plane as the symmetry plane. Thus, the dispersion equation derived in the following holds also for layers of monoclinic materials. The displacement components in the ith and (i + 1)th laminae will be approximated by interpolation polynomials. To this end, it will be convenient to divide each laminate into N sublayers of equal thickness 2h without loss of generality, where N is a positive

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Elastic Waves in Composite Media and Structures

integer. The displacement components in the mth sublayer will be approximated by the following cubic polynomials (omitting the prime.) D D ∂w  u = um f1 + um+1 f 2 +  44 χm − 45 τ m − m  f3 ∆ ∂x   ∆



D D ∂w  +  44 χm+1 − 45 τ m+1 − m+1  f 4 ∆ ∂x   ∆

(3.2.3a)

D  D υ = υm f1 + υm+1 f 2 +  55 τ m − 45 χm  f 3 ∆  ∆ 



D  D +  55 τ m+1 − 45 χm+1  f 4 ∆ ∆  

(3.2.3b)

σ D ∂u D ∂υ  w = wm f1 + wm+1 f 2 +  m − 13 m − 36 m  f 3 D D ∂ x D33 ∂x   33 33



σ D ∂u D ∂υ  +  m+1 − 13 m+1 − 36 m+1  f 4 ∂ D D x D33 ∂x   33 33

(3.2.3c)

where fn (n = 1, 2, 3, 4) are cubic polynomials in the local coordinate z and are given by 1 f1 = (2 − 3η + η 3 ) 4 1 f 2 = (2 + 3η − η 3 ) 4 h f 3 = (1 − η − η 2 + η 3 ) 4

(3.2.4)

h f 4 = (−1 − η + η 2 + η 3 ) 4

and ∆ = D44 D55 − D45 2

(3.2.5) 1 1 h = (z m+1 − z m ) = (z m+1′ − z m ′ ) 2 2 In the above, h = z/h, z = x3′, and um, υm, wm, χm, t m, s m are the values of u′, υ′, w′, s zx′, syz′, s zz′, respectively, at the mth interface (m = 1, 2, … , N). Note that 2Nh = 2h(i). This choice of interpolation preserves the continuity of the displacement and traction at the interface between adjacent layers. Similarly, the (i + 1)th laminate is divided into N′ sublayers of equal thickness, and the displacement components within each sublayer are expressed in (i +1) terms of the same polynomials of the local variable h = z/h′, with h ′ = h . N′

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Periodic Layered Media

The equations governing the nodal generalized coordinates um, υm, wm, χm, t m, and s m (m = 1, … , N + 1) are obtained by the application of Hamilton’s principle. For motion in the plane of x1′–x3′, these generalized coordinates are functions of x1′ and t. Now, assuming that the generalized coordinates have the form



um ( x1′ , t )     χ (x ′ , t )    m 1   υm ( x1′ , t )   = {qm0 } exp(ikx1′ − iωt )  τ m ( x1′ , t )    w ( x ′ , t )    m 1   σ m ( x1′ , t ) 

(3.2.6)

where {qm0 } is a 6 × 1 constant matrix. The potential and kinetic energies for the mth layer are obtained by integrating over the layer thickness 2h and the wavelength L. These are V



(m )

1 = 2

L h

∫ ∫ ({e } [D]{e})dzdx ′ T

1

0 −h

ω2 T (m ) = 2

(3.2.7)

L h

∫ ∫ ρ({u} {u})dzdx ′ T

1

0 −h



where   {u}T = 〈u υ w〉   {e}T = 〈exx eyy ezz γyz γzx γxy〉



 D11   D13 [D] =  0  0   D16

D16   D36  0 D44 D45 0   0 D45 D55 0   D36 0 0 D66 

D13 0 0

D33 0 0

(3.2.8)

The overbar denotes complex conjugate, and the superscript T denotes transpose. Using equation (3.2.6) in the displacement interpolation relations (3.2.3a–c), and in turn in the strain-displacement relations, the strain vector {e} is obtained in terms of amplitude vectors of the generalized nodal quantities at the mth and (m + 1)th interfaces of the mth layer. Substituting this in the expression for V(m) in equation (3.2.7), one obtains V (m ) =

Lh {r }T  k 4 [e1 ] − ik 3 ([e2 ] − [e2 ]T ) − k 2 ([e3 ] + [e3 ]T − [e4 ]) + ik([e5 ] − [e5 ]T ) + [e6 ] {rm } 2 m 



(3.2.9)

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Elastic Waves in Composite Media and Structures

where {rm }T = 〈um χm υm τ m wm σ m um+1 χm+1 υm+1 τ m+1 wm+1 σ m+1 〉 are the amplitudes of the nodal displacement-stress vector. Similarly, equation (3.2.7) gives T (m ) =



Lh 2 ω {rm }T  k 2[c2 ] − ik [c3 ] − [c3 ]T + [m] {rm } 2

)

(

(3.2.10)

The matrices [e1], …, [e6] and [c2], [c3] are given below. We first define the first six columns of the following matrices   0   [N1 ] =  0   − f D13  3 D33





  f1   [N 2 ] =  0  0  



 0  0   [a] =  0    f1′   0 

  f1   − f ′ D13  3D 33  [b] =  0    0    0 

f3

 …   …  … 

0

0

0

− f3

0

0

0

0

0

0

0

0

0

0

0

0

0

f1

f3 D33

0

D44 ∆ D45 − f3 ∆ f3

0

− f3

D36 D33

D45 ∆ D55 f3 ∆

− f3

0 f1 0

0

 …   …  …    …  …   …   …   … 

0

0

0

0

0

0

0

0

f1′

f 3′ D33

0

0

0

0

0

0

0

D45 ∆

0

0

0

0

0

D − f 3′ 45 ∆ D44 f 3′ ∆

f1′ 0

0

D44 ∆

D f 3′ 55 ∆ D45 − f 3′ ∆

0 0

− f3

D36 D33

0

− f 3′

0

0

0

0

0

0

0

0

( f1 − f 3′)

f3 D33

0

0

− f3

D45 ∆

f1

f3

D55 ∆

 …  …   …   …   … 

(3.2.11)

(3.2.12)

(3.2.13)

(3.2.14)

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  0   0   [d] =  0   D  − f 3 13 D33    0 

0

0

0

− f3

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

− f3

0

D36 D33

0

 …  …   …   …   … 

(3.2.15)

where superscript prime (′) denotes differentiation with respect to z and ∆ = D44 D 55 − 2 . Note that columns 7 to 12 in the matrices [N ], [N ], [a], [b], and [d] can be obtained D45 1 2 from corresponding columns 1 to 6 by replacing f1 and f3 by f2 and f4, respectively. Then, [c2], [c3], [m], and [e1], … , [e6] are given by 1

1

∫

1

∫

∫

[c2 ] = ρ[N1 ]T [N1 ]dη , [c3 ] = ρ[N1 ]T [N 2 ]dη , [m] = ρ[N 2 ]T [N 2 ]dη

−1

−1

(3.2.16)

−1

and 1

∫

1

∫

1

∫

[e1 ] = [d] [D][d]dη , [e2 ] = [d] [D][b]dη , [e3 ] = [d]T [D][a]dη ,

T

T

−1

−1

−1

1

1

1

∫

∫

(3.2.17)

∫

[e4 ] = [b]T [D][b]dη , [e5 ] = [b]T [D][a]dη , [e6 ] = [b]T [D][b]dη

−1

−1



−1

Thus, for the mth sublayer, we can write



V (m ) − T (m ) =

L h[r ]T [k(m ) − ω 2m(m ) ][rm ] 2 m

(3.2.18)

where k(m) and m(m) are given by the quantities within the large square brackets in equations (3.2.9) and (3.2.10). Now, assembling all the elements of the N layers in the ith laminate, we obtain the global expressions for the potential and kinetic energies of the ith laminate as



V (i ) − T (i ) =

L h{r (i ) }T [K (i ) − ω 2 M (i ) ]{r (i ) } 2

(3.2.19)

where

{r (i ) }T = 〈u1 χ1 υ1 τ 1 w1 σ 1 … uN +1 χ N +1 υ N +1 τ N +1 w N +1 σ N +1 〉



(3.2.20)

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Elastic Waves in Composite Media and Structures N

K (i ) =

∪k

(3.2.21)

(m )



m =1 N

M (i ) =

∪m m =1

(3.2.22)

(m )



Using the same procedure as described above, the global expressions for the potential and kinetic energies of the (i + 1)th laminate can be derived. It is found that



V ( j) − T ( j) =

L h ′{r ( j ) }T [K ( j ) − ω 2 M ( j ) ]{r ( j ) } (j = i + 1) 2

(3.2.23)

{r ( j ) }T = uN +1 χ N +1 υ N +1 τ N +1 w N +1 σ N +1 …

uN + N ′ +1 χ N + N ′+1 υ N + N ′+1 τ N + N ′+1 w N + N ′+1 σ N + N ′ +1

(3.2.24)

In writing equations (3.2.19) and (3.2.23), use has been made of the continuity of displacement and traction at the interfaces between the sublayers and the ith and (i + 1)th laminates. To obtain the dispersion equation for a plane wave propagating in the periodic laminate, the Floquet periodicity condition must be imposed. This requires uN + N ′+1 χ N + N ′+1 υ N + N ′+1 τ N + N ′+1 w N + N ′+1 σ N + N ′ +1 uN + N ′+ 2 χ N + N ′+ 2 υ N + N ′+ 2 τ N + N ′+ 2 w N + N ′ + 2 σ N + N ′+ 2

= u1 χ1 υ1 τ 1 w1 σ 1 u2 χ 2 υ2 τ 2 w 2 σ 2

T

e ikz d

(3.2.25)

where d = 2(h(i) + h(i+1)) and kz is the Floquet wave number. Applying Hamilton’s principle to the total V – T of the unit cell and using equation (3.2.24), the equilibrium equations for nodes 2 to N + N′ + 1 can be written. This yields the dispersion relation as an algebraic eigenvalue problem: [ A]{R} = ω 2[B]{R}



(3.2.26)

where

{R}T = 〈u1 χ1 υ1 τ 1 w1 σ 1 … uN + N ′ χ N + N ′ υ N + N ′ τ N + N ′ w N + N ′ σ N + N ′ 〉

(3.2.27)

and [A] and [B] are 6(N + N′) × 6(N + N′) complex matrices, whose elements depend on the material properties and thickness of the unit cell and on the wave numbers k and kz. Equation (3.2.26) simplifies when the wave is moving in the x1–x3- or x2–x3-plane. This occurs when θ = 0 or p/2. In either case, waves polarized in the plane of symmetry (plane strain) and perpendicular to the plane of symmetry (antiplane strain) are

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Periodic Layered Media

uncoupled. These were studied by Delph et al. (1978, 1979, 1980) and Shah and Datta (1982). These special cases are discussed in the following subsections. 3.2.1.1 Antiplane-Strain Motion Consider the case when θ = 0, u = w = 0, and υ(x, z) ≠ 0, where (x, z) = (x1, x3). Then, equation (3.2.3b) takes the form



υ = υm f1 + υm+1 f 2 +

1 1 fτ + fτ D44 3 m D44 4 m+1



(3.2.28)

Now, assuming that υm    = q10m exp(ikx1 − iω t ) τ m 

{ }



(3.2.29)

where {q01m} is a constant 2 × 1 matrix, and the potential and kinetic energies—see equation (3.2.7)—are found to be V (m ) =

T (m ) =

1 2

L h

∫ ∫ ({e } [D ]{e})dzdx T

1

1

0 −h

ω 2

2

(3.2.30)

L h

∫ ∫ ρυυdzdx

1



0 −h

The matrices [D1] and {e} are given by  D44 0    D1  =  0 D66    γ yz    e =  γ xy 

(3.2.31)

{}



Equations (3.2.9) and (3.2.10) now take the forms



V (m ) =

Lh {r }T [k 2[k1 ] + [k2 ]]{r1m } 2 1m

T (m ) =

Lh 2 ω {r1m }T [m]{r1m } 2

(3.2.32)

where {r1m }T = 〈υm τ m υm+1 τ m+1 〉

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Elastic Waves in Composite Media and Structures

are the amplitudes of the nodal displacement stress vector and 1

∫

 k1  = D66 {N }{N }T dη −1 1

∫

[k2 ] = D44 {N ′}{N ′}T dη −1 1

∫

[m] = ρ{N }{N }T dη −1

{N }T = 〈 f1 f 3/D44 f 2 f 4 /D44 〉



As before, for the mth sublayer, we can write V (m ) − T (m ) =



Lh {r }T [[k(m ) ] − ω 2[m(m ) ]]{r1m } 2 1m

(3.2.33)

where [k(m)] is the expression within the large square brackets in equation (3.2.32). Now, assembling the elements of the N layers in the ith laminate, we get the global matrix expressions for V(i) − T(i) as V (i ) − T (i ) =



L h r1 (i ) 2

{ }

T

{ }

(3.2.34)

 K1(i ) − ω 2 M1(i )  r1(i )

where

{r } (i ) 1



T

= 〈υ1 τ 1 υ2 τ 2 … υ N τ N υ N +1 τ N +1 〉

(3.2.35)



and K1(i) and M1(i) are 4(N + 1) × 4(N + 1) matrices. In the same manner, it is found that, for the jth laminate, V ( j) − T ( j) =



{ }

L h ′ r1 ( j ) 2

T

{ }

(3.2.36)

 K1( j ) − ω 2 M1( j )  r1( j )  

where

{r } ( j) 1

T

= 〈υ N +1 τ N +1 υ N + 2 τ N + 2 … υ N + N ′ τ N + N ′ υ N + N ′+1 τ N + N ′+1 〉



(3.2.37)

The dispersion equation for a plane SH wave propagating in the periodic medium is now obtained by imposing the Floquet condition in equation (3.2.25) in the form [ A1 ]{R1 } = ω 2[B1 ]{R1 }



(3.2.38)



where

{R1 }T = 〈υ1 τ 1 … υ N + N ′ τ N + N ′ 〉



(3.2.39)

and [A1] and [B1] are 2(N + N′) × 2(N + N′) matrices.

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3.2.1.2 Plane-Strain Motion If the wave is propagating in the x1–x3-plane and the polarization is also in the same plane, then we have u = u( x1 , x 3 , t ), υ = 0, w = w( x1 , x 3 , t )



(3.2.40)



The expressions for the potential and kinetic energies are obtained from equations (3.2.7) and (3.2.8) by taking

{u}T = 〈u, w 〉 {e}T = 〈e xx e zz γ yz 〉

(3.2.41)

 D11 D13 0    [D] =  D13 D33 0  0 0 D  55  



The matrices [c2], [c3], [m], and [e1], …, [e6], are as defined by equations (3.2.16) and (3.2.17), where  0 0 − f3 0 0 0  [N1 ] =  D D 13 0 0 − 13 f 4 0  − D f3 0 D 33 33 



  f1 [N 2 ] =  0 

  0  0  [a] =  0 0   f 3′ f′  1 D55  f1    D [b] =  − f 3′ 13 D  33   0 

0 f1′ 0 f3

f3 D55

0 f1

0

f2

f4 D55

0

f3 D33

0

0

f2

0

0

0

0

0

f 2′

0

f 2′

f 4′ D55

0

0

0

0

0

0

0

( f1 − f 3′ )

D55

0

0

0 f 3′

D33

− f4

f3

D33

f2 − f 4′

D13

D33

0

  0  f 4′   D33   0   f4 D55

0   0   0   f 4  D33 

0

0

0

0

( f 2 − f 4′)

 0    0    f4  D33 

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Elastic Waves in Composite Media and Structures

   0 [d] =  0  D13   − f3 D 33 



0 0

− f3 0

0 0

0

0

0

− f4

0 0 D13

D33

0 0

− f4 0

0

0

  0 0   0 

where the superscript prime (′) denotes differentiation with respect to z and the displacement–traction vector is {rm }T = 〈um χm wm σ m um+1 χm+1 wm+1 σ m+1 〉



The dispersion equation is of the same form as equation (3.2.26) with

{R}T = 〈u1 χ1 w1 σ 1 … uN + N ′ χ N + N ′ w N + N ′ σ N + N ′ 〉



(3.2.42)

The matrices [A] and [B] in equation (3.2.6) are now 4(N + N′) × 4(N + N′). These are obtained in pretty much the same way as outlined above for the antiplane case.

3.2.2 Effective Modulus Method For long wavelengths in comparison with the thickness of the unit cell, the periodic structured medium can be treated as an effective homogeneous orthotropic medium having symmetry planes coincident with the symmetry planes (x1–x3 and x2–x3) of the layers comprising the unit cell.* Postma (1955) developed a technique for deriving the effective static moduli of a periodic stratified medium with isotropic layers. In the following, a derivation generalizing Postma’s approach to orthotropic layers is presented. Consider an elementary rectangular parallelepiped of n unit cells having height H = n(d1 + d2), where n is an integer, and length and width are L and B, respectively (see Fig. 3.1). Note that d1 = 2h(1) and d2 = 2h(2). Let a normal traction s zz be applied to faces perpendicular to the x3 (= z)-axis and there are no shear tractions acting on these faces. 2 , Simultaneously, let there be normal tractions (but no shear tractions) σ 1xx and σ xx respectively, applied to the faces perpendicular to the x1 (= x)-axis of the layers 1 and 2 of 2 are applied to the faces perpendicular the unit cell. Also, normal tractions σ 1yy and σ yy 2 =e 1 2 to the x2 (= y)-axis. These latter tractions are such that e1xx = e xx xx and e yy = e yy = e yy , so that displacement is continuous at the interfaces. For the anisotropic layer 1, equation (3.2.1) gives



1 1 σ 1xx  c11 c12 c113  e xx        1   1 1 1   = c c c σ  yy  12 22 23 e yy       1 1  1 c123 c33 σ 1zz  c13  e zz 

(3.2.43a)

* Note that if the layers do not have x – x and x – x planes as common symmetry planes, then the 1 3 2 3 effective medium will have more general anisotropy.

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Periodic Layered Media

Corresponding relations for layer 2 is obtained by replacing the superscript 1 by 2 in equation (3.2.43a). That set of equations will be denoted as equation (3.2.43b) (not shown). The weighted averages of the normal tractions acting on the faces perpendicular to the x- and y-axes are

σ xx = σ yy =



2 d σ 1xx d1 + σ xx 2 d1 + d2

(3.2.44)

2 d σ 1yy d1 + σ yy 2

d1 + d2



Since the traction normal to the plane of xy remains the same s zz, we obtain by using equations (3.2.43a) and (3.2.43b) in equation (3.2.44)



e xx  1 + d c2 1 2  1 + d c2  σ xx  d1c11 d1c12 2 12 d1c13 d2 c13 2 11     e yy     1 2 1 2  1 + d c2 (d1 + d2 ) σ yy  = d1c12  2 12 d1c 22 + d2 c 22 d1c 23 d2 c 23    1 ezz      2 1 2 1 2 1 σ zz  d1c13 + d2 c13 d1c23 + d2 c23 d1c33 d2 c33    ezz2 

(3.2.45)

Defining ezz =



d1e1zz + d2 ezz2 d1 + d2

(3.2.46)



and using the last two equations of (3.2.43a) and (3.2.43b), it is found that



e1zz = ezz2 =

(c

2 13

)

(

)

1 e + c 2 − c1 e + (1 + d )c 2 e − c13 xx 23 23 yy 33 zz

)

(

1 + dc 2 c33 33

(

)

(3.2.47a)

1 e − d c 2 − c1 e + (1 + d )c 1 e −d c132 − c13 c33 xx 23 23 yy zz

c + dc 1 33

where d=



2 33

(3.2.47b)

d1 . d2

Substituting the relations in equations (3.2.47a,b) in equation (3.2.45), we obtain the normal-stress–normal-strain relationship given by



σ xx   c11 c12 c13  e xx            c c σ = c  yy  12 22 23 e yy       σ zz   c13 c23 c33  ezz 

(3.2.48)

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Elastic Waves in Composite Media and Structures

where



c11 =

2 1 1 1 + dc 2 − d c 2 − c1 dc11 + c112 c33 33 13 13    D

c12 =

1 1 2 1  2 + d c1 − c 2 dc + c 2 c1 + dc33 13 c23 − c23  13 D  12 12 33

c13 =

1 dc1 c 2 + c 2 c1 (1 + d ) D 13 33 13 33

c23 = c33 =

) (

)(

(

c22 =



)(

(

)

) (

)(

)

(

)( c

(dc

1 22

(c

c + dc c

+c

2 22

1 2 33 23

1 33

1 2 23 33

D

+ dc

2 33

D

) − d (c

1 31

−c

2 31

)

2

)



(3.2.49)

)(1 + d )

(1 + d ) c c D

2 1 2 33 33

where

(

1 + dc 2 D = (1 + d ) c33 33



).

The derivation for the effective shear coefficients follows the same steps as for the case of isotropic layers derived by Postma (1955). If a tangential traction s xz is applied to the faces perpendicular to the z-axis, we obtain (d1 + d2 )γ xz = d1γ 1xz + d2γ xz2





(3.2.50)

1 γ 1 = c 2 γ 2 , it is found that Since σ 1xz = c55 xz 55 xz

σ xz =



1 c2 (d + 1)c55 55 γ xz 2 1 dc55 + c55

(3.2.51)

Thus, c55 =



1 c2 (d + 1)c55 55 2 + c1 dc55 55

(3.2.52)

In the same way, the shear stress s yz is related to γyz by the equation

σ yz = c44γ yz

(3.2.53)



where



c44 =

2 (d + 1)c144 c44 2 + c1 dc44 44



(3.2.54)

Finally, to find the effective shear coefficient c66, a tangential force σ 1xy Bd1 is applied 2 Bd is applied to the to the face perpendicular to the y-axis of layer 1, and a force σ xy 2

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Periodic Layered Media

2 corresponding face of layer 2. Since γ 1xy must be equal to γ xy in order to maintain continuity of the displacement, it follows that 1 γ σ 1xy = c66 xy 2 = c2 γ σ xy 66 xy



(3.2.55)

Now, by taking s xy as the average tangential traction, we have



σ xy =

1 + c2 dc66 66 γ xy d +1

(3.2.56)

1 + c2 dc66 66 d +1

(3.2.57)

Thus,

c66 =

This completes the derivation of the stiffness coefficients of the effective homogeneous elastic medium representing the two-layer periodic medium. As noted before, these effective static moduli are valid within the limit of long wavelengths. Within that limit, the effective density is given by

ρ=

d ρ1 + ρ2 d +1

(3.2.58)

The velocities of plane-wave propagation in the effective homogeneous orthotropic medium can be calculated using the formulae derived in Chapter 2. Note that the velocities are frequency independent, and so the medium is nondispersive. As will be shown in the following section, the numerical results obtained by solving the dispersion equation (3.2.26) tend to these limiting values when the wavelength becomes very large. However, depending upon the contrast between the material properties of the layers, dispersion can be exhibited earlier or later. In the following section, numerical results are presented for two particular cases as examples. First, we consider a laminated medium composed of isotropic laminae. This case was considered by an effective stiffness model by Sun et al. (1968b) and Achenbach (1976). An exact solution for the antiplane-strain case was presented by Sun et al. (1968a). Later, Delph et al. (1978, 1979, 1980) presented exact solutions to the antiplane- and plane-strain cases using Floquet theory. Their results are found to agree with the solutions of equation (3.2.26). As a second example, we consider a periodic layer of aluminum sandwiched between boron-fiber-reinforced aluminum layers.

3.3 Numerical Results Consider the unit cell composed of two layers of orthotropic materials having x–z- and y–z-planes as planes of symmetry. For given material and geometric parameters, equation (3.2.26) relates frequency to three wave numbers: k x, ky, and kz . The roots of this

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Elastic Waves in Composite Media and Structures

equation define a surface in frequency–wave-number space, which is the dispersion surface. This surface is in general discontinuous at kz = np/d, n = 1, 2, …. These planes divide the surface into Brillouin zones. The equation admits complex and real wave numbers kz . At the ends of the Brillouin zones, except when k = k 2 + k 2 = 0, there are complex x y branches. Thus the surface can be interpreted in terms of passing and stopping bands. When k = 0, the dispersion equation yields three 2(N + N′) × 2(N + N′) equations. One of these equations is the dispersion equation for longitudinal (P) waves propagating normal to the layers, while the other two are for two shear waves (SV and SH) propagating normal to the layers.

3.3.1 Isotropic Laminates The material and geometric parameters of this two-layered periodic composite are: ε = 1 / c 2 = 50, and r /r = 3. The Poisson ratios are taken to be v1 = 0.35 h(1)/h(2) = 4, γ = c55 1 2 55 and v2 = 0.3. Each laminate was subdivided into two and then three laminae, thus creating fourand six-layered periodicity. It was found that the results for the lower modes did not appreciably change by increasing the number of layers. Results of four-layered periodicity are shown below. These results are found to agree with those obtained by Delph et al. (1978, 1980), except for higher modes at high frequencies. Nondimensional quantities used for the following figures are defined as 2 2h( 2 ) kz 2h( 2 ) kx 2h( 2 )ω c55 , ζ= , Ω= / π π π ρ2

(3.3.1) Figure 3.2 shows the variation of Ω with Re(h) for different values of z for antiplanestrain motion. First four Brillouin zones are shown. Note the discontinuities in the dispersion curves at the ends of the Brillouin zones. Figure 3.3 shows both real and complex branches for a portion of the surface in the z = 0 plane. Here, the imaginary h-axis has been plotted on the real plane for clarity. Along the complex branches, the real part of h remains constant, while the imaginary part varies. Dispersion curves for P and SV waves propagating normal to the layering are sketched on an extended zone scheme in Fig. 3.4. The complex branches originating at the ends of the Brillouin zones are also included. The real branches for the SV and P waves for different values of z are shown in Figs. 3.5 and 3.6, respectively. These figures show that the interpretation of a plane wave propagating in an arbitrary direction in a periodic medium is not meaningful except in the first Brillouin zone for small values of h and z. Figure 3.7 depicts variation of the normalized phase velocities of the lowest SV and P waves propagating normal to the layers as functions of the normal2 /ρ and x = k(2h(1)). Results predicted by the effectiveized wavelength. Here, β = c/ c55 2 modulus method and the effective-stiffness model proposed by Sun et al. (1968b) are also shown on this plot. It is seen that the results derived by the present method and the exact solutions are the same. Waves become more dispersive with increasing rigidity ratio (γ). The effective-stiffness model predicts less dispersion. The range of validity of both the effective-modulus and the effective-stiffness models is restricted to very low values of x.

η=



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Present analysis Delph et al. (1978)

5.0

ζ = 1.5 1.0

4.9 ζ = 1.5 3.0

1.0

Ω

ζ = 1.5

0.5

1.0 2.0

0.5

0.5

ζ = 1.5

0

1.0 1.0

0

0.5

0

0 0

0.2

0.4

0.6

0.8

Re(η)

FIGURE 3.2  Curves of constant z on the antiplane-strain dispersion surface. (Reprinted by permission from Shah and Datta 1982, Fig. 3.)

3.3.2 Anisotropic Laminates The second example considered is a boron–aluminum composite that was studied by Datta et al. (1983). This is a stack of alternating layers of uniaxial boron-fiber-reinforced aluminum and thin aluminum layers. Material and geometric properties of the layers are listed in Table 3.1. TABLE 3.1  Material and Geometric Properties of the Layers in the Unit Cell Elastic Constants (1011 N/m2) Layer 1 2 3 4

Thickness 6.0 6.0 0.5 0.5

Density (g/cm3)

c11

c22 = c33

c12 = c13

c23

c55 = c66

2.6907 2.6907 1.1070 1.1070

1.8860 1.8860 1.1070 1.1070

0.5850 0.5850 0.5730 0.5730

0.7634 0.7634 0.5730 0.5730

0.6019 0.6019 0.2670 0.2670

r 2.52 2.52 2.702 2.702

Note: Periodicity N + N′ = 4, d = 12.

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Elastic Waves in Composite Media and Structures 2.5

Real branch Imaginary branch ζ=0

2.0

Ω

1.5 0

1.0

0.1

0.2

Im(η)

0.5

0

Im(η)

0

0.1

0.2

0.2

0.3

0.4 Re(η)

0.1

0.5

0.6

0.7

0.8

FIGURE 3.3  Antiplane-strain dispersion in z = 0 plane with complex branches. (Reprinted by permission from Shah and Datta 1982, Fig. 5.)

Real branch Imaginary branch

2

0.05 0.1

Ω

ζ=0

1

P 0

S

0.05 0.1 0.2

0.4

0.6

Re(η)

FIGURE 3.4  P and S surfaces in z = 0 plane with complex branches. (Reprinted by permission from Shah and Datta 1982, Fig. 7.)

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Periodic Layered Media Present analysis Delph et al. (1980)

3.0

ζ = 1.0

0.5

2.0 Ω

ζ = 1.0

0.3

0.5 0.3

1.0

0.1 0

0.1 0 0

0.2 Re(η)

0.4

FIGURE 3.5  Curves of constant z on the S-surface. (Reprinted by permission from Shah and Datta 1982, Fig. 8.)

Here, the elastic constants are in units of 1011 N/m2 and the density is in g/cm3. The boron-fiber-reinforced aluminum layer is modeled as transversely isotropic. The dispersion of the lowest SH, SV, and P waves propagating normal* to the layers is depicted in Fig. 3.8. Also shown in this figure are the predictions of the effective modulus and effective stiffness methods. It appears that in this example the range of validity of the effective modulus method is broader than observed in the first example. This is because the boron-fiber-reinforced aluminum layer can be modeled as a homogeneous transversely isotropic medium over a broad range of frequency, and dispersion in the layered composite considered here is dominated by the reinforced layer. This observation is validated by the results shown in the following figures (Figs. 3.9–3.12), where propagation in different directions is considered. These figures show the variation of Ω with k , where k = ξ/π d . For waves propagating normal to the layers, Fig. 3.9 shows that only the P wave shows some dispersion when k > 0.08. The phase velocities of SV and SH waves are For the definition of the angles θ and a, see Fig. 2.2.

*

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Elastic Waves in Composite Media and Structures

ζ = 1.0 Present analysis Delph et al. (1980)

3.0

0.5 ζ = 1.0

Ω

2.0

0.3

0.5

0 0.1

0.3 1.0 0.1 0 0

0.2 Re(η)

0.4

FIGURE 3.6  Curves of constant z on the P-surface. (Reprinted by permission from Shah and Datta 1982, Fig. 9.)

constant over the range 0 < k < 0.1. The effect of the thin aluminum layer is to cause a small difference in the velocities of SV and SH waves. When the wave propagates at an angle to the fiber direction in the plane of x–z (Fig. 3.10), the P wave shows much stronger dispersion as k increases, but the SV and SH waves move with constant speeds. For propagation in the plane of the layers (Figs. 3.11 and 3.12), P waves are found to show slight dispersion with increasing k , but SV and SH waves do not. In summary, waves are strongly dispersive when they propagate normal to the layers. Otherwise, dispersion is weak, and the periodic laminate can be modeled approximately by the effective-modulus model.

3.4 Remarks A stiffness method has been developed in Section 3.2 to analyze wave motion in a periodically laminated composite medium. Results are presented for waves propagating in a periodic orthotropic medium composed of repetitive two-layer unit cells. It has been

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Periodic Layered Media 3

Longitudinal γ = 100 γ = 50 γ = 10

2

β

Transverse γ = 100 γ = 50 1

γ = 10

Eff. modulus method Eff. stiffness method Finite element method and exact analysis 1

2

3

4

ξ

FIGURE 3.7  Lowest transverse (SV) and longitudinal (P) mode propagating normal to the layering. (Reprinted by permission from Yeo 1983, Fig. 4.4.)

assumed that the symmetry axes of the orthotropic layers making up the unit cell are coincident. The Floquet wave analysis shows the existence of Brillouin zones, at the edges of which pass and no-pass bands occur. In this section, a brief analysis is presented for a more general case where layers comprising the unit cell are transversely isotropic, having symmetry axes making arbitrary angles with the x1-axis. Details of the Floquet wave analysis can be found in Braga and Herrmann (1988, 1992), Ting and Chadwick (1988), Potel and Belleval (1993), Safaeinili and Chimenti (1995), Safaeinili et al. (1995), Nayfeh (1995), Wang and Rokhlin (2002b), and references therein. It is assumed that two adjacent layers comprising the unit cell are transversely isotropic, with symmetry axes aligned at angles q1 and q 2 to the global x-axis (see Fig. 2.2). A local system of coordinates (X, Y, Z) is chosen for each layer, with the origin at the middle plane and the X-axis aligned with the symmetry direction. The governing elasticity equations in each layer can be solved exactly for the displacements and tractions (details

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Elastic Waves in Composite Media and Structures 3 P

Eff. modulus method Eff. stiffness Finite element method

2

β

SV

SH 1

0

2

4

6

8

10

12

ξ

FIGURE 3.8  Lowest SH, SV, and P mode propagating normal to the layers in boron–aluminum composite (q = 0°, a = 0°). (Reprinted by permission from Yeo 1983, Fig. 4.6.)

are given* later in Chapter 4, Section 4.2.1). The displacement−traction vector



ui    υi  σ  zzi  [Ξi ] =  σ zxi    σ yzi  w   i 

(3.4.1)



* We are making a little digression here to discuss waves in a periodically layered anisotropic medium. Some of the equations needed for this discussion are fully developed in Chapter 4.

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0.3 P

Ω

0.2

SH SV 0.1

0

0.02

0.04

0.06

0.08

0.10

k

FIGURE 3.9  Lowest SH, SV, and P mode propagating in the fiber direction (q = 0°, a = 90°). (Reprinted by permission from Yeo 1983, Fig. 4.7.)

at the ith interface between adjacent layers is related to that at the (i + 1)th interface by the transfer matrix [Pi] by the equation [Ξi +1 ] = [Pi (2hi )][Ξi ]



(3.4.2)



where

[Pi ( Z )] = [Hi ]−1[T ( Z )][Hi ]

(3.4.3)



where T(z) is the 6 × 6 matrix defined in equation (4.2.19) and [Hi] is defined in equation (4.2.27). Note that [Pi(Z)] has the following properties (see Braga and Herrmann 1988): [Pi (0)] = I , [Pi ( Z1 + Z 2 )] = [Pi ( Z1 )][Pi ( Z 2 )]

[Pi ( Z )]−1 = [Pi (− Z )], det[Pi ( Z )] = 1

(3.4.4)

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Elastic Waves in Composite Media and Structures 0.3

P

Ω

0.2

SV SH 0.1

0

0.02

0.04

0.06

0.08

0.10

k

FIGURE 3.10  Lowest SH, SV, and P mode propagating in the x-z plane (q = 0°, a = 60°). (Reprinted by permission from Yeo 1983, Fig. 4.8.)

For the unit cell composed of two layers i and (i + 1), the transfer matrix is then [P(d )] = [Pi +1 (d2 )][Pi (d1 )]



(3.4.5)



According to the Floquet theory, [Ξi + 2 ] = e iκ d [Ξi ]



(3.4.6)



where k is the Floquet wave number. We also have [Ξi +1 ] = [P(d )][Ξi ]





(3.4.7)

It follows from equations (3.4.6) and (3.4.7) that eiκd is an eigenvalue of [P(d)]. Since T (Z) = T(−Z) = T−1(Z), it can be shown that the eigenvalues of [P(d)] and [P(d)]−1 are the same. Thus, the eigenvalues of [P(d)] are eiκd and e−iκd. Now, expanding the determinant |P − lI| in terms of the invariants of [P], we obtain the characteristic equation T



λ 6 − I1 λ 5 + I 2 λ 4 − I 3 λ 3 + I 4 λ 2 − I5 λ + I 6 = 0



(3.4.8)

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61

Periodic Layered Media 0.3 P

0.2

Ω

SH SV 0.1

0

0.02

0.04

0.06

0.08

0.10

k

FIGURE 3.11  Lowest SH, SV, and P mode propagating in the x–y-plane (q = −30°, a = 90°). (Reprinted by permission from Yeo 1983, Fig. 4.16.)

Clearly, the following relations hold among the invariants Ik(k = 1, …, 6); I1 = I5, I2 = I4, I6 = 1. Equation (3.4.8) can be written as

λ 6 − I1 (λ + λ 5 ) + I 2 (λ 2 + λ 4 ) − I 3 λ 3 + 1 = 0



(3.4.9)

Using l = eiκd in equation (3.4.9) we get



1 cos(3κ d ) − I1 cos(2κ d ) + I 2 cos(κ d ) − I 3 = 0 2

(3.4.10)

The characteristic equation (3.4.10) can also be written in the form (Safaeinili and Chimenti 1995)

χ 3 − I1 χ 2 + I 2 χ − I 3 = 0

(3.4.11) 1 where χ = λ + . Thus, in general, there are three Floquet wave components correspondλ ing to the three roots of equation (3.4.11). If both layers of the unit cell have symmetry axes aligned with the x-axis, then the equation separates into two uncoupled equations, one corresponding to the SH wave and the other corresponding to the coupled qP and

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Elastic Waves in Composite Media and Structures 0.3 P

0.2

Ω

SH SV 0.1

0

0.02

0.04

0.06

0.08

0.10

k

FIGURE 3.12  Lowest SH, SV, and P mode propagating in the y–z-plane (q = −90°, a = 90°). (Reprinted by permission from Yeo 1983, Fig. 4.11.)

qSV waves. The equation has to be solved numerically given the properties of the materials of the layers, the frequency ω, and the wave numbers K and L. Results for special cases can be found in Braga and Herrmann (1988), Nayfeh (1995), and Safaeinili and Chimenti (1995). As has been discussed in Section 3.3, results obtained by the stiffness method agree with the exact solutions for low and moderate values of the frequency.

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4 Guided Waves in Fiber-Reinforced Composite Plates 4.1 Introduction .............................................................................. 64 4.2 Governing Equations ............................................................... 64 Analytical Solution • Stiffness Method 4.3 Numerical Results .....................................................................76 4.4 Application to Materials Characterization .......................... 85 4.5 Thin Layers ................................................................................ 88 Ultrasonic Waves in Paper • First-Order Plate Equations • Numerical and Experimental Results 4.6 Guided Waves in Plates with Thin Coating and Interface Layers......................................................................... 98 • Isotropic Plate with Thin Coating Layers • Isotropic Plate with a Thin Interface Layer • Numerical Results and Discussion 4.7 Transient Response due to a Concentrated Source of Excitation .............................................................................114 Transient Waves in a Multilayered • Plate: Plane-Strain Motion • Stiffness Method and Modal Expansion of Green’s Function: Plane Strain • Stiffness Method and Modal Expansion of Green’s Function: Three-Dimensional Problem 4.8 Laminated Plate with Interface Layers................................ 142 4.9 Remarks ................................................................................... 146 4.10 Laser-Generated Thermoelastic Waves................................147 Analytical Solution 4.11 Results for Thermoelastic Dispersion and LaserGenerated Waves .................................................................... 157 63

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4.1 Introduction As summarized in Chapter 1 (Section 1.1.4), a large volume of work on wave propagation in composite platelike structures has accumulated. These studies have been motivated by the increasing use of ultrasonics to inspect these structures for defects and for material characterization. Although the initial impetus came from aerospace structural applications, ultrasonic techniques are being explored for nondestructive evaluation of concrete slabs, wood, gas pipelines, pressure vessels, and many other structural applications. Various approximate theories and exact solutions for wave propagation in laminated media have been developed. References to relevant works are given in Section 1.1.4. The dispersive behavior of guided waves in laminated plates of finite thickness is the subject of this chapter. As mentioned in Section 1.1.4, dispersive modal propagation behavior is strongly influenced by the anisotropic properties of each lamina and the stacking sequence used. So, it is necessary to model accurately and efficiently the effect of different layer properties on the modes of propagation in laminated plates. In this chapter, an exact method of solution as well as a semi-analytic finite element (SAFE) method is developed in Sections 4.2.1 and 4.2.2. Numerical results presented in Section 4.3 show that there is excellent agreement between model solutions by the SAFE method and the exact solutions. In Section 4.4, comparison of experimental observations for a particular case of a periodic layered plate with the model calculations using the SAFE method illustrates the utility of the method in material characterization. Section 4.5 deals with simplified approximate plate theories for flexural and longitudinal wave propagation in a thin orthotropic plate. Predictions of these approximate theories compare favorably with available observations on ultrasonic waves in copy papers. Approximate boundary and interface conditions for wave propagation in a plate with thin anisotropic coating and interface layers are derived in Section 4.6. Solutions obtained by using the approximate boundary conditions are compared with those predicted by an exact analysis. It is found that the approximate boundary conditions can be used with confidence when the wavelengths are long compared with the thickness of the layer. Waves generated in composite plates by an external force are studied in Section 4.7. Both time-harmonic and transient solutions are considered. Section 4.8 deals with the effect of thin interface layers on guided wave dispersion. A brief discussion of the effect of a protective viscoelastic coating layer is given in Section 4.9. The effect of thermal loading on guided waves in anisotropic plates has been studied in several investigations during the last two decades. Laser thermal excitation provides an efficient means of noncontact generation of ultrasonic waves. This is investigated in Sections 4.10 and 4.11.

4.2 Governing Equations Time-harmonic elastic-wave propagation in an infinite plate composed of perfectly bonded fiber-reinforced layers (laminae) of possibly distinct thicknesses is considered here. The two plane faces of the plate at z = 0 and z = H are taken to be traction-free, and the global rectangular Cartesian coordinate system (x, y, z) is as shown in Fig. 4.1. The direction of wave propagation is assumed to be coincident with the x-axis. Since the

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65

Guided Waves in Fiber-Reinforced Composite Plates x y Fibers

z

X

1st Sublayer

αfiber x

2nd Sublayer

zi zi+1

ith Sublayer Nth Sublayer

ZN+1

ith Sublayer y

z

(a) Fibers

W av e

X

s θ

y´

φ

Obs er va (r, θ) poi tion nt

x´ W av e

ith Sublayer y

z

(b)

x

dir ec

tio

n

FIGURE 4.1  (a) Geometry of the plate and the coordinate system; (b) coordinate transformation in wave-propagation direction.

plate is infinite in the y-direction, all field quantities (displacement, strain, and stress) are independent of the y-coordinate. Each layer is assumed to be reinforced by continuous unidirectional elastic fibers in an elastic matrix. Thus, the layer can be modeled as homogeneous transversely isotropic. If the fibers are parallel to the x–y-plane, then the axis of symmetry of the layer lies along the direction of the fibers in this plane. For the sake of numerical accuracy, each layer is subdivided into a number of sublayers. As shown in Fig. 4.1(a), a local coordinate system (X, Y, Z) with the origin in the midplane of the sublayer is chosen such that the X-axis is

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Elastic Waves in Composite Media and Structures

along a symmetry axis, the Y-axis is parallel to the plane of the layer, and the Z-axis is coincident with the z-axis. The stress–strain relation for the sublayer is given, in matrix notation, by the following equation (see also equation [2.2.28]): [σ ] = [c][e]



(4.2.1)

where the stiffness matrix [c] is given by equation (2.2.41). The stress and strain components are given in the matrix forms as



e XX  σ XX      eYY  σ YY      e  σ  ZZ  ZZ  , [e] =  [σ ] =      γ YZ  σ YZ      γ ZX  σ ZX      γ XY  σ XY 

(4.2.2)

It may be noted that the symmetry axes can be in different directions from lamina to lamina. As shown in Fig. 4.1(a), the symmetry axis for the ith sublayer makes an angle −a fiber with the global x-axis. This angle may vary from one layer to another. Then, the stress–strain relation for the ith sublayer is given in the global coordinates system (x,y,z) by (see equation [2.2.63])

σ ij = Dijkl ekl



(4.2.3)

where i, j, k, l take the values x, y, z. In writing equation (4.2.3), we have replaced x′, y′, z′ by x, y, z, respectively, and cijkl ′ by Dijkl. The equations of motion for a particle in the ith sublayer in the local coordinate system are

σ IJ , J = ρU��I



(4.2.4)

Here, I, J take the values X, Y, Z; r is the density; and UX (= U), U Y (= V), and UZ (= W) are the components of the displacement u(X, Y, Z, t) in the local coordinate directions.

4.2.1 Analytical Solution To solve the governing equations (4.2.4), it would be convenient (Buchwald 1961) to introduce the potential functions Θ, Φ, and Ψ such that



U=

∂Θ ∂X

(4.2.5a)

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Guided Waves in Fiber-Reinforced Composite Plates

∂Φ ∂ Ψ + ∂Y ∂Z ∂Φ ∂ Ψ W= − ∂Z ∂Y V=



(4.2.5b) (4.2.5c)

The stress components are then given by equation (4.2.1), with the strain components given in terms of the potentials as e XX =

∂  ∂Φ ∂ Ψ  ∂2Θ ∂  ∂Φ ∂ Ψ  − , eYY = , e ZZ = +   2 ∂X ∂Y  ∂Y ∂Z  ∂Z  ∂ Z ∂Y 

1 1  ∂2 Φ ∂2 Ψ ∂2 Ψ  − + eYZ = (γ YZ ) =  2 2 2  ∂Y ∂Z ∂Z 2 ∂Y 2  eZX



(4.2.6)

1 1 ∂  ∂Θ ∂ Φ ∂ Ψ  = (γ ZX ) = + − 2 2 ∂X  ∂Z ∂Z ∂Y 

1 1 ∂  ∂Θ ∂ Φ ∂ Ψ  e XY = (γ XY ) = + + 2 2 ∂X  ∂Y ∂Y ∂Z 



Using equations (4.2.1) and (4.2.6) in the equations of motion (4.2.4), and after some algebraic manipulations of the resulting equations, we obtain the following equations governing the potentials Θ, Φ, and Ψ



c55

 ∂2 ∂2 ∂2 ∂2  ∂2Θ =0 (∇ 2 Θ ) + (c13 + c55 ) 2 (∇ 2 Φ ) +  c11 2 − ρ 2  2 ∂X ∂X ∂t  ∂X 2  ∂X

(4.2.7a)

∂2 ∂2 ∂2 (∇ 2 Θ ) + c55 2 (∇ 2 Φ ) + c33 ∇ 4 Φ − ρ 2 (∇ 2 Φ ) = 0 2 ∂X ∂X ∂t

(4.2.7b)

(c13 + c55 )

 ∂2 ∂2  c44 ∇ 4 Ψ +  c55 2 − ρ 2  ∇ 2 Ψ = 0 ∂t   ∂X

(4.2.7c) ∂2 ∂2 where ∇ 2 = + 2. 2 ∂Y ∂Z It is seen that the potentials Θ and Φ satisfy two coupled equations (4.2.7a) and (4.2.7b), whereas equation (4.2.7c) involving Ψ is uncoupled. This simplifies considerably the problem of solving for the potentials and, hence, for the displacement components in the local coordinates. Assuming simple harmonic wave propagation in the global x-direction, the solutions to equations (4.2.7a–c) can be taken as Θ( X , Y , Z , t ) = f1 ( Z )exp (iψ ) Φ( X , Y , Z , t ) = f 2 ( Z )exp (iψ )

Ψ( X , Y , Z , t ) = f 3 ( Z )exp (iψ )

(4.2.8)

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Elastic Waves in Composite Media and Structures

Here y = kx − wt and fi(Z) (I = 1,2,3) are three unknown functions of Z that are to be found such that equations (4.2.7a–c) are satisfied by the functions Θ, Φ, and Ψ. Note that y can be written as

ψ = KX + LY − ωt , K = k cosθ , L = k sinθ



(4.2.9)



Without any cause for confusion, we have used θ for a fiber for convenience. Thus, the wave numbers in the X and Y directions are given by K and L, respectively. Substitution of equation (4.2.9) in equations (4.2.7a–c) leads to the following system of ordinary differential equations governing f i(Z) (I = 1,2,3).  d2  d2   2 2 2 2  dZ 2 − L − λ K + k2  f1 + δ  dZ 2 − L  f 2 = 0 (4.2.10)

  d2  − δ K 2 f1 +  k22 − K 2 + β  2 − L2   f 2 = 0  dZ  





 2  d2   k2 − K 2 + ε  2 − L2   f 3 = 0  dZ  



(4.2.11)

Equation (4.2.10) is a coupled system of two linear second-order equations in f1 and f2, whereas equation (4.2.11) is a single second-order ordinary differential equation in f3. Here,

λ=

c c c11 c , β = 33 , ε = 44 , δ = 1 + 13 , k2 = c55 c55 c55 c55

ρω 2 c55

(4.2.12)

Now, solutions to equations (4.2.10) and (4.2.11) can be written in the form f j = Fj exp (irZ )



(4.2.13)



where Fj (j = 1, 2, 3) are constants. Substituting equation (4.2.13) in equations (4.2.10) and (4.2.11), it is found that r must satisfy the equation

{

}

)(

(

)

 β (r 2 + L2 )2 + η K 2 − k22 (1 + β ) (r 2 + L2 ) + λ K 2 − k22 K 2 − k22   

× ε (r 2 + L2 ) + K 2 − k22  = 0

(4.2.14)

Here, h = 1 + lb − d  . Writing r + L = s , the roots of equation (4.2.14) can be expressed as 2

2

s12 , s22 =

2

2

k22 (1 + β ) − K 2η ± Γ 2β 2

(

)(

Γ =  k22 (1 + β ) − K 2η  − 4β λ K 2 − k22 K 2 − k22

s32 =

k22 − K 2 ε

(4.2.15)

)

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Guided Waves in Fiber-Reinforced Composite Plates

Then, the constants Fj can be chosen as F2(1)

F1(1) = 1,

F

(1)

= A=

1

F F

(2) 1 (2) 2



=B=

k22 − λ K 2 − s12 δ s12 (4.2.16)

k22 − K 2 − β s22 , F2( 2 ) = 1 δK2

F3 = 1



The displacement components U, V, and W are now given by

)

(

U = ik W1+ + BW 2+ exp(iψ ) V = iL( AW1+ + W 2+ ) − ς W 3+  exp(iψ )

W =  Ar1 W1− + r2 W 2− − iLW 3−  exp(iψ )

(4.2.17)

where

r1 = s12 − L2 , r2 = s22 − L2 , and ς = s32 − L2



(4.2.18)

Furthermore, W ±j are defined below.



 W1+  cos(r1Z ) 0 0 sin(r1Z ) 0 0   A11        W +2   0 sin(r2 Z ) 0   A21  0 cos(r2 Z ) 0      W+   0 cos(ς Z ) 0 0 sin(ς Z )  A31  0 3  =    (4.2.19)  −    0 0 cos(r1Z ) 0 0   A12   W1   − sin(r1Z )  −    0 0 cos(r2 Z ) 0   A22  0 − sin(r2 Z ) W2        0 cos(ς Z )   A32  0 0 − sin(ς Z ) 0  W 3−  

Here, Aij (i = 1,2,3; j = 1,2,3) are arbitrary constants for the sublayer. Using equation (4.2.19) in equation (4.2.17), we get



U  iK iKB    V  = iLA iL    W   0 0   

0

0

−ς

0

0

r1 A

0   0 0 [T ( Z )][C]exxp(iψ )  r2 − iL  0

(4.2.20)

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Elastic Waves in Composite Media and Structures

where the matrices [T(Z)] and [C] are, respectively, the 6 × 6 and 6 × 1 matrices on the right-hand side of equation (4.2.19). The strain vector in the XYZ coordinates system can now be expressed as e XX     2 −K 2B 0 0 0 0  eYY   − K  2   −L A  − L2 −iLς 0 0 0  e   2 −r22 iLς 0 0 0   ZZ  =  −r1 A     0 0 0 2iLr1 A 2iLr2 ( L2 − ς 2 ) γ  YZ   0 0 0 iKr1 (1 + A) iKr2 (1 + B) KL      γ ZX   − KL(1 + A) − KL(1 + B) −iKς 0 0 0      γ XY  × [T ( Z )][C]exp(iψ )

(4.2.21)

Now, using equation (2.2.56), we obtain the stress vector in the global xyz-system as [σ i ] = [ M ][c][e I ]





(4.2.22)

where [M] is given by equation (2.2.62). The displacement vector components in the global xyz-system is obtained from equation (4.2.20) as U  u      v  = [a] V      W  w     



(4.2.23)

Here, [a] is given by equation (2.2.61) after replacing q with −q. The displacement–stress vector in the xyz-system for the ith sublayer is found to be



u    v    σ  [a][R] [0]   zz  =  [T ( Z )][C]exp(iψ ) σ    [ 0 ] [ a ][ S ] zx      σ yz    w 

(4.2.24)

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Guided Waves in Fiber-Reinforced Composite Plates

Here, iK   R  = iLA  R  31

  iL − ς   R32 2c55 εiLς 

iKB

0

ic55 Kr1 (1 + A)   S  =  2ic55 ε Lr1 A   rA 1 

  c55 ε ( L2 − ς 2 )   − iL 

ic55 Kr2 (1 + B)

c55 KL

2ic55 ε Lr2 r2

(4.2.25)

R31 = c55 (1 − δ )K 2 − Aβ s12 + 2 Aε L2 

R32 = c55 (1 − δ )K 2 B − β s22 + 2ε L2 



The next step in the process of setting up the total matrix for the layered plate is to calculate the transfer matrix from one layer to the next. This is done by evaluating the coefficient matrix [Ci] in terms of the displacement–stress vector at the interface between the ith and (i−1)th sublayers. This gives

[Ci ] = [T (−hi )]−1[Hi ][Ξi ]



(4.2.26)

where ui    v  i   σ   zzi  [Ξi ] =   σ zxi    σ yzi    wi 

[Ri ]−1[a i ]T [0]  , [H i ] =   [0] [Si ]−1[a i ]T  



(4.2.27)

Note that [T(Z)] has the property

[T (− Z )][T ( Z )] = [I ]

(4.2.28)

Now, evaluating [Ξ] at the (i+1)th interface and using equation (4.2.27), we obtain

[Ξi +1 ] = [Pi ][Ξi ]



(4.2.29)

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Elastic Waves in Composite Media and Structures

Here, [Pi] is the transfer matrix for the ith sublayer and is given by

[Pi ] = [Hi ]−1[T (2hi )][Hi ]

(4.2.30)



Using equation (4.2.29) successively for sublayers 1, …, N, we obtain the relation between the displacement–stress vectors at the bottom and top faces of the plate as

[Ξ N +1 ] = [P][Ξ1 ], [P] = [PN ][PN −1 ]…[P1 ]



(4.2.31)

In deriving the above equation relating the displacement–stress vector at the top surface of the plate to that at the bottom of the plate, it has been assumed that all the laminae comprising the plate are in welded contact so that the displacements and tractions are continuous across the interfaces. If the interface conditions change, equation (4.2.31) can be modified appropriately. In addition, equation (4.2.31) can be modified to include body forces acting at points within the plate. Now, for a free-free plate (where the top and bottom surfaces are traction-free), the boundary conditions are



σ zz ( N +1)    σ  = 0,  zx ( N +1)    σ yz ( N +1) 

σ zz 1    σ  = 0  zx1  σ   yz 1 

(4.2.32)

Using these boundary conditions in equation (4.2.32), we obtain



 P31 P32 P36  u1      P41 P42 P46  v1  = [0]     P P P  w   51 52 56   1 

(4.2.33)

For nontrivial solution, the determinant of the 3 × 3 matrix on the left-hand side must be zero. Thus, P31 P32 P36 P41 P42 P46 = 0

P51 P52 P56

(4.2.34)

Equation (4.2.34) is the exact dispersion equation governing the relationship for frequency (w) and wave number (k) for waves propagating in the x-direction in the plate. For a fixed value of w or k, equation (4.2.34) is a transcendental equation for finding k or w, respectively. It is possible to find the roots by some search method (see Press et al. 1988). This is a rather time-consuming and complex process, especially when real

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73

as well as imaginary and complex roots are needed to model accurately the representation of the field variables as modal sums for forced motion of the plate or for scattering of guided waves by defects. Herein Muller’s method is employed to find the roots accurately. Approximate roots obtained by using the stiffness method (described in the next section) are used as initial guesses in Muller’s method. If the roots are required over a range of k (or w), approximate roots using the stiffness method are needed only at the first step as initial guesses. At the next step, k (or w) is changed by a small amount, and equation (4.2.34) is solved taking the exact roots from the previous step as the initial guesses for the current step. The process is repeated until the range of interest is covered. Once the exact roots are determined, the exact wave functions are obtained as discrete eigenvectors using equation (4.2.29) at successive interfaces. If the plate is symmetrically laminated, then the boundary-value problem discussed above can be separated into two problems, symmetric and antisymmetric. In both cases, it is necessary to model only half of the plate. Then, the boundary conditions at the middle surface of the plate, z = H/2, are

w = 0, σ zx = 0, σ yz = 0



(4.2.35)

when the problem is symmetric, and for the antisymmetric case, we have u = 0, v = 0, σ zz = 0





(4.2.36)

This reduces the size of the problem to half. In the following subsection, we present stiffness methods that have proved to be efficient and versatile for obtaining the dispersion relations governing guided waves in layered systems.

4.2.2 Stiffness Method As discussed in Chapter 3 (Section 3.2), in the stiffness method the layered medium is divided into several sublayers, and the displacements in a sublayer are expressed in terms of polynomials in the thickness variable, z, with coefficients (generalized coordinates) that depend upon x and t in the form exp(iy), where y has been defined in the previous section. In this approach, the displacement components are given by equations (3.2.3a–c), where cubic polynomials in the local coordinate Z are used, as seen in equation (3.2.4). The generalized coordinates are the displacement and traction components at the nodes on the interfaces between two adjacent sublayers. In this approach, the displacement and traction continuity at the interfaces is maintained. The equations governing the nodal generalized coordinates ui, vi, wi, χ i, t i, and s i (i = 1, 2, …, N + 1) are obtained by using Hamilton’s principle. For this purpose, the Lagrangian Li per unit length in the y direction for the ith sublayer is written as



  hi 1  ( ρ{u� }T {u�} − {e }T [D]{e})dz  dx Li =  2   − hi 

∫ ∫

(4.2.37)

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Elastic Waves in Composite Media and Structures

where the overdot denotes the derivative with respect to time. Using equations (3.2.3a–c) in the strain-displacement relations in equation (2.1.15), and in turn in equation (4.2.37), Li can be written as {q� ′}[c2 ]{q� ′} + {q� ′}T [c3 ]{q�} + {q� }T [c3 ]{q� ′} + {q� }T [m]{q�} − {q� ′′}T [e1 ]{q ′′}    1   dx (4.2.38) Li = −{q ′′}T [e2 ]{q ′} − {q ′′}T [e3 ]{q} − {q ′}T [e2 ]T {q ′′} − {q ′}T [e4 ]{q ′}  2   −{q ′}T [e ]{q} − {q }T [e ]T {q ′′} − {q }T [e ]T {q ′} − {q }T [e ]{q}  5 3 5 6  

∫

Here, the matrices [c2], [c3], [m], and [e1], …, [e6] are given by equations (3.3.16) and (3.3.17). The generalized coordinate vector {q} is defined as

{q}T = 〈ui χi vi τ i wi σ i ui +1 χi +1 vi +1 τ i +1 wi +1 σ i +1 〉

(4.2.39)



and the primes denote derivatives with respect to x. The Lagrangian for the entire plate is obtained by summation over all the sublayers, and its first variation leads to an approximate governing equation for the plate. Now, assuming the time dependence in the form exp(−iwt), the ordinary differential equation in x governing the assembled generalized coordinate vector {Q} is found to be

ω 2 (−[C2 ]{Q ′′} − [C1 ]{Q ′} + [ M ]{Q}) − ([E1 ]{Q iv }

+[E2 ]{Q ′′′} + [E3 ]{Q ′′} + [E4 ]{Q ′} + [E5 ]{Q}) = 0

(4.2.40)



The matrices [C1], [C2], [M], and [E1], …, [E5] are as follows:

[C1 ] = ∪ [c3 ] − [c3 ]T  , [C2 ] = ∪[c2 ], [ M ] = ∪[m]

(4.2.41a)



[E1 ] = ∪[e1 ], [E2 ] = ∪ [e2 ] − [e2 ]T  , [E3 ] = ∪ [e3 ] − [e4 ] + [e3 ]T  ,

[E4 ] = ∪ [e5 ]T − [e5 ] , [E5 ] = ∪[[e6 ]

(4.2.41b)

Note that the matrices [M], [C2], [E1], [E3], and [E5] are symmetric, and [C1], [E2], and [E4] are antisymmetric. The interpolation polynomials used in equations (3.2.3a–c) to represent approximately the displacement within a sublayer are cubic, and the coefficients of expansion are the displacement and traction components at the top and bottom interfaces of the sublayer. This way, displacement and traction continuity is maintained at the interfaces. An alternative approximation was used by Dong and Huang (1985). There, quadratic interpolation polynomials were used. Each sublayer was divided into two equal thickness layers, and coefficients of interpolation (generalized coordinates) were taken as the

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Guided Waves in Fiber-Reinforced Composite Plates

displacements at the bottom, middle, and top of the sublayer. Thus, the displacement components are approximated as



u    n1 v  =  0    0 w 

0 n1 0

n2 0 0

0 0 n1

0 n2 0

n3 0 0

0 0 n2

0 n3 0

0 0  {q} n3 

(4.2.42)

The interpolation functions n1, n2, and n3 are given in terms of the local Z-coordinate as

1 n1 = (−η + η 2 ), 2

1 n3 = (η + η 2 ) 2

n2 = (1 − η 2 ),

(4.2.43)

and {q} is the vector of generalized displacement components, i.e., u1    v1  w   1 u2    {q} = v 2  w   2 u3    v 3  w   3

(4.2.44)

and h = Z/h, 2h being the thickness of the sublayer. Then, the strain vector {e} is obtained as {e} = [a]{q ′} + [b]{q}



(4.2.45)



with the 6 × 9 matrices [a] and [b] given by n1  0 0 [a] =  0 0   0



0  0 d 0 [b] =  dz  0 n  1  0

0 0 0 0 0 n1

n2

0 0 0 0 n1 0 0 0 0 n1 0 0

0 0 0 0 0 n2

0 0 0 0 0

0 0 n1 0 0 0

0 0 0 0 n2 0

n3

0 0 0 0 n2 0 0 0 0 n2 0 0

0 0 0 0 0

0 0 n2 0 0 0

0  0 0  0 n3   0 

0 0 0 0 0 n3 0 0 0 0 n3 0

0 0 0 n3 0 0

0  0 n3   0 0  0 

(4.2.46)



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Elastic Waves in Composite Media and Structures

The prime (′) in equation 4.2.45 denotes derivative with respect to x. Now, using the Lagrangian as in the above and taking the first variation, the governing equation for the entire plate is found to be

��} = 0 [K1 ]{Q ′′} + [K 2 *]{Q} − [K 3 ]{Q} − [ M ]{Q

(4.2.47)



Here, [K1], [K3], and [M] are symmetric, whereas [K2*] is skew-symmetric. The vector {Q} is the vector of nodal displacements. Equation (4.2.47) is a second-order partial differential in x and t. If {Q} is assumed to be of the form {Q 0}exp[i(kx − wt)], then the eigenvalue problem for w and k is given by

 − K1 k 2 + iK 2 * k − K 3 + Mω 2  {Q0 } = 0

(4.2.48)

For nontrivial solution of {Q 0}, the determinant of the matrix within the square brackets must be zero. This gives the dispersion equation for w, k. On the other hand, the dispersion equation obtained from equation (4.2.40) is the determinant of the matrix given below equated to zero.

[ω 2 (k 2[C2 ] − ik[C1 ] + [ M ]) − (k 4 [E1 ] − ik 3[E2 ] − k 2[E3 ] + ik[E4 ] + [E5 ])]



(4.2.49)

Both of these approximate dispersion equations have been used by Datta et al. (1988c) to study dispersion of waves in a cross-ply laminated plate, with each lamina being a continuous graphite-fiber-reinforced epoxy layer. Thus, each lamina can be modeled as transversely isotropic, with the symmetry axis aligned with the X- or Y-axis. This study was confined to the case when the propagation direction coincided with the fiber direction (symmetry axis). In this special case, the problem decouples into plane-strain and antiplane-strain cases. Numerical results are presented in the following section.

4.3 Numerical Results Equations (4.2.34), (4.2.48), and (4.2.49) apply to any fiber-reinforced laminated plate having properties that can be quite different from lamina to lamina. It may also be noted that equations (4.2.48) and (4.2.49) apply more generally to laminated plates with laminae having monoclinic symmetry, with the symmetry plane coincident with the x–yplane. Results for some particular cases of engineering importance are presented in this section as illustrative examples. First, we will consider a graphite-fiber-reinforced epoxy plate. The material will be modeled as homogeneous transversely isotropic. The effective properties are listed in Table 4.1. The X-axis is taken as the symmetry axis and is assumed to coincide with the direction of wave propagation (q = 0°). As mentioned above, the problem can be modeled as either plane strain (displacement lying in the X–Z-plane) or antiplane strain (displacement in the Y-direction). Note that if the wave propagates in the Y-direction (q = 90°), then the plane-strain and antiplane-strain problems also decouple. This case is the same as that for an isotropic plate, which has been studied extensively since the

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Guided Waves in Fiber-Reinforced Composite Plates TABLE 4.1  Material Properties of Uniaxial Graphite-Fiber-Reinforced Epoxy Composite Elastic Stiffness (GPa) Material

Density (kg/m3)

c11

c22

c12

c44

c66

0o lamina

1600

160.73

13.92

6.44

3.5

7.07

early works of Lamb (1917) (see Achenbach 1973). Figure 4.2 shows the dispersion of plane-strain waves propagating in the X-direction. Results obtained by using equation (4.2.49) as well as those from the exact equation are displayed. The plot shows both real and imaginary branches as functions of frequency. Excellent agreement between the two model predictions are found. Here, W = wH/[2p (c55 /ρ )0� ], and γ = kH/(2π). Fifteen sublayers were used in the stiffness method. Because of the high stiffness in the fiber direction, the phase velocity of the first symmetric mode (S 0) (longitudinal wave) remains constant at finite but low frequencies. It then drops off rapidly and approaches the Rayleigh velocity. At low frequencies, the phase velocity of the S 0 mode tends to the plate wave velocity, cp, which is given by (c11 − c132 /c33 )/ρ (see, for example, Mukdadi et al. 2001). Thus, cp is less than the velocity of the longitudinal wave along the symmetry axis in the medium. For this material, the second term under the square root is very small, so the plate velocity is slightly below the longitudinal velocity. The Rayleigh wave

Symmetric Antisymmetric

2.0

Exact

Stiffness method I

1.6 1.2

0.8 0.4

0.0

2 γI

1.5

1 0.5 Imaginary

0

0.5

1 Real

1.5

2 γR

FIGURE 4.2  Dispersion curves for propagation in the fiber direction in a homogeneous graphite-fiber-reinforced plate. (Reprinted with permission from Datta et al. 1988c, Fig. 4.)

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Elastic Waves in Composite Media and Structures

Stiffness method I Stiffness method II

16

and

12

6

4

0

4 γI

3

2 Imaginary

1

0

1

2 Real

3

4 γR

FIGURE 4.3  Dispersion curves for propagation in a four-layer 0°/90°/0°/90° plate in the 0° direction. (Reprinted with permission from Datta et al. 1988c, Fig. 6.)

velocity along the symmetry axis in this medium is somewhat below c55 /ρ . Note that the velocities of the higher branches drop rapidly to a plateau at the velocity cp. To see the comparison between the predictions of the two stiffness methods discussed above, Fig. 4.3 shows the results obtained for a four-layer (0°/90°/0°/90°) plate (Dong and Huang 1985) with the elastic stiffnesses in units of psi given by (c11 )0° = 21.289 × 106 , (c13 )0° = 0.592 × 106 ,

(c33 )0° = 2.3186 × 106 ,

(c55 )0° = 0.85 × 106

The density r is taken to be 1. Solid lines represent the results predicted by the second method, and open circles and dashed lines are the predictions of the first one. Twenty sublayers were used to obtain these results. The agreement is found to be very good except when the frequency is large and when considering modes with large imaginary wave numbers. For the same composite plate, both real and complex branches are shown in Fig. 4.4. Solid dots are the results of using the second stiffness method. Again, the agreement is found to be excellent at moderate frequencies. Figure 4.5 shows the dispersion of SH waves propagating along the symmetry axis in the graphite-fiber-reinforced composite plate. This figure has a lot of similarity with the dispersion curves for SH waves in an isotropic plate (see, for example, Achenbach 1973). Results for propagation at an angle of 45° to the X-axis in a multilayered cross-ply graphite-fiber-reinforced composite plate are shown in Fig. 4.6. In this and the following ωH figures, Ω = , g = kH/2, and the normalized phase velocity c = Ω/g . 2 (c55 )0° /ρ

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Guided Waves in Fiber-Reinforced Composite Plates Ω

4 I d II ho od t e m eth ss m e s n s iff e St tiffn S

3

2

1 γR 1

1

2 Real

2 3 4 γI

Im

ry

na

i ag

FIGURE 4.4  Complex frequency branches corresponding to Fig. 4.3. (Reprinted with permission from Datta et al. 1988c, Fig. 7.)

The plate considered has a symmetric lay-up, 90°/0°/…/0°/90°, and the properties of the 0° lay-up laminae are given in Table 4.1. The number of laminae is 35, and two sublayers were used in each lamina. The second stiffness method and the exact solution were used to compare their predictions. First, it is noted that both predictions agree well. Second, it was found that predictions of the multilayered model agree well with those of the effective-medium model discussed in Chapter 3. In a study reported by Karunasena et al. (1991b), it was found that for this particular system increasing the number of laminae more than 35 did not change the dispersion results appreciably. Later in this chapter, we will present results for time-dependent forced motion of this plate, and it will be shown that the effective-medium approximation gives very good results. Wang and Rokhlin (2002b, 2002c) have studied recently homogenized effective-medium approximations of periodic layered media using Floquet theory. Their study extends the results presented here to higher frequencies. Using equations

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Symmetric Antisymmetric 2.0

1.6

1.2

0.8

0.4 0.0

2 γI

1.5

1 Imaginary

0.5

0

0.5

1 Real

1.5

2 γR

FIGURE 4.5  Dispersion curves for SH waves in the plate as in Fig. 4.2. (Reprinted with permission from Datta et al. 1988c, Fig. 8.)

Exact

Stiffness method II

Symmetric Antisymmetric

8

Effective modulus method

Normalized Phase Velocity

7 6 5 4 3 2 1 0

0

1

2

3

4

5

6

7

8

Normalized Frequency

FIGURE 4.6  Dispersion of waves propagating at an angle of 45° to the X-axis. (Reprinted with permission from Karunasena et al. 1991b, Fig. 1.)

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Guided Waves in Fiber-Reinforced Composite Plates

(3.2.49), (3.2.52), (3.2.54), and (3.2.57), the long-wavelength effective moduli of the periodic cross-ply plate are given below. c11 = c22 =

2c33 (c11 + c33 ) − (c13 − c23 )2 4c33

1 c13 = c23 = (c23 + c13 ) 2 c33 = c33 c12 =

4c12c33 + (c13 − c23 )2 4c33

c66 = c66 , c44 = c55 =



(4.3.1)

2c44c55 c44 + c55



Thus, the homogeneous medium is tetragonal. The effective properties of the medium are, in units of 1011 N/m2,

c11 = 0.8732,

c13 = 0.0668,

c33 = 0.1392,

c12 = 0.0644 ,

c66 = 0.0707 ,

c55 = 0.0468

Since the propagation angle is different from a symmetry direction, both in-plane (S and A) and out-of-plane (SH) modes are coupled. The effect of this coupling is in the mode interchanges that take place between SH and S or A modes at certain frequencies. First, mode interchange occurs between qSH0 and qS0 modes at a frequency slightly below the cutoff frequency of the qS1 mode. At low frequencies, the qS0 mode has the plate velocity of 2.6 (normalized), and it is almost independent of frequency (normalized) until W ≈ 1.7, when the velocity starts dropping and becomes very close to the velocity of the qSH0 mode (2.4). Then, the velocity of the qSH0 drops rapidly toward the Rayleigh velocity, whereas the qS0 mode propagates at the velocity of 2.4 independent of frequency until W ≈ 5.9, when it starts dropping to the quasi-shear wave speed in the medium. Note also the coupling between the qSH1 and qA1 modes at about W = 3.7 and between qA1 and qSH3 modes at about W = 7.8. This coupling phenomenon is a feature of a strongly anisotropic medium for off-axis propagation and can be used to determine the in-plane properties. The effect of anisotropy and layering on guided wave propagation in different directions in homogeneous, cross-ply, and angle-ply plates are further illustrated in Figs. 4.7–4.12. All the laminae are assumed to have the properties listed in Table 4.1. The first stiffness method was used to obtain the dispersion results. In Figs. 4.11 and 4.12, the results of an analytical solution are also shown. Note that when the wave propagates in a direction different from a symmetry direction, the in-plane and out-of-plane motions are coupled, i.e., longitudinal, flexural, and SH waves are coupled. So, these figures show the dispersion behavior of all these three types of motion. It is seen from Figs. 4.7 and 4.8 that the phase velocities in a homogeneous plate increase much more steeply as the wave propagation angle becomes smaller, showing the strongly anisotropic nature of the material. This feature is also manifested in the three-layer angle-ply composite plate, as seen in Figs. 4.9 and 4.10. Finally, Figs. 4.11 and

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Symmetric Antisymmetric 8 7

Ω

6 5 4 3 2 1 0

0

1

2

3

4 γ

5

6

7

8

FIGURE 4.7  Frequency versus wave-number plot for a homogeneous graphite-epoxy plate for propagation at an angle of 30° to the X-axis. (Reprinted with permission from Karunasena 1992, Fig. 2.5(b).)

Symmetric Antisymmetric 8 7

Ω

6 5 4 3 2 1 0

0

1

2

3

4 γ

5

6

7

8

FIGURE 4.8  Frequency versus wave-number plot for a homogeneous graphite-epoxy plate for propagation at an angle of 60° to the X-axis. (Reprinted with permission from Karunasena 1992, Fig. 2.5(c).)

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Symmetric Antisymmetric 8 7 6 Ω

5 4 3 2 1 0

0

1

2

3

4

5

6

7

8

γ

FIGURE 4.9  Frequency versus wave-number plot for propagation in the 30° direction in a 0°/60°/0° plate. (Reprinted with permission from Karunasena 1992, Fig. 2.6(b).)

Symmetric Antisymmetric 8 7 6

Ω

5 4 3 2 1 0

0

1

2

3

4 γ

5

6

7

8

FIGURE 4.10  Frequency versus wave-number plot for propagation in the 60° direction in a 0°/60°/0° plate. (Reprinted with permission from Karunasena 1992, Fig. 2.6(c).)

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Elastic Waves in Composite Media and Structures Exact

Stiffness method II

Symmetric Antisymmetric

8 7 6

Ω

5 4 3 2 1 0

0

1

2

3

4 γ

5

6

7

8

FIGURE 4.11  Dispersion curves for a three-layer 0°/90°/0° plate for propagation in the 45° direction. (Reprinted with permission from Karunasena 1992, Fig. 2.3.)

Exact

Stiffness method II

Symmetric Antisymmetric

8 7 6

Ω

5 4 3 2 1 0

0

1

2

3

4 γ

5

6

7

8

FIGURE 4.12  Dispersion curves for a 35-layer 0°/90°/…90°/0° plate for propagation in the 45° direction. (Reprinted with permission from Karunasena 1992, Fig. 2.4.)

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4.12 show the effect of increasing the number of layers in a cross-ply (90°/0°/90°/…) plate when the wave propagates at an angle of 45° to the fiber direction. These show that as the number of laminae increases, the phase velocities decrease for the same frequencies.

4.4 Application to Materials Characterization As shown in the previous section, the theoretical formulation given in Section 4.2 can be used to gain fundamental understanding of guided waves in multilayered composite plates. This knowledge can be used to estimate elastic properties of the laminae making up the plate. In this section, a particular example of modeling and measuring guided wave dispersion for the determination of anisotropic elastic properties is discussed as an illustration. Shull et al. (1994) studied the dispersion of a fluid-loaded layered plate composed of alternating layers of aluminum bonded by layers of epoxy impregnated by aramid fibers (ARALL). Both experimental observations and theoretical predictions of the dispersion characteristics were presented. The experimental technique involved a single-side access immersion reflection method. This has been described in Shull et al. (1994). The use of guided waves for material characterization has been widely used with considerable success in the last two decades. We will not make any attempt to list all the works that are in the literature. Only a few that are related to the example discussed in this section will be given here. Many others can be found in the cited references and in the reviews and books that have been published. Some of these books and reviews are by Mal and Ting (1988), Datta et al. (1990a), Rokhlin et al. (1993), Kinra et al. (1994), Chimenti (1997), Datta (2000), Nayfeh (1995), Rose (1999), Liu and Xi (2002), Rose (2002), and Kundu (2004). Other works that are related to the example considered here are by Rokhlin and Wang (1989), Rokhlin and Chimenti (1990), Chimenti and Rokhlin (1990), Karim et al. (1990), Chimenti and Martin (1991) Bar-Cohen et al. (1993), Datta et al. (1999), Safaeinili and Chimenti (1995), Safaeinili et al. (1995, 1996), Lobkis et al. (2000), Rokhlin and Wang (2002), and Wang and Rokhlin (2003). Consider a periodic multilayered Al/ARALL plate with aramid fibers in the bond layers aligned along the X-axis (Fig. 4.1). We will consider waves propagating along the global x-axis that makes an arbitrary angle q to the fiber direction. The properties of aluminum (isotropic) are given in Table  4.2. This table also shows the estimated properties of the aramid-epoxy layers referred to the local XYZ-axes. As discussed in Shull et al. (1994), c11 and c12 were determined by ultrasonic measurements and approximate thin-plate calculations. They were found to be 60 and 5.8 GPa, respectively. Also, resonance spectra measurements give an estimate of c33 to be 8.5 GPa. Other moduli of ARALL are taken from the TABLE 4.2  Geometrical and Material Properties of Aluminum and Aramid–Epoxy Layers

Material Aluminum Aramid– epoxy

Elastic Stiffness (GPa)

Thickness (mm)

Density (kg/m3)

c11

c22

c33

c12

c13

c23

c44

c55

c66

0.31 0.22

2700 1600

109 60

109 5.8

109 8.5

56 1.7

56 5.0

56 5.0

27 1.8

27 2.3

27 2.1

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Theoretical Experimental 10 9

Phase Velocity (km/sec)

8 7 6 5 4 3 2 1 0

0

1

2

3

5 6 4 fd(MHz-mm)

7

8

9

10

FIGURE 4.13  Theoretical dispersion curves and experimental points for propagation in the fiber direction in the 2/1 aluminum/aramid–epoxy/aluminum plate. (Reprinted with permission from Datta et al. 1999, Fig. 2.)

literature. Although the values of the diagonal elements of the [c] matrix (except c22 and c66) can be obtained by through-the-thickness ultrasonic measurements, the evaluation of the other elements is a difficult task. Leaky Lamb wave measurements as described in Shull et al. (1994), along with model calculations (using the stiffness method II) of the dispersion of the modes, provide an accurate way of determining the cIJ matrix. Figure 4.13 shows the comparison of the measured dispersion results for propagation in q = 0° direction and the model results after optimizing the appropriate elements of cIJ (c11 = 59, c33 = 8.6, c55 = 2.3, and c31 = 6.0, in units of GPa). Thus, the values shown for these constants in Table 4.2 are reasonably good. Now, to obtain the other elements, we compared the model and experimental results for propagation in q = 90° direction. Figure 4.14 depicts the best-fit model and experimental results. It is found that the values for c22, c33, c44, and c23 are 8.6, 8.6, 1.8, and 5.0 GPa, respectively. Here, the greatest discrepancy is between the values for c22 as shown in Table 4.2 and the model prediction. Finally, Fig. 4.15 shows the experimental and model results for propagation in q = 45° direction. The best-fit values for c66 and c12 are found to be 2.3 and 6.0 GPa, respectively. Thus, the aramid–epoxy material can be treated as transversely isotropic. Figures 4.13–4.15 have some interesting features that are caused by the very different material properties of the three layers making up the composite plate. The two outer

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Theoretical Experimental

10 9

Phase Velocity (km/s)

8 7 6 5 4 3 2 1 0

0

1

2

3

5 6 4 fd (MHz-mm)

7

8

9

10

FIGURE 4.14  Theoretical and experimental dispersion results for propagation at 90° to the fibers in the 2/1 plate. (Reprinted with permission from Datta et al. 1999, Fig. 3.)

layers of aluminum have longitudinal and shear wave velocities of cl = 6.35 km/s and ct = 3.16 km/s, respectively. The middle layer can be modeled as transversely isotropic, as discussed above. The longitudinal wave velocities in this medium are given by c1l = 6.07 km/s, c2l = c3l = 2.32 km/s, along the symmetry axes. On the other hand, the shear wave velocities in these directions are



c1t =

c55 c c = 66 = 1.44 km/s and c3t = 44 = 1.06 km m/s ρ ρ ρ

The Rayleigh wave velocity in aluminum is about cR = 2.93 km/s. It is seen from Fig. 4.13 that at low frequencies, the S0 mode has the velocity cp = 5.7 km/s, which is less than the longitudinal wave velocities in both materials in the 0° direction. It is nearly frequency independent for fd < 1 MHz × mm. The velocity then drops rapidly to a minimum of 2.5 km/s at about the cutoff frequency of the S1 mode in the plate and then rises slowly toward the Rayleigh wave velocity in aluminum, but starts dropping at about fd = 8 MHz  × mm toward the shear wave velocity, c1t , because the bond layer is much softer than Al. The velocity of the A0 mode rises rapidly first and then more slowly toward cR, but starts dropping at around fd = 4 MHz × mm toward c1t . The low shear wave velocity of the bond layer causes the higher modes, A1 and S1, to have two plateaus in their phase velocities, one near cp and one near cR. These features are exhibited (in a more pronounced manner) by the

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Theoretical Experimental 10 9

Phase Velocity (km/s)

8 7 6 5 4 3 2 1 0

0

1

2

3

5 6 4 fd (MHz-mm)

7

8

9

10

FIGURE 4.15  Theoretical and experimental dispersion results for propagation at 45° to the fibers in the 2/1 plate. (Reprinted with permission from Datta et al. 1999, Fig. 4.)

waves propagating in the 90° direction (Fig. 4.14). Figure 4.15 shows the strong coupling effects between the in-plane and out-of-plane (SH) modes for propagation in the 45° direction. Note that the qSH0 and qS0 modes are strongly coupled, causing a mode interchange at around 1 MHz × mm (slightly below the cutoff frequency of the qS1 mode). It is seen that the qSH0 mode starts at the shear wave velocity in Al and stays fairly flat (independent of frequency) until it meets the qS0 mode, which has dropped from the plate velocity to ct. Then, the qS0 mode takes the character of the qSH0 mode, while the latter behaves like the former from that point on. This strong coupling phenomenon allows the determination of the in-plane properties of the bond material. Another interesting feature of a periodic stack of Al/ARALL layers is the pass and nopass bands exhibited by the guided waves. The use of Floquet wave theory (discussed in Chapter 3) to explain this phenomenon can be found in Safaeinili et al. (1995), Safaeinili and Chimenti (1995), and Wang and Rokhlin (2002b, 2002c).

4.5 Thin Layers In many applications of guided wave nondestructive evaluation (NDE), the thickness of the plate is much smaller than the wavelengths of propagating waves used in interrogating the medium. In other applications, such as coatings, thin-layer depositions on

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89

substrates, and adhesive bonds, the layer thickness can be very small compared to predominant wavelengths of guided waves. In such circumstances, the equations governing the motion in the layer can be approximated to simplify the analysis. Here, we will discuss some of these applications. Online measurement of material properties is a goal of many manufacturers of sheet material to allow them to improve production and quality through feedback and control. One example is the paper industry, where the parameters that have significant influence on the quality are the stiffness properties. Ultrasonic methods provide a means of determining stiffness from the characteristics of propagating dispersive elastic waves. Studies have shown a high correlation between the quality (strength, fiber orientation, and other mechanical properties) of paper and its elastic constants. Both contact and noncontact measurement methods have been used to measure the elastic constants in a wide range of studies, including Habeger et al. (1979, 1989), Mann et al. (1980), Baum et al. (1981), Baum (1985), Habeger and Wink (1986), Dewhurst et al. (1987), Hutchins et al. (1989), Nakano and Nagai (1991), Bobbin et al. (1992), Brodeur et al. (1993, 1997), Safaeinili et al. (1996), Castaings and Cawley (1996), Johnson et al. (1996), Cheng and Berthelot (1996), Mukdadi et al. (2001), Telschow and Deason (2001, 2002), Lafond et al. (2002), and others. One of the noncontact methods that has proven to be effective is the full-field imaging technique that provides a complete measurement of the elastic waves in paper traveling in all planar directions simultaneously, without single-pointmeasurement scanning devices (Telschow and Deason 2001, 2002; Lafond et al. 2002). This technique, coupled with modeling of ultrasonic waves in paper, has been shown (Mukdadi et al. 2001) to be capable of measuring orthotropic properties of paper. In the following discussion, modeling results are presented along with comparison with experimental observations.

4.5.1 Ultrasonic Waves in Paper Paper microstructure is usually modeled by orthotropic elastic anisotropy with nine elastic constants, the symmetry axes being aligned with the machine direction (MD), cross-machine direction (CD), and thickness direction (ZD). The symmetry axes have been chosen to coincide with the rectangular coordinate axes (x,y,z). Let u, v, and w be the dynamic displacement components along x-, y-, and z-axes, respectively. Then, the linear elastic constitutive equation for paper may be written as



σ xx  c c c . .    11 12 13 σ yy   c22 c23 . .    σ   c33 . .  zz    =  c44 . σ yz      σ zx   c55    σ   symmetric  xy 

.  e xx    .  e yy      .  ezz     .  γ yz    .  γ zx    c66  γ xy   

(4.5.1)

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The equation of motion is given by equation (2.1.56). It will be assumed that there are no body forces acting in the plate, so that fi = 0. If the plate is infinite in the x–y-plane, then one can use the Fourier transformation in space and time as ˆ (k , k , z ; ω ) = Φ x y

∞ ∞ ∞

∫ ∫ ∫ Φ(x , y , z ,t )e

i ( kx x + k y y −ωt )

dx dy dt

(4.5.2)

−∞ −∞ −∞





Here, k x, ky are the wave numbers in the x and y directions, respectively, and w is the circular frequency. Applying the Fourier transformation to equation (4.5.1) and the equation of motion will yield the eigenvalue problem [ A]{S, z } = [B]{S}



(4.5.3)



where {S} is the displacement–stress vector

{S(z )} = 〈uˆ vˆ wˆ σˆ zx σˆ yz σˆ zz 〉T

(4.5.4)



The subscript comma (,) in equation (4.5.3) denotes differentiation with respect to z, and the superscript T in equation (4.5.4) denotes transpose. The matrices [A] and [B] are given below.

 ikx c13  .   . [B] =  2 2 k c + k c − ρω 2  x 11 y 66  kx k y (c12 + c66 )  . 

.  .  c 55 [ A] =  .  .   .

.  c44 . . . .   . . . . .    . ikx c13 − 1 . .   . ik y c23 . − 1 .   . . . . − 1 .

c33 .

(4.5.5a)

ik y c23

.

.

.

.

ik y c44

.

1

.

ikx c55

1

.

.

.

.

.

.

−ikx

−ik y

kx k y (c12 +c66 ) k c + k c − ρω 2 x 66

.

2 y 22

.

.

2

− ρω

2

1  .  . (4.5.5b) .  .  . 

The general solution to equation (4.5.3) can be obtained by determining the eigenvalues and eigenvectors, which are found numerically by using appropriate LAPACK

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(Linear Algebra PACKage) subroutines. Solving for the displacement–traction vector S(z) across each layer (if the plate is laminated) and applying the continuity of displacements and tractions at the interfaces between adjacent layers, one can write a global matrix [Q], which is banded and has order of 6N × 6N for N-layered plate (Ju and Datta 1992a; Mal 1986). We have {S(z )} = [Q(m)][E(z , m)]{C(m)}, z m ≤ z ≤ z m+1





(4.5.6)

where [Q(m)] is the matrix of eigenvectors for the mth layer, and [E(z,m)] is the diagonal matrix, [E(z , m)] = Diag[e iς1 (m )( z − zm ) , e iς 2 (m )( z − zm ) , e iς3 (m )( z − zm ) , e iς1 (m )( zm+1 − z ) , e iς 2 (m )( zm+1 −zz ) , e iς3 (m )( zm+1 − z ) ]



and iςp(m) (p = 1,2,3) are the eigenvalues of the characteristic equation of equation (4.5.5) with Im Vp ≥ 0. {C(m)} is a 6 × 1 matrix of constants. Now, to apply the continuity condition for the displacement–traction vector {S(z)} at the interface between the mth and (m+1)th layers, equation (4.5.6) is evaluated at z = z − , giving m +1 {S(z m− +1 )} = [Q(m)][E(z m− +1 )]{C(m)}



(4.5.7a)



where [E(z m− +1 )] = Diag[e1 (m) e2 (m) e3 (m) 1 1 1], ei (m) = e iςi (m )h(m )



Here, h(m) is the thickness of the mth layer. In the same way, for the (m+1)th layer, we find {S(z m+ +1 )} = [Q(m + 1)][E(z m+ +1 )]{C(m + 1)}





(4.5.7b)

where  E(z m+ +1 ) = Diag 1 1 1 e1 (m + 1) e2 (m + 1) e3 (m + 1)



The continuity condition is satisfied if {S(z m− +1 )} = {S(z m+ +1 )}





(4.5.8)

Partitioning the displacement–stress vector as



[Q11 (m)] [Q12 (m)]  [E + (z , m)] [0]  C + (m)   {S(z )} =   [Q21 (m)] [Q22 (m)]  [0] [E − (z , m)] C − (m)  

(4.5.9)

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where [E − (z m+1 )] = Diag [e1 (m) e2 (m) e3 (m)] [E + (z m+1 )] = Diag [e1 (m + 1) e2 (m + 1) e3 (m + 1)]



(4.5.10)

equation (4.5.8) can be written in the form [Q11 (m)] [Q12 (m)]  [E − (z m+1 )]   [Q21 (m)] [Q22 (m)]  [0]  

[0] C + (m)  = [I ] C − (m)

 Q11 (m + 1)     Q21 (m + 1)

Q12 (m +11)  [I ]  C + (m + 1) [0]     Q22 (m + 1)  [0] [E + (z m+1 )] C − (m + 1) 



(4.5.11)

Now, assembling all the matrices from m = 1 and m = N, we obtain the global equation for the unknown coefficients C(m) (m = 1, …, N) in terms of the tractions applied at the top and bottom surfaces as [Q]{C} = {F }



(4.5.12)

where  [Q21 (1)]  Q  = [Q11 (1)][E − (z 2 )]  … 

[Q22 (1)][E − (z 2 )]

[−Q11 (2)]

[Q12 (1)] …

… [Q21 ( N − 1)][E (z N )]

[Q22 ( N − 1)]

[0]

[−Q12 (2)][E (z 2 )] [0] +

…

…

  [−Q21 ( N )] [−Q22 ( N )][E (z N )]  [Q21 ( N )][E − (z N +1 )] [Q22 ( N )]  …

[0] [0]

[0]

…

… −

[0]

…

+

(4.5.13)

{C}T = {C + (1)}T {C − (1)}T … {C + (N)}T {C − (N)}T

and

{F }T = {− F1 }T {0}T … {0}T {FN +1 }T

where {F1} and {FN+1} are the forces acting on the bottom and top surfaces, respectively. If there are no forces acting on the top and bottom surfaces of the plate, then for nontrivial solution, the determinant of [Q] must vanish, i.e.,

|Q| = 0



(4.5.14)

This equation governs the frequency w as a function of the wave numbers k x and ky.

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For a thin plate, like paper, the thickness is much smaller than the wavelengths at which the imaging measurements (Mukdadi et al. 2001) are made. It is useful to derive the first-order plate equations following Mindlin (1951a, 1951b) and Kane and Mindlin (1956). These are discussed in the following section.

4.5.2 First-Order Plate Equations 4.5.2.1 Symmetric (Including Extensional) Motion Symmetric wave motion in an elastic plate of thickness 2h is considered in this section using a first-order approximation. In this, the symmetric modes of a plate bounded by two planes at z = 2h are described by assuming that the displacement components (Kane and Mindlin 1956) are u�( x , y , z , t ) = u( x , y , t ) v�( x , y , z , t ) = v( x , y , t )

(4.5.15)

z w� ( x , y , z , t ) = w( x , y , t ) h



Thus, the corresponding engineering strain components are



e xx = u,x ( x , y , t ),

z γ yz = w , y ( x , y , t ) h

e yy = v , y ( x , y , t ),

z γ zx = w ,x ( x , y , t ) h

ezz =

1 w( x , y , t ), h

γ xy = u, y ( x , y , t ) + v ,x ( x , y , t )

(4.5.16)



where the subscript commas (,) in equation (4.5.16) denotes derivative with respect to the spatial variables. Using the stress–strain relation in equation (4.5.1) in the equation of motion (2.1.56), integrating the first two equations of (2.1.56) and multiplying z times the third equation with respect to z through the thickness of the plate, we obtain the approximate displacement equations of motion of the plate. These are k c11u,xx + c66 v , yy + (c12 + c66 )v ,xy + c13 w ,x = ρu�� h k (c12 + c66 )u,xy + c22 v , yy + c66u,xx + c23 w , y = ρv�� h

(4.5.17)

k �� −3 (c13u,x + c23v , y ) + c55 w ,xx + c44 w , yy − 3k 2 h −2 c33w = ρw h

where the overdots denote the derivative with respect to time.

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Note that a correction factor k has been introduced in the expressions for ezz and s zz to account for the correct cutoff frequency of the first thickness-stretch mode. Equation (4.5.17) is the generalization of the isotropic case considered by Kane and Mindlin (1956). This gives the equations governing the Fourier transforms of the displacement components as h 2 (kx2 c11 + k y2 c66 )  h 2 k k (c + c )  x y 12 66   ikhkx c13

 uˆ     vˆ  h 2 (kx2 c66 + k y2 c22 ) − ikhk y c23     ikhk y c23 k 2 c33 + h 2 (kx2 c55 + k y2 c44 ) / 3  wˆ 

h 2 kx k y (c12 + c66 )

− ikhkx c13

 ρh  −ω2  .   . 



2

(4.5.18)

 uˆ     ρh 2 .  vˆ  = {0}   . ρh 2 / 3  wˆ  .

.

For nontrivial solution, the determinant of the coefficient matrix above must vanish. This gives the dispersion equation as h 2 (kx2 c11 + k y2 c66 ) − ρω 2 h 2 h 2 kx k y (c12 + c66 )

h 2 kx k y (c12 + c66 )

− ikhkx c13

h 2 (kx2 c66 + k y2 c22 ) − ρω 2 h 2

− ikhk y c23

=0

ikhkx c13 ikhk y c23 k 2 c33 + h 2 (kx2 c55 + k y2 c44 )/3 − ρω 2 h 2 /3 (4.5.19) This equation shows the coupling between the out-of-plane motion ( vˆ ) and the inˆ If either ky = 0 or k x = 0, then these two motions are uncoupled. plane motion ( uˆ, w). This equation is cubic in w 2 for given k x and ky. Thus, there are three propagating waves if w is above the cutoff frequency of the S1 mode. The S 0 and SH0 modes propagate at all frequencies. When ky = 0, the dispersion equation becomes    c c2 1c c 2  k 2c 2 (c66 /ρ − c 2 ) (c11 /ρ − c 2 )  k 2 33 2 + 55 −  − (c13 /ρ )2 ] = 0  (4.5.20) 2 3  (ω h)  ρ(ω h) 3 ρ  



Here, c is the phase velocity in the x-direction. It is seen that the SH0 mode is uncoupled from the extensional motion in the xz-plane. Furthermore, as wh → 0, the S 0 mode has the plate velocity c p = [(c11 − c132 / c33 )/ρ]1/2





(4.5.21)

The velocities of the two extensional modes are governed by a quadratic equation

α c 4 − 2β c 2 + γ = 0



(4.5.22)

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where

α=

c33 1 − k2 3 ρ(ω h)2

(4.5.23)

c c − c2 1 c +c  2β =  11 55  − k 2 11 33 213 3 ρ  ( ρω h)

γ =

1 c11c55 3 ρ



Equation (4.5.21) has both roots positive if a > 0, i.e., wh > k 3c33 /ρ . Thus, the cutoff frequency of the S1 mode is

ω h = k 3c33 /ρ



(4.5.24)



From the exact analysis presented in Section 4.5.1, it is easily shown that this cutoff frequency is given by

ωh =

c33 ρ

π 2

(4.5.25)



Thus, the correction factor k can be chosen such that the two frequencies given by equations (4.5.24) and (4.5.25) are the same. This gives k=

π

(4.5.26)

2 3

This value of k has been chosen when comparing the numerical results obtained by this approximate theory and the exact analysis. 4.5.2.2 Antisymmetric Motion In this section, we consider the antisymmetric (A) or bending motion of the plate. The displacement components (Mindlin 1951a, 1951b) will be chosen as u� = zψ ( x , y , t ) v� = zφ( x , y , t ) w� = w( x , y , t )



(4.5.27)

As can be seen from these approximate expressions, it has been assumed that the displacement normal to the plate does not vary through the thickness. The symbols f and ψ represent the rotations of the normal to the midplane about the x- and y-axes, respectively. The strain components are obtained as e xx = zψ ,x , e yy = zφ, y , ezz = − z (c13ψ ,x + c23φ, y )/c33

γ yz = k1 (φ + w , y ), γ zx = k2 (ψ + w ,x ), γ xy = z (φ,x + ψ , y )

(4.5.28)

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Correction factors k1 and k2 are introduced so that the cutoff frequencies of the A1 and SH1 modes predicted by the approximate and exact analyses coincide. Now, integrating the third equation and multiplying z times the first two equations of motion through the thickness of the plate gives h2 h2 h2 ( c11ψ ,xx + c66ψ , yy ) − c55 k22ψ + ( c12 + c66 )φ,xy − c55 k22 w ,x = ρ ψ�� 3 3 3 (4.5.29)

h2 h2 h2 ( c12 + c66 )ψ ,xy + ( c22φ, yy + c66φ,xx ) − c44 k12φ − c44 k12 w , y = ρ φ�� 3 3 3 �� c55 k22 (ψ ,x + w ,xx ) + c44 k12 (φ, y + w , yy ) = ρw





where

2 /c c11 = c11 − c132 /c33 , c12 = c12 − c13c23 /c33 , c22 = c22 − c23 33

Taking the Fourier transforms of the three equations (4.5.29), we obtain the linear eigenvalue problem  2 h2 h2 2 h2 2 2 k k (c + c )  k2 c55 + 3 (kx c11 + k y c66 ) − ρω 3 3 x y 12 66   h2 h2 h2  kx k y ( c12 + c66 ) k12 c44 + (k y2 c22 + kx2 c66 ) − ρω 2 3 3 3   2 2 − ik2 kx c55 − ik1 k y c44    ψˆ  0       φˆ  = 0  ik12 k y c44         k22 kx2 c55 + k12 k y2 c44 − ρω 2  wˆ  0  ik22 kx c55



(4.5.30)

Solving for the eigenvalues w when k x and ky are assigned values leads to the dispersion relation for the A0, A1, and SH1 modes. The correction factors are found by equating the cutoff frequencies of A1 and SH1 modes obtained by solving the approximate and exact equations. They are



k1 = k2 =

π 2 3

Numerical results are presented in the following section for two kinds of paper, RSK 59 and LNR42, that were studied by Telschow and Deason (2001, 2002) using a full-field laser-based ultrasonic imaging technique.

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Guided Waves in Fiber-Reinforced Composite Plates TABLE 4.3  Geometrical and Material Properties of RSK59 and LNR42 Papers Elastic Stiffness (GPa)

Material

Thickness (mm)

Density (kg/m3)

c11

c22

c33

c12

c13

c23

c44

c55

c66

RSK59 LNR42

110 288

818 742

9.85 10.3

4.95 3.79

0.14 0.10

1.77 1.82

0.10 0.10

0.15 0.15

0.20 0.14

0.21 0.16

2.53 2.23

4.5.3 Numerical and Experimental Results The exact dispersion equation (4.5.14) and the approximate equations for the symmetric and antisymmetric motion were solved for two kinds of paper, RSK59 and LNR42, that were studied by Telschow and Deason (2001, 2002). The properties of these two papers are given in Table 4.3. Figures  4.16 and 4.17 show the variations of the nondimensional frequency Ω (= 2wh/ c66 /ρ ) vs. the nondenominational wave number K (= 2kh) for LNR42. The direction of propagation of the wave, a, is defined by the relations k x = K cosa, ky = K sina. Figure 4.16 shows the dispersion behavior for propagation along the machine direction (MD) of the paper (a = 0°). Because this is a principal direction, it is seen that the SH modes are uncoupled from the other two modes. An important feature of this figure is that the cutoff frequencies of the A1 and SH1 modes are higher than that of the S1 mode because the stiffness c33 is smaller than both c44 and c55. It is found that the longitudinal wave speed is cp = 3.72 mm/ms. The approximate and exact results are found to agree very well for the A0 and SH0 modes. For the S 0 mode, the agreement is very good when K is 2.0

Symmetric Antisymmetric Exact

Ω

1.5

1.0

0.5

0.0 0.0

0.5

1.0 K

1.5

2.0

FIGURE 4.16  Exact and approximate dispersion curves for wave propagation along MD-axis for LNR42 paper. (Reprinted with permission from Mukdadi et al. 2001, Fig. 1.)

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Symmetric Antisymmetric Exact

Ω

1.5

1.0

0.5

0.0 0.0

0.5

1.0 K

1.5

2.0

FIGURE 4.17  Exact and approximate dispersion curves for wave propagation along 45° off MD-axis for LNR42 paper. (Reprinted with permission from Mukdadi et al. 2001, Fig. 2.)

small, and it is reasonably good even when K has moderate values. The same is true for the S1 mode. The comparison of the results for the A1 mode shows that the approximate results are good only for small K. For propagation at an angle of 45° (shown in Fig. 4.17), there is evidence of strong coupling between the in-plane and out-of-plane motions. Figure 4.18 shows the variation of the wavelength with frequency for flexural waves propagating along MD and cross-machine direction (CD). Also shown in this figure are the values determined by experiments described in Mukdadi et al. (2001) and Telschow and Deason (2002). Very good agreement is found. Similar numerical results were obtained for the RSK59 paper, which is similar to the LNR42 paper, except that it has half the thickness. This means that the resonance frequencies are higher, and stronger coupling between the in-plane and out-of-plane motions is observed in this case. Figure 4.19 shows the variation of the wavelength with frequency for the A0 mode propagating in the MD and CD in RSK59 paper. Very good agreement with experimental data is seen.

4.6 Guided Waves in Plates with Thin Coating and Interface Layers Ultrasonic guided-wave propagation in thin coating layers on a thick elastic substrate has been investigated by many researchers for applications in seismology, electronic devices, thermal barrier coatings, and many others. In most of these studies, the substrate is assumed to be much thicker than the wavelengths of interest so that it can be effectively modeled as a semi-infinite medium. A good review of early works was published by Farnell

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MD - 0° (theoretical) MD - 0° (experimental) CD - 90° (theoretical) CD - 90° (experimental)

7 6

λ (mm)

5 4 3 2 1 0

0.0

50.0

f (kHz)

100.0

150.0

FIGURE 4.18  Wavelength vs. frequency of A0 modes of propagation along MD and CD directions in LNR42 paper. (Reprinted with permission from Mukdadi et al. 2001, Fig. 4.)

8

MD - 0° (theoretical) MD - 0° (experimental) CD - 90° (theoretical) CD - 90° (experimental)

7 6

λ (mm)

5 4 3 2 1 0 0.0

50.0

f (kHz)

100.0

150.0

FIGURE 4.19  Wavelength vs. frequency of A0 modes of propagation along MD and CD directions in RSK59 paper. (Reprinted with permission from Mukdadi et al. 2001, Fig. 6.)

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and Adler (1972). Additional studies of guided waves in an elastic layer over an anisotropic half-space have revealed many interesting features of the dispersion characteristics (Bouden and Datta 1990; Sotiropoulos 1999; Shuvalov and Every 2002). Wu and Wu (2000) studied surface waves in a coated anisotropic medium loaded with viscous fluid. In this section, we discuss guided waves in a plate with thin superconducting coating and interface layers. The investigation is relevant to ultrasonic characterization of superconducting tapes that are fabricated for commercial high-current applications. The tapes are composites consisting of a brittle superconducting phase and a ductile metal phase. They are in the form of (a) a superconducting oxide layer (such as BSCCO) sandwiched in a high-conductivity metal like silver, (b) a dip-coated silver tape in which the oxide layers are outside the core conducting layer, (c) a silver-sheathed dip-coated tape, and (d) a deposition of biaxially aligned oxide (YBCO) coating on nickel tapes. The degree of current-carrying capacity in the oxide layer is a strong function of its integrity. Ultrasonic waves provide an efficient means of characterizing the thin oxide layers. As mentioned previously, the tapes may consist of either an elastic isotropic metallic core coated with thin anisotropic oxide layers or a thin oxide layer sandwiched between two metallic layers. Both of these problems are studied here. It is assumed that the thickness of the oxide layer is small compared to all wavelengths in the coatings or layers. In such cases, one approximation is to model the thin layer as a thin plate, which was done in the previous section. In this section, a different approach is taken, which involves expanding the field quantities in the (small) thickness and obtaining approximations by truncating the infinite series. Such an approach was taken by Bövik (1994, 1996), Niklasson et al. (2000a, 2000b), Niklasson and Datta (2002a, 2002b), Benveniste (2006), and Ting (2007). In the following discussion, we first consider the case of a layered anisotropic plate consisting of a thick isotropic core and two identical anisotropic thin coating layers. The second problem considered is a thin anisotropic layer sandwiched between two identical thick isotropic layers.

4.6.1 Isotropic Plate with Thin Coating Layers To obtain an approximate solution to this problem, we first derive an effective boundary condition for a thin layer overlying an elastic half-space. This problem is similar to that considered by Bouden and Datta (1990), who solved it by using the global matrix technique discussed in Section 4.5.1. Let the displacement and stress in the half-space (x3 > 0) (Fig. 4.20) be denoted by uj and s jm, respectively, and let those in the layer (−h < x3 < 0) be denoted by Uj and Σjm, respectively. Note that for this purpose the origin is taken at the interface between coating and the substrate. The equation of motion in the layer—see equation (4.2.4)—is

Σ jm , j = ρcU��m

(4.6.1)



For simplicity, it is assumed that the x–y-plane is a plane of symmetry. Thus, the stress–strain relation given in equation (3.2.2) is given by

Σ jm = D jmkl Ekl



(4.6.2)

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Guided Waves in Fiber-Reinforced Composite Plates h

CJK, ρc

a x2

x1

a

λ, µ, ρ

x3

h

FIGURE 4.20  Cross-section of the coated plate. (Reprinted with permission from Niklasson et al. 2000a, Fig. 2.)

where Ekl are the engineering strain components. The displacement and traction are assumed to be continuous across the interface, z = 0. On the surface z = −h, it is assumed that Σ m3 = Tm , x 3 = −h



(4.6.3)



The goal is now to replace the effect of the coating layer on the half-space by an effective boundary condition. We do this by introducing the notation ( f )a for f(x,y,a,t) and then expanding the traction Σm3 in a Taylor series ( Σ m3 )− h = ( Σ m3 )0 − h( Σ m3,3 )0 + O(h 2 )



(4.6.4)



The equation of motion (4.5.30) can now be written as Σ m3,3 = ρcU��m − Σ m1,1 − Σ m 2 ,2



(4.6.5)



Using the continuity of displacement and traction across x3 = 0, we have (U j )0 = (u j )0 ( Σ 33,3 )0 = ρc (u��3 )0 − (σ 13,1 )0 − (σ 23,2 )0



(4.6.6)

Now, by differentiating Σ11, Σ12, and Σ22 with respect to x and y, we get

 Σ11,1   D   11  Σ  12 ,1   D16 =    Σ12 ,2   0  Σ   0  22 ,2 

D12

D26 0 0



D13

D36 0 0

0 0 D16 D12

0 0 D26 D22

0 0 D36 D23

D16 D66 0 0

 E11,1     E22 ,1    0   E33,11   0   E11,2    D66   E22 ,2   D26   E33,2    Γ12 ,1    Γ12 ,2 

(4.6.7)

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where Eij = U i , j (i = j ), Γ ij = U i , j + U j ,i (i ≠ j )

We also have

E33 = ( Σ 33 − D13 E11 − D23 E22 − D36 Γ12 )/D33



(4.6.8)

Here DIJ = CIJ. Now, using the continuity conditions at x3 = 0 and equation (4.6.5), all of the components of (Σj3,3)0 are obtained. Thus, equation (4.6.4) can be written as

τ + h( Aσ τ + Au u ) = T + O(h 2 ), x 3 = 0





(4.6.9)

where t = (s13,s23,s33)T, u = (u1,u2,u3)T, T = (T1,T2,T3)T, and the superscript T denotes transpose. The nonzero elements of As are ( Aσ )13 =

D13 D ∂ + 36 ∂ D33 1 D33 2

( Aσ )23 =

D36 D ∂1 + 23 ∂ 2 D33 D33

(4.6.10)

( Aσ )31 = ∂1 ( Aσ )32 = ∂ 2





Also, the nonzero elements of the symmetric matrix Au are    D2  D D  D2  ( Au )11 = − ρc ∂t2 +  D11 − 13  ∂12 + 2  D16 − 13 36  ∂1 ∂ 2 +  D66 − 36  ∂ 22 D33  D33  D33        D D  D D + D326  D23 D36  2 ( Au )12 =  D16 − 13 36  ∂12 +  D12 + D66 − 13 23  ∂1 ∂ 2 +  D26 − D  ∂ 2 D33  D33     33     D2  D2  D D  ( Au )22 = − ρc ∂t2 +  D66 − 36  ∂12 + 2  D26 − 23 36  ∂1 ∂ 2 +  D22 − 23  ∂ 22 D33  D33  D33     ( Au )33 = − ρc ∂t (4.6.11) ∂ ∂ where ∂m = and ∂t = . ∂xm ∂t If the thickness h is small compared to all wavelengths in the coating, an accurate approximation of the effect of wave propagation in the coating on the half-space is given by the boundary condition 2



τ + h( Aσ τ + Au u ) = T , x 3 = 0



(4.6.12)

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The boundary condition derived above agrees with that derived by Bövik (1996) for the isotropic layer. This boundary condition correct to O(h) for a monoclinic coating layer on a substrate is valid for any substrate material, isotropic or anisotropic. Thus, this can be used to study reflection of body waves from a coating layer, propagation of surface waves on a coated substrate, and forced motion of the coated medium. In the following, equation (4.6.12) will be used to derive, as an example, the dispersion equation for an isotropic plate with coatings of the same material and thickness on the top and bottom surfaces of the plate. The geometry of the plate is shown in Fig. 4.20. This has been chosen for the convenience of algebra. To derive the approximate dispersion relation, we replace the thin coatings by the truncated series expansion in equation (4.6.12). The governing equations are ∂mσ mj = ρu��j , − a < x 3 < a

τ ∓ h( Aσ τ + Au u ) = 0, x 3 = ±a



(4.6.13)

where

σ jm = λδ jmum ,m + µ(u j ,m + um , j )



(4.6.14)



and where l and m are the Lamé constants for the isotropic plate. The general solution to equation (4.6.13) may be written in terms of the longitudinal and shear wave potentials (see Achenbach 1973) as u = ∇Φ + ∇ × Ψ , ∇.Ψ = 0 Φ = ( A0 sin px 3 + B0 cos px 3 )e i ( kx1 −ωt ) Ψ j = ( A j sin qx 3 + B j cos qx 3 )e i ( kx1 −ωt )

(4.6.15)

ik ik A3 = − B1 , B3 = A1 q q p = kl2 − ω 2 , kl = ω / cl , cl = (λ + 2µ )/ρ

q = ks2 − ω 2 , ks = ω / cs , cs = µ /ρ



Here, j = 1,2,3. Note that, since wave propagation is in the plane of x1–x3, the field quantities do not depend on x2. If the coating is isotropic, then P–SV motion will be uncoupled from the SH motion. However, because of the anisotropy of the coating layer, all these three are coupled. The dispersion relation is obtained by applying the effective boundary condition in line 2 of equation (4.6.13). Due to the symmetry of the problem, it can be split into two: symmetric and antisymmetric. The dispersion equation for the symmetric mode is G11h cos qa + F11 sin qa

G12 h cos pa + F12 sin pa

G21h cos qa

G31 cos qa + F31h sin qa

G22 h cos pa

G13h cos qa

(4.6.16)

G23h cos qa + F23 sin qa = 0

G32 cos pa + F32 h sin pa

0



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and that for the antisymmetric mode is −G11h sin qa + F11 cos qa

G12 h sin pa − F12 cos pa

− G21h sin qa

− F31h cos qa + G31 sin qa

G22 h sin pa

G13h sin qa

G23h sin qa − F23 cos qa = 0

F32 h cos pa − G32 sin pa

0

(4.6.17)

Here,   D (D + 2 µ )  2  G11 =  ρcω 2 −  D11 − 13 13 k q D33        D (λ − D13 )  D − (λ + 2 µ ) 13 p 2  k G12 = −i  ρcω 2 −  D11 + 13  D33 D33      D D  G13 = −i  D16 − 13 36  ks2 k D33    D (D + 2 µ )  3 G21 = −  D16 − 36 13 k q D33     D D (λ − D13 )  2  2 G22 = i  (λ + 2 µ ) 36 p 2 +  D16 + 36 k k D33 D33    

(4.6.18)

  D2   G23 = i  ρcω 2 −  D66 − 36  k 2  ks2 D33     G31 = 2i µ kq G32 = −(λ k 2 + (λ + 2µ ) p 2 ) F11 = µ(q 2 − k 2 ) F12 = −2i µ kp F23 = i µ ks2 q F31 = i( µ(k 2 − q 2 ) − ρcω 2 )k

F32 = ( ρcω 2 − 2µ k 2 ) p



When h = 0, equations (4.6.16) and (4.6.17) decouple into equations for the antiplane and in-plane motions of the isotropic plate. However, for nonzero h, these two motions are coupled, except when the propagation direction coincides with a symmetry direction of the coating material of symmetry higher than monoclinic. In that case, the elements G13, G 21, and G 22 vanish. Then, both determinants in equations (4.6.16)

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105

and (4.6.17) decouple into 2 × 2 and 1 × 1 determinants corresponding to the in-plane and antiplane motions. It is seen that the use of effective boundary conditions leads to 3 × 3 determinantal equations for the symmetric and antisymmetric cases instead of 9 × 9 determinantal equations obtained by the global matrix method discussed earlier. Another important feature is that the thickness h appears explicitly as a simple factor in the elements of the determinants in equations (4.6.16) and (4.6.17). Numerical results for a plate with high-temperature superconducting coating layers will be discussed later in this section. In the following subsection, we will consider the case of a thin monoclinic layer sandwiched between two isotropic layers.

4.6.2 Isotropic Plate with a Thin Interface Layer Because interface properties between two adjacent layers in composite structure play a significant role in the mechanical behavior of the structure, there have been many investigations of nondestructive evaluation techniques for the characterization of the interface layers. Techniques based on ultrasonic wave propagation and scattering have been widely explored. Mechanical models of the interfaces are an essential ingredient of these techniques. Here, we will use the series expansion method to develop an effective interface condition in terms of the thickness and anisotropic mechanical properties of the layer. For various other modeling and experimental investigations, reference may be made to the works by Jones and Whittier (1967), Nayfeh and Nassar (1978), Schoenberg (1980), Rokhlin et al. (1980), Baik and Thompson (1984), Pilarski and Rose (1988), Mal (1988b), Paskaramoorthy et al. (1988), Datta et al. (1988a, 1988b), Mal and Xu (1989), Martin (1990, 1992), Olsson et al. (1990), Xu and Datta (1990), Rokhlin and Wang (1991, 1992), Boström et al. (1992), Rokhlin and Huang (1992, 1993), Huang and Rokhlin (1992), and Ju and Datta (1992a). Critical examination of the studies using a spring model for the layer, an exact O(h) (or higher) approximation of the layer, and the exact solution shows that the spring model is only good if the interface layer is much weaker (low shear modulus) than the surrounding media. The rational O(h) approximation provides a more accurate picture of the dynamic behavior of the layered medium as long as the wavelengths of the propagating waves are longer than the thickness h of the interface layer. Consider a multilayered plate as shown in Fig. 4.1. As before, the plate consists of N anisotropic layers, each of thickness h(m), m = 1, 2, …, N. Under the assumptions that the adjacent layers are perfectly bonded to one another and that the top and bottom surfaces of the plate are stress free, the dispersion equation for harmonic waves propagating in the x-direction is obtained using the global matrix method discussed in Section 4.5. As noted before, unless the material symmetry of each layer is transversely isotropic or higher, the eigenvalues and eigenvectors of equation (4.5.3) have to be obtained numerically. Then, the dispersion equation is solved using some numerical search routines. In the following, we consider a special material system, for which we introduce an approximation that enables us to write down the dispersion relation explicitly. First, we derive effective interface conditions (IC) for wave propagation in a thin anisotropic layer of thickness 2h sandwiched between two elastic bodies (Fig. 4.21). Let the displacement and stress be denoted by uj+ and σ +jm , respectively, in the lower body

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a

b

λ, µ, ρ X1

CJK, ρL

X2

h h a

X3

b

FIGURE 4.21  Cross-section of the plate with an interface layer. (Reprinted with permission from Niklasson et al. 2000b, Fig. 2.)

(x3 > h), by uj− and σ −jm , respectively, in the upper body (x3 < −h), and by Uj and Σjm, respectively, in the layer (|x3| < h). We assume that equations (4.6.1) and (4.6.2) hold in the layer, with the density of the layer denoted by rL . Since the layer is assumed to be perfectly bonded to the surrounding bodies, the following conditions must hold. u ±j = U j , σ ±j 3 = Σ j 3 , x 3 = ±h



(4.6.19)



Using the notation fa for f(x1, x2, a, t) and expanding the displacement Uj and traction Σj3 in Taylor series (U j )0 = (U j )± h ∓ h(∂3U j )± h + O(h 2 ) ( Σ j 3 )0 = ( Σ j 3 )± h ∓ h(∂3 Σ j 3 )± h + O(h 2 )



(4.6.20)

we get from the stress–strain relation (4.6.2) ∂3U1 = (D44 Σ13 − D45 Σ 23 )/δ − ∂1U 3

(4.6.21)

∂3U 2 = (D55 Σ 23 − D45 Σ13 )/δ − ∂ 2U 3

∂3U 3 = ( Σ 33 − D13 E11 − D23 E22 − D36 Γ12 )/D33



where d = D44 D55 − D . Using equation (4.6.21) and the interface conditions in equation (4.6.19) in equation (4.6.20), we obtain the following relation between the displacements and tractions at x3 = ±h. 2 45



(u + )h + h( Bu u + + Bσ τ + )h = (u − )− h − h( Bu u − + Bσ τ − )− h + O(h 2 )

(4.6.22)

± σ ± 〉T Here, u± = 〈u1± u2± u3± 〉T , τ± = 〈σ 13± σ 23 33 , and the nonzero elements of Bu and B s are

( Bu )13 = ∂1 , ( Bu )23 = ∂ 2 ( Bu )31 = (D13 ∂1 + D36 ∂ 2 )/D33

(4.6.23)

( Bu )32 = (D36 ∂1 + D23 ∂ 2 )/D33 ( Bσ )11 = − D44 / δ , ( Bσ )22 = − D55 /δ

( Bσ )33 = −1 / D33 , ( Bσ )12 = ( Bσ )21 = D45 /δ



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Following the same procedure as in the case of coating, equation (4.6.20) leads to the equation giving the relation between (τ+)h and (τ−)−h as

(τ + )h + h( Au u + + Aσ τ + )h = (τ − )− h − h( Au u − + Aσ τ − )− h + O(h 2 )



(4.6.24)

where Au is given by equation (4.6.11), with r c replaced by rL . Also, As = (Bu)T. By combining equations (4.6.22) and (4.6.24), we obtain the set of interface conditions as

(T + )h + h(C T + )h = (T − )− h − h(C T − )− h + O(h 2 )

(4.6.25)



± σ ± 〉T where T± = 〈u1± u2± u3± σ 13± σ 23 and the matrix C is given by 33

 Bu Bσ   C=  A BT   u u 



(4.6.26)

If the thickness of the thin layer, 2h, is small compared with the wavelengths, then a good approximation to the interface condition is given by

(T + )h + h(C T + )h = (T − )− h − h(C T − )− h

(4.6.27)



It is clear from the effective boundary conditions in equations (4.6.12) and (4.6.27), which replace the coating and the interface layers, that the effect of the inertia of the layer (represented by the terms in Au containing the density of the layer) is of the same order of magnitude as that of the stiffness coefficients of the layers. The interface conditions derived above simplify to the ones derived by Bövik (1994) for isotropic layers. If it is assumed that fields are in the specific form T± = S±(x3) e i ( kx1 −ωt ), then the interface conditions reduce to those derived by Rokhlin and Huang (1993) correct to O(h). They derived the effective interface conditions using a transfer matrix approach. Again, it may be emphasized that the above conditions were derived for a thin monoclinic interface layer between two media, which could be finite or infinite, isotropic or anisotropic. In the following, we will derive the dispersion equations for guided waves in a plate with an interface layer. The plate, shown in Fig. 4.21, consists of a thin anisotropic layer sandwiched between two identical isotropic layers. We will use the effective interface conditions in equation (4.6.27) to derive analytical expressions for the dispersion equations. The choice of geometry and material properties of the surrounding layers has been made for algebraic convenience and practical applications. The governing equations of motion in the two adjoining isotropic layers (see equation (4.6.13)) are

± = ρ ∂ 2u ± , ± h < x ∂mσ mj t j > 3

< >

±b

(4.6.28)



The stress–strain relation in the isotropic layers is expressed as

σ ±jm = λδ jm ∂nun± + µ(∂mu ±j + ∂ j um± ), ± h x 3

< >

±b



(4.6.29)

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The interface and boundary conditions are given by (T + )h + h(CT + )h = (T − )− h − h(CT − )− h



σ ±j 3 = 0, x 3 = ±b



(4.6.30)



(4.6.31)



The solution to these equations will be assumed to be in the form u ±j (x1, x2, x3, t) = f j± ( x 3 )e i ( kx1 −ωt ) (see equation (4.6.15)). u ± = ∇Φ ± + ∇ × Ψ ± , ∇.Ψ ± = 0 Φ ± = ( A0± sin px 3 + B0± cos px 3 )e i ( kx1 −ωt ) Ψ ±j = ( A ±j sin qx 3 + B ±j cos qx 3 )e i ( kx1 −ωt )

(4.6.32)

ik ik A = − B1± , B3± = A1± q q ± 3

p = kl2 − ω 2 , kl = ω / cl , cl = (λ + 2µ )/ρ q = ks2 − ω 2 , ks = ω / cs , cs = µ/ρ





The dispersion equation is obtained by satisfaction of the interface and boundary conditions. Due to the symmetry of the problem, we can split the solution into a symmetric and an antisymmetric part. These two cases will be treated separately in the following discussion. 4.6.2.1 Symmetric Motion In this case, (u + )x3 = (u − )− x3 , (v + )x3 = (v − )− x3 , and (w + )x3 = − (w − )− x3 . Thus, A0− = − A0+ , B0− = B0+ , Ar− = Ar+ , Br− = − Br+



(r = 1, 2)



(4.6.33)

The displacement–stress vector at x3 = h is given by



ik sin( ph)   0   (T + )h =  p cos ph  2 µikp cos ph   0  2 2  − µ(q − k ) sin ph 0

ik cos ph 0 − p sin ph − 2 µikp sin ph 0

0 − µks2 sin qh

− µ(q 2 − k 2 )cos ph

− q cos qh

ks2 0 sin qh q 0 ik sin qh µ(q 2 − k 2 ) sin qh 0 − µ ks2 cos qh 0

−

0

0 ks2 cos qh q 0

2i µ kq cos qh

0

   A0+    + 0   B0    + ik cos qh  ×  A1  (4.6.34) µ(q 2 − k 2 ) cos qh   B1+   0   A2+     − 2i µ kq sin qh   B2+  q sin qh

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Using the symmetry properties of (T +)h and substituting equations (4.6.23) and (4.6.26) in equation (4.6.30), we find that the interface conditions lead to the matrix equation  D  hik 13  D33     D132   h  ρLω 2 − k 2  D11 −  D33        D13 D36   2  − hk  D16 − D  33 

hik

D36 D33

1

 D D  − hk 2  D16 − 13 36  D33  

0

  D2   h  ρLω 2 − k 2  D66 − 36   D33    

0



0

0

1

0

0

1

− h/D33   hikD13 /D33  (T + )h = 0  hikD36 /D33 

(4.6.35)

This gives three equations in the six unknown coefficients appearing on the righthand side of equation (4.6.34). The other three equations are obtained by imposing the stress-free boundary condition of equation (4.6.31) at x3 = b. This gives  2ikp cos pb  0    −(q 2 − k 2 )sin pb

− 2ikp sin pb

0 − k sin qb

0

2 s

− (q 2 − k 2 )cos pb

0

0

(q 2 − k 2 )sin qb

− ks2 cos qb 0

0 2ikq cos qb

(q 2 − k 2 )cos qb   0  {c} = 0  − 2ikq sin qb 



(4.6.36)

where {c} is the 6 × 1 unknown constant matrix appearing on the right-hand side of equation (4.6.34). Combining equations (4.6.35) and (4.6.36), we obtain six equations in the six unknowns {c}. The dispersion equation is obtained by equating the determinant of the coefficients to zero. 4.6.2.2 Antisymmetric Motion For the antisymmetric motion, we have (u + )x = −(u − )− x , (v + )x = −(v − )− x , and (w + )x = 3 3 3 3 3 (w − )− x3. These are satisfied if

A0− = A0+ , Ar− = − Ar+ , B0− = − B0+ , Br− = Br+

(r = 1, 2)



(4.6.37)

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The interface condition now becomes 1  0  0 



0

ikh

−hD44 / δ

hD45 / δ

1

0

hD45 / δ

− hD55 / δ

0

ρLω 2h

ikh

0

0  0  (T + )h = 0  1 

(4.6.38)

Equation (4.6.38) provides three equations involving the 6 × 1 constant matrix {c}. The stress-free boundary condition at x3 = b leads to three more equations given by equation (4.6.36). Equating the determinant of the 6 × 6 coefficients matrix to zero gives the dispersion equation for antisymmetric guided waves in the plate. The use of the effective interface conditions leads to explicit expressions for the dependence on the interface thickness and the interface anisotropic properties, as seen in equations (4.6.35) and (4.6.38). The symmetric and antisymmetric modes are found to depend upon different elastic constants of the layer. The former modes show explicit dependence upon D11, D13, D16, D33, D36, and D66, whereas the latter depend only upon D44, D45, and D55.

4.6.3 Numerical Results and Discussion For illustration of the use of the effective boundary and interface conditions to obtain dispersion results, we will show some numerical results for two cases of superconducting plates. The two cases considered are: (a) a nickel plate with thin layers of superconducting YBa 2Cu3O7-d (YBCO) material of equal thickness deposited on the top and bottom surfaces, and (b) a thin layer of YBCO sandwiched between two nickel plates of equal thickness. In both cases, the total thickness of the YBCO layer(s) is taken as 2h = 10 mm, and the total thickness of the nickel plate(s) is 2a = 100 mm. The material properties of nickel and YBCO are given in Table 4.4. Note that YBCO is orthotropic. An exact solution to either problem can be obtained by the global matrix method discussed earlier. In the following, results are presented showing the comparison of the exact results and those obtained by the approximate O(h) boundary or interface conditions, as appropriate. Some interesting features due to anisotropy of the layer are discussed. Also, the differences between the results for two cases are pointed out. First, dispersion curves for a homogeneous nickel plate of 110-mm thickness and the sandwich plate with nickel layers of total 100-mm thickness and 10-mm YBCO interface layer are shown in Fig. 4.22. The propagation is along the q = 0° direction. Since this is a principal direction, the in-plane P-SV and out-of-plane SH modes are uncoupled. TABLE 4.4  Geometrical and Material Properties of Nickel and YBCO Material

Thickness (mm)

Density (kg/m3)

Nickel YBCO

2a = 100 2h = 10

8910 6333

Elastic Stiffness (GPa) c11

c22

c33

c12

c13

c23

c44

c55

c66

299 268

299 231

299 186

129 132

129 95

129 71

84.7 37

84.7 49

84.7 95

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c (km/s)

8

6

4

2

0

With YBCO layer With Ni layer 0

20

40

60

80

100

f (MHz)

FIGURE 4.22  Exact dispersion curves for nickel plate with and without the YBCO interface layer for propagation in the q = 0° direction. (Reprinted with permission from Niklasson et al. 2000b, Fig. 4.)

Thus, the curves for the SH modes intersect those for the longitudinal (symmetric) and flexural (antisymmetric) modes. The SH0 propagates with the shear wave velocity of 3.1 km/s in nickel. The effect of the YBCO layer is not significant. In fact, the velocities of the S 0 and A0 modes are not significantly affected by the layer. However, the effect is clearly seen on the higher modes, especially on the SH modes. Because the velocity of the shear wave polarized in the two-direction and propagating in the three-direction (= c44 /ρ ) is 2.4 km/s, which is much lower than the shear wave velocity in nickel, the cutoff frequencies of SH modes in the sandwich plate are lowered more. Figures 4.23 and 4.24 show the comparison of the approximate results with the exact solution in the two cases for propagation along q = 45° direction. In both cases, the approximate and exact results agree very well up to about f = 60 MHz. At this frequency, the shortest wavelength of the quasi-S waves in the thin layers is about 40 mm. Because of the anisotropy of the layers, the in-plane and out-of-plane modes are now coupled. The coupling between qS 0 and qSH0 modes occurs at about 33.5 MHz and 29.5 MHz in cases (a) and (b), respectively. Magnified views of these zones are shown in Figs. 4.25 and 4.26 for different propagation directions. It is noted that interchanges between the two modes occur around these frequencies. The locations of these points on the dispersion curves and the sizes of the gaps between the curves are fairly sensitive to the anisotropy and thickness of the layers. Furthermore, frequencies are lower in case (b) than in case (a). This coupling can also be observed between higher modes. In the foregoing sections, various methods have been developed to accurately describe the dispersion of guided waves in layered anisotropic composite plates. It has been shown that these dispersive waves have features that are quite distinct in nature and are caused

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c (km/s)

8

6

4

2

0

Exact solution O(h) Bcs 0

20

40

f (Mhz)

60

80

100

FIGURE 4.23  Comparison of the exact solution and the approximate O(h) solution for propagation in q = 45° direction in the YBCO/Ni/YBCO plate. (Reprinted with permission from Niklasson et al. 2000a, Fig. 6.) 10

c (km/s)

8

6

4

2

0

Approximate solution Exact solution 0

20

40

f (MHz)

60

80

100

FIGURE 4.24  Comparison of the exact and approximate O(h) solutions for propagation in q = 45° direction in the Ni/YBCO/Ni plate. (Reprinted with permission from Niklasson et al. 2000b, Fig. 6.)

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θ = 0° θ = 15° θ = 45° q(Longitudinal)

3.2 c (km/s)

q(SH)

3.1

q(SH) 3

28

q(Longitudinal) 30

32

34

36

f (MHz)

FIGURE 4.25  A magnification of the exact dispersion curves for the YBCO/Ni/YBCO plate for different propagation directions q. (Reprinted with permission from Niklasson et al. 2000a, Fig. 8.)

3.4

θ = 0° θ = 15° θ = 45°

c (km/s)

3.3

3.2

3.1

3 20

24

26 f (MHz)

28

30

FIGURE 4.26  A magnification of the exact dispersion curves for the Ni/YBCO/Ni plate for different propagation directions q. (Reprinted with permission from Niklasson et al. 2000b, Fig. 7.)

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by anisotropic properties of the layers. Our attention has been focused so far on timeharmonic waves. The following section will be devoted to transient waves in layered anisotropic plates caused by time-dependent line loads applied on one of the surfaces of the plate. Two particular methods of analyzing time-dependent problems will be discussed. One of these is the method of residues to evaluate the integral over the wave numbers, and the other involves direct numerical integration of the wave-number integrals. Both of these have been widely used in earlier studies. These lead to expressions for the field quantities in terms of the spatial variables and frequency. In the final step, time dependence is derived by performing the inverse Fourier transform in frequency. Generalized rays have also been used to describe the time-dependent behavior. However, generalized ray methods get extremely complicated in the presence of anisotropy and layering. Theoretical and experimental results for frequency- and time-dependent field variables have been used to inversely determine the source location and its nature. They have also been used for the determination of material and geometric properties of the medium of propagation as well as for characterization of defects. There are many excellent books, edited volumes, and papers that have dealt with these methods, e.g., Ewing et al. (1957), Brekhovskikh (1960), Miklowitz (1978), Rose (1999), Liu and Xi (2002), Mal and Ting (1988), Datta et al. (1990a), Forrestal et al. (1992), Kinra et al. (1994), Pao and Grajewski (1977), Santosa and Pao (1989), Green and Green (2000), Mal (2002), and Velichko and Wilcox (2007).

4.7 Transient Response due to a Concentrated Source of Excitation Ultrasonic nondestructive evaluation of cracks, delaminations, and other localized defects in a composite or layered plate requires a clear understanding of the complex interaction of guided waves with such defects. A general analytical solution of scattering by defects in a composite plate is an extremely difficult problem. Thus, efficient numerical techniques are needed to analyze such problems. Several numerical methods, including finite element (FEM), boundary element (BEM) or boundary integral (BIE), volume integral equation (VIE), strip element (SEM), and hybrid methods combining localized finite element (LFEM) with either boundary integral (BIE) or modal expansion techniques, have been used to study scattering by defects in composite or layered media. Those methods that use boundary or volume integrals rely upon the construction of Green’s functions appropriate for the host medium. This section deals with three accurate and efficient methods for calculating the transient response of anisotropic layered plates due to a force acting on the plate. The first of these uses Cauchy’s residue theorem to evaluate the infinite integrals involving the wave number k to get the response in the frequency domain. Then, the time-domain response is obtained by the inverse Fourier transformation combined with an exponential window method (Liu and Achenbach 1994; Vasudevan and Mal 1985; Kausel and Roesset 1992). In the second, the wave-number integral is performed using an adaptive integration scheme (Xu and Mal 1985). The convergence of the integral can be aided by deformation of the line of integration so that the poles of the integrand on the real k-axis are avoided. Note that the latter can also be achieved by adding a small material dissipation.

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The third approach is to use the stiffness method (Dong and Huang 1985), discussed in Section 4.2, and the modal summation (Zhu et al. 1995a, 1995b; see also Kausel 1999). As in the first method, the time-dependent response is obtained by performing the inverse Fourier transform in frequency or by a fast Fourier transform (FFT). These methods are developed in the following subsections, and numerical results are presented.

4.7.1 Transient Waves in a Multilayered Plate: Plane-Strain Motion Figure 4.27 shows the cross section of the multilayered plate in the x–z-plane. The plate is of thickness H and is made up of N parallel, homogeneous, orthotropic layers. We place the Cartesian coordinates with origin O at the top surface, with the z-coordinate pointing down into the plate. The jth layer is bounded by the interfaces z = zj−1, zj. Thus, we have z0 = 0 and zN = H. It is assumed that the x-, y-, z-axes are parallel to the symmetry axes of the layers. The stress–strain relation is given by equation (4.5.1). Using the Fourier transforms in space and time, as defined in equation (4.5.2), we obtain the solutions for the transforms of the displacement components (Pan and Datta 1999) as follows. See also equations (4.2.17–4.2.20). uˆ(k , z ; ω ) = ik ( A11 cosh r1′z − iA12 sinh r1′z ) + b( A21 cosh r2′z − iA22 sinh r2′z )



wˆ (k , z ; ω ) = ar1′( A12 cosh r1′z + iA11 sinh r1′z ) + r2′( A22 cosh r2′z + iA21 sinh r2′z )

(4.7.1)

where a = (λ k 2 − k22 − r1′ 2 )/δ r1′ 2

b = (k22 − k 2 + β r2′ 2 )/δ k 2 and l, b , d, and k2 are defined in equation (4.2.12). Also, r1′ = −is1 and r2′ = −is2. z0 z1

Free surface

0

x hj

Layer 1

zs–1 Layer 2 zs zN–1 zN

(4.7.2)

Layer N

Layer s1

hs1

Layer s2

hs2

hN

(z = h)

(z = H)

Free surface z

FIGURE 4.27  Geometry of a multilayered plate in the x–z-plane. (Reprinted with permission from Pan and Datta 1999, Fig. 1.)

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Transformed stresses can be obtained from the transformed displacement solutions given by equation (4.7.1). This transformed displacement–stress vector for each layer can be expressed in terms of a propagator matrix relation (see equation (4.2.29)) [Q(z j )] = [Pj (z j − z j −1 )][Q(z j −1 )]



(4.7.3)



where [Q] = 〈uˆ wˆ σˆ zz σˆ xz 〉T , and [Pj] is the propagator matrix for the jth layer. This equation holds if there are no body forces acting in the plate. Suppose now that there is a line force with time history f(t) located on the z-axis at the depth z = h, which belongs to layer s (see Fig. 4.27), i.e., Fm ( x , z ; t ) = δ ( x )δ (z − h) f (t )nm



(4.7.4)



where nm is the mth (m = 1,3) component of the unit vector along the x- or z-axis. Note that the force is uniform along the y-axis. Then, the transforms of the traction discontinuity are ∆σˆ xz = σˆ xz (h + 0) − σˆ xz (h − 0) = F (ω )nx ∆σˆ zz = σˆ zz (h + 0) − σˆ zz (h − 0) = F (ω )nz



(4.7.5)

Introducing this in the propagator relation between [Q(zN)] and [Q(z0)], we obtain [Q(z N )] − [PN ][PN −1 ] ... [P1 ][Q(z 0 )] = [PN ][PN −1 ] ... [Ps2 ][∆Q]





(4.7.6)

with [∆Q] = 〈0 0 − F (ω )nx − F (ω )nz 〉T





(4.7.7)

Now, substituting the traction-free boundary conditions at the top and bottom surfaces into equation (4.7.6), we can solve for the Green’s displacement (in the transformed domain) at either the top or the bottom surface. Thus, for a line force acting in the x-direction, we find uˆ(k , z 0 ; ω )  F (ω )  = wˆ (k , z ; ω ) a31a42 − a41a32 0  



b34 a42 − b44 a32    a31b44 − a41b34   

(4.7.8)

Similarly, for a line force in the z-direction, we get



uˆ(k , z 0 ; ω )  F (ω )  = wˆ (k , z ; ω ) a31a42 − a41a32 0  

b33a42 − b43a32    a31b43 − a41b33   

(4.7.9)

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where [a] = [PN ][PN −1 ]…[P1 ] [b] = [PN ][PN −1 ]…[Ps2 ]

(4.7.10)

∞

F (ω ) =

∫ f (t )dt



−∞

Note that the matrix [a] is the same as [P] in equation (4.2.31) when the wave propagation is in the symmetry direction (X). In the next step, inverse Fourier transforms with respect to x are carried out by using Cauchy’s theorem. For example, we obtain ux(x, z; w) as ux ( x , z ; ω ) =

1 2π

= −i

∞

∫ U (k, z ;ω )e x

ikx

dk (4.7.11)

−∞ M

∑U (k , z ;ω )e x

m

m =1

ikm x



The poles of the integrand in equation (4.7.11) are the roots (real and complex) of the equation a31a42 − a41a32 = 0



(4.7.12)



These roots can be found accurately, as described in Section 4.2. Since all the poles are of first order, the residue summation can be carried out exactly by finding the derivative of the left-hand side of equation (4.7.12) with respect to k. For a given frequency, there are only a finite number of propagating modes, while all other modes are evanescent, and their amplitudes decay exponentially to zero in the selected half plane of the complex k-plane. Thus, the finite number M in the summation will be enough to give accurate results, with a suitable range being 40 ≤ M ≤ 60. It should be emphasized here that, for a given frequency, we need to find the poles only once. With these poles, the frequency-domain Green’s functions for any pair of source and field points can be evaluated accurately. Next, to find a time-domain solution, we evaluate the following integrals numerically. ∞



u( x , z ; t )  1 u ( x , z ; ω )  −iωt  =   e dω w( x , z ; t ) 2π −∞ w( x , z ; ω )

∫

(4.7.13)

The difficulties in the numerical integration occur due to the singularities of the frequency-domain solutions at w = 0 and at the cutoff frequencies. To overcome these difficulties, a small imaginary part is introduced in w in the above integrals. By doing so, equation (4.7.13) can be written as



u( x , z ; t )  e ηt  = w( x , z ; t ) 2π

∞

u ( x , z ; ω + iη )    e −iωt dω   x z i w ( , ; ω + η ) −∞

∫

(4.7.14)

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Note that td

∫

F (ω + iη ) = e −ηt f (t )e iωt dt

0

(4.7.15)

in which td is the duration of the force and h is the shifting constant. For the numerical examples presented in the following discussion, the normalized h is chosen to be 0.1. 4.7.1.1 Transient Waves in a Three-Layered Superconducting Plate We now apply this formulation to the examples considered in Section 4.6, namely, (a) a homogeneous nickel plate of thickness H = 110 mm, (b) a three-layered YBCO/Ni/YBCO plate, and (c) a three-layered Ni/YBCO/Ni plate. In case b, each YBCO layer is 5 mm thick and the nickel layer is 100 mm thick. Thus, the total thickness of the plate is 110 mm. In case c, a symmetric configuration is also considered, where the thickness of each Ni layer, H1, is 50 mm, and the YBCO layer has the thickness, 2h = 10 mm. The material properties of Ni and YBCO are given in Table 4.4. Frequency spectra for u and w at the observation point (10H1, 0) due to forces acting in the x- and z-directions, respectively, at (0,0) are shown in Figs. 4.28 and 4.29. Here, H1 is the total thickness of the nickel layer. The most notable features of Fig. 4.28 are the 2.0

Ni only YBCO/Ni/YBCO Ni/YBCO/Ni

Amplitude of Ux (by x - force)

1.6

1.2

0.8

0.4

0.0

0

20

40 60 Frequency (Mhz)

80

100

FIGURE 4.28  Frequency-domain amplitude of u at (10H1, 0) due to the force in the x-direction at (0,0). (Reprinted with permission from Pan and Datta 1999, Fig. 5(a).)

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Ni only YBCO/Ni/YBCO Ni/YBCO/Ni

Amplitude of Uz (by z - force)

1.2

1.0

0.8

0.6

0.4

0.2

0.0

0

20

60 40 Frequency (MHz)

80

100

FIGURE 4.29  Frequency-domain amplitude of w at (10H1, 0) due to the force in the x-direction at (0,0). (Reprinted with permission from Pan and Datta 1999, Fig. 5(b).)

sharp peaks in amplitude at frequencies that correspond to certain cutoff frequencies of the three plates. Figure 4.29 also shows peak amplitudes at certain frequencies, but these are not as pronounced now (except the peaks at the cutoff frequencies of the S1 modes). Normalized values of the first few cutoff frequencies are given in Table 4.5. Here, the normalized frequency w’ = wH1/cs, cs being the shear wave speed in nickel. TABLE 4.5  Comparison of Cutoff Frequencies for Nickel, YBCO/Ni/YBCO, and Ni/YBCO/ Ni Plates Nickel (Exact) p = 3.14159265 pcl/cs = 5.90176362 2p 3p 2pcl/cs 4p 5p 3pcl/cs 6p

Nickel (Numerical) 3.14159265 5.90176363 6.28318531 9.42477796 11.8035273 12.5663706 15.7079633 17.7052909 18.8495559

YBCO/Ni/YBCO

Ni/YBCO/Ni

2.93210164 5.50844412 5.85805496 8.77136624 11.00674801 11.66493129 14.53100467 16.48432576 17.36170711

2.68261224 5.09063922 5.85805496 8.13176395 11.00674801 11.66493129 13.74952922 15.39956710 17.36170711

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To calculate the time-domain Green’s functions, a time history for the line force is assumed as f (t ) =

2

σ π

2 /( 2σ 2 )

e − ( t − t0 )

sin(ω c t )



(4.7.16)

where s is a parameter controlling the width of the pulse, t0 determines the time delay of the pulse, and w c is the center frequency (angular). As an example, we choose s = 0.8, t0 = 3.0 H1/cs, and w c = 3.14 cs/H1. For these parameters, the time and frequency dependence of the pulse are shown, respectively, in Figs. 4.30(a,b). The center frequency is seen to be about 15 MHz, and the amplitude of the force approaches zero at 40 MHz. Time-domain Green’s displacements are shown in Figs. 4.31(a–c) and 4.32(a–c) for the three cases. The observation point is at (10 H1, 0) and the source is at (0, 0). Note that the plate wave speed in all the cases is about 5.2 mm/ms, and the Rayleigh wave speed is 2.87 mm/ms in nickel. These figures show that, while the plate wave speeds in the three cases do not change appreciably, the arrival times of the A1 mode and Rayleigh wave are altered measurably because of the coating and interface layers. Also, there are large differences in the time response of the plate with coating and interface layers. The results presented above show that even at this relatively low frequency, the dynamic behaviors of the homogeneous nickel, YBCO/Ni/YBCO, and Ni/YBCO/Ni plates show measurable differences. So far, our attention has been focused on the plane-strain problem. In the following discussion, we will examine the case of off-axis propagation and the effects of coupling of in-plane and out-of-plane displacements. 4.7.1.2 Transient Waves in a Multilayered Cross-Ply Graphite–Epoxy Plate In Section 4.3, dispersion of guided waves in a multilayered cross-ply plate was discussed. It was pointed out that if the number of laminae is sufficiently large, then the dispersion characteristics of the layered plate can be predicted well by an effective homogeneous anisotropic plate. In this section, the validity of the effective-medium approach is explored further by examining the transient response of such a plate due to a vertical line load acting on the surface of the plate. This was studied by Datta et al. (1992a). The geometry of the problem is shown in Fig. 4.33. A global matrix method, as discussed in Section 4.5.1, is adapted to obtain the Green’s displacement and traction components in the Fourier-transformed domain. The governing 4N equations obtained from equation (4.5.12) for the unknown coefficients are solved for prescribed line load acting on the top surface. Then, the transforms of the displacements and tractions at any point in the plate can be found by using equation (4.5.6). Once the transforms are known, then the time-domain Green’s displacement–traction vector is obtained by first evaluating the inverse Fourier transform with respect to the spatial variable using an adaptive Clenshaw-Curtis scheme and then using an FFT. In the following discussion, numerical results are presented for a particular example. Numerical results are presented for a cross-ply graphite-fiber-reinforced epoxy plate. The properties of the 0° ply are given in Table 4.1. The ply lay-up is [0°/90°]sn when n is

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f(t)

0.5

0.0

–0.5

–1.0 0.00

0.05

0.10

0.15 0.20 0.25 Time (microsec) (a)

0.30

0.35

1.0

f (frequency)

0.5

0.0

–0.5

–1.0

0

10

20 30 Frequency (MHz)

40

50

(b)

FIGURE 4.30  (a) Time history of the applied force; (b) Real (solid line) and Imaginary (dashed line) parts of the Fourier transform of the applied load. (Reprinted with permission from Pan and Datta 1999, Fig. 6.)

taken to be 4 and 8. The total thickness of the plate is H = 5.08 mm. The time dependence of the vertical line load applied at x = 0 and z = 0 is taken to be a Ricker pulse with a center frequency of 0.21 MHz. The form of the pulse is

f (t ) = (2π 2 f c 2τ 2 − 1)e −π

2 f 2τ 2 c



(4.7.17)

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Elastic Waves in Composite Media and Structures 0.15

0.15

Point force in x – direction Ni only 0.10 YBCO/Ni/YBCO

Displacement Uz

Displacement Ux

Point force in x – direction Ni only 0.10 YBCO/Ni/YBCO

0.05 0.00 –0.05 –0.10 –0.15 0.0

0.05 0.00 –0.05 –0.10

0.1

0.2 0.3 0.4 0.5 Time (microsec)

–0.15 0.0

0.6

0.1

0.2 0.3 0.4 0.5 Time (microsec)

0.6

(b)

(a)

Displacement Uz

0.15 Point force in z – direction Ni only YBCO/Ni/YBCO

0.10 0.05 0.00 –0.05 –0.10 –0.15 0.0

0.1

0.2

0.3

0.4

0.5

0.6

Time (microsec) (c)

FIGURE 4.31  Time-domain displacement components at (10H1, 0) due to applied loads at (0, 0) in nickel and YBCO/Ni/YBCO plates. (a) u for force in the x-direction; (b) w for force in the x-direction; (c) w for force in the z-direction. (Reprinted with permission from Pan and Datta 1999, Fig. 7.)

where t = tcs/H and cs = c55 /ρ . Figure 4.34 shows the time response of the horizontal strain exx at (5H, 0) for 8-ply and 16-ply symmetric cross-ply plates. The propagation is in the 0° direction. Also shown in these figures are the results for a homogeneous plate with effective properties given by equation (4.3.1). Clearly, the response of the homogeneous plate reproduces closely that of the 16-ply plate. The behavior of the 8-ply plate is similar at early times but deviates markedly as the time increases. Note the center frequency is such that fcH = 1.07 MHz×mm, which is low. The shear wavelength is about 1 mm at this frequency, which is about three times the thickness of a ply. Thus, it appears that if the wavelength is greater than about three times the thickness of a layer in a periodically layered medium, the effective-medium approximation works well.

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0.15

0.10 Displacement Uz

Displacement Ux

Point force in x – direction Ni only 0.10 Ni/YBCO/Ni

0.05 0.00 –0.05

0.05 0.00 –0.05 –0.10

–0.10 –0.15 0.0

Point force in x – direction Ni only Ni/YBCO/Ni

0.1

0.2 0.3 0.4 0.5 Time (microsec)

0.6

–0.15 0.0

0.1

0.2 0.3 0.4 0.5 Time (microsec)

(a)

0.6

(b)

Displacement Uz

0.15 Point force in z – direction Ni only Ni/YBCO/Ni

0.10 0.05 0.00 –0.05 –0.10 –0.15 0.0

0.1

0.2 0.3 0.4 0.5 Time (microsec)

0.6

(c)

FIGURE 4.32  Time-domain displacement components at (10H1, 0) due to applied loads at (0, 0) in nickel and Ni/YBCO/Ni plates. (a) u for force in the x-direction; (b) w for force in the x-direction; (c) w for force in the z-direction. (Reprinted with permission from Pan and Datta 1999, Fig. 8.)

4.7.1.3 Transient Waves due to Forces Acting along a Line Inclined to an Axis of Symmetry In the previous discussion, the line of action of the forces was assumed to be perpendicular to a symmetry axis. Next, we will consider the line of action to be the y-axis that makes an angle −q to the Y-axis, which is an axis of symmetry. Thus the guided waves are propagating along the x-axis. Dispersion characteristics of such waves in an isotropic plate with thin anisotropic coating or interface layers were considered in Section 4.6. Here, we will deal with the transient waves caused by a line force. As shown in Section 4.6, for waves propagating in a direction off a symmetry axis, the in-plane and out-of-plane

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Elastic Waves in Composite Media and Structures

X

Y Z

FIGURE 4.33  Geometry of the cross-ply plate with the vertical line force. (Reprinted with permission from Datta et al. 1992a, Fig. 1.)

displacements are coupled. The coupling between the qS 0 and qSH0 modes for thin coating layers of YBCO on a nickel layer occurs at about 33.5 MHz; in contrast, for a thin interface layer of YBCO in a nickel plate, this occurs at about 29.5 MHz. These frequencies are higher than the center frequency considered in Section 4.7.1, so we will extend the analysis presented in Section 4.7.1 to higher frequencies and to coupled modes. 1.0

Homogeneous 8 crossply (0) 16 crossply (0) Fc = 0.21 MHz

stxx

0.5

0.0

–0.5

–1.0

0

10

20 30 Time (microsec)

40

50

FIGURE 4.34  The time response of the horizontal strain at (5H, 0) for a Ricker pulse with center frequency of 0.21 MHz. Comparison of the homogeneous medium model with the 8-ply and 16-ply plates. (Reprinted with permission from Datta et al. 1992a, Fig. 8.)

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125

We present some numerical examples using the exact and approximate solutions derived in Section 4.6 for the line force Green’s function. Let G33(x1, x3; 0, −b) and g33(x1, x3; 0, −b) denote the displacement in the z-direction at (x1, x3) due to a line force acting in the z-direction at (0, −b) obtained by using, respectively, the exact and approximate boundary or interface conditions. Numerical results will be presented for these two quantities only. Other components of the Green’s displacement tensor can be obtained by using the method described here. To calculate the expressions for the Green’s functions, the inverse Fourier transforms with respect to k and w must be performed. As mentioned before, great care must be taken to perform the k-integral. The integrands have a large number of poles on the real k-axis when the frequency is high. In Section 4.7.1.1, this difficulty was avoided by keeping the frequency low and using the method of residues to evaluate the integral. One way to remove the singularities (poles) from the integrands is to deform the integration path in the complex k-plane. Here, the integrals (Niklasson and Datta 2002a, 2002b) are performed along k(s ) = s(1 − iα e − β|s| ), 0 ≤ s < ∞ (4.7.18) ˆ ˆ g instead of along the real k-axis. Symmetries of G jm and jm with respect to k have been used to reduce the integration from (−∞, ∞) to [0, ∞). Suitable values of the parameters a, b have been determined from numerical tests, and a = 0.1 and b = ½|k s| have been used in the examples below. When the integration contours have been deformed as described above, the singularities are no longer on the integration path. The integrands do, however, still experience highly irregular behavior, which makes the numerical evaluation very difficult. As discussed in Section 4.7.1.2, we use an adaptive integration scheme (see Xu and Mal 1985) for the finite part, 0 ≤ s ≤ sc, and the scheme by Xu and Mal (1987) for the remaining semi-infinite part, sc ≤ s ≤ ∞. For the finite part, we use polynomial approximations of fourth and eighth orders, and for the semi-infinite part, we approximate the integrands by polynomials of fourth order. The time dependence of the force is taken as (somewhat different than equation (4.7.16))





f (t ) =

2 − cs 2 (t −t0 )2 /8a2 e sin(ω c t ) π

(4.7.19)

with t0 = 0.16 ms, a = 50 mm, and cs is the shear wave speed in nickel. The center frequency is varied depending upon the frequency where the coupling and mode interchange between the SH0 and S 0 modes occurs. For fc = 32 MHz, the time response f3(t) and its Fourier transform F3(w) are shown in Fig. 4.35. Note that the line force is essentially zero outside of an interval of width 30 MHz centered around fc. This center frequency is close to where the mode interchange of the above-mentioned two modes occurs in the YBCO/Ni/YBCO plate. Finally, we employ an exponential windowing technique (Pan and Datta 1999) when inverting the temporal Fourier transform. First, we consider a homogeneous nickel plate of thickness 110 mm. Figures 4.36(a–c) show the vertical displacement at a point x1 = ξ = 10 mm on the top surface x3 = −b = −a − h due to a vertical line load applied at (0,−b). The center frequency has been varied

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Elastic Waves in Composite Media and Structures 1

1

0.5

0.5 F3 (ω)

f3(t)

126

0 –0.5 –1

Real part Imaginary part

0 –0.5 –1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 t(µs)

0

10

20

30 40 f (MHz)

50

60

(b)

(a)

0.08 0.06

0.08 0.06

0.04 0.02

0.04 0.02

u3 (ξ, –a – h, t)

u3 (ξ, –a – h, t)

FIGURE 4.35  The line force when fc = 32 MHz in the (a) time domain and (b) frequency domain. (Reprinted with permission from Niklasson and Datta 2002b, Fig. 2.)

0 –0.02 –0.04

–0.06

–0.06 –0.08

0 –0.02 –0.04

2

2.5

3

3.5

4

4.5

5

5.5

–0.08

6

2

2.5

3

3.5

4

4.5

t (µs)

t (µs)

(a) fc = 25 MHz

(b) fc = 32 MHz

5

5.5

6

u3 (ξ, –a – h, t )

0.08 0.06 0.04 0.02 0 –0.02 –0.04 –0.06 –0.08

2

2.5

3

3.5

4 4.5 t (µs)

5

5.5

6

(c) fc = 35 MHz

FIGURE 4.36  The vertical displacement on the top surface of the homogeneous nickel plate at 10 mm from a line vertical force for different values of fc. (Reprinted with permission from Niklasson and Datta 2002b, Fig. 4.)

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0.08 0.06 0.04 0.02

0.08 0.06 u3 (ξ, –a – h, t)

u3 (ξ, –a – h, t)

Guided Waves in Fiber-Reinforced Composite Plates

0 –0.02 –0.04 –0.06 –0.08

2

2.5

3

3.5

4 4.5 t (µs)

5 5.5

0.04 0.02 0 –0.02 –0.04 –0.06 –0.08

6

2 2.5

3

(a) φ = 0°

6

5

6

0.08 0.06

0.04 0.02 0

u3 (ξ, –a – h, t)

u3 (ξ, –a – h, t)

5 5.5

(b) φ = 30°

0.08 0.06

–0.02 –0.04 –0.06 –0.08

3.5 4 4.5 t (µs)

0.04 0.02 0 –0.02 –0.04 –0.06

2

2.5

3

3.5

4 4.5 5 5.5 t (µs)

(c) φ = 60°

6

–0.08

2

2.5

3

3.5

4 4.5 t (µs)

5.5

(d) φ = 90°

FIGURE 4.37  The vertical displacement on the top surface of the YBCO/Ni/YBCO plate at 10 mm from the source with fc = 32 MHz for different propagation angles. (Reprinted with permission from Niklasson and Datta 2002b, Fig. 6.)

to show the difference in responses due to different pulse shapes. Figure 4.36(a) clearly shows the arrival of the A1 mode at 2.9 ms, followed by the arrival of the A0 mode having large amplitude. In Fig. 4.36(b), small amplitude S1 mode appears first, followed by A1, A0, and then S2 (3.9 ms) and S 0 (4.3 ms). The last two are prominent in Fig. 4.36(c), with arrival times a little earlier than in Fig. 4.36(b). Vertical displacements at the top surface of the YBCO/Ni/YBCO plate for different propagation angles are shown in Figs. 4.37(a–d). Figure 4.36(b) is to be contrasted with the response of the YBCO/Ni/YBCO plate for propagation in different directions. The center frequency of 32 MHz is below that at which the mode interchange occurs. However, the SH0 mode now has a large vertical displacement component. It is seen that this mode arrives shortly after the A0 mode and is followed by the S2 and S 0 modes. Figures 4.38(a–c) show, respectively, the phase and group velocities of the dominant modes and the vertical components of displacements associated with them. Exact solutions were used for the results presented. It is found that the solutions obtained for this case agree well with those obtained using approximate boundary conditions (see Niklasson and Datta 2002b).

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Elastic Waves in Composite Media and Structures

8 7

S2

6 5 4 3 2 20

SH3

A2 v (km/s)

c (km/s)

Sh2

S1

S0 SH 1 A1

SH0

25

30

A0

35 40 f (MHz) (a)

45

50

5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 20

A1 A0 SH0

SH1

S2

S0 S1

SH2 25

30

A2

35 40 f (MHz) (b)

45

SH3 50

0.8

S0

0.6

A0 S2

|u3|

|u3| (normalized)

1

0.4

A1

S1

0.2 SH SH 0 1 0 20

25

SH2 30

35 40 f (Mhz)

SH3 A2 45

50

(c)

FIGURE 4.38  (a) Phase velocity, (b) group velocity, and (c) vertical displacement on the top surface for the YBCO/Ni/YBCO plate when q = 30°. Solid lines are symmetric modes and dashed lines are antisymmetric modes. (Reprinted with permission from Niklasson and Datta 2002b, Fig. 3.)

Finally, results for the sandwich (Ni/YBCO/Ni) plate are shown in Figs. 4.39(a,b) using the exact and approximate solutions for the vertical displacement on the top surface at a distance of 5 mm from the source. Clearly, the approximate effective interface conditions lead to fairly accurate results. The figures show the arrival of the qSH0 mode at about 2 ms. The group velocity of this mode is vg ≈ 3000 m/s (see Fig. 4.40). Note that the center frequency of 29.4 MHz corresponds to the interchange of qSH0 and qS 0 modes.

4.7.2 Stiffness Method and Modal Expansion of Green’s Function: Plane Strain In this section, the Green’s displacements due to a time-harmonic line load acting on a laminated composite plate are derived by using the stiffness method (Section 4.2.2) and a modal expansion using the eigenvectors obtained by solving the eigenvalue problem (see Zhu and Shah 1997). As described in Section 4.2.2, the dispersion equation governing

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Guided Waves in Fiber-Reinforced Composite Plates 0.08

0.08

0.06

0.06

0.04

0.04

0.02

G33

0.02

0

g33

0 –0.02

–0.02

–0.04

–0.04 –0.06

–0.06 –0.08

–0.08

1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 t (µs) (a) Exact Solution.

1 1.25 1.5 1.75 2 2.25 2.5 2.75 t (µs) (a) Approximate Solution.

3

FIGURE 4.39  G33 and g33 for a 30° Ni/YBCO/Ni plate when fc = 29.4 MHz. (Reprinted with permission from Niklasson and Datta 2002a, Fig. 10.)

the frequency and wave number is obtained as an eigenvalue problem by expressing the thickness variation of the displacement u in terms of interpolation polynomials involving the nodal values of the displacements and tractions or just nodal values of the displacements. For this purpose, the plate is divided into several parallel sublayers, and the displacements within the ith layer are expressed in terms of polynomials in the thickness variable z (see equations (4.2.42) and (4.2.43)). Thus, u is expressed (assuming plane strain) using quadratic polynomials as u( x , z , t ) = N( Z/h)q( x , t )



(4.7.20)



where the shape-function matrix N is given by n 0 n2 0 n3 0  N= 1  0 n1 0 n2 0 n3 



1

6000

vg (m/sec)

gS2

4000

gSH0

3000 2000

gA0

1000 0

gSH1 gS1 10

gSH2

20

30 40 50 f (MHz) (a) Group Velocity.

60

|u3| (normalized)

gS0

5000

0

(4.7.21)

0.8 0.6 0.4 0.2 0

0

10

20

30 40 f (MHz)

50

60

(b) |u3| on the Top Surface.

FIGURE 4.40  (a) Group velocity and (b) vertical displacement on the top surface for a 30° Ni/ YBCO/Ni plate. Solid lines are for the symmetric modes, and the dashed lines are for the antisymmetric modes. (Reprinted with permission from Niklasson and Datta 2002a, Fig. 7.)

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Elastic Waves in Composite Media and Structures

The interpolation polynomials n1,n2,n3 are given by equation (4.2.43). If the origin of the local coordinates is taken on the top of the ith layer and h is its thickness, then we have

n1 = (1 − 3Z/h + 2( Z / h)2 ), n2 = 4( Z/h − ( Z /h)2 ), n3 = 2( Z / h)2 − Z/h



(4.7.22)

The nodal displacement matrix is given by q = 〈u1 w1 u2 w2 u3 w3〉T. The stress vector s = 〈s xx s xz〉T at any point along a plane x = constant in the sublayer can be expressed as s = D xx



∂u ∂u + D xz ∂x ∂z

(4.7.23)

where



0 D13   D11 0    , D xz =  D xx =   D55 0  0 D55     

(4.7.24)

Using equations (4.7.20), (4.7.21), and (4.7.23), we obtain the interaction force vector f composed of the forces acting at the top, middle, and the bottom points of the sublayer as



f = 〈f Z =0 f Z =h/2 f Z =h 〉T = R1 q + R 2

∂q ∂z

(4.7.25)

with



 −3D xz 4D xz − D xz   4D xx 2D xx − D xx      1 h R1 = −4D xz 0 4D xz  ; R 2 =  2D xx 16D xx 2D xx    6 30   D − 4D   −D 2D  3 3 D D xz xz  xx xx   xz  xx

(4.7.26)

By applying the principle of virtual work to each sublayer, a set of differential equations can be obtained. The governing differential equation for the entire plate is obtained by assembling all the equations for the sublayers. It is found that (see equation [4.2.47])



  d2 d T =  − K1 2 − K *2 + K 3 − ω 2 M Q dx dx  



(4.7.27)

Here, the vectors Q and T represent the nodal displacement and the traction applied at the interfaces of the plate, respectively. The sizes of these vectors are M = 2(2N + 1), and those of Ki (i = 1,3), K*2, and M are M × M. Now, introducing the Fourier transform of a function f(x) to be ∞

f (k ) =

∫ f (x )e

−∞

− ikx

dx

(4.7.28)

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we obtain, by taking the Fourier transform of equation (4.7.27), the equation � )Q T = (k 2 K1 + ikK *2 + K

Here,

� = K − ω 2M K 3



(4.7.29)



(4.7.30)



For the free motion of the plate, we get the eigenvalue problem (see equation (4.2.48)) � )QR = 0 (km2 K1 + ikm K *2 + K m



(4.7.31)



Equation (4.7.31) is to be solved for the eigenvalues km and the right eigenvectors QmR . Then, the left eigenvectors are obtained by solving the equation � )=0 QmL (km2 K1 − ikm K *2 + K



(4.7.32)



where QmL is 1 × M. Equations (4.7.31) and (4.7.32) can be arranged in the form



QmR   0  QmR  I     = km  � − iK −1 K *  k QR   − K1−1 K km QmR  1 2  m m   Q

L m

km Q

L m



(4.7.33)

0 K1−1    = km QmL km QmL � − iK * K −1  K 2 1  

(4.7.34)

Following the modal summation technique (G.R. Liu et al. 1991) and making use of the orthogonality of the left and right eigenvectors, we obtain the displacement in the transformed domain as 2M

Q=

∑ m =1

km QmL TQmR (km − k )am

(4.7.35)

where

am = QmL QmR − km2 QmL K1QmR

(4.7.36)



The Green’s displacement functions along the thickness of the plate in the spatial domain due to a line load acting at (x0, z0) that is obtained by taking the inverse Fourier transform of equation (4.7.35). This gives Q( x ; x 0 , z 0 ) =

1 2π

∞ 2M

km QmL T0 QmR ik( x − x ) 0 dk e m − k )am

∫ ∑ (k

−∞ m =1

(4.7.37)

where, T0 is a constant vector representing the amplitude of the external force. It is noted that QmL , QmR , T0, and am are independent of k. Applying Cauchy’s residue theorem,

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choosing M out of 2M modes for each of the equations (4.7.33) and (4.7.34) that decay with distance from the source, or that propagate away from it, we obtain M

Q( x ; x 0 , z 0 ) =



∑A q e m

m

ikm ( x − x0 )

(4.7.38)

m =1

where Am = −



ikm QmL T0 , q m = QmR am

(4.7.39)

Using equation (4.7.25), the interaction forces along the thickness can be represented as M



F( x ; x 0 , z 0 ) =

∑A f e m m

m =1

(4.7.40)

ikm ( x − x0 )



where fm is the mth nodal force vector that is obtained by substituting qm into equation (4.7.25). In the following discussion, some numerical results are presented for the Green’s displacement and stress vector in the plate due to a line vertical load applied on the top surface at (0,0) of a uniaxial graphite-fiber-reinforced plate of thickness 2H = 5.08 mm. The plate can be modeled as transversely isotropic, with the axis of symmetry aligned with the x-axis in this example. The stiffness properties of the material are given in Table 4.1. The density is taken as 1.8 g/cm3. The results are compared with the analytical wavefunction expansion technique discussed in Section 4.2 and by Zhu et al. (1995a). The plate was divided into 50 sublayers to accurately compute the eigenvectors. Forty modes (20 symmetric and 20 antisymmetric) are used in the modal representation of the wave field. Two normalized frequencies (W = wH/cs) are used, W = 1.0 and W = 5.0. Here, cs = D55 /ρ . Note that in this case, cij = Dij. Figures 4.41 and 4.42 show, respectively, the distribution of w and s xz through the thickness of the plate for x = H and x = 10H. It is seen that the solution obtained by the method developed here agrees well with that predicted by the wave-function method. So far, our attention has been focused on the response of a composite plate to a line force of excitation. The method discussed in Section 4.7.2 can be extended to derive the Green’s function due to a point force. This is given in the following section.

4.7.3 Stiffness Method and Modal Expansion of Green’s Function: Three-Dimensional Problem This section describes a modal representation of the three-dimensional time-harmonic Green’s function in a laminated composite plate due to a point load oriented arbitrarily. The governing equations for the nodal variables are derived using a stiffness method (Section 4.2.3). The eigenvectors (normal modes) are obtained by solving the resulting eigenvalue problem. The steady-state response in the wave-number domain is found by a double sum (Bai et al. 2004b): (a) an inner sum over normal modes and (b) an outer sum over the plane-waves’ propagation directions (plane-wave superposition). This approach

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Guided Waves in Fiber-Reinforced Composite Plates 0.5 0.4

0.8

0.3

σxz

W

1

freq = 1, present result freq = 1, wave function expansion freq = 5, present result freq = 5, wave function expansion

0.2

0.4 0.2

0.1 0

0.6

0

0.5

1.5

1 z/H (a)

0

2

0

0.5

1 z/H (b)

1.5

2

FIGURE 4.41  (a) Green’s displacement component w and (b) the traction component σxz along the thickness z at x = H. (Reprinted with permission from Zhu and Shah 1997, Fig. 3.)

can be contrasted with that of G.R. Liu et al. (1991), where the Fourier inversions were carried out by using a double FFT. Consider a laminated composite plate of thickness 2H consisting of N number of layers, as shown in Fig. 4.1(a). The governing stress–strain relation for the ith sublayer in the coordinate system (x,y,z) is given below. (See Fig. 4.1(b). Note that θ in this figure denotes the observation angle.)

σ = De



(4.7.41)

The equations governing the nodal generalized coordinates ui, vi, and wi (i = 1, 2, …, 2N + 1) are obtained using Hamilton’s principle. The Lagrangian for the steady-state problem for the layer is written as L= 0.5

  

 ( ρω 2 uT u − e T De )dz dxdy − hi 

∫∫ ∫ x y

hi



0.8 σxz

0.3 0.2

0.6 0.4 0.2

0.1 0

(4.7.42)

1

freq = 1, present result freq = 1, wave function expansion freq = 5, present result freq = 5, wave function expansion

0.4 W

1 2

0

0.5

1 z/H (a)

1.5

2

0

0

0.5

1 z/H (b)

1.5

2

FIGURE 4.42  (a) Green’s displacement component w and (b) the traction component σxz along the thickness z at x = 10H. (Reprinted with permission from Zhu and Shah 1997, Fig. 4.)

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In each lamina, the displacement u is expressed in the quadratic form as z u = N   q( x , y )  hi 



(4.7.43)

where the shape function matrix N is given by equation (4.2.42). The interpolation functions n1, n2, and n3 are given in terms of the local coordinate h = z/hi by equation (4.2.43). The vector q is the vector of generalized displacement components (equation [4.2.44]), and 2hi is the thickness of the sublayer. Then, the strain vector {e} is obtained as where q,x =



e = aq , x + gq , y + bq

(4.7.44)



∂q ∂q , q,y = , and a and b are defined in equation (4.2.46) and ∂x ∂y 0  0 0 g= 0 0  n1

0 n1 0 0 0 0

0 0 0 n1 0 0

0 0 0 0

0 n2 0 0

0 n2

0 0

0 0 0 n2 0 0

0 0 0 0

0 n3 0 0

0 n3

0 0

0  0 0  n3  0  0 

(4.7.45)



Now, using the Lagrangian in equation (4.7.42) and taking the first variation, the governing equation for the ith sublayer (G. R. Liu et al. 1991) is found to be

a1 q ,xx + a 2 q ,xy + a 3 q , yy − a 4 q ,x − a 5 q , y − a 6 q + ω 2 mq = 0



(4.7.46)

where the matrices ai are defined as hi

a1 =

hi

∫ a Dadz ,

a3 =

T

− hi

∫ g Dgdz T

− hi

hi

a2 =

∫ (g Da + a Dg )dz T

T

− hi hi

a4 =

∫

(4.7.47)

( bT Da − a T Db) dz

− hi hi

a5 =

∫ (b Dg − g Db)dz T

T

− hi hi

a6 =

∫ b Dbdz , T

− hi

hi

m=ρ

∫ a adz T

− hi



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After performing the integration, we obtain  4D1 hi  2D 30  1  −D  1  4D 2 hi  a 2 =  2D 2 30  −D  2

2D1 16D1 2D1

−D1   2D1  4D1 

2D 2 16D 2 2D 2

 4D 3 hi  a 3 =  2D3 30   −D3

2D3 16D3 2D3

−D 2   2D 2  4D 2  −D3   2D3  4D3 

a1 =



a4 =

3D − 3D7 1 4 4D 6  −D4 4 

− 4D 4 32D 4 − 32D7 4D 4

 D4  − 4D 4 −3D 4 4 + 3D7 

a5 =

3D − 3D8 1 5 4D 6  −D5  5

− 4D5 32D5 − 32D8 4D5

 D5 − 4D5   −3D5 + 3D8 





 7D 6 −8D 6 D6  1  −8D 6 16D 6 −8D 6  3hi  D −8D 6 7D 6   6 1 −I  2I ρh  4I m = i  2I 16I 2I  , I= 0 30  −I 0 2I 4I    a6 =



(4.7.48) 0 1 0

0 0 1 

a1, a2, a3, a6, and m are symmetric, while a4 and a5 are antisymmetric. The matrices Di appearing in equation (4.7.48) are given in terms of the material constants Dij as  D11 D1 =  D16   0  D66 D3 =  D26   0

D16 D66 0 D26 D22 0

 0 D5 =  0   D36 + D45  0 D7 =  0   2D13

0 0 2D36

 2D16 0   0 , D 2 =  D12 + D66   D55  0   0  0 0  , D4 =  0   D44   D13 + D55 0 0 D23 + D44

D36 + D45  D23 + D44  ,  0 

2D55  2D45  , D8 =  0 

 0  0   2D36

D12 + D66 2D26 0

0  0   2D45 

0 0 D45 + D36

D13 + D55  D45 + D36  (4.7.49)  0 

 D55 D6 =  D45   0 0 0 2D23

D45 D44 0

2D45  2D44   0 

0  0   D33 



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The governing differential equation for the entire plate is obtained by assembling all the equations for the sublayers. It takes the form

A1Q, xx + A 2 Q, xy + A 3Q, yy − A 4 Q, x − A 5 Q, y − A 6 Q + ω 2 MQ = − T

(4.7.50)



where Q = {u1, v1, w1, …, ui, vi, wi, …, u2N+1, v2N+1, w2N+1}T, and T represents the applied nodal traction. The sizes of vectors Q and T are M = 3(2N + 1), and of Ai and M are M × M. This equation is the generalization to three dimensions of equation (4.7.27). We now define the Fourier transforms of a function f(x, y, w) and its inverse with respect to x, y as ∞ ∞

f (kx , k y , ω ) =

∫ ∫ f (x , y ,ω )e

− i ( kx x + k y y )

dxdy

−∞ −∞



1 f (x , y ,ω ) = (2π )2

(4.7.51)

∞ ∞

∫ ∫ f (k , k ,ω )e x

y

i ( kx x + k y y )

dkx dk y , where i = −1

−∞ −∞

Now, the wave numbers kx and ky in the x- and y-directions, respectively, can be written as

kx = k cos φ and k y = k sin φ

(4.7.52)



where k is the wave number of a wave propagating in a direction (x′) (see Fig. 4.1[b]) making an angle f with the x-direction. Then, the Fourier transform of equation (4.7.50) gives

(k 2 K1 + ikK 2 + K )Q = T

(4.7.53)



Here, [K1 ] = [A1 ]cos 2 φ + [A 2 ]cos φ sin φ + [A 3 ]sin 2 φ

[K 2 ] = [A 4 ]cos φ + [A 5 ]sin φ , and [K] = [A 6 ] − ω 2[M]

(4.7.54)

For the free motion of the plate, we get the right and left eigenvalue problems (see equations (4.7.31) and (4.7.32)) (km2 K1 + ikm K 2 + K )QmR = 0

QmL (km2 K1 − ikm K 2 + K ) = 0

(4.7.55)

Following the procedure described in the previous subsection—see equations (4.7.35) and (4.7.36)—we get 2M

Q=

∑ m =1



km QmL 2 TQmR 1 (km − k )Bm

Bm = QmL 1QmR 1 − km2 QmL 2 K1QmR 2

(4.7.56)

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where  QR  QmR =  Rm1  , Qm 2 



{Q } = {Q L m

L m1

QmL 2

}

Consider a steady-state unit load located at the origin of a plane at the ith interface, a distance zi from the plate’s top exterior surface. The spatial representation of this load, f(x, y), takes the form f ( x , y ) = f0δ ( x )δ ( y ), where f0 = ( f x , f y , f z )T





(4.7.57)

Applying Fourier transform to equation (4.7.57) gives the load, in the transformed domain, as f = f0



(4.7.58)



Now, f can be expressed as



 f x ′   f x cos φ + f y sin φ          f =  f y ′  = − f x sin φ + f y cos φ        f z   fz

(4.7.59)

so that the global load vector T appears as

T = {0…0, 0…0, f x ′ , f y ′ , f z ′ , 0…0, 0…0}T



(4.7.60)

Here, x ′ = x cos φ + y sin φ , y ′ = − x sin φ + y cos φ , z ′ = z . Then,

QmL 2 T = ψ 1 f x ′ + ψ 2 f y ′ + ψ 3 f z



(4.7.61)

L . where y1, y2, and y3 are the corresponding elements in Qm2 The displacement response u at a nodal point, due to T , can be written from equations (4.7.56) and (4.7.60) as (see Mahmoud et al. 2006)

U m′  (ψ 1cosφ + ψ 2 sinφ + ψ 3 )   u′ =  Vm′  (km − k )Bm W ′  m =1  m 2M

∑



(4.7.62)

where

ψ1 = ψ1 fx + ψ 2 f y (4.7.63)

ψ 2 = −ψ 2 f x + ψ 1 f y

ψ 3 = ψ 3 fz



and Um′, Vm′, and Wm′ are the elements of Q . R m1

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Transforming equation (4.7.62) in cylindrical coordinates (r, q, z), followed by an inverse Fourier transform given by the second line of equation (4.7.51) leads to ur    u′′ = uθ  w   

U ′ cosϕ − Vm′ sinϕ  ψ 1 cos φ + ψ 2 sin φ + ψ 3  m  U m′ sinϕ +Vm′ cosϕ e ikr cosϕ kdkdφ ( k − k)B m m   ( k,φ ) m =1 Wm′   M   ψ 1 cos φ +ψ 2 sin φ +ψ 3 π   (km − k)Bm 2 ∞ M   m=1 1  U m′ cosϕ − Vm′ sinϕ  e ikr cosϕ kdkdϕ =−   (2π )2 km    π − ∞ m=1  × 1 + U m′ sinϕ +Vm′ cosϕ   −   2   k − km    Wm′   

1 = (2π )2

M

∫∫∑

∫ ∫∑

∑

(4.7.64) where j = f − q and df = dj. Note that only half the 2M modes associated with the waves propagating from the source toward the field points are relevant to the plane wave problem. For transversely isotropic material, numerical quadrature will be required to carry out the integrals. Dividing the domain [− π , π ] into Nj subdomains with equal 2 2 intervals leads to ∞ M  Nϕ ψ 1 cos(ϕ + θ ) + ψ 2 sin( ϕ + θ ) + ψ 3     (km − k)Bm ur   1  1 ∆ϕi −∞ m=1   u = −  θ U m′ cosϕ − Vm′ sinϕ   2  2 ( π ) w  km   ikr cosϕ   kdk∆ϕ     ×(1 + k − k ) U m′ sinϕ +Vm′ cosϕ  e m     Wm′    

∑ ∫ ∫∑



(4.7.65)

It is possible to simplify equation (4.7.65) if a unit load is in the x- or z-direction. First, we note that ψ 1 , ψ 2 , ψ 3 , Bm, Um′, Vm′, and Wm′ are independent of k. Then we use Cauchy’s residue theorem and the following expressions for Delta function ∞

∫e

ikr cos ϕ

dk = 2πδ (r cos ϕ )

−∞

  1 π   δ (r cos ϕ )dϕ = − r ± 2 ∈∆ϕ   0 otherwise  ∆ϕ δ (r cos ϕ )cos ϕ dϕ =0

∫ ∫

∆ϕ

 1 − r  1 δ (r cos ϕ )sin ϕ dϕ =   r ∆ϕ  0 

∫



 π ∈∆ϕ  2  π  − ∈∆ϕ  2  otherwise  

(4.7.66)

−



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For z-direction unit load, by applying the middle point integration formula, we obtain from equation (4.7.65) U m′ cosϕ i − Vm′ sinϕ i  ur   0  N   i ϕ M ψ 3 km    1 M ψ 3* km  nϕ i + Vm′ cosϕ i  e ikmr cosϕi ∆ϕ ∑ *  0 − ∑∑ U m′ sin uθ  =   w  π r m=1 Bm W ′*  2π i =1 m=1 Bm  Wm′      m 

(4.7.67)

where the superscript asterisk (*) represents quantities corresponding to the wave traveling in the direction f = j − q = p/2. For x-direction unit load, the resulting expression becomes ur  Vm′*    1 M ψ 1* sinθ − ψ 2* cosθ  *  ∑ uθ  = U m′  Bm*  w  π r m=1  0      (4.7.68)

U m′ cosϕ i − Vm′ sinϕ i  i Nϕ M ψ (ϕi )km   ik r cosϕ − ∑∑ U ′ sinϕ i +Vm′ cosϕ i  e m i ∆ϕ 2π i=1 m=1 Bm  m  Wm′









where ψ (ϕ i ) = ψ 1 cos(ϕ i + θ ) + ψ 2 sin(ϕ i + θ ) . For isotropic materials, the plane-stain and antiplane waves exist independently. The eigenvalues and associated eigenvectors of isotropic material are independent of the direction j of the propagation. For this case, following the procedure described previously, the result is (see equation [4.7.56]) Qα =

2 Mα

∑ m =1

kα m Qα mL 2 Tα QαRm1 (kα m − k )Bαm

(4.7.69)

where

Bαm = QαL m1QαRm1 − kα2m QαL m 2 K α 1QαRm1



(4.7.70a)

and  QR  QαRm =  Rαm1  , Qαm 2 

 QαL m1   L  αm 2 

{Q } = Q L αm

α = p for plane-strain problem

(4.7.70b)

α = a for antiplane problem

M p = 2( N + 1), M a = ( N + 1)



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and

{ = {V ′, … , V ′ , … , V ′ }

Q p = U1′, W1′, … , U m′ , Wm′ , … , U N′ +1 , WN′ +1 T

Qa



1

N +1

m

}

T

(4.7.70c)

The resulting equations, similar to equation (4.7.64), can be integrated over f. The results can be presented in compact forms for unit loads acting in z- and x-directions (see Bai et al. 2004b) as ur    1 uθ  = w  2  

Mp

ψ 3p

m =1

pm

∑B

  k pm Hˆ 1 (k pm r )U m′     0      2 ˆ   − ik p H 0 (k pm r )Wm′     π r

for z -direction load, and

ur  1   uθ  = − 2 w   

−



1 2

Mp

ψ1p

m =1

pm

∑B

Ma

ψ 2a

m =1

am

∑B

  ik pm  Hˆ 0 (k pm r ) − Hˆ 2 (k pm r ) cosθU m′    2        ik pm 2  Hˆ 0 (k pm r ) + Hˆ 2 (k pm r )  sinθU m′    sinθ −  2     π r    ˆ k pm H 0 (k pm r )cosθWm′  (4.7.71)        ik pm  Hˆ 0 (kam r ) + Hˆ 2 (kam r )  sinθVm′   − 2 +   2    π r    ik pm   ˆ ˆ   − H ( k r ) + H ( k r ) sin θ V ′   2 am  m  0 am 2   0        

for x -direction unit load



where

ψ 1 p = ψ 1 p f x , ψ 2a = ψ 2a f x , ψ 3 p = ψ 3 p f z (4.7.72) L and where y1p and y3p are corresponding elements in Q pm2 and ψ2a is the corresponding L element in Qam2 . It should be noted that the Hˆ n appearing in equation (4.7.71) are not Hankel functions. Rather, they take the form e −nπ Hˆ n (kαm r ) = π

2

∫

π 2

(4.7.73) In the following, numerical results for two examples are presented for Green’s displacement in the plate due to unit load applied on the top surface at (0, 0, 0). The first

−π 2

e ikαmr cos ϕ +inϕ dϕ

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Vertical Displacement

0.10

0.05 0.00

–0.05

–0.10 0.0

0.5

1.0 z/H

1.5

2.0

(a)

Vertical Displacement

0.05

Re (w) from hybrid method Im (w) from hybrid method Re (w) from superposition method Im (w) from superposition method

0.00

–0.05

–0.10

0.0

0.5

1.0

1.5

2.0

z/H (b)

FIGURE 4.43  Vertical-load-induced vertical displacements for a three-ply laminated plate when Ω = 10: (a) x = 2H, (b) x = 10H. (Reprinted with permission from Bai et al. 2004b, Fig. 8.)

example is a plate consisting of three isotropic layers. It is symmetric about the middle plane. The outer layers are assumed to be aluminum, and the middle layer is steel. The material and geometric properties are taken as Kgg N , ρ1 = ρ3 = 2.7 × 103 3 , H1 = H 3 = 0.2H m m2 Kg N ν 2 = 0.28, µ2 = 75 × 109 2 , ρ3 = 7.8 × 103 3 , H 2 = 1.6H m m

ν1 = ν 3 = 0.32, µ1 = µ3 = 25 × 109

For this example, the normalized frequency Ω = wH/ µ2 ρ2 is used. Figures 4.43(a,b) compare the vertical displacement (w) through the thickness at sections x = 2H and

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Vertical Displacement, w

0.8

Re (w) Im (w)

0.6 0.4 0.2 0.0 0 –0.2

2.0

4.0

6.0

8.0

10.0

–0.4 –0.6 –0.8

x/H

FIGURE 4.44  Vertical displacement distribution along top surface (z = 0) of a three-ply laminated plate due to a vertical load when Ω = 10. (Reprinted with permission from Bai et al. 2004b, Fig. 11.)

x = 10H for a frequency of Ω = 10. Both the hybrid (Bai et al. 2004b) as well as the planewave superposition techniques are used. The figures show that the data from the two approaches agree well, both in the near and far fields. Figure 4.44 shows the variation of vertical displacement along the top surface due to the vertical point load. Since r = 0 is a singular point of the solutions, the results are shown for 0.05 < x/H < 10. In the second example, we consider a homogeneous graphite–epoxy fiber-reinforced plate of thickness 2H. The effective material properties are listed in Table 4.1. The global x-axis is taken as the symmetry axis. For this example, the normalized frequency Ω = wH/ c66 ρ . Frequency spectra for this plate are shown in Figs. 4.7 and 4.8 for the propagation directions 30° and 60°, respectively. It is observed that the phase velocities increase steeply as the wavepropagation angle becomes smaller. Figures 4.45(a,b) show the vertical displacement along the top surface for 0.3 < r/H < 10 for the observation directions 30° and 60°.

4.8 Laminated Plate with Interface Layers In Sections 4.6.2 and 4.7.2, solutions to the dynamic response of a three-layered plate with a thin interface layer have been presented. The dispersion of guided waves and the forced response of the plate obtained by using an exact method and a rational approximate O(h) approximation show that the approximate effective interface conditions predict fairly accurate results unless the frequency of excitation is high. This O(h) approximation includes effects of both the stiffness and the density of the layer. In this section, we will analyze the effect of replacing the interface layer by a spring model (not including the effect of the density of the layer) on the dynamic response of a laminated cross-ply graphite–epoxy plate. We consider a laminated cross-ply composite plate as shown in Fig. 4.33. Since each lamina is assumed to be a uniaxial fiber-reinforced composite, it can be modeled as transversely isotropic. A lamina with fibers aligned parallel to the x-axis (q = 0°) has

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Re (w) Im (w)

0.2 0.0 –0.2 –0.4 –0.6 –0.8

(a)

0.4 0.3 0.2 0.1 0.0 –0.1 –0.2

2

4

6

8

–0.3 –0.4 –0.5

(b)

FIGURE 4.45  Vertical displacement distribution along top surface (z = 0) of a homogeneous graphite–epoxy plate when Ω = 5: (a) observation direction = 30°, (b) observation direction = 60°.

its symmetry axis along this axis. For one with fibers aligned along the y-axis (q = 90°), the symmetry axis coincides with that axis. We will restrict our attention to planestrain motion in the plane of x–z. The solution for the displacement–stress vector in the Fourier-transformed domain in the mth layer can be written as (see equation (4.5.6))

{S(z )} = [Q(m)][E(z , m)]{C(m)}, z m ≤ z ≤ z m+1 (m = 1, 2,,… , N )



(4.8.1)

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where {S(z )} = 〈uˆ wˆ σˆ zz σˆ xz 〉T is the Fourier-transformed displacement–stress vector, [E(z , m)] = Diag[e is1 ( z − zm ) e is2 ( z − zm ) e is1 ( zm+1 − z ) e is2 ( zm+1 − z ) ],



{C(m)} is the vector of unknown coefficients, and 1  r1 [Q(m)] =  P  1 Q  1

where r1,2 =

c55 (k22 − s12,2 ) − c11 k 2 c55δ ks1,2

1  r2 − r1 − r2   P2 P1 P2   Q2 − Q1 − Q2 

1

1

(4.8.2)



,

P1,2 = i(c33r1,2 s1,2 + c13 k ), Q1,2 = ic55 (kr1,2 + s1,2 ), and all the other quantities have been defined in Section 4.2.1 (equations (4.2.12) and (4.2.15)). Now, enforcing the continuity of the vector {S(z)} across the interfaces between adjacent layers (i.e., perfect bonding) and the boundary conditions at the top and bottom surfaces of the plate, we can write the 4N equations in the 4N unknown coefficients in the global matrix form (equation (4.5.12)). If there are interface layers between the adjacent laminae, they are incorporated in the global equation (assuming that the interface layers are elastic). This will give the exact equation for a cross-ply laminate with interface layers. In the following, we will derive the equations governing a spring model for the interface layers. Consider an elastic isotropic interface layer between the mth and (m+1)th laminae of thickness h0 and elastic stiffnesses l 0 and m 0. The thickness of this layer is assumed to be much smaller than the characteristic wavelength. Then, in the spring model, the interface conditions are approximated as



 w+ − w−   u+ − u−  σ zx =  µ0 , σ zz =   ( λ0 + 2 µ0 )   h0   h0 

Note that these are not the exact O(h) approximations that have been derived in Section 4.6. Since the stresses are assumed to be constant, the interface conditions now become

[B][Q(m)][E(z m− , m)][C(m)] = [Q(m + 1)][E(z m+ , m + 1)][C(m + 1)]



(4.8.3)

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where the matrix [B] is given by 1  0 [B] =   0 0 

h0 / µ0   h0 0  1  λ0 + 2 µ0  0 1 0  0 0 1  0

0

(4.8.4)

The interface condition in equation (4.8.3) is now easily incorporated in the global matrix equation. Two extreme cases can be easily identified from the equations above. First, when the off-diagonal terms are neglected, the perfect bond condition is obtained. If the off-diagonal terms are very large, then the finiteness of the displacements leads to the zero tractions at the interface (i.e., delamination). Dispersion of waves in a free-free plate using the spring model as well as a density model has been studied by Xu and Datta (1990), who used the delta matrix technique to do a parametric study of the various limiting cases. We consider the response of a cross-ply composite plate having eight layers due to a vertical line load applied at (0,0). Results for the normal stress sxx at the point (1.75H, 0) are shown for different properties of the bond layers. The properties of the graphite-fiberreinforced lamina are given in Table 4.1. The properties of the interface layers are r = 1.2 kg/m3, l0 + 2m0 = 8.65 GPa, m0 = 1.95 GPa, and h0 = 0.0584 mm. The thickness of each lamina is 0.584 mm. Thus, thickness of the plate is 5.08 mm. Frequency spectra due to an impulsive line load are shown in Fig. 4.46. These results for the perfect bond, with interface

400

Interface layer Perfect bond Spring model Spring model (–2, –2)

stxx

300

200

100

0

0.0

0.2

0.4 0.6 Frequency (MHz)

0.8

1.0

FIGURE 4.46  Comparison of the spectra of s xx predicted with different models. Source and receiver are on the same side of the plate surface and 1.75H apart. (Reprinted with permission from Ju and Datta 1992a, Fig. 5.)

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Elastic Waves in Composite Media and Structures 150

Central frequency = 0.21 MHz Interface layer Perfect bond Spring model Spring model (–2, –2)

100

stxx

50 0 –50 –100 –150

0

10

20

30

40

Time (microsec)

FIGURE 4.47  Comparison of the transient response when the source is a Ricker pulse with center frequency 0.21 MHz. The locations of the source and receiver are as in Fig. 4.43. (Reprinted with permission from Ju and Datta 1992a, Fig. 6.)

layer, and the spring model agree reasonably well below the cutoff frequency of the first thickness-stretch mode (0.26 MHz). Beyond that, both the perfect bond and spring models start deviating from the exact interface results, the latter showing very different behavior than the other two. This figure also shows the results when spring stiffnesses are very low (10−2 times the actual values). Clearly, in this case the amplitude of the response is much larger than in the other three cases. Furthermore, there are more peaks at low frequencies. This is because the cutoff frequencies are much lower and closer to one another. The time responses of the plate in these four cases are shown in Fig. 4.47. The time dependence of the pulse is given by equation (4.7.17). It is seen that the spring model predictions for the early times deviate from the other three. However, the long-time behaviors predicted by the three models agree well and show a periodic oscillation at a frequency corresponding to the cutoff frequency of the first thickness-stretch mode. As expected, weak bonding causes larger amplitudes and different time behavior.

4.9 Remarks In this chapter, a complete treatment of free and forced guided waves in a multilayered composite plate has been presented. Both analytical and numerical (stiffness) techniques have been developed and used to study dispersion of free time-harmonic guided waves and forced motion in the frequency and time domains. These techniques are quite general and are applicable to layered plates when each layer has monoclinic symmetry in the plane of the plate. For plates with thin coating or interface layers, exact effective-boundary

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147

and interface conditions, correct to first order in the thickness of the layers, have been derived. It has been shown that these effective-boundary (or interface) conditions lead to simplified boundary-value problems that can be solved analytically. Results of approximate boundary-value problems agree well with those obtained by much more complicated matrix methods. Since, for thin layers, so-called spring models have been widely used, results using a spring model have been presented in Section 4.8. It has been shown that such models are useful only at low frequencies and do not predict correct dynamic behavior at early times. A first-order plate theory has also been presented for thin anisotropic plates, and comparison of the results with exact numerical predictions shows that, at low frequencies, such a simplified theory may be preferable for practical use. All the illustrative problems discussed here are for elastic layers. However, the techniques developed are easily extended to layers with viscoelastic properties. Pan et al. (1999) presented a detailed study of a bilayered plate in which one layer is thick and dissipative (such as found in pipelines and other structures having insulating and protective layers.) It was found that there are certain modes that were primarily guided by the elastic layer and are less affected by the thick protective coating layer. Thus, mode selection for increased sensitivity can be important for particular applications. Wave propagation in a plate with a thick coating layer (elastic or viscoelastic) gives rise to complex and interesting propagation characteristics. Some detailed studies were done by Jones (1964), Bratton and Datta (1992), and Laperre and Thys (1993) (see also Yapura and Kinra 1995). Dispersion equations governing time-harmonic guided waves in an elastic isotropic plate with a thick isotropic soft elastic (or viscoelastic) coating can be analyzed by the stiffness method developed in Section 4.2.1. The effect of damping can be included in this model. Figure 4.48 shows the coupling of the symmetric and antisymmetric modes. The longitudinal- and shear-wave velocities in the coating layer are, respectively, 2.5 km/s and 1.00 km/s, and those in the steel plate are 5.96 km/s and 3.23 km/s. The values for w′ and k′ are defined as

w′ = whsteel/Cssteel



k′ = khsteel

The subscript s denotes shear wave. It is found that, in a given frequency range, there are certain modes in the steel plate that are relatively unaffected by the coating. These would be particularly suited for characterizing defects using ultrasonic guided waves. Wave propagation in a viscoelastic orthotropic plate was studied by Deschamps and Hosten (1992) among others.

4.10 Laser-Generated Thermoelastic Waves So far, attention has been focused on the isothermal dynamic response of composite plates. There is also a large body of literature on the effect of thermal loading on the dynamic response of elastic media. Laser-generated ultrasonic waves are widely used for material property determination and characterization of defects. They have the advantage that both generation and reception are accomplished in a noncontact manner. Both narrow- and broad-band signals can be generated and detected. Since the early work by White (1963) demonstrating the generation of ultrasonic waves by laser irradiation,

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Normalized Frequency

Elastic Waves in Composite Media and Structures 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

0.0 cm 0.3 cm 0

1

2

3

4

5

6

7

8

9 10

11 12 13 14 15

Normalized Wavenumber 15

Normalized Frequency

14 13 12 11 10

0.0 cm 0.3 cm 0

5

10

15

20

25

30

35

40

45

50

55 60

Normalized Wavenumber

FIGURE 4.48  The dispersion of a 1-cm steel plate with and without a 0.3-cm soft elastic coating: (a) range of 0 ≤ w′ ≤ 15 and 0 ≤ k′ ≤ 15 and (b) range of 10 ≤ w′ ≤ 15 and 0 ≤ k′ ≤ 60. (Reprinted with permission from Pan et al. 1999, Fig. 2.)

there have been many publications dealing with laser ultrasonic techniques for nondestructive evaluation. Scruby and Drain (1990) provide a good description of the principles and techniques of laser-based ultrasonics. Among the recent works dealing with the modeling of the laser source and propagation and measurement of laser-generated waves, we mention Telschow and Conant (1990), Noui and Dewhurst (1990), Dubois et al. (1994), Cheng and Berthelot (1996), Johnson et al. (1996), Sanderson et al. (1997), Mukdadi et al. (2001), Arias and Achenbach (2003), Hurley (2004), Hurley and Spicer (2004), Achenbach (2005), and Al-Qahtani et al. (2005). Equations governing generalized dynamic thermoelasticity were derived in Chapter 2 (see equations (2.1.54) and (2.1.55)). Equation (2.1.54) is a generalization of the classical

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linear theory of heat conduction by the inclusion of a relaxation time τ0, which makes the equation hyperbolic, so that there is a finite speed of the thermal wave. Chandrasekharaia (1986, 1998) has given a thorough review of the large volume of work on generalized thermoelasticity (see also Ignaczak 1989; Hetnarski and Ignaczak 1999). In this section, hyperbolic thermoelastic equations will be used to describe thermal- and elastic-wave propagation in an anisotropic plate due to laser irradiation of one surface of the plate. A laser beam focused on a small region of the surface of the plate generates thermal and elastic waves. As mentioned above, such waves generated by a pulsed laser operating in the thermoelastic regime are ideally suited for noncontact nondestructive evaluation. We consider an infinite, homogeneous, transversely isotropic, thermally conducting plate at a uniform temperature T0 in the undisturbed state. The geometry of the plate is shown in Fig. 4.1. Note that the symmetry axis can be taken in the fiber direction (X-axis). A global coordinates system (x,y,z) is chosen such that the x-axis makes an angle q with the symmetry axis. The thickness of the plate is assumed to be H, and for computational purposes the plate will be divided into N sublayers. As discussed in Section 4.2, the symmetry axes may be oriented in different directions from one sublayer to another. For simplicity of analysis, it is assumed here that all the sublayers have the same symmetry direction.

4.10.1 Analytical Solution The solutions to equation (2.1.55) can be assumed to be in the form given by equations (4.2.5a–c). Then, in the absence of external body forces, the equations governing Θ, Φ, and y (see Bouden and Datta 1990) are



 ∂FY ∂FZ   ∂2 ∂2  2 2 c55 ∂X 2 + c44 ∇ − ρ ∂t 2  ∇ Ψ + ρ  ∂Z − ∂Y  = 0     (c13 + c55 )

∂F  ∂2 2 ∂2 ∂2  ∂2Θ ∇ Φ + c55 ∇ 2 + c11 2 − ρ 2  2 + ρ X = 0 2 ∂X ∂X ∂X ∂t  ∂X 

(4.10.1) (4.10.2)

 ∂FY ∂FZ   ∂2 ∂2  2 ∂2 2 2 c33 ∇ + c55 ∂X 2 − ρ ∂t 2  ∇ Φ + (c13 + c55 ) ∂X 2 ∇ Θ+ρ  ∂Y + ∂Z  = 0 (4.10.3)    

Here,

ρ FX = − β XX T, X , ρ FY = − βYY T,Y , ρ FZ = − β ZZ T,Z



(4.10.4)

with T being the temperature field above the ambient temperature, T0. The subscript comma (,) denotes the derivative with respect to the space variable. Using equation (4.10.4) in equations (4.10.1–4.10.3), we obtain



 ∂2 ∂2  2 2 c55 ∂X 2 + c44 ∇ − ρ ∂t 2  ∇ Ψ = 0  

(4.10.5)

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 ∂2 2 ∂2 ∂2  ∂2Θ 2 +c − β XX T, XX = 0 ∇ + c ∇ − ρ Φ  55 11 ∂X 2 ∂X 2 ∂t 2  ∂X 2  2 2  2  ∂ ∂ ∂ 2 2 2 2 c33 ∇ + c55 ∂X 2 − ρ ∂t 2  ∇ Φ + (c13 + c55 ) ∂X 2 ∇ Θ − βYY ∇ T = 0  

(c13 + c55 )



(4.10.6) (4.10.7)

∂2 ∂2 In the above equations, ∇ 2 = + 2 . It is seen that the above equations are the 2 ∂Y ∂Z same as equations (4.2.7a–c) if there are no thermal gradients. Note that, as in the case of the isothermal problem, the equation governing Ψ is uncoupled from equations (4.10.6) and (4.10.7) governing Θ and Φ. These equations are supplemented by the coupled heatconduction equation (2.1.54), which takes the form kXX

�  ∂2Θ ∂ 2T ∂ 2T − T 1 + τ �  − ρc  1 + τ ∂  T� = −Q (4.10.8) + βYY ∇ 2 Φ + ∇ k β 0 0 0 YY XX      ∂X 2 ∂X 2 ∂t   ∂t  

In equation (4.10.8), the overdot denotes a time derivative, and Q is the heat-generation term. Let us introduce nondimensional variables X′ =

υx υ υ υ2 X , Y ′ = x Y , Z ′ = x ,t ′ = x kx kx kx kx

Θ′ =

ρυ 4 ρυ 4 ρυ x4 Θ, Φ ′ = 2 x Φ, Ψ ′ = 2 x Ψ kx β xx T0 kx β xx T0 k β xx T0 2 x

T ′ = T / T0 , c1 =



(4.10.9)

c13 c c c , c2 = 55 , c3 = 33 , c4 = 233 c11 c11 c11 c11

c5 =

k yy β yy c44 , δ = c1 + c2 , β = , K= kxx β xx c11

τ 0′ =

υ x2 β2 T τ 0 , ε = 2xx 02 kx ρ cυ x



where υ x = c11 /ρ is the longitudinal velocity, k x = k xx/rc is the thermal diffusivity in the x-direction, and e is the thermoelastic coupling constant. Then, equations (4.10.5– 4.10.8) take the forms



 ∂2 ∂2  2 2 c2 ∂x 2 + c5 ∇ ′ − ∂t 2  ∇ ′ Ψ ′ = 0 ′ ′  

δ

 ∂2 ∂2 ∂ 2  ∂ 2 Θ ′ ∂ 2T ′ 2Φ + c ∇ 2 + =0 ∇ − − ′ ′ ′  2 ∂x ′ 2 ∂x ′ 2 ∂t ′ 2  ∂x ′ 2 ∂x ′ 2 

(4.10.10)



 ∂2 ∂2  2 ∂2 2 2 2 c3 ∇ ′ + c2 ∂x 2 − ∂t 2  ∇ ′ Φ ′ + δ ∂x 2 ∇ ′ Θ ′ − β ∇ ′ T ′ = 0 ′ ′  ′ 

(4.10.11) (4.10.12)

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TABLE 4.6  Values of the Thermoelastic Coupling Constant e for Some Materials Material

ux (m/s)

kx (m2/s)

K

β

e

Aluminum Zinc Silicon nitride

6.42 × 103 4.77 × 103 1.34 × 104

9.8 × 10−5 4.45 × 10−5 2.58 × 10−5

1 1 0.786

1 0.882 0.843

2.97 × 10−2 2.16 × 10−2 2.49 × 10−3

2  ∂   ∂2 � � ′  − ∂ T ′ − K ∇ ′ 2T ′ +  1 + τ ′ ∂  T� ′ = −Q ′ ε  1 + τ 0′   2 Θ ′ + β∇′2Φ  0  2 ∂t ′  ∂t ′   ∂x ′   ∂x ′



(4.10.13)

Qkx . T0 ρcυ x2 In the following discussion, we will drop the prime (′). Note that, in many cases of interest, the coupling parameter e is small and can be neglected. Table 4.6 shows the values of e for some materials. Neglecting thermoelastic coupling would allow the thermalwave equation to be solved independently with the appropriate boundary and initial conditions. Then, equations (4.10.11) and (4.10.12) can be solved as nonhomogeneous equations, with the forcing terms due to the temperature gradient being known. In the following discussion, we will present an exact solution taking into account the coupling effect. To solve the homogeneous equations (4.10.10–4.10.13) when Q′ is zero, we assume propagation of guided waves in the plate, given by

where Q’ =

Θ = g 1 (z )e iψ Φ = g 2 (z )e iψ

(4.10.14)

T = g 3 (z )e iψ Ψ = g 4 (z )e iψ





Here, y = kx − wt and gi(z) (i = 1, …, 4) are four unknown functions of z that are to be found such that equations (4.10.10–4.10.13) are satisfied by the respective functions Θ, Φ, T, and Ψ. Note that y can be written as

ψ = KX + LY − ωt , K = k cosθ , L = k sinθ





(4.10.15)

Substitution of equation (4.10.14) in equations (4.10.10–4.10.13) leads to the following system of ordinary differential equations to solve for gi(z) (i = 1, …, 4).  d2  d2   2 2 2 2 c2 dz 2 − c2 L − K + ω  g 1 + δ  dz 2 − L  g 2 − g 3 = 0      d2  d2  d2  d2    − δ K 2  2 − L2  g 1 + c3 2 − c3 L2 − c2 K 2 + ω 2   2 − L2  g 2 − β  2 − L2  g 3 = 0 dz dz dz      dz     d2  d2   ετ K 2 g1 − ετβ  2 − L2  g 2 + [K 2 − K  2 − L2  − τ ]g 3 = 0 dz dz    



(4.10.16a)

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   d2   −c2 K 2 + c5  2 − L2  + ω 2  g 4 = 0  dz   



(4.10.16b)

where t = iw + t 0w 2. Solutions to equations (4.10.16a,b) can be written—see equation (4.2.13)—as g i = Gi e irz (i = 1, …, 4) where Gi ’s are constants and r satisfies the equation −c2 s 2 − K 2 + ω 2 [c5 s 2 + c2 K 2 − ω 2 ] ×

−δK2

− δ s2

−1

− c3 s 2 − c 2 K 2 + ω 2

−β

ετ K 2



(4.10.17)

βετ s 2

= 0 (4.10.18)

K 2 + Ks 2 − τ



Here we have written r2 = s2 − L2. Equation (4.10.18) is a quartic equation in s2, one of whose roots is s42 =



ω 2 − c2 K 2 c5

(4.10.19a)

The other three roots are the solutions to the cubic equation obtained by expanding the determinant on the left-hand side of equation (4.10.18). The cubic equation can be written as s 6 + A2 s 4 + A1 s 2 + A0 = 0



(4.10.19b)



The coefficients A0, A1, and A2 are given by A0 =

1 [( K 2 − τ )(c2 K 2 − ω 2 )( K 2 − ω 2 ) + ετ K 2 (c2 K 2 − ω 2 )] c 2 c3 K

A1 =

1 [( K 2 − τ ){(c3 + c22 − δ 2 )K 2 − (c2 + c3 )ω 2 } c 2 c3 K

(4.10.19c)

+K ( K − ω )(c2 K − ω ) + ετ {(2δβ + c3 − β )K + β ω }] 2



A2 =

2

2

2

2

2

2

2

1 [c c ( K 2 − τ ) + K {(c3 + c22 − δ 2 )K 2 − (c2 + c3 )ω 2 } − τεβ 2 c2 ] c 2 c3 K 2 3



If the coupling term in the third equation of (4.10.16a) is neglected (e = 0), then the determinant in equation (4.10.18) factors into ( K 2 + Ks 2 − τ )[c2 c3 s 4 + {K 2 (c3 + c22 − δ 2 ) − ω 2 (c2 + c3 )}s 2 +(K 2 − ω 2 )(c2 K 2 − ω 2 )] = 0 (4.10.20)

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Thus, the thermal- and elastic-wave equations are uncoupled. For the former, we get s32 =



τ − K2 K

(4.10.21)

Since t is complex, it is seen that the thermal wave is attenuated. For the elastic wave, equation (4.10.20) leads to a quadratic equation in s2 given below:

c2 c3 s 4 + {K 2 (c3 + c22 − δ 2 ) − ω 2 (c2 + c3 )}s 2 +(K 2 − ω 2 )(c2 K 2 − ω 2 ) = 0 (4.10.22)

This equation is the same as that derived from equation (4.2.14) for the elastic wave propagating in the plane of xz. This is not attenuative. Now, the three roots of equation (4.10.19b) are given below. We define 9 A A − 27 A0 − 2 A23 3A1 − A22 , q= 1 2 2 27 p q R = , S = , D = R3 + S 2 3 2 p=



L= 3 S+ D , M= 3 S− D

(4.10.23)



Then, the three roots of equation (4.10.19b) are



1 s12 = − A2 + (L + M) 3 1 1 1 2 s2 = − A2 − (L + M) + i 3 (L − M) 3 2 2 1 1 1 s32 = − A2 − (L + M) − i 3 (L − M) 3 2 2

(4.10.24)

The corresponding constants Gi( j ) (i = 1, 2, 3; j = 1, 2, 3) are chosen as G1(1) = 1, G2(1) =



G3(1) =

(c2 s12 + K 2 − ω 2 )(c3 s12 + c2 K 2 − ω 2 ) − δ 2 K 2 s12 ∆1

G2( 2 ) = 1, G1( 2 ) =



δ K 2 − β (c2 s12 + K 2 − ω 2 ) ∆1

c s + c2 K − ω − δβ s ∆2 2 3 2

2

2

βδ s32 − (c3 s32 + c2 K 2 − ω 2 ) G3(3) = 1, G1(3) = ∆3

δ K 2 − β (c2 s32 + K 2 − ω 2 ) ∆3



2 2

δ 2 K 2 s22 − (c2 s22 + K 2 − ω 2 )(c3 s22 + c2 K 2 − ω 2 ) G3( 2 ) = ∆2

G2(3) =

(4.10.25a)

(4.10.25b) (4.10.25c)



Also, G4 = 1.

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Here, ∆1 = βδ s12 − (c3 s12 + c2 K 2 − ω 2 ), ∆ 2 = β (c2 s22 + K 2 − ω 2 ) − δ K 2 ∆ 3 = (c2 s32 + K 2 − ω 2 )(c3 s32 + c2 K 2 − ω 2 ) − δ 2 K 2 s32



(4.10.25d)

The displacement components and the temperature can now be written as U = iK (W1+ + G1( 2 ) W 2+ + G1(3) W 3+ )exp(iψ ) V = [iL(G2(1) W1+ + W 2+ + G2(3) W 3+ ) − r4 W 4+ ]exp(iψ )

(4.10.26)

W = [G2(1)r1 W1− + r2 W 2− + G2(3)r3 W 3− − iLW −4 ]exp(iψ )

T = (G3(1) W1+ + G3( 2 ) W +2 + W 3+ )exp(iψ )



where

ri = si2 − L2 (i = 1,…, 4 )

(4.10.27)



Furthermore, Wi± are defined below. W1+  cos(r1 Z ) 0 0 0 sin(r1 Z )   A11  0 0 0        A21  W +2   0 sin(r2 Z ) 0 cos(r2 Z ) 0 0 0 0       A  W +   0 0 cos(r3 Z ) 0 0 0 sin(r3 Z ) 0 3   31      W +  0 0 cos(r4 Z) 0 0 0 sin(r4 Z )   A41   4  0  =    0 0 0 0 0 cos(r1Z) W1−   − sin(r1Z) 0   A12       cos(r2 Z ) 0 0 W −2   0 − sin(r2 Z) 0 0 0   A22       W 3−   0 0 cos(r3 Z ) 0 0 − sin(r3 Z ) 0 0   A32       W −   0   A42  0 0 cos ( r Z ) 0 0 − sin(r Z) 0 4 4  4  



(4.10.28)

Here, Aij (i = 1, …, 4; j = 1, …, 4) are arbitrary constants. Using equation (4.10.28) in equation (4.10.26), we can write 0 0 U  iK iKG1( 2 ) iKG1(3)    V  iLG2(1) iL iLG2(3) − r4 0  =  1 0 0 T   G3(1) G3( 2 )    W  0 0 0 0 r1G2(1)   

0

0

0

0

0

0

r2

r3G2(3)

  0   × [T ( Z )]{C}exp(iψ ) 0    − iL  0

(4.10.29)

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The matrices [T(Z)] and {C} are, respectively, the 8 × 8 and 8 × 1 matrices appearing on the right-hand side of equation (4.10.28). The strain vector in the XYZ-coordinates system takes the form  − K2 − K 2G1( 2 ) − K 2G1(3) e XX   − L2 − L2G2(3)   − L2G2(1)  eYY   2 (1) − r22 − r32G2(3)   − r1 G2  e  ZZ   0 0 0 =   γ YZ   0 0 0    γ ZX   − KL(1 + G (1) ) − KL(G ( 2 ) +1) − KL(G (3) +G (3) ) 1 1 2 2 γ    XY   

0 − iLr4 iLr4 0 0 − iKr4

  0 0 0 0   0 0 0 0  ×[T ( Z )]{C}exp(iψ ) 2iLr1G2(1) 2iLr2 2iLr3G2(3) (L2 − r42 )  iKr1 (1+G2(1) ) iKr2 (1 + G1( 2 ) ) iKr3 (G1(3) +G2(3) ) KL    0 0 0 0 0

0

0

0



(4.10.30)

The stress vector in the global xyz-system is

[σ i ] = [ M ][c]([e I ] − T[α ])



(4.10.31)

where [a] = 〈α XX α ZZ α ZZ 0 0 0〉T is the thermal expansion vector. This is the generalization of equation (4.2.22) including the thermal effect. The displacement–stress–temperature–temperature gradient vector in the xyz-system is given by



T    1 0 0 0      u    0     [R] υ    0 [a]      σ    0    zz     = 1 0 0 T,Z       0 σ yz       [0] 0 [a] σ zx       0 w      

   [0]      [T ( Z )][C]exp(iψ ) 0      [S]        

(4.10.32)

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Here, [a] is given by equation (2.2.61), with θ replaced by –q and G3(1) G3( 2 )  iK iKG ( 2 ) 1 [R] =  iLG (1) iL  3   R41 R42



0

 r1G3(1) r2G3( 2 ) r3   2c44iLG2(1) 2c44iLr2 2c44iLr3G2(3) [S] =  c iKr 1 + G (1) c iKr 1 + G ( 2 ) c iKr G (3) + G (3) 2 55 2 1 55 3 1 2  55 1  r2 r3G2(3) r1G2(1) 

(



  iKG1(3) 0   iLG2(3) 0    R43 2c44iLr4  1

)

(

)

(

(4.10.33a)

  c44 ( L2 − r42 )  (4.10.33b) c55 KL   − iL  0

)

R41 = −c13 K 2 − c33 s12G2(1) + 2c44 L2G2(1) − β33G3(1) R42 = −c13 K 2G1( 2 ) − c33 s22 + 2c44 L2 − β33G3( 2 )

R43 = −c13 K 2G1(3) − c33 s32G2(3) + 2c44 L2G2(3) − β33

(4.10.33c)

As in Section 4.2, it is now possible to set up a transfer matrix for the layer by evaluating the coefficient matrix [C] in terms of the displacement–stress–temperature– temperature gradient at the interface Z = −H/2. This gives

[C] = [T (− H/2)]−1[ℵ][Ξ]− H /2



(4.10.34)

where



[R]−1[a ]T [0]   [ℵ] =   [0] [S]−1[a ]T 

(4.10.35a)

and [a ] is the augmented matrix



1 0 0 0    0   [a ] =  0 [a]    0   

(4.10.35b)

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157

and [Ξ]− H /2 is the displacement–stress–temperature–temperature gradient vector at Z = −H/2. Now, evaluating [Ξ] at Z = H/2 and using equation (4.10.34), we obtain [Ξ]H /2 = [P][Ξ]− H /2



(4.10.36)



with the transfer matrix [P] given by [P] = [ℵ]−1[T ( H )][ℵ]



(4.10.37)

The dispersion equation for free thermoelastic simple harmonic wave propagation is obtained by using the boundary conditions



σ ZZ    σ YZ    = 0 at Z = ± H/2 σ   ZX    T,Z 

(4.10.38)



Thus, the dispersion equation is P41 P42 P43 P48 P51 P52 P53 P58 P61 P62 P63 P68

=0

P71 P72 P73 P78

(4.10.39)

The exact dispersion equation given by equation (4.10.39) was solved by Al-Qahtani and Datta (2004) to obtain complex wave numbers k for a range of frequency w. They also generalized the stiffness method presented in Section 4.2, including the thermal effect. Details of the equations can be found in the reference given above. Some numerical results are presented in the following section.

4.11 Results for Thermoelastic Dispersion and Laser-Generated Waves The material of the plate is chosen to be silicon nitride (Si3N4), the properties of which are given in Table 4.7. The dimensional speed of the thermal wave in the X-direction is



υt =

K11 ρCE τ 0



(4.11.1)

In most of the reported work, υt has been taken as the longitudinal wave speed in the X-direction or less. If it is chosen as υX, then t 0 = 1.44 × 10−13 s (nondimensional t 0 = 1.0)

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Elastic Waves in Composite Media and Structures TABLE 4.7  Physical Properties of Si3N4 Property

Units

Numerical Value

r c11 c13 c33 c23 c55 T0 b 11 b 33 CE K11 K33 e

kg/m N/m2 N/m2 N/m2 N/m2 N/m2 K N/m2×K N/m2×K J/kg×K W/m×K W/m×K …

3.20 × 103 5.74 × 1011 1.27 × 1011 4.33 × 1011 1.95 × 1011 1.08 × 1011 296 3.22 × 106 2.71 × 106 0.67 × 103 55.4 43.5 2.49 × 10−3

3

for silicon nitride. As shown by Al-Qahtani and Datta (2004), a choice of nondimensional t 0 > 1 does not have significant influence on the elastic-wave dispersion. However, it does have significant effect on the thermal-wave dispersion. A three-dimensional plot of the dispersion curves for propagation in the X-direction is shown in Fig. 4.49 for nondimensional t 0 = 1.0. The value of ε is shown in Table 4.7. The corresponding phase velocity (= w/Re[k]) vs. frequency plots are shown in Fig. 4.50.

4 3.5 3

ω

2.5 2 1.5 1 0.5 0 4 3 Im (

ξ)

2 1 0

0

2

4

6

8

10

ξ) Re(

FIGURE 4.49  3-D frequency spectrum for t 0 = 1.0. (Reprinted with permission from Al-Qahtani and Datta 2004, Fig. 2.)

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5

Cp

4

3

2

1

0

0

1

2 ω

3

4

FIGURE 4.50  Normalized phase velocity vs. normalized frequency for propagation along the symmetry axis. (Reprinted with permission from Al-Qahtani and Datta 2004, Fig. 8.)

These figures show that the elastic modes are very close to the isothermal case. The thermal-wave modes are highly attenuated since Im(k) is large. The first thermal mode shows a similar behavior as the lowest elastic modes; however, it has high attenuation. Higher thermal modes originate with large imaginary values of the wave numbers and approach the first thermal mode as the frequency increases. It is seen from Fig. 4.50 that the second thermal mode has a very large phase velocity as the frequency approaches zero, then drops rapidly as the frequency increases, eventually approaching the phase velocity of the first mode. The effect of anisotropy is shown in Fig. 4.51 for propagation angle θ = 45°. The phase velocities decrease as the direction of propagation deviates from the X-axis. Also, mode coupling is clearly seen to occur. The effect of the coupling parameter e on dispersion plots for e = 0 and e = 2.94 × 10−3 is shown in Fig. 4.52. The two spectra are indistinguishable. Thus, the thermal-wave equation can be simplified by neglecting the coupling effect. This would simplify the analysis considerably because the thermal-wave equation can be solved independently of the elastic-wave equation. The latter then can be solved including the thermal expansion effect as the body-force term. In the following, we consider the forced motion of the plate due to a pulsed-laser irradiation of one side of the plate. If the coupling constant e is taken to be zero, then equation (4.10.8) simplifies to



T, XX + KT,ZZ =

1 � 1 �� � T + 2 T − Q ; −H < z < 0 vt kx

(4.11.2)

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Elastic Waves in Composite Media and Structures 6

5

Cp

4

3

2

1

0

0

1

3

2 ω

4

FIGURE 4.51  Normalized phase velocity vs. normalized frequency for propagation along the line q = 45°. (Reprinted with permission from Al-Qahtani and Datta 2004, Fig. 9.)

where vt = kx /τ 0 and Q� (= Q/k xx). In writing the above equation, we have assumed that the wave propagation is in the plane of XZ. This assumption is not necessary, but it is made to keep the analysis simple. Note that equation (4.11.2) is similar to the SH wave motion if material dissipation is taken into account. This can be solved independently of equations (4.10.5–4.10.7) once the thermal input Q� due to the laser irradiation is known. This will be modeled (see Cheng and Berthelot 1996) as

Q� = I 0 f (t ) g ( X )h(z )



(4.11.3a)

Here, I0 is the energy absorbed, and f(t), g(X), and h(z) are given by f (t ) =

g(X ) =

1 − t /t 0 e t02



1 − X 2 /a2 e 2π a 2

h(z ) = γ e γ z



(4.11.3b) (4.11.3c) (4.11.3d)

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ε=0 ε≠0

4 3.5 3

ω

2.5 2 1.5 1 0.5 0 4 3

Im

(k)

2 1 0

0

2

4

6

8

10

) Re(k

FIGURE 4.52  Effect of e on the dispersion of thermoelastic waves. (Reprinted with permission from Al-Qahtani and Datta 2004, Fig. 7.)

Equation (4.11.2) is to be solved subject to the boundary and initial conditions

T ,z = 0 at z = 0, − H

(4.11.4a)

T ( X , z , 0) = T� ( X , z , 0) = 0

(4.11.4b)

Taking the Fourier transform of equations (4.11.2), (4.11.3a), and (4.11.3c), we get mTˆ + kz Tˆ,zz = − kx Qˆ (4.11.5) kx 2 2 w , and kz = k x K . The Fourier transform where m = −k k x + t, kz = kzz/rCE , t = iw + v t2 of a function F(X, z, t) is defined as

Fˆ(k , z , ω ) =

∞ ∞

∫ ∫ F(X , z ,t )e

− ikX −iωt

dXdt

(4.11.6) The solution to equation (4.11.5) that satisfies the boundary conditions in equation (4.11.4a) is found to be (Al-Qahtani et al. 2008)

−∞ −∞

γ k I fˆ (ω ) gˆ(k )  −iγ  {e −irz (e −γ H +irH − 1) − e ir ( H + z ) (e irH − e −γ H )} − e γ z  Tˆ (k , z , ω ) = x 0  2 2 irH m + kz γ  r (1 − e )  (4.11.7) where r = m/kz .

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Now, to solve the elastic-wave equations (4.10.6) and (4.10.7), we take the Fourier transforms and obtain

δ

ˆ  d2 d 2Φ ω 2  ˆ β xx ˆ +  c2 2 − k 2 + 2  Θ = T 2 υx  c11 dz  dz

(4.11.8)

 d2 ω2  ˆ ˆ βzz Tˆ  c3 2 − c 2 k 2 + 2  Φ − δ k 2 Θ = υx  c11  dz



These equations are to be solved subject to the boundary conditions c3



ˆ d 2Φ ˆ − βzz Tˆ = 0 at z = 0 and − H − k 2 c1Θ 2 dz c11

d ˆ ˆ (Θ + Φ ) = 0 at z = 0 and − H dz

(4.11.9)

This problem can be solved in a manner similar to that presented in Section 4.7, so the details will be omitted (see Al-Qahtani et al. 2007). For the computation of the numerical results, the thickness of the plate is taken as 0.1 mm, and the parameters defining the time and spatial dependence of the laser pulse—equations (4.11.3a–c)—are taken as t0 = 8 ns, a = 100 mm, and g = 105/m. After obtaining the complete solution to equations (4.11.8) satisfying the boundary conditions in equation (4.11.9), inverse Fourier transforms with respect to k are obtained by using Cauchy’s residue theorem. The residues are calculated at the poles, which are the solutions of the thermal- and elastic-dispersion equations at fixed frequencies. These are simple poles and, thus, the residues can be calculated in a simple manner. Then, the time-domain response is obtained by numerical integration with respect to w. The displacement in the time domain is given by ∞



1 ηt u( X , t ) = e uˆ( X , ω − iη )e iωt dω 2π

∫

−∞



(4.11.10)

where a small imaginary part ih is added to w to avoid difficulties near cutoff frequencies, and w = 0. Figure 4.53 shows the temperature-rise history at different distances from the origin on z = 0. It is seen that as the distance from the origin increases, the temperature pulse takes some time (quite small but finite) to arrive. Furthermore, the temperature rises quickly to its peak and then decays slowly by diffusion, as expected. The normal displacements of the surface z = 0 at different distances from the Y-axis were calculated by using both the SAFE and the exact methods when g(X) = d(X). Figure 4.54 shows the comparison between the two predictions at a distance X = 80H. Clearly, both agree very well. In the SAFE method, a coupling term was included. As discussed in the foregoing, the coupling effect is negligible. At early times, the S 0 mode arrives first, at about 0.62 ms. At later times, the A0 mode is seen to dominate. An important conclusion from this example is that the response of the plate far away from the laser

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Temp. Rise (a. U.)

0H 0.1 H 0.3 H 0.5 H 0.7 H

1

0.5

0

0.5

0

2

1.5

1

FIGURE 4.53  Temperature-rise history at different points on the X-axis near the laser source.

1 Exact FEM

ω (a.u.)

0.5

0

–0.5

–1

0

0.5

1

1.5 t (µs)

2

2.5

3

FIGURE 4.54  Comparison of the predictions of the SAFE and exact models at x = 80H for g(x) = d(x). (Reprinted with permission from Al-Qahtani and Datta (2008), Fig.6.)

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ω (a.u.)

0.5

0

–0.5

0

0.5

1

1.5 t (µs)

2

2.5

3

FIGURE 4.55  Transient response of the plate at x = 80H using g(x) given by equation (4.11.3c). (Reprinted with permission from Al-Qahtani and Datta (2008), Fig.8.)

source can be modeled as purely elastic. The effect of the width of the pulse g(X) as given by equation (4.11.3c) is shown in Fig. 4.55. The distance from the center of the source is 80H. It is interesting to note that when the width goes to zero, it causes a much sharper flexural response at the time of its arrival. However, the wider pulse causes larger flexural motion than a narrower pulse. So, it is important to take into account the width of the laser pulse for quantitative prediction.

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5 Wave Propagation in Composite Cylinders 5.1 Introduction .............................................................................165 5.2 Governing Equations ............................................................. 166 5.3 Analytical Solution for Transversely Isotropic Composite Cylinder ............................................................... 168 5.4 Stiffness Method I ...................................................................174 5.5 Stiffness Method II ..................................................................179 5.6 Numerical Results—Circular Cylinder ...............................181 5.7 Guided Waves in a Cylinder of Arbitrary Cross Section ...................................................................................... 186 Six-Node Element Formulation • Nine-Node Quadratic Element Formulation 5.8 Numerical Results—Cylinder with Rectangular and Trapezoidal Cross Sections ................................................... 195 5.9 Harmonic Response of a Composite Circular Cylinder due to a Point Force ............................................... 205 5.10 Forced Motion of Finite-Width Plate ...................................213

5.1 Introduction Ultrasonic wave propagation in composite plates and cylinders has been studied extensively during the last half century. The subject of guided waves in composite plates has been dealt with in Chapter 4. This chapter will be devoted to the study of free and forced motion of composite cylinders of circular as well as noncircular cross sections. References to related studies are Bartoli et al. (2006); Nagaya (1981); and Gisell and Dual (2004). In the first part of this chapter, dispersion of waves propagating in composite circular cylinders will be considered. As evidenced from the discussion in Chapter 4, interfaces have considerable influence on the guided-wave behavior. The influence of interface properties on guided waves is also discussed here. This is followed by a detailed study of the characteristics of dispersive wave propagation in cylinders of rectangular and trapezoidal cross sections. As is to be expected, the 165

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cross-sectional geometry also plays a significant role in the dispersion behavior. Forced motion of cylinders due to applied loads is discussed next. Three methods of solution are presented for the analysis of guided waves in a composite solid or hollow circular cylinder. The first is an analytical method that is applicable when the cylinder is composed of coaxial circular cylinders of transversely isotropic materials having the symmetry axes aligned with the cylinder axis. When the material properties of the cylinders making up the composite cylinder have lower symmetries, then it is not possible to obtain a solution using analytical techniques, and one must resort to numerical methods. Two stiffness methods (applicable to circular cylinders) are presented, as in the case of laminated plates. In both of these, each coaxial cylinder is discretized in the radial direction (like the discretization in the thickness direction of a plate) into coaxial subcylinders (sublayers). Then, in the first method, the radial dependence within a sublayer is approximated by cubic polynomials involving the displacement and traction components at the nodes on the two bounding interfaces of the sublayer. Here, continuity of both displacement and traction components is maintained. This will be called the stress continuity method or the stiffness method I. In the second method, the displacement variation with respect to r (the radial coordinate) in each sublayer is approximated by quadratic interpolation functions of r involving displacements at three nodes (two on the two bounding interfaces and one in the middle) in the sublayer. Thus, continuity of displacements at the interfaces (assuming perfect bond) between two adjacent sublayers is maintained. This will be called the displacement continuity method or the stiffness method II. For cylinders with noncircular cross section, the cross section is divided into finite elements, while an analytical form for wave propagation in the axial direction is assumed. Forced motion of cylinders is analyzed by using the expansion in terms of guided modes. In the following discussion, each of these methods is described, and the governing dispersion equations for free time-harmonic guided waves are derived.

5.2 Governing Equations Time-harmonic elastic wave propagation in an infinite laminated cylinder is considered. The geometry of the cylinder and the cylindrical coordinates (r,q,z) are shown in Fig. 5.1. It is assumed that the direction of propagation is along the z-axis. Consider that the cylinder is composed of coaxial cylindrical layers of orthotropic elastic properties having one of the symmetry axes parallel to the radial direction (er) referred to the cylindrical coordinates system. The other two make angles of f (or −f) with ez and eq , respectively. In terms of the local coordinate axes aligned with these symmetry axes, the stress–strain relation is given by



σ 11  c11 c12 c13 .    c22 c23 . σ 22   σ   c33 . 33  =  σ  23   symmetric c44 σ    31   σ 12  

. . . . c55

.  e11    .  e22  .  e33    .  γ 23   .  γ 31    c66  γ 12 

(5.2.1)

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y

H hk

rk+1

rk

r

θ

x

k - th sublayer

z

Typical layer

FIGURE 5.1  Geometry of laminated cylinder. (Reprinted with permission from Rattanawangcharoen 1993, Fig. 2.1.)

Here, the x1-axis is parallel to the radial direction, and the x2- and x3-axes are along the other two symmetry directions. Referred to the global (r,q,z) coordinates, the stress– strain relation is then found as

σ ij′ = Dijkl ekl′



(5.2.2)



where σ ij′ = 〈srr s qq szz s θz szr srq 〉T and eij′ = 〈err eqq ezz eqz ezr erq 〉T. The transformed stiffness coefficients, Dijkl, are found from equation (2.2.63) by replacing the subscripts 1, 2, 3, and the angle q by 2, 3, 1, and j, respectively. As mentioned before, the transformed stiffness matrix has the form of a monoclinic material, the symmetry plane now coinciding with the x2x3-plane. In the sequel, the primes will be dropped without causing any confusion. The relation between the strain e and displacement u (r, q, z, t) are related by the equation e = ( Lr + Lθ + Lz )u



(5.2.3)



where Lr, Lq , Lz are the 6 × 3 matrices given below.



 ∂ . . .   .  ∂r . .    1 ∂  .   . .    1 . .  θ r ∂    .  r . . .   . . .  1 ∂  , Lz =  , Lθ =  Lr =     . . . . .  .   r ∂θ    . . ∂ . . .  ∂  ∂r   1 ∂  ∂z  ∂ 1   . .   . − . . θ r ∂      ∂r r 

.  .   ∂ . ∂z   ∂ . ∂z  . .  . .  . .

(5.2.4)



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The equations of motion in cylindrical coordinates are



∂σ rr 1 ∂σ rθ ∂σ rz 1 + + + (σ rr − σ θθ ) = ρu��r r ∂θ r ∂r ∂z ∂σ rθ 1 ∂σ θθ ∂σ θ z 2 + + + σ rθ = ρu��θ r ∂θ r ∂r ∂z ∂σ rz 1 ∂σ θ z ∂σ zz 1 + + + σ rz = ρu��z r ∂θ r ∂r ∂z



(5.2.5a) (5.2.5b)



(5.2.5c)



Here, r is the density and the overdot denotes a time derivative. In general, equations (5.2.5a–c) for the monoclinic material described by equation (5.2.2) and the strain–displacement relation given by equations (5.2.3) and (5.2.4) cannot be solved by analytical means. For the particular case when each lamina is transversely isotropic with the axis of symmetry aligned with the z-axis, these equations admit analytical solutions. This is discussed in the following section.

5.3 Analytical Solution for Transversely Isotropic Composite Cylinder Consider a composite cylinder, solid or hollow, where each coaxial cylindrical layer is transversely isotropic, having the symmetry axis parallel to the z-axis. In this case, the strain–displacement relation in each layer is given by equation (5.2.1), with c11 = c22, c44 = c55, and c66 = ½(c11 − c12). Similar to the case of a transversely isotropic plate (see Chapter 4, Section 4.2.1), the displacement u can be represented in terms of three potentials (Niklasson and Datta 1998) as u=



∂A e + ∇ ⊥ B + ∇ × (Ce z ) ∂z z

(5.3.1)



where ez is the unit vector in the z-direction, and ∇⊥ is the transverse part of the gradient operator. Using the representation in equation (5.3.1) in the strain–displacement relation in equation (5.2.3), in turn in the stress–strain relation in equation (5.2.1), the stress–displacement relation can be written in terms of the scalar potentials A, B, and C. Now, using the equations of motion (5.2.5a–c) yields the equations governing A, B, and C, which are found as

δ

 ∂2 A  ∂2 +  2 + λ∇ 2⊥ + k22  B = 0 2 ∂z  ∂z 

 ∂2  2 2 2  β ∂z 2 + ∇ ⊥ + k2  A + δ ∇ ⊥ B = 0  ∂2  2 2  ∂z 2 + ε ′∇ ⊥ + k2  C = 0





(5.3.2a) (5.3.2b) (5.3.2c)

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Here, e′ = c66/c55, and other constants appearing in the above equations have been defined by equation (4.2.12). Note that these equations are very similar to equations (4.2.10) and (4.2.11) for transversely isotropic plates. As in that case, the equation governing C is uncoupled from those governing A and B. The solutions to equations (5.3.2a–c) can be expressed in terms of elementary cylindrical waves  J m (qr )  cos mθ  e ikz     Y (qr )  sin mθ  m



(5.3.3)

where Jm(qr) and Ym(qr) are Bessel functions of the first and second kinds, respectively. For a solid cylinder, admissible solutions involve Jm(qr) only. The variable q satisfies the equation

λ q 4 +  k 2η − k22 (λ + 1) q 2 + (k 2 − k22 )(β k 2 − k22 ) = 0

(5.3.4)



for the functions A and B, and the equation

ε ′q 2 + k 2 − k22 = 0



(5.3.5)



for C. Here, h = 1 + lb − d 2. These equations are the same as those obtained from equation (4.2.14) for s2 if b and l are interchanged and e is replaced by e′. Equation (5.3.4) gives two roots for q2, and the third root is given by equation (5.3.5). These are independent of m. In the isotropic case, l = b, d = l − 1, and e′ = 1. Thus, equations (5.3.4) and (5.3.5) have the roots q1 = k12 − k 2 , q2 = q3 = k22 − k 2



(5.3.6)



Here, k1 = ρω 2 /c11 Let the potentials have the following forms (see Niklasson and Datta 1998): ∞

A=

2

∑∑ m = 0 j =1



∞

B=−

2

∑∑ m = 0 j =1



∞

C=

( f j1m cos mθ + f j2m sin mθ ) J m (q j r ) +  q   j e ikz ( g 1m cos mθ + g 2m sin mθ )Y (q r )  k j m j  j 

( f



2m j

cos mθ ) J m (q3r ) +   e ikz 1m sin mθ − g 2m cos mθ )Y (q r )  3 3 m 3 

1m 3

∑ ( g m=0

( f cos mθ + f sin mθ ) J m (q j r ) +    s e ikz ( g 1m cos mθ + g 2m sin mθ )Y (q r )  j j m j  j  1m j

(5.3.7a)

sin mθ − f

2m 3

(5.3.7b) (5.3.7c)

where



sj =

q 2j + β k 2 − k22 q j kδ



(5.3.7d)

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and q j (j = 1, 2) are obtained from equation (5.3.4), and q3 is found from equation (5.3.5). These potentials satisfy the equations of motion, and the unknowns are the coefficients lm lm lm lm f jlm , g lm j and f 3 , g 3 (l = 1, 2). For a solid cylinder, the coefficients g j and g 3 are set equal to zero. The displacement vector can be calculated as



)

(

)

(

 U r1m cos mθ + U r2m sin mθ e r + Uθ1m sin mθ − Uθ2m cos mθ eθ      1 2 m m m = 0  + U z cos mθ + U z sin mθ e z   ∞

U=

∑ (

)

(5.3.8)

with



 U rlm =   



 Uθlm =   

2

∑( f j =1 2

∑( f

J + g lm j Ym

lm j m

j =1

2

U zlm =

∑i( f

(5.3.9a)

)

 − f 3lm J m′ + g 3lmY ′ q3  e ikz r  

(5.3.9b)

ms j

)

)

(

(

)

ikz J + g lm j Ym q j e

lm j m

j =1



 m lm f 3 J m + g 3lmYm  e ikz r  

)

J ′ + g lm j Ym′ s j q j +

lm j m

(5.3.9c)



Here, the prime (′) denotes a derivative with respect to the argument q jr or q3r, as appropriate. The traction components at the cylindrical surface with radius r is given by



)

(

(

)

 Σ1rrm cos mθ + Σ rr2m sin mθ e r + Σ1rθm sin mθ − Σ r2θm cos mθ eθ    (5.3.10)   1 2 m m m = 0  + Σ rz cos mθ + Σ rz sin mθ e z   ∞

e r .Σ =

∑ (

)

with   Σ lm = c rr 55  

  q j s j  m2 s j   J′  [ f jlm   λ q 2j s j − (δ − 1)q j k − 2ε ′ 2  J m + 2ε ′ r m r    j =1   2

∑

 q j s j   m2 s j  2 Y ′  + Y + g lm 2 ε ′  m j  λ q j s j − (δ − 1)q j k − 2ε ′ r m  r2    + 2ε ′

 m lm { f 3 (q3rJ m′ − J m ) + g 3lm (q3rYm′ − Ym )}  e ikz 2 r 

(5.3.11a)



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  Σ lm rθ = c55  

2

ms j

∑ 2ε ′ r j =1

ε′

2

 f jlm (q j rJ m′ − J m ) + g lm  j (q j rYm′ − Ym )  + 

1 f lm (2q3rJ m′ +((q3r )2 − 2m 2 )J m ) r2 3

(5.3.11b)

{

}

+ g 3lm (2q3rYm′ +((q3r )2 − 2m 2 )Ym )  e ikz

  Σ lm rz = ic55  



2

∑ j =1

 mk lm f J + g 3lmYm  e ikz (5.3.11c) q j (q j − ks j ) f jlm J m′ + g lm j Ym′ + r 3 m  

)

(

)

(

Thus, the mth Fourier coefficients of the displacement–traction vector υlm,N (l = 1, 2; m = 0, …, ∞) in the Nth sublayer of the cylinder can be written in the matrix form

υlm,N = D m ,N F lm ,N

(5.3.12)

= 〈U U U Σ Σ Σ 〉 and F = 〈f f where υ g 2lm ,N g 3lm ,N 〉T . The elements of the 6 × 6 matrix D are given as follows. lm,N

lm , N θ

lm , N r

lm , N z

lm , N rr

lm , N rθ

lm , N T rz m,N

lm , N

lm , N 1

f

lm , N lm , N 2 3

g 1lm ,N

D1mj ,N = − s Nj q Nj J m′ (q Nj r ) m J (q N r ) r m 3 ms Nj J (q N r ) = r m j = −q3N J m′ (q3N r )

D13m ,N = D2mj,N m ,N D23

D3mj,N = iq Nj J m (q j r ) m ,N = 0 D33

 m 2 s Nj  q Nj s Nj N  λ N (q N )2 s N − (δ N − 1)q N k − 2ε N N J m′ q Nj r D4mj,N = c55 ′  J m q j r + 2ε ′  j j j 2 r  r    m m , N = c N 2ε D43 ′ 2 q3N rJ m′ q3N r − J m q3N r  55  r  

( )

(

( )

( ))

(

( ) (( )

= c iq (q − s k ) J m′ (q r ) imk m ,N = c N D63 J (q r ) 55 r m 3 D

N 55

N j

N j



( )

  ms Nj N 2ε D5mj,N = c55  ′ 2 q Nj rJ m′ (q Nj r ) − J m q Nj r  r   2  1 N m ,N = c N ε D53 ′ q3N r + q3N r − 2m 2 J m q3N r 55  ′ 2  2q3 rJ m  r  m ,N 6j



( )

N j

)

( )  

N j

(5.3.13)

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N In the above, j takes the values 1, 2. The elements Dim,k,+3 (i = 1, …, 6; k = 1, 2, 3) are m ,N obtained from Dik (i = 1, …, 6; k = 1, 2, 3) by replacing the Bessel function of the first kind, Jm, by the Bessel function of the second kind, Ym. The continuity of the displacement–traction vector at the interface between the Nth and (N + 1)th sublayers can be written as



υlm,N = υlm,N+1 at r = rN

(5.3.14)

where r = rN is the outer radius of the Nth sublayer. Using equation (5.3.12) in equation (5.3.14), we obtain

F lm ,N +1 = (D m ,N +1 )r−=1r (D m ,N )r =rN F lm ,N N



(5.3.15)

Thus,

(υlm,N+1)+ = (D m ,N +1 )+ (D m ,N +1 )−−1 (υlm,N+1).

(5.3.16)

Here, the subscripts + and – refer to quantities evaluated at the outer and inner boundaries of the (N + 1)th sublayer, respectively. Equation (5.3.16) relates the displacement– traction vector at the outer boundary of the (N + 1)th layer to that at the inner boundary of the same layer by the transfer matrix Pm,N+1 given by

P m ,N +1 = (D m ,N +1 )+ (D m ,N +1 )−−1



(5.3.17)

Equation (5.3.16) then is written as

(υlm,N+1)+ = Pm,N+1( υlm,N+1)−

(5.3.18)

Repeated application of equation (5.3.18) to every sublayer of the composite cylinder having M sublayers gives



{U lm , M }  [P11 ] [P12 ]  {U lm ,0 }   =    {S lm , M }  [P21 ] [P22 ] {S lm ,0 } 

(5.3.19)

P = P m , M .P m , M −1 . ... .P m ,1

(5.3.20)

where

Note that the inner radius of the first sublayer is r = r0 and the outer radius of the Mth sublayer is rM. In writing equation (5.3.19), the displacement–traction vector has been partitioned into two 3 × 1 matrices: the displacement vector {U lm,N} and the traction vector {Slm,N}. If the composite cylinder is hollow with inner and outer surfaces being traction free, then equation (5.3.19) gives



{U lm , M }  [P11 ] [P12 ]  {U lm ,0 }    =     [P21 ] [P22 ] {0} {0}

(5.3.21)

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where the 6 × 6 matrix P has been partitioned into four 3 × 3 submatrices. For nontrivial solution, the determinant of [P21] must vanish, i.e., | P21 | = 0





(5.3.22)

Equation (5.3.22) is the exact dispersion equation for free harmonic waves propagating along the axis of the cylinder. The solution of this equation for a circumferential wave number, m, provides the relationship between the frequency w and the axial wave number k for the different branches of guided modes. For a given w, there are a finite number of real roots k and an infinite number of imaginary or complex roots. The real roots k describe the propagating modes. When the composite cylinder has a solid core, equation (5.3.22) has to be modified. In this case, since the displacement must be finite at r = 0, the coefficients g lm (j = 1, 2, j 3) must vanish. So, the matrix Dm,1 reduces to a 6 × 3 matrix having elements given by equation (5.3.13), and the column matrix Flm,1 becomes F lm ,1 = f1lm f 2lm f 3lm



T

(5.3.23)



Equation (5.3.12) can be modified to the form {U lm ,1 }  [D11m ,1 ]  F lm,1 =   lm , 1 m , 1  {S }  [D21 ]



Equation (5.3.24) can be used to write {S

(5.3.24)

} in terms of {U lm,1} as

lm,1

{S lm ,1 } = [K ]{U lm ,1 }



(5.3.25)

where the stiffness matrix [K] is given by m ,1   D m ,1  [K ] =  D21   11 



−1



(5.3.26)

Thus, equation (5.3.19) takes the form



{U lm , M }  [P11 ] + [P12 ][K ]   {U lm ,1 } =   lm , M }   P K { S [ P ] + [ ][ ]   21  22 

(5.3.27)

Now, imposing the traction-free boundary condition at r = rM, we obtain the dispersion equation

|[P21 ] + [P22 ][K ]| = 0



(5.3.28)

Equations (5.3.22) and (5.3.28) are the exact dispersion equations for simple harmonic guided-wave propagation in layered transversely isotropic hollow and solid cylinders, respectively. In the following discussion, approximate equations are presented. These equations are derived using the generalizations of the stiffness methods for plates (presented in Chapter 4) to composite cylinders. Note that the exact equations presented above apply

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only to transversely isotropic materials having the symmetry axis parallel to the cylinder axis. On the other hand, the stiffness methods can be used for cylinders having arbitrarily anisotropic properties.

5.4 Stiffness Method I In this method, the equations governing the dynamic behavior of composite cylinders are approximated by dividing the cylinder into a number of coaxial cylinders (sublayers) and representing the displacement distribution through the thickness of each sublayer by polynomial interpolation functions in the radial coordinate. These functions are chosen to satisfy the displacement and traction continuity at the interfaces between the adjacent sublayers. By applying Hamilton’s principle, the dispersion equation is obtained as a standard algebraic eigenvalue problem. As in the previous section, time-harmonic elastic wave propagation in an infinite circular cylinder composed of perfectly bonded laminae with possibly distinct mechanical properties, as shown in Fig. 5.1, is considered. It is assumed that the cylinder is hollow,* having the inner and outer lateral surfaces traction free. Let the kth sublayer have thickness hk with inner and outer radii rk and rk+1, respectively. The stress–strain relation for the material occupying the sublayer is given by {σ } = [C]{e}



(5.4.1)

where [C] is the matrix of elastic stiffness constants and

{σ } = 〈σ rr σ θθ σ zz σ θz σ zr σ rθ 〉T



{e} = 〈err eθθ ezz γ θz γ zr γ rθ 〉T



(5.4.2a)



(5.4.2b)

Now, the displacement components (ur, uq , uz) are assumed in the forms

ur (r ,θ , z , t ) = u�(r , z , t )e imθ

(5.4.3a)



uθ (r ,θ , z , t ) = υ� (r , z , t )e imθ

(5.4.3b)



uz (r ,θ , z , t ) = w� (r , z , t )e imθ

(5.4.3c)

The strain–displacement relation is given by equations (5.2.3) and (5.2.4). The displacement components at a point in the kth sublayer are approximated as (suppressing the factor eimq) {U } = [N1 (r )]{q ′} + [N 2 (r )]{q}



(5.4.4)



where {U } = 〈u� υ� w� 〉T {q} = 〈u� k σ� k υ� k τ�k w� k χ� k u� k +1 σ� k +1 υ� k +1 τ�k +1 w� k +1 χ� k +1 〉T



(5.4.5)

This condition can be relaxed to include a solid cylinder.

*

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The nonzero elements of 3 × 12 matrices [N1] and [N2] are N1 (1, 3) =

f 2 P3 f P f P f P , N1 (1, 5) = 2 2 , N1 (2, 3) = 2 6 , N1 (2, 5) = 2 5 ∆ ∆ ∆ ∆

f P f P N1 (3,1) = − f 2 , N1 (3, 3) = 2 9 , N1 (3, 5) = 2 8 ∆ ∆



N 2 (1,1) = f1 +



(5.4.6a)

f 2 P1 f ∆ f P , N 2 (1, 2) = 2 11 , N 2 (1, 3) = im 2 1 ∆rk +1 ∆rk ∆

N 2 (1, 4 ) =

f 2 ∆13 f P f ∆ , N 2 (1, 5) = im 2 3 , N 2 (1, 6) = 2 12 ∆ ∆rk ∆

N 2 (2,1) =

f ∆ f 2 P4 f f f P (5.4.6b) − im 2 , N 2 (2, 2) = 2 13 , N 2 (2, 3) = f1 + 2 + im 2 4 ∆rk rk rk ∆ ∆rk

N 2 ( 2, 4 ) =

f 2 ∆ 33 f P f ∆ , N 2 (2, 5) = im 2 6 , N 2 (2, 6) = 2 23 ∆ ∆rk ∆

N 2 (3,1) =

f 2 P7 f P f ∆ , N 2 (3, 2) = 2 12 , N 2 (3, 3) = im 2 7 ∆rk ∆rk ∆

N 2 (3, 4 ) =

f 2 ∆ 23 f P f ∆ , N 2 (3, 5) = f1 + im 2 9 , N 2 (3, 6) = 2 22 ∆ ∆rk ∆



The quantities appearing in the above equations are given by C1 j C15 C16 Pl = − C5 j C55 C56

(5.4.6c)

for l = 1, 2, 3; j = l + 1

C6 j C65 C66



C11 C1 j C15 Pl = C51 C5 j C55

C61 C6 j C65

for

(5.4.6d)

l = 4 , 5, 6; j = l − 2

C11 C1 j C16 Pl = − C51 C5 j C56

C61 C6 j C66

(5.4.6e)

for l = 7,…,10; j = l − 5

where ∆ = P10 and ∆pq is the cofactor of the element Cpq of ∆. Functions fn (n = 1, 2, 3, 4) are cubic polynomials given below:



1 1 f1 = (2 − 3η + η 3 ), f 2 = (2 + 3η − η 3 ) 4 4 h hk 3 2 f 3 = (1 − η − η + η ), f 4 = k (−1 − η + η 2 + η 3 ) 8 8

(5.4.7)

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where

η=

1 (2r − rk +1 − rk ) hk

hk = rk +1 − rk



The prime sign (′) in equation 5.4.4 denotes derivative with respect to z. The generalized coordinates u� k , υ� k , w� k , σ� k , τ�k , and χ� k are the values of u� , υ� , w� , σ� rr , σ� rθ , and σ� rz , respectively, at the kth interface. Subscript k + 1 denotes the same quantities at the (k + 1)th interface. These nodal values of the displacement and traction components are functions of z and t. The equations governing the generalized coordinates u� k , υ� k , w� k , σ� k , τ�k , and χ� k (k = 1, …, N) are obtained by using Hamilton’s principle. The Lagrangian, Lk, for the kth sublayer is calculated as 1 Lk = 2



 rk +1  � }T {U� } − {e }T [C]{e})rdr  dzdt ( ρ { U   k z   rk 

∫∫ ∫ t

(5.4.8)

The overbar and overdot denote complex conjugate and time differentiation, respectively. Using equation (5.4.4) in the strain–displacement relation in equation (5.2.3) and in turn in equation (5.4.8), then summing the contributions from all the sublayers and setting the first variation to zero, leads to the following equation for the assembled generalized coordinate vector {Q}. iv �� �� �� [K1 ]{Q } + [K 2 ]{Q ′′′} + [E1 ]{Q ′′} + [E2 ]{Q ′} + [E3 ]{Q} − [C1 ]{Q ′′} − [C2 ]{Q ′} + [ M ]{Q} = 0 (5.4.9) Equation (5.4.9) is a linear fourth-order homogeneous partial differential equation in z and t. Matrices [K1], [K2], [E1]–[E3], [C1], [C2], and [M] are defined as follows: H

H

∫

∫

[K1 ] = [d]T [C][d]rdr , [K 2 ] = ([d]T [C][b] − [b ]T [C][d])rdr 0 H

0

∫

[E1 ] = ([d]T [C][a] − [b ]T [C][b] + [a ]T [C][d])rdr 0 H

(5.4.10)

∫

[E2 ] = ([a ]T [C][b] − [b ]T [C][a])rdr 0 H

∫

[E3 ] = [a ]T [C][a]rdr 0 H

[C1 ] =

∫ ρ[N ] [N ]rdr , 1

T

1

0 H

[M ] =

∫ ρ[N ][N ]rdr 2

H

[C2 ] =

∫ ρ([N ] [N ] − [N ] [N ])rdr 1

T

2

2

T

1

0

2

0



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The nonzero elements of the 6 × 12 matrix [a] are a(1,1) = f1,r + a(1, 4 ) = a(2,1) =



f 2 ,r ∆13 ∆

, a(1, 5) = im

f 2 ,r ∆11 ∆

f 2 ,r P3 ∆rk

, a(1, 3) = im

, a(1, 6) =

f 2 ,r P1 ∆rk

f 2 ,r ∆12 ∆

f ∆ f1 m f 2 f 2 P1 f P f ∆ + im 2 4 , a(2, 2) = 2 11 + im 2 13 + + r rk r ∆rk r ∆rk r ∆r ∆r 2

m 2 f 2 P6 im  f ∆ f ∆ f f P +  f1 + 2 + 2 1  , a(2, 4) = 2 13 + im 2 33 ∆rk r r  rk ∆rk  ∆r ∆r

a(2, 5) = −

f P f ∆ m 2 f 2 P6 f ∆ + im 2 3 , a(2, 6) = 2 12 + im 2 23 ∆rk r ∆r ∆r ∆rk r

a(4,1) = im

f 2 P7 m 2 f 2 P7 f ∆ , a(4, 2) = im 2 12 , a(4, 3) = − ∆rk r ∆rk r ∆r

a(4 , 4 ) = im

f 2 ∆ 23 m 2 f 2 P9 f f ∆ , a(4, 5) = − + im 1 , a(4 , 6) = im 2 22 r ∆r ∆r ∆rk r

a(5, 4 ) =

f 2 ,r P7 ∆rk

, a(5, 2) =

f 2 ,r ∆ 23 ∆

f 2 ,r ∆12 ∆

, a(5, 3) = im

, a(5, 5) = f1,r + im

f 2 ,r P9 ∆rk

f 2 ,r P7 ∆rk

, a(5, 6) =

f 2 ,r ∆ 22

a(6,1) =

f f  f  f P f P4  f 2,r − 2  + im  1 + 2 1 + 2 − 2,r  ∆rk  r  r ∆rk r rk r rk 

a(6, 2) =

f  f ∆ ∆13  f 2,r − 2  + im 2 11  ∆  r ∆r

a(6, 3) = f1,r + a(6, 4) =

a(6, 6) =

∆

f 2,r f1 f 2 m 2 f 2 P1 f  P  − − − + im 4  f 2,r − 2  rk r rk r ∆rk r ∆rk r  r

(5.4.11a)

f  f ∆ ∆ 33  f 2,r − 2  + im 2 13  ∆  r ∆r

a(6, 5) = −



∆rk

, a(1, 2) =

a(2, 3) = −

a(5,1) =



f 2 ,r P1

m 2 f 2 P3 f  P  + im 6  f 2,r − 2  ∆rk r r ∆rk 

f  f ∆ ∆ 33  f 2,r − 2  + im 2 12  r ∆  ∆r



where fn,r represents ∂fn/∂r.

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The nonzero elements of the 6 × 12 matrix [b] are b(1, 3) =



f 2 ,r P3 ∆

, b(1, 5) =

f 2 ,r P2 ∆

, b(2, 3) =

f 2 P3 f P + im 2 6 ∆r ∆r

b(2, 5) =

imf 2 P7 f P f P f ∆ f 2 P2 + im 2 5 , b(3,1) = 2 7 , b(3, 2) = 2 12 , b(3, 3) = ∆rk ∆ ∆rk ∆r ∆r

b(3, 4 ) =

f 2 ∆ 23 f P f ∆ , b(3, 5) = f1 + im 2 9 , b(3, 6) = 2 22 ∆ ∆rk ∆

b(4,1) =

 1 1 f ∆ f f P P  f 2 P4 − imf 2  +  , b(4, 2) = 2 13 , b(4, 3) = f1 + 2 + im 2  4 + 9  ∆rk ∆ rk ∆  rk r  rk r 

b(4, 4) =

f 2 ∆ 33 f ∆ f P P  , b(4 , 5) = im 2  6 + 8  , b(4 , 6) = 2 23 r ∆ ∆  rk ∆

b(5,1) = f1 + f 2 ,r +

f P f ∆ f 2 P1 f P , b(5, 2) = 2 11 , b(5, 3) = 2 ,r 9 + im 2 1 ∆ ∆ ∆rk ∆rk

b(5, 4) =

f P f P f 2 ∆13 f ∆ , b(5, 5) = 2 ,r 8 + im 2 3 , b(5, 6) = 2 12 ∆ ∆rk ∆ ∆

b(6, 3) =

P6 ∆

 f 2 P3 P5  f2  f2  f 2 P2  f 2 ,r − r  + im ∆r , b(6, 5) = ∆  f 2 ,r − r  + im ∆r



(5.4.11b)

The nonzero elements of the 6 × 12 matrix [d] are d(3,1) = − f 2 , d(3, 3) =



f 2 P9 f P , d(3, 5) = 2 8 ∆ ∆

f P f P f P f P d(4 , 3) = 2 6 , d(4 , 5) = 2 5 , d(5, 3) = 2 3 , d(5, 5) = 2 2 ∆ ∆ ∆ ∆

(5.4.11c)

The remaining six columns of matrices [N1], [N2], [a], [b], and [d] are obtained from the first six columns by replacing f1 by f3, f2 by f4, and rk by rk+1. Note that the matrices [C1] and [K1] are real and symmetric; [M], [E1], and [E3] are Hermitian; and [C2], [K2], and [E2] are skew-Hermitian. The solution to equation (5.4.9) can be assumed to be in the form {Q(z , t )} = {Q0 }e −iωt −γ z



(5.4.12)



where {Q 0} is the amplitude vector, w is the circular frequency, and g is the complex wave number. Substitution of equation (5.4.12) in equation (5.3.9) results in a system of linear algebraic equation for the amplitude vector as

(γ 4 [K1 ] − γ 3[K 2 ] + γ 2[K 3 ] − γ [K 4 ] + [K 5 ]){Q0 } = 0



(5.4.13)

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The 6(N + 1) × 6(N + 1) matrices [K3], [K4], and [K5] are given by

[K 3 ] = [E1 ] + ω 2[C1 ] (5.4.14)

[K 4 ] = [E2 ] + ω 2[C2 ] [K 5 ] = [E3 ] − ω 2[M ]



For a nontrivial solution {Q 0} of equation (5.4.13), the determinant of the coefficient matrix must be zero. This results in an eigenvalue problem for the determination of the wave number g for a given frequency w. There are complex eigenvalues of the form

γ = ±γ R ∓ iγ I , (γ R , γ I ≥ 0)

(5.4.15)

For waves propagating in the positive z-direction, g must be –ig I (g R = 0). Alternatively, for a given g = ∓ ig I, equation (5.4.13) can be rewritten as

where

([K γ ] − ω 2[ Mγ ]){Q0 } = 0

(5.4.16)

[K γ ] = γ 4 [K1 ] − γ 3[K 2 ] + γ 2[E1 ] − γ [E2 ] + [E3 ]

(5.4.17)

[ Mγ ] = [ M ] + γ [C2 ] − [C1 ]



This defines an eigenvalue problem for the determination of the frequency w of the propagating modes.

5.5 Stiffness Method II As in the case of stiffness method I, the cylinder is subdivided into several sublayers, and the displacement distribution within a sublayer is approximated by polynomial interpolation functions in the radial coordinate. In the displacement-based approximation, equation (5.4.4) is replaced by {U } = [N (r )]{q}



(5.5.1)



where {U } = 〈u� υ� w� 〉T

{q} = u� kb υ� kb w� kb u� km υ� km w� km u� kf υ� kf w� kf

(5.5.2)

T



f In equations (5.5.1) and (5.5.2), the generalized coordinates u kb , u m k , and u k are the displacements at the back (inner), middle, and front (outer) nodal surfaces of the kth

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sublayer. The interpolation polynomials ni (i = 1, 2, 3) are quadratic functions of the radial variable defined as n1 = 1 − 3η + 2η 2 , n2 = 4η − 4η 2 , n3 = −η + 2η 2



(5.5.3)



where h = (r − rk)/hk, hk being the thickness of the sublayer, and rk being the radial coordinate of the inner surface of the kth sublayer. The Lagrangian for the kth sublayer is given by (see equation (5.4.8)) 1 Lk = 2



 rk +1   �  ( ρ{U }T {U� } − {e }T [C]{e})rdr  dzdt z   rk

∫∫ ∫ t

(5.5.4)

Using equation (5.5.1) in the strain–displacement relation in equation (5.2.3) and in turn in equation (5.5.4), then summing the contributions from all the sublayers and setting the first variation to zero, the following equation for the assembled generalized coordinate vector {Q} is obtained: ��} = −{P} [K1 ]{Q ′′} + [K 2 ]{Q ′} − [K 3 ]{Q} − [ M ]{Q



(5.5.5)



The matrices [M], [K1], [K2], and [K3] are defined below. {P} is the nodal force vector composed of forces acting at the boundary nodes. H

[M ] =

∫ ρ[N ] [N ]rdr , T

0

H

∫

H

∫

[K1 ] = [b]T [C][b]rdr 0

(5.5.6)

H

∫

[K 2 ] = [[b]T [C][a] − [a ]T [C][b]]rdr , [K 3 ] = [a ]T [C][a]rdr

0

0



The nonzero elements of the 6 × 9 matrix [a] are as follows: dn dn1 dn , a(1, 4) = 2 , a(1, 7) = 3 dr dr dr n1 n1 n a(2,1) = , a(2, 2) = im , a(2, 4 ) = 2 r r r n n n a(2, 5) = im 2 , a(2, 7 ) = 3 , a(2, 8) = im 3 r r r n n2 n1 a(4, 3) = im , a(4, 6) = im , a(4, 9) = im 3 r r r dn3 dn1 dn2 , a(5, 6) = , a(5, 9) = a(5, 3) = dr dr dr n n dn n a(6,1) = im 1 , a(6, 2) = 1 − 1 , a(6, 4 ) = im 2 r r r dr dn3 n3 n3 dn2 n2 − a(6, 5) = − , a(6, 7) = im , a(6, 8) = r dr r dr r a(1,1) =



(5.5.7a)

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The nonzero elements of [b] are b(3, 3) = b(4 , 2) = b(5,1) = n1 b(3, 6) = b(4, 5) = b(5, 4) = n2 b(3, 9) = b(4, 8) = b(5, 7) = n3



(5.5.7b)

It is noted that [K1] and [M] are real and symmetric, [K2] is skew-Hermitian, and [K3] is Hermitian. In the absence of the nodal forces {P}, we obtain the homogeneous equation ��} = 0 [K1 ]{Q ′′} + [K 2 ]{Q ′} − [K 3 ]{Q} − [ M ]{Q



(5.5.8)



The solution to equation (5.5.8) can be assumed to be in the form {Q} = {Q 0 }e −iωt −γ z



(5.5.9)



Then, equation (5.5.8) leads to the system of homogeneous linear algebraic equations (γ 2[K1 ] − γ [K 2 ] − [K 3 ] + ω 2[ M ]){Q0 } = 0



(5.5.10)



For a nontrivial solution for {Q 0}, the determinant of the 3(N + 1) × 3(N + 1) coefficient matrix must vanish. For fixed g , this gives the eigenvalue problem for determining the frequency w. On the other hand, for a given frequency w, the determinantal equation can be solved for the complex frequency g .

5.6 Numerical Results—Circular Cylinder Equations (5.3.22) and (5.3.28) apply to a cylinder having transversely isotropic symmetry about its central axis, whereas equations (5.4.13), (5.4.16), and (5.5.10) can be used to obtain the dispersion behavior of cylinders having arbitrary anisotropy. In the following discussion, some numerical results are presented for particular cases of composite cylinders. The first case considered is axisymmetric (m = 0) guided-wave propagation in a solid transversely isotropic cylinder made of graphite. This problem was considered by Xu and Datta (1991), who used the exact equation (5.4.16) as well as equation (5.5.10). Material properties and the radial thickness of the graphite cylinder are given in Table 5.1. The TABLE 5.1  Material Properties and Outer Radius of the Composite Cylinder of Graphite Fiber in Magnesium with and without a 0.1-mm Interface Elastic Stiffness (GPa)

Outer Radius (mm)

Density, r (g/cm3)

c11

c33

c12

c13

c55

1.00 (graphite) 11.00 (magnesium) 1.1 (interface layer)

2.27 1.74 1.74

20.02 109.4 10.94

234.77 109.4 10.94

9.98 57.0 5.7

6.45 57.0 5.7

24.00 26.2 2.62

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Phase Velocity (mm/µ second)

V33 = 10.17 15

Stiffness method II Exact V33 = 10.17

10

5

0

V55 = 3.25 Vrf = 2.87 0

1

2

3

4

Frequency (MHz)

FIGURE 5.2  Phase velocity (mm/ms) vs. frequency (MHz) for axisymmetric guided-wave propagation in a graphite cylinder. (Reprinted with permission from Xu and Datta 1991, Fig. 4.)

cylinder was subdivided into 10 sublayers of equal thickness for the stiffness method. It is seen from Fig. 5.2 that there is very good agreement between the exact and approximate results except when the frequency is high (and when the phase speed changes sharply.) Since the longitudinal phase velocity along the fiber axis is much higher than in other directions, it is seen that the first branch is fairly flat at low frequencies at the phase speed of 10.17 mm/ms, then drops rapidly and approaches the Rayleigh wave speed of 2.87 mm/ms. It is interesting to note that the higher branches rapidly reach a plateau at the longitudinal wave speed for a wide range of frequency and then approach asymptotically the shear wave speed (v55 = c55 /ρ ) of 3.25 mm/ms. Xu and Datta (1991) considered also axisymmetric wave propagation in a composite cylinder having a core of graphite fiber surrounded by an isotropic cylinder (magnesium or epoxy) of outer radius 11 mm. They also considered the effects of a thin (0.1 mm) interface layer between the fiber and the surrounding matrix having different material (isotropic) properties. Properties of magnesium are also listed in Table 5.1. They used both the stiffness method II and a hybrid method combining the stiffness method II for the inner core and the exact solution for the outer magnesium layer to study the dispersion behavior of the composite cylinder. Figure 5.3 shows the dispersion behavior of the composite cylinder (without the interface layer). These results are obtained by using the stiffness method II. The composite cylinder is divided into a total of 75 sublayers, 10 for the core and 65 for the outer layer. The following critical values of wave speeds play a significant role in the dispersion behavior. The longitudinal wave in the fiber in the axial direction (10.17 mm/ms), shear wave speed v55 in the fiber (3.25 mm/ms), compressional wave speed v p in magnesium (7.93 mm/ms), shear wave speed vs in magnesium (3.88 mm/ms), longitudinal wave speed vl in the magnesium rod (6.34 mm/ms), and the Rayleigh wave speed v r of 3.63 mm/ms in magnesium. Examination of Fig. 5.3(a) shows that the lowest branch starts at the longitudinal wave speed in magnesium (since the radius of the core is much smaller than the thickness of the magnesium layer), pinches the second branch at about the frequency of 3.2 MHz (at the speed of 3.62 mm/ms), and then approaches asymptotically v55 (because this is lower than the Rayleigh wave speed in magnesium). Figure 5.3(b) shows the dispersion curves

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Phase Velocity (mm/µsec)

15 VI = 6.34 V33 = 10.17

10

Vp = 7.95 Vs = 3.68 Vf = 3.63 V55 = 3.25

5

0

0

2

Frequency (MHz)

6

4

(a) 3.650 3.640 3.630 3.620 3.610 3.600 3.00

3.10

3.20

3.30

3.40

3.50

Frequency (MHz) (b)

FIGURE 5.3  Dispersion curves for the composite graphite fiber (core) in magnesium cylinder (outer): (a) axisymmetric radial–axial mode and (b) an enlarged view of the first two branches near the pinch frequency. (Reprinted with permission from Xu and Datta 1991, Fig. 4.)

for the first and second branches near the pinch frequency. It is noted that this pinching occurs between the second and third branches at about a frequency of 2.2 MHz (shear wave speed in magnesium, 3.88 mm/ms) and also approaches v r (3.63 mm/ms) at higher frequencies. Higher branches are seen to reach plateaus at the compressional and shear wave speeds in magnesium as well as at v33 for much higher frequencies. Dispersion curves for the first ten branches when the circumferential wave number m = 1 are shown in Fig. 5.4. It may be noted that the lowest branch corresponds to the flexural wave. The speed of this wave rises sharply to a plateau at the Rayleigh wave speed v r and then drops to the value of v55. Note that the phenomena of pinch and terrace are observed here also. These phenomena of pinch and terrace are typical of layered structures (Whittier and Jones 1967; Crampin 1977; Ju and Datta 1992a; Niklasson et al. 2000a, 2000b). As mentioned before, interfaces play a significant role in the failure mechanisms of composite structures. The effect of a soft interface layer between the graphite fiber and

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Phase Velocity (mm/µs)

15

10

5

0

0

1

2 3 Frequency (MHz)

4

5

FIGURE 5.4  Dispersion curves for graphite fiber in magnesium cylinder when the circumferential wave number m = 1.

the magnesium cylinder is examined in Figs. 5.5 and 5.6 for m = 0 and 1. Properties of the interface layer are given in Table 5.1, and it is assumed that the thickness of the layer is 0.1 mm. In all these cases, the outer radius of the magnesium cylinder is taken to be 11 mm. The noticeable features of these curves are the lower frequencies at which the pinching between branches occurs. Also, it is found that the speeds of the lowest branches drop as the frequency increases toward the shear wave speed in the interface layer. Both stiffness methods I and II are applied to obtain the dispersion curves for wave propagation in a cylindrical shell composed of layers of angle-ply laminae. These problems are not amenable to exact solutions except when the ply angles are 0° or 90° to the

Phase Velocity (mm/µs)

15

10

5

0

0

1

2 3 Frequency (MHz)

4

5

FIGURE 5.5  Dispersion curves for graphite fiber in magnesium cylinder with a 0.1-mm interface layer for m = 0.

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Phase Velocity (mm/µs)

15

10

5

0

1

0

2 3 Frequency (MHz)

4

5

FIGURE 5.6  Dispersion curves for graphite fiber in magnesium cylinder with a 0.1-mm interface layer for m = 1.

axis of the cylinder. It is assumed that each lamina is a graphite-fiber-reinforced epoxy composite. Engineering elastic constants of each lamina relative to its symmetry axes (Huang and Dong 1984) are: EL = 139.274 GPa , ET = 15.160 GPa , GLT = GTT = 5.861 GPa, ν LT = νTT = 0.21 (5.6.1) The density is taken to be 1. Here, L refers to the fiber direction and T is transverse to it. The transformed elastic constants referred to the x-, y-, and z-axes for ±30° orientation of the L-direction to the z-axis are given by the matrix (in GPa)

[C]±30�



15.986 3.608 3.924 ± 0.273 … …     23.730 27.589 ± 13.492 … …     86.233 ± 40.630 … …   =  symmetric 29.368 … …      5.861 …     5.861 

(5.6.2)

Figures 5.7 and 5.8 show the variation of the normalized frequency Ω (= wH/vL) with ξ (= ig H/p) predicted by the two methods for a cylinder having (+30°/−30°/+30°/−30°) ply lay-up when m = 1 and m = 3, respectively. Here, vL = EL /ρ and H/R = 0.667, R being the mean radius of the cylinder. It is seen that, as the frequency increases, the results predicted by method II become less accurate. However, for lower modes, the differences are negligible. The phenomenon of pinching is seen to occur for higher branches.

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5

Ω

4

3

2

1

Stiffness method II Stiffness method I 0 0.0

0.5

1.0

1.5

2.0 ξ

2.5

3.0

3.5

4.0

FIGURE  5.7  Dispersion curves for a four-layer angle-ply (+30°/−30°/+30°/−30°) composite cylindrical shell for m = 1.

Figures 5.9 and 5.10 show the results for symmetric lay-ups ([+30°/−30°]s). Although the results look similar, there are differences in details.

5.7 Guided Waves in a Cylinder of Arbitrary Cross Section So far, the discussion in this chapter has been confined to a circular cylindrical structure. Guided waves in cylinders of noncircular cross sections are of interest in many structural applications, but have received limited attention. The semi-analytic finite

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5

Ω

4

3

2

1

Stiffness method II 0 0.0

Stiffness method I 0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

ξ

FIGURE  5.8  Dispersion curves for a four-layer angle-ply (+30°/−30°/+30°/−30°) composite cylindrical shell for m = 3.

element (SAFE) method, as outlined in Chapter 4 for plates and for circular cylinders in Sections 5.4 and 5.5, has been generalized to obtain the dynamic response of cylinders of arbitrary cross section. In this section, a formulation is given for analyzing dispersion of free guided waves in a plate of rectangular cross section. The formulation is general enough for application to plates of arbitrary cross section and will be discussed in the following section. Time-harmonic waves in a multilayered anisotropic plate of rectangular cross section are considered. It is assumed that the plate consists of N parallel and homogeneous

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5

Ω

4

3

2

1 Stiffness method II Stiffness method I 0 0.0

0.5

1.0

1.5 γ

2.0

2.5

3.0

FIGURE 5.9  Dispersion curves for a four-layer symmetric angle-ply (+30°/−30°)s composite cylinder for m = 1. (Reprinted with permission from Rattanawangcharoen 1993, Fig. 2.9(a).)

orthotropic layers, which are perfectly bonded together along the entire length of the plate. Global rectangular coordinates (X,Y,Z) with origin on the central axis of the plate are chosen. The X-axis coincides with the central axis and the Y- and Z-axes are parallel, respectively, to the width and thickness directions. Two different finite-element discretizations of the cross section are used for the purpose of comparisons and accuracy. In one, six-noded elements are used such that at each node displacement and traction continuities are maintained in the thickness direction (as in the stiffness method I presented in Section 5.4), while only displacement continuity is imposed in the width direction. Since one of the objectives is to

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5

Ω

4

3

2

1 Stiffness method II 0 0.0

Stiffness method I 0.5

1.0

1.5 γ

2.0

2.5

3.0

FIGURE  5.10  Dispersion curves for a four-layer symmetric angle-ply (+30°/−30°)s composite cylinder for m = 3. (Reprinted with permission from Rattanawangcharoen 1993, Fig. 2.9(b).)

study the effect of layering and different aspect ratios of the plate on guided-wave dispersion, maintaining traction continuity in the thickness direction would be desirable for analyzing interface stresses. In the other scheme, nine-noded quadratic elements that satisfy displacement continuity at the nodes in both the thickness and width directions are used. The relative merits of the two schemes will be pointed out in the discussions. For each element, local coordinates (x,y,z) are taken to be parallel, respectively, to the global (X,Y,Z) coordinates. Typical six-node and nine-node elements with thickness 2h (in the z-direction) and width 2b (in the y-direction) are shown in Fig. 5.11.

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Elastic Waves in Composite Media and Structures z 6

y h h

x

5 3

4 2

1

b

b z

3

y h h

x

7 9 5

6 2 b

4 8 1 b

FIGURE 5.11  Geometry of the six- and nine-node elements. (Reprinted with permission from Mukdadi et al. 2002b, Fig. 1.)

5.7.1 Six-Node Element Formulation The displacement u(x,y,z,t) at a point within an element is written in the product form

u( x , y , z , t ) = N( y , z )q� ( x , t )



(5.7.1)

where N is the matrix of interpolation functions and q� is the displacement–traction vector. The stress–strain relation is taken to be

σ = De



(5.7.2)

Here, s and e are the stress and strain column matrices, and D is the matrix of elastic constants. They are



σ = 〈σ xx σ yy σ zz σ xy σ zx σ yz 〉T e = 〈e xx e yy ezz γ xy γ zx γ yz 〉T  D11    D=  symmetricc   

D12 D22

D13 D23 D33

(5.7.3a)



(5.7.3b)



D14 D24 D34

D44

. . . . D55

.   .  .   .  D56   D66 

(5.7.3c)



The global displacement components U, V, and W along the global X-, Y-, and Z-directions within an element can be written as

U = 〈U V W 〉T = B1 q� ,x + B 2 q�



(5.7.4)

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The matrices Bj (j = 1, 2) and q� are given by B j = [N j1 N j 2 N j 3 N j 4 N j 5 N j 6 ], (j = 1, 2) (5.7.5)

q� = 〈q1 q2 q3 q4 q5 q6 〉T qi = 〈U i Vi Wi τ i χi σ i 〉T , (i = 1, … , 6)





where τi = σ yzi , χi = σ xzi , and s i = σ zzi. The nodal values of the displacement–traction vectors are functions of x and t. It will be assumed that the time dependence is of the form e−iwt, w being the circular frequency. The factor e−iwt is dropped in the sequel. The matrices Nji are given as follows:    . . − g i f p . . .   N1i =  . . . . . .  , i = 1, …, 6    − g f D13 − g f D34 . . . .  i p i p   D33 D33   D D . . g i f p 66 − g i f p 56 .   g i fq ∆ ∆     D56 D55 N 2i =  . g i fq hi f p − g i f p gi f p .  , i = 1, …, 6 ∆ ∆   gi f p   D34 D23 . . hi f p D hi f p D g i f q D33  33 33 



(5.7.6)



Here, p = 3, q = 1 for i = 1, 2, 3; and p = 4, q = 2 for i = 4, 5, 6. The interpolation functions are given below: 1 f1 = (2 − 3ξ + ξ 3 ), 4

1 f 2 = (2 + 3ξ − ξ 3 ), 4

h f 3 = (1 − ξ − ξ 2 + ξ 3 ), 4



h f 4 = (−1 − ξ + ξ 2 + ξ 3 ) 4



η η g 1 = g 4 = (η − 1), g 2 = g 5 = 1 − η 2 , g 3 = g 6 = (1 + η ) 2 2



h1 = h4 =

(5.7.7a) (5.7.7b)

1 2 1 (3 g + g 2 − g 3 ), h2 = h5 = ( g 3 − g 1 ), h3 = h6 = ( g 1 − g 2 − 3 g 3 ) (5.7.7c) 2b 1 2b b

The nondimensional coordinates in the above representations are ξ = z/h, h = y/b. Here, ∆ = D55D66 − D562 . The interpolation polynomials fi and gi are such that the displacement–traction vector qi is continuous at the nodes lying at the interface z = zi, and the displacement is continuous at the nodes lying at the interface y = yi. As was shown in Chapter 4 (see also Figs. 5.7

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Elastic Waves in Composite Media and Structures

and 5.8), the enforcement of traction continuity at the interface between two sublayers in the plate gives better results at high frequencies with a smaller number of nodes in the z-direction. Also, this way, the traction at the interfaces between two layers in the plate is determined directly as members of the eigenvector array. Since there are no material discontinuities in the width direction, and when the width is considered much larger than the thickness, it is economical to use quadratic interpolation in y. As discussed before, the differential equations of motion for an element can be obtained by applying Hamilton’s principle. The governing equations for the plate are obtained as

E 5 Q + E 4 Q ′ + E 3Q ′′ + E 2 Q ′′′ + E1Qiv = ω 2 (MQ − C1Q ′′ − C 2 Q ′ )

(5.7.8)



Matrices C1, M, E1, E3, and E5 are symmetric, whereas, the others are skew-symmetric. The assembled matrices are defined below. C1 = ∪ c 1 ,

C 2 = ∪(c 2 − c T2 ),

M = ∪m

E1 = ∪e1 ,

E 2 = ∪(e T2 − e 2 ),

E 3 = ∪(eT3 + e 3 − e 4 ),

E 4 = ∪(e T5 − e 5 ),

E 5 = ∪e 6

(5.7.9)

The matrices c1, c2, e1, …, e6, and m are given by

∫∫ ρB B dA, = ∫∫ d DddA, = ∫∫ b DbdA,

c1 = e1

e4

T 1

1

∫∫ ρB B dA, m = ∫∫ ρB B dA = ∫∫ d DbdA, e = ∫∫ d DadA = ∫∫ b DadA, e = ∫∫ a DadA

c2 =

T

e2

T

e5

T 1

T 2

2

T

3

T

6

2

T

(5.7.10)

T

Matrices d, b, and a are computed from the strain–displacement relations as

l = [l1 l2 l3 l4 l5 l6 ],

l = d , b, and a

where the integrals are over the rectangular cross section of the plate and



 . . − gi f p   . . .   . . .  di =  . . .    − g f D13 − g f D34 . i p  i pD D33 33   . . . 

. . .  . . .  . . .   . . .  , i = 1, 2, … , 6  . . .   . . . 

(5.7.11)

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Wave Propagation in Composite Cylinders

  gi f p   .    − g i f p D13  D33  bi =  .     h f D34  i p D33   D  − g i* f p 13 D33 

   . . . . .    D . . . .  g i f p* 34  D33   , i = 1, 2, …, 6 D56 D55 * gi f p .  g i f q (hi − g i ) f p − g i f p ∆ ∆  g i f p  D hi f p 23 g i (f q -f p* ) . . D33 D33    D − g i* f p 34 . . . .  D33  .

.

gi f p

D66 D − g i f p 56 ∆ ∆

 . . . .   g i* f q hi* f p − g i* f p  .   D h f * D34 hi f p* 23 g i f q* .  i p D D 33 33  ai =   g i* f q . . g i* f p    * g i f p* . .  g i fq   D D hi* f p 34 hi* f p 23 + g i f q* g i* f q + hi f p* − g i f p* D33 D33 

.

.

(5.7.12)

.    D55 * gi f p .  ∆   g i f p*  . D33    , i = 1, 2, … , 6 D − g i* f p 56 .  ∆   D  − g i f p* 56 .  ∆   D g i f p* 55 .  ∆  .

D56 ∆ . D66 ∆ D66 ∆ D56 ∆



(5.7.13)

Here, p = 3, q = 1 for i = 1, 2, 3; and p = 4, q = 2 for i = 4, 5, 6. Further,

f1* =



3 (−1 + ξ 2 ), 4h

f 2* =

1 f = (−1 − 2ξ + 3ξ 2 ), 4 * 3

1 1 g 1* = g 4* =  η −  , 2 b

3 (1 − ξ 2 ) 4h

(5.7.14a)

1 f = (−1 + 2ξ + 3ξ 2 ) 4 * 4

g 2* = g 5* = −

2η , b

1 1 g 3* = g 6* =  η +  b 2

−2h1* = h2* = −2h3* = −2h4* = h5* = −2h6* =

2 b2



(5.7.14b) (5.7.14c)

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Equation (5.7.9) is a fourth-order linear differential equation in the axial coordinate (same as equation (5.4.9)). If it is assumed that Q is of the form Q = Q0 e ikX



(5.7.15)



where Q 0 is the amplitude vector then substituting equation (5.7.15) in equation (5.7.9), the following system of linear homogeneous equations is obtained.

)

(

 k 4 E1 − ik 3E 2 − k 2 E 3 + ikE 4 + E 5 − ω 2 M − ikC 2 + k 2 C1  Q0 = 0  

(5.7.16)

Equation (5.7.16) defines a generalized eigenvalue problem that can be solved for the wave number k (real, imaginary, and complex) for a given frequency w or, for a given k (real), it can be solved for the frequency w.

5.7.2 Nine-Node Quadratic Element Formulation As in equation (5.6.1), the displacement at a point within an element is written in the product form

u( x , y , z , t ) = N( y , z )u e ( x , t )

(5.7.17)



where N(y,z) is the nine-node isoparametric element interpolation function. The strain vector is then expressed in the form e = B1 u e ,x + B 2 u e



(5.7.18)



with





N .  B1 = . .  . .  .  B 2 = .  .

.

.

.

.

.

N

.

.

.

N

.

Ny . .

Nz

T

.  .  . 

(5.7.19a)

T

N y Nz .   . . N z   . . N y 

(5.7.19b)

The subscripts refer to partial derivatives. Now, using Hamilton’s principle as before, the governing equations for the nodal displacements are obtained as, assuming harmonic time dependence e−iωt,

k 1(e ) u e , XX − k 2(e ) u(e ), X − k 3(e ) u(e ) + ω 2 m (e ) u(e ) = 0



(5.7.20)

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Coefficient matrices appearing in the above equation are given by



∫∫ B DB dydz , = ∫∫ B DB dydz ,

k 11(e ) =

T 1

k 22(e )

T 2



1

2

k 12(e ) = ( k 21(e ) )T = m(e ) =

∫∫

∫∫ B DB dydz T 1

2

NT ρ Ndydz

(5.7.21a)

k 1(e ) = k 11(e ) , k 2(e ) = k 21(e ) − k 12(e ) , k 3(e ) =k 22(e )



(5.7.21b)

Next, the global mass and stiffness matrices are assembled in the usual manner to obtain the equation for the plate as

K1 U , XX − K 2 U , X − K 3 U + ω 2 MU = 0



(5.7.22)

Now, assuming ˆ (k , ω )e ikX U( X , ω ) = U



(5.7.23)

the following eigenvalue problem is obtained

ˆ =0  k 2 K1 + ikK 2 + K 3 − ω 2 M  U

(5.7.24)

ˆ for a given real k. These This can be solved for the eigenvalue w and the eigenvector U correspond to the propagating modes. Alternatively, equation (5.7.24) can be solved for the propagating and evanescent modes at a given frequency w. Numerical results are presented in the following section for propagating waves in a rectangular waveguide. These results are compared with those for a plate having infinite width in some particular cases.

5.8 Numerical Results—Cylinder with Rectangular and Trapezoidal Cross Sections Results presented in the following discussion include propagation in a homogeneous isotropic plate of different width-to-thickness ratios. Results for a Ni plate with B/H = 8 are compared with those for an infinite width (B/H ã ∞). It is found that the dispersion characteristics of guided waves in an infinite-width plate differ significantly from those in the high-aspect-ratio plate considered. Properties of Ni are listed in Table 4.4. Figure 5.12 depicts the dispersion curves for an isotropic rectangular waveguide (B/H = 2) obtained by the stiffness methods I and II. The Poisson ratio of the material is 0.3. In this case, advantage is taken of the doubly symmetric geometry. Thus, only one-quarter of the cross section needs be discretized. Results are compared with those presented by Taweel et al. (2000). There are four waveforms: (a) extensional (sym/sym), (b) torsion (antisym/antisym), (c) flexure about the Y-axis (antisym/sym), and (d) flexure about the Z-axis (sym/antisym). Both types of elements were used: six-node elements

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Elastic Waves in Composite Media and Structures 10.0

Extensional

Torsional

Flexural about y-axis

Flexural about z-axis

8.0 6.0

Normalized Frequency

4.0 2.0 0.0 10.0 8.0 6.0 4.0 2.0 0.0 0.0

2.0

4.0

6.0

0.0

2.0

4.0

6.0

Normalized Wavenumber, K

FIGURE 5.12  Comparison of the results obtained by the SAFE method for an isotropic rectangular bar with B/H = 2 with those of Taweel et al. (2000). Results using nine-node elements (----) and those from Taweel et al. (2000) are shown. Those obtained by using the six-node elements fall on the solid lines and are not shown. (Reprinted with permission from Mukdadi et al. 2002b, Fig. 2.)

(5 × 5 with 66 nodes) and nine-node elements (5 × 5 with 121 nodes), as discussed in Sections 5.4 and 5.5. Both give results that agree with one another and with those presented by Taweel et al. (2000). Here, Ω = wH/cs and K = kH. Comparison of the SAFE predictions using nine-node elements with the experimental results reported by Morse (1948) for a brass rod with B/H = 2 is shown in Fig. 5.13. The Poisson ratio of brass is 0.35, and cs = 2 mm/ms. The agreement between the two is found to be excellent. Next, comparison of the model results using six- and nine-node elements for a nickel rod of rectangular cross section with B/H = 8 is presented in Fig. 5.14. In Fig. 5.14, the normalized frequency is Ω {= wH/[2p(1.1)cs]}, and the normalized phase speed is c

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Theoretical Experimental

c (mm/µs)

4

3

2 200

300

400

500

600

700

f (KHz)

FIGURE  5.13  Comparison of the results using nine-node elements with the experiments of Morse (1948) for the first two branches of the extensional mode. (Reprinted with permission from Mukdadi et al. 2002b, Fig. 3.)

(= c/cs). The thickness of the plate is H (= 110 µm). For the six-node elements, one-quarter of the plate was discretized into 5 × 11 (138 nodes) elements, whereas there were 5 × 5 nine-node elements (121 nodes). Both predictions are in excellent agreement. Also shown in this figure are exact results for an infinite-width plate. Comparison of the dispersion behavior of the finite- and infinite-width plates shows several interesting features: (a) dispersion curves for the lowest flexural modes in the two cases are almost identical, indicating that this mode is not sensitive to the width B; (b) the phase speed of the lowest extensional mode in the infinite plate is higher than that in the finite-width plate. Note that as w ã 0, the former tends to the plate velocity, cp (= 1.693 cs), and the latter tends to the rod velocity, c b (= 1.614 cs), for nickel plate; (c) cutoff frequencies for the guided modes in a finite-width plate are lowered more and more as the width increases, and there are more propagating modes at any frequency than in an infinite plate; (d) phase speeds of the lowest flexural and extensional modes tend to the Rayleigh wave velocity as the frequency increases (as in the case of the infinite plate); (e) phase velocities of the low extensional modes decrease rapidly as frequency increases, tend to a plateau at the plate velocity, and then decrease gradually to the Rayleigh wave velocity in nickel; (f) phase velocities of the low flexural modes decrease rapidly with increasing frequency, tending to the Rayleigh wave velocity. (Note that the phase velocity of the second flexural mode drops below that of the lowest mode and then increases to the Rayleigh wave velocity.) A problem of particular interest in the context of guided waves in a rectangular cylinder is the dispersion of acoustic phonon modes in homogeneous and bilayered nanowires. Precise knowledge of acoustic phonon modes in these wires is necessary to model heat-transport properties of the wires (see, for example, Lü and Chu 2006). Acoustic phonon modes in nanowires of rectangular and trapezoidal cross sections were studied

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Elastic Waves in Composite Media and Structures 4.00 Normalized Phase Velocity

3.50 3.00 2.50 2.00 1.50 1.00 Six node elements Nine node elements Infinite plate

0.50 0.00 0.00

0.25

0.50

0.75 1.00 1.25 Normalized Frequency

1.50

1.75

0.25

0.50

0.75 1.00 1.25 Normalized Frequency

1.50

1.75

4.00 Normalized Phase Velocity

3.50 3.00 2.50 2.00 1.50 1.00 0.50 0.00 0.00

FIGURE 5.14  Comparison of the results using six-node elements with those using nine-node elements for the extensional and flexural (flexure about the Y-axis) motion in a nickel plate with B/H = 8. Also shown are the corresponding results for an infinite plate having the same thickness H = 110 µm. (Reprinted with permission from Mukdadi et al. 2002b, Fig. 4.)

by Mukdadi et al. (2002a, 2005) using the SAFE method. Dispersive guided modes in a GaAs wire having width B = 100 nm and height H = 120 nm are shown in Fig. 5.15. Results for a bilayered GaAs–Nb wire having the same H/B ratio are shown in Fig. 5.16. Here, the thickness of the Nb layer is 20 nm and that of the GaAs layer is 100 nm. Properties of GaAs and Nb are given in Table 5.2. Note that GaAs and Nb are cubic. In this example, the Z-axis is aligned with the central axis of the wire, and the X- and Y-axes are aligned with the symmetry axes of the materials and are in the cross-sectional plane. To illustrate the effect of the finite width of the wires, results for an infinite-width plate are also shown in these figures. These figures are for motion symmetric about the Y-axis, which is along the height direction. Thus, they show the longitudinal and flexural (about

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Wave Propagation in Composite Cylinders f (GHz) 6.94

13.87

20.81

Finite width plate (H/B = 1.2)

27.74 6.658

Infinite plate (exact)

1.5

L1

L0

1

3.329

Bx0

0.5

0

4.993

Bx1

0

0.25

c (mm/µs)

Nondimensional Phase Velocity, C /Cs

2

0

1.665

0.5 0.75 Nondimensional Frequency

1

0

FIGURE 5.15  Exact dispersion curves of infinite plate (including SH modes) and the results for a finite-width plate of H/B = 1.2 (120 nm × 100 nm) GaAs nanowire. (Reprinted with permission from Mukdadi et al. 2002a, Fig. 1.)

Nondimensional Phase Velocity, C /Cs

2

Finite width plate (H/B = 1.2) Infinite plate (exact)

1.5

Bx1 L0

1

Bx0

0.5

0

L1

0

0.25

0.5 0.75 Nondimensional Frequency

1

FIGURE 5.16  Exact dispersion curves of infinite plate and the results for a finite-width plate with H/B = 1.2 ([100 + 20] × 100 nm) GaAs–Nb nanowire. (Reprinted with permission from Mukdadi et al. 2002a, Fig. 2.)

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Elastic Waves in Composite Media and Structures TABLE 5.2  Material Properties of GaAs, Nb, and Si Velocity (mm/µs)

ρ

c33 ρ

c44 ρ

c66 ρ

c12 ρ

c13 ρ

4.71 5.29 8.43

4.71 5.29 8.43

3.33 1.81 5.84

3.33 1.81 5.84

3.17 3.83 5.24

3.17 3.83 5.24

c11

GaAs Nb Si

Density, ρ (g/cm3) 5.36 8.57 2.33

GaAs /ρ ) in GaAs, the X-axis) wave dispersion. The variable cs is the shear wave speed ( c66 and the normalized frequency is F = wH/2pcs. Results for the infinite plate also include the SH-wave dispersion. As Fig. 5.15 shows, the SH0 mode in the infinite GaAs plate is nondispersive. However, it is dispersive in the bilayered plate, as seen from Fig. 5.16. It is interesting to note that the lowest flexural and longitudinal modes in the homogeneous and bilayered plates of nearly square cross sections show almost the same dispersive behavior as in the corresponding infinite plates. Presence of the Nb layer lowers the cutoff frequencies, as seen in Fig. 5.16. Also, because the shear wave speed in Nb is lower than that in GaAs, the phase speed of the lowest flexural mode is seen to approach the Nb shear wave speed as the frequency increases. Effect of increasing the width of the GaAs–Nb wire is shown in Fig. 5.17(a). Here, the height-to-width ratio is 0.5. This changes the dispersion behavior quite dramatically. The cutoff frequencies become much

Nondimensional Phase Velocity, C/Cs

0.82 GaAs - Nb L1 0.81

Bx1 0.8

0.79 0.86

0.87

0.88 0.89 0.9 0.91 Nondimensionalized Frequency, F

0.92

FIGURE 5.17(a)  Close-up view of the pinching between the L1 and BX1 branches shown in Fig. 5.17(a). (Reprinted with permission from Mukdadi et al. 2002a, Fig. 4.)

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Nondimensional Phase Velocity, C /Cs

2

1.5 L1 L0

1

0.5

Bx1

Bx0

Finite width plate (H/B = 0.5) Infinite plate (exact)

0

0

0.25

0.5 Nondimensional Frequency

1.5

1

FIGURE 5.17(b)  Exact dispersion curves of infinite plate and results for a finite-width plate of H/B = 0.5 ([100 + 20] × 240 nm) GaAs–Nb nanowire. (Reprinted with permission from Mukdadi et al. 2002a, Fig. 3.)

lower than seen in Fig. 5.16. Now, the longitudinal mode dispersion in the finite-width wire is not close to that in the infinite-width plate. The L0 branch in the wire is found to approach the lowest flexural branch in the infinite plate, which approaches the SH0 branch. The pinching of the L1 and BX1 branches is seen to occur at high frequency, as seen in Fig. 5.17(b). Group-velocity dispersion curves are plotted in Figs. 5.18(a–c) for the GaAs and GaAs–Nb nanowires. Lower group velocities and cutoff frequencies are observed for the composite wire. In addition, mode coupling is clearly seen in the dispersion curves for the composite wire. This mode interchange is illustrated in the close-up view (Fig. 5.18(c)) of the second bending BX1 and L1 modes. Another nanodevice studied is a silicon (Si) nanowire used for microelectromechanical systems (MEMS). A rectangular Si wire is modified using wet etching, which yields a trapezoidal cross section. This Si wire has been studied by Namazu et al. (2000), who used atomic force microscopy (AFM) to report on the size effect on mechanical properties. Here, we present dispersion results for trapezoidal Si wires having 200 nm height and different widths. These are important for understanding acoustic phonon modes in such wires. They can also be used to measure mechanical properties. Properties of Si are given in Table 5.2. Figure 5.19 shows the geometry of the trapezoidal cross-sectioned wire. The effect of aspect ratio on dispersion is shown in Fig. 5.20. Note that, in all these cases, the angle between the two inclined sides with the Y-axis is fixed, as shown in Fig. 5.19. It is interesting to note that as B1/H increases from 0 (triangular cross section) to 2, the cutoff frequencies are lowered so that more modes appear at a given frequency. Corresponding group-velocity dispersion curves are shown in Fig. 5.21. These figures

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Elastic Waves in Composite Media and Structures

1.5

1.5 L0

L0

Bx0

0.5 0

Cg /Cs

Cg /Cs

1

Bx0 0.5

L1

Bx1 0

1

1

2

3

4 5 6 Ω (a) GaAS (H/B = 1.2)

7

0

8

Bx1 0

1

2

L1 3

4 5 6 7 Ω (b) GaAS-Nb (H/B = 1.2)

8

Cg /Cs

0.6 0.4

Bx1

L1

0.2 0 3.3

3.32

3.34 Ω (c) Close-up view of (b)

FIGURE 5.18  Group-velocity dispersion curves for (a) (120 × 100 nm2 ) GaAs nanowire and (b) ([100 + 20] × 100 nm2) GaAs–Nb composite wire; (c) close-up view of the indicated region in (b) that shows the mode interchange between BX1 and L1. (Reprinted with permission from Mukdadi et al. 2005, Fig. 3(a–c).)

Y

Y B1 GaAs X H

H

Si 35.26°

X

Nb Z

B

Z

B2

FIGURE 5.19  Schematic diagram of the trapezoidal cross section. (Reprinted with permission from Mukdadi et al. 2005, Fig. 1.)

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Wave Propagation in Composite Cylinders 5

5

4

4 L1

3

3 L1

Ω

Ω

Bx1

2 1 0

2

L0

1

Bx0 0

1

Bx1

4 3 K (a) B1/H = 0

Bx0

0

1

3 4 K (b) B1/H = 1

4 3 K (c) B1/H = 2

5

6

5

2

0

L0

6

2

5

6

5 4

Ω

3 2 1 0

L1 Bx1

L0 Bx0

0

1

2

FIGURE  5.20  Dispersion curves for Si nanowire having B1/H ratios: (a) 0, (b) 1, and (c) 2. (Reprinted with permission from Mukdadi et al. 2005, Fig. 8.)

show that the longitudinal and bending modes become strongly coupled as the width increases. Also, as the frequency increases, the phase velocity of the BX0 mode falls below the Rayleigh wave speed. Finally, Fig. 5.22 shows the comparison between the dispersion curves for trapezoidal and rectangular cross-sectional cylinders when B1/H takes the values 1 and 2. Note that, for the same B1/H ratio, the cross-sectional area of the trapezoid is larger than the rectangle. It is found that the cutoff frequencies for the cylinder with the rectangular cross section are lower than those for the cylinder with trapezoidal cross section. As would be expected, the dispersion curves for the two cylinders become closer as B1/H increases.

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204

Elastic Waves in Composite Media and Structures 1.5 0.9 0.6

Bx0

0.3 0

Bx1 0

1

L0

1.2 0.9

Cg /Cs

Cg /Cs

1.5

L0

1.2

0.6

L1

2

3 4 Ω (a) B1/H = 0

0

6

5

1.5

Bx 1 L1 0

2

3 4 Ω (b) B1/H = 1

1

5

6

L0

1.2 Cg / Cs

Bx 0

0.3

0.9 0.6

Bx0

0.3 0

Bx1 L1 0

2

3 4 Ω (c) B1/H = 2

1

5

6

4

4

3

3

2

1

L1

Ω

Ω

FIGURE  5.21  Group-velocity dispersion curves for Si nanowires corresponding to Figs. 5.20(a–c). (Reprinted with permission from Mukdadi et al. 2005, Fig. 9.)

L1

Bx1

1 L0

0

Bx1

Bx0 Rectangle Trapezoid

0

2

1

2 (a)

3

4

0

0

Bx0

L0

Rectangle Trapezoid 1

2

3

4

(b)

FIGURE 5.22  Comparison of the dispersion curves for the rectangular and trapezoidal nanowires having B1/H ratios: (a) 1 and (b) 2. (Reprinted with permission from Mukdadi et al. 2005, Fig. 12.)

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Wave Propagation in Composite Cylinders

5.9 Harmonic Response of a Composite Circular Cylinder due to a Point Force In this section, attention is focused on the steady-state elastodynamic response (Green’s function) of a composite circular cylinder due to a time-harmonic point force acting upon the cylinder. As in Chapter 4 (Section 4.7) for the plate, the steady-state Green’s function is constructed using the Fourier integral transform method. One major difference is that a double series is needed for a cylinder as opposed to a single series for plates. Stiffness method II is used to derive the Green’s function. Equation (5.5.5) is rewritten to show explicitly the derivative with respect to q (see Zhuang et al. 1999) as �� = F K1Q + K 2 Q ,θ + K 3Q ,z − K 4 Q ,θθ − K 5 Q ,θ z − K 6 Q ,zz + MQ



(5.9.1)



where F=−

1 {P} 2π

∫ = ( B CB − B CB )rdr , ∫ = ( B CB + B CB )rdr , ∫

K1 = B1T CB1rdr , K3

K5

T 1

3

T 3

1

T 2

3

T 3

2

∫ (B CB − B CB )rdr K = B CB rdr ∫ K = B CB rdr ,M = ρ N Nrdr ∫ ∫ K2 =

T 1

T 2

2

4

T 2

2

6

T 3

3

1

(5.9.2)

T

Note that the displacement within each sublayer is given by equation (5.5.1). Thus, in the assembled form, the displacement and strain vectors are u = N(r)Q(q, z, t) and e = B1Q + B2Q,q + B3Q,z . It will be assumed that F is a time harmonic of the form F(q,z)e−iwt. Furthermore, F(q,z) will be expanded in a Fourier series in q. Thus, we will write m =+∞

F(θ , z ,t ) = e −iωt F(θ , z ) = e −iωt

∑e

imθ

Fm (z )

m =−∞

(5.9.3)

m= + ∞

Q(θ , z ,t ) = e −iωt Q(θ , z ) = e −iωt

∑e

m =−∞



imθ

Qm ( z )

Substituting equation (5.9.3) in (5.9.1), we obtain a system of linear ordinary differential equations in z for the Fourier coefficient functions Qm (z ) as

(K

1

)

+ imK 2 + m 2 K 4 − ω 2 M Qm + (K 3 − imK 5 )Qm ,z − K 6 Qm ,zz = Fm

(5.9.4)



Taking the Fourier transform of equation (5.9.4) with respect to z yields

(K

1

)

ˆ ˆ ˆ ˆ + imK 2 + m 2 K 4 − ω 2 M Qm + ikm (K 3 − imK 5 )Qm + km2 K 6 Qm = Fm



(5.9.5)

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Elastic Waves in Composite Media and Structures

where the Fourier transform of Qm is ∞

ˆ Qm = Qme −ikmz dz

∫

−∞



(5.9.6)

Equation (5.9.5) is the governing equation for the mth circumferential Fourier coefficient in the transformed domain. The homogeneous form of equation (5.9.5) in the absence of any external forces is obtained by setting the right-hand side equal to zero. This defines an eigenvalue problem that has been discussed in Section 5.5. Equation (5.9.5) can be recast into first-order form (analogous to that for the plate studied in Chapter 4, Section 4.7) as

� = P� [A(m, ω ) − km B(m, ω )]Q m m



(5.9.7)

where







 0 (K1 + imK 2 + m2 K 4 − ω 2 M)  A(m, ω ) =  (K1 + imK 2 + m2 K 4 − ω 2 M) i(K 3 − imK 5 )  

(

)

 K1 + imK 2 + m 2 K 4 − ω 2 M 0   B(m,ω ) =    0 − K 6  ˆ Q  0  � =  m  , P� =   Q m m  ˆ   Fˆ   km Qm   m

(5.9.8a)

(5.9.8b)

(5.9.8c)

By extracting the eigendata from the homogeneous form of equation (5.9.7), the response to the mth circumferential mode in the Fourier series representation of the applied loads can be constructed by modal summation. The eigenvalue problem defined by the homogeneous form of equation (5.9.7) gives the eigenvalues kmn (n = 1, 2, …, 2M) for given frequency w and mode number m. These may be real, imaginary or complex, as discussed before. Associated with each eigenR and Φ L . They satisfy the equations value, kmn, are right and left eigenvectors, Φmn mn

R =0 [A(m, ω ) − kmn B(m, ω )]Φ mn

(5.9.9a)

L =0  A T (m, ω ) − kmn BT (m, ω ) Φmn

(5.9.9b)

These left and right eigenvectors satisfy the bi-orthogonality relations LT BΦ R = diag( B ); Φmn mp mn

LT AΦ R = diag(k B ) Φmn mn mn mn

(5.9.10) ˆ � is of dimension 2M, in which the upper half is If the dimension of Qm is M, then Q m ˆ ˆ Qm and the lower half is km Qm . An eigenvector can be partitioned into upper and lower



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Wave Propagation in Composite Cylinders

halves as follows: Φ

R mn



R  R  Φmnu   Φmnu  ;   = = ΦR  k ΦR   mnl   mn mnu 

Φ

L  Φmnu    = k ΦL   mn mnu 

L mn

(5.9.11)

� in equation (5.9.7) can be represented by a sum of the right Now, the solution vector Q m eigenvectors as � = Q m



2M

∑Q

mn

n =1

R Φmn

(5.9.12)

The coefficients Qmn are found by substituting equation (5.9.12) into equation (5.9.7) and then using the bi-orthogonality relations in equation (5.9.10) to give LT P � Φmn m (kmn − km )Bmn

Qmn =



(5.9.13)



� is given by Thus, the explicit form of Q m � = Q m



2M

∑ n =1

LT P � Φmn m ΦR (kmn − km )Bmn mn

(5.9.14)

ˆ In view of equations (5.9.8c), (5.9.11), and (5.9.14), the solution vector Qm that occu� pies the upper half of Qm can be written as ˆ Qm =



2M

∑ n =1

ˆ LT F kmn Φmn u m ΦR (kmn − km ) mnu

(5.9.15)

The inverse Fourier transform of equation (5.9.15) yields the mth circumferential harmonic coefficient 1 Qm ( z ) = 2π



2M ∞

∑∫

n =1 −∞

ˆ LT F kmn Φmn u m Φ R e ikmz dkm (kmn − km ) mnu

(5.9.16)

ˆ Equation (5.9.16) can be evaluated by Cauchy’s residue theorem. In particular, if Fm is independent of km, then evaluating the integral yields the modal response as M



Qm (z ) = −i

∑ n =1

2M ˆ ˆ LT F LT F kmn Φmn kmn Φmn u m u m R e ikmn z − i R e − ikmn z Φmn Φmn u u B Bmn mn n = M +1

∑

(5.9.17)

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Elastic Waves in Composite Media and Structures

Now, let F(q,z) appearing in equation (5.9.3) be concentrated at q = 0 in the crosssectional plane z = 0 of the cylinder at some radial distance r0 from its axis. Moreover, let r0 coincide with a finite-element nodal surface. The spatial representation of this is assumed to be of the form



 F0q0δ (z )e F , F(θ , z ) =  0,

− θ0 ≤ θ ≤ θ0 | θ| > θ0

(5.9.18a)



such that θ0

∫ q r dθ = 1 or q 0 0

0

=

−θ 0



1 2r0θ0

(5.9.18b)

Here, eF is the unit vector in the direction of F, i.e., e F = α Fr e r + α Fθ eθ + α Fz e z



(5.9.19)



and er, eq , and ez are the unit vectors in the r, q, and z directions, respectively. The Fourier series representation of F(q,z) is given by ∞

F(θ , z ) =

∑e

m =−∞

imθ

1 sin mθ0 F e δ (z ) 2π r0 mθ0 0 F

(5.9.20)

Thus, Fm (z ) appearing in equation (5.9.3) is given by Fm (z ) =



1 sin mθ0 F e δ (z ) 2π r0 mθ0 0 F

(5.9.21)



Taking the Fourier transform of the above equation with respect to z gives 1 sin mθ0 ˆ Fe Fm = 2π r0 mθ0 0 F



(5.9.22)



Using equation (5.9.22) in equation (5.9.17) and considering wave propagation in the positive z-direction only, we obtain



Qm ( z ) = −

i 2π r0

M

∑ n =1

LT F sin mθ kmn Φmnu 0 0 R e ikmn z e Φ mnu F Bmn mθ0



(5.9.23)

Substitution of Qm (z ) from equation (5.9.23) in equation (5.9.3)2 gives the steady-state expression for Q(q,z,t). Numerical results are presented in Figs. 5.23(a) and 5.24(a) for the distribution of magnitudes of the displacement components through the thickness of a hollow isotropic

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209

Wave Propagation in Composite Cylinders 0.1 |M| = 200 |M| = 210 |M| = 220 |M| = 300

|u|

0.09

0.08

0.07 0.95

|M| = 100

0.96

0.97

0.98

0.99

1

1.01

1.02

1.03

1.04

1.05

r/R 0.09

|M| = 200 |M| = 210 |M| = 220 |M| = 300

|v|

0.06

0.03

0 0.95

|M| = 100

0.96

0.97

0.98

0.99

1

1.01

1.02

1.03

1.04

1.05

r/R 2.5 |M| = 200 |M| = 210 |M| = 220 |M| = 300

|w|

2 1.5 1 0.5 0.95

|M| = 100 0.96

0.97

0.98

0.99

1

1.01

1.02

1.03

1.04

1.05

r/R

FIGURE 5.23(a)  Distribution of the displacement components through the thickness of the isotropic cylinder, R/H = 10, Ω = 15. (Reprinted with permission from Zhuang et al. 1999, Fig. 3(a).)

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210

Elastic Waves in Composite Media and Structures 0.08 |M| = 100

|σzr|

0.06 0.04

|M| = 280 |M| = 290 |M| = 300

0.02 0 0.95

0.96

0.97

0.98

0.99

1

1.01

1.02

|M| = 200 1.03

1.04

1.05

r/R 1.5 |M| = 200

|M| = 280 |M| = 290 |M| = 300

|σzθ|

1

0.5

0 0.95

|M| = 100

0.96

0.97

0.98

0.99

1

1.01

1.02

1.03

1.04

1.05

1.04

1.05

r/R 4 |M| = 100

|σzz|

3 2

|M| = 200 |M| = 280 |M| = 290 |M| = 300

1 0.5 0.95

0.96

0.97

0.98

0.99

1

1.01

1.02

1.03

r/R

FIGURE 5.23(b)  Distribution of stress components through the thickness of the isotropic cylinder, R/H = 10, Ω = 15. (Reprinted with permission from Zhuang et al. 1999, Fig. 3(b).)

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211

Wave Propagation in Composite Cylinders 0.10 |M| = 100 |M| = 240 |M| = 250 |M| = 260 |M| = 300

|u|

0.09

0.08

0.07 0.95

0.96

0.97

0.98

0.99

1

1.01

1.02

1.03

1.04

1.05

r/R 0.03

|M| = 240 |M| = 250 |M| = 260 |M| = 300

|v|

0.02

0.01

0 0.95

|M| = 100

0.96

0.97

0.98

0.99

1

1.01

1.02

1.03

1.04

1.05

1.03

1.04

1.05

r/R 0.02

|M| = 240 |M| = 250 |M| = 260 |M| = 300

|w|

0.015 0.01

|M| = 100

0.005 0 0.95

0.96

0.97

0.98

0.99

1

1.01

1.02

r/R

FIGURE 5.24(a)  Distribution of the displacement components for the two-layer (+30°/−30°) composite cylinder, R/H = 10, Ω = 15. (Reprinted with permission from Zhuang et al. 1999, Fig. 5(a).)

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Elastic Waves in Composite Media and Structures 1.2

0.8 |σzr|

|M| = 100 0.4

0 0.95

|M| = 200 |M| = 210 |M| = 300 0.96

0.97

0.98

0.99

1

1.01

1.02

1.03

1.04

1.05

1.02

1.03

1.04

1.05

1.02

1.03

1.04

1.05

r/R 2 |M| = 100 |M| = 200 |M| = 210 |M| = 300

1.5 |σzθ|

1 0.5 0 0.95

0.96

0.97

0.98

0.99

4

1.01

|M| = 100 |M| = 200 |M| = 210 |M| = 300

3 |σzz|

1 r/R

2 1 0 0.95

0.96

0.97

0.98

0.99

1

1.01

r/R

FIGURE 5.24(b)  Distribution of the stress components for the two-layer (+30°/−30°) composite cylinder, R/H = 10, Ω = 15. (Reprinted with permission from Zhuang et al. 1999, Fig. 5(b).)

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Wave Propagation in Composite Cylinders

cylinder and a two-layer angle-ply (+30°/−30°) composite cylinder. For the isotropic cylinder, the Poisson ratio ν is taken to be 1/3. Elastic properties of the fiber-reinforced ply referred to its symmetry axes are given by equation (5.6.1). In the case of the isotropic cylinder, nondimensional frequency Ω = wR/ µ/ρ , where R is the mean radius, m is the shear modulus, and r is the density. The nondimensional frequency is Ω = wR/ ET /ρ for the composite cylinder. In both cases, the thickness of the cylinder is taken to be 1 and the mean radius-to-thickness ratio, R/H, was taken to be 10. All the results are for a section at a distance H/4 from z = 0 and for q = p/4. The force is applied over the angular half-width q 0 = 0.001 at (rout, 0, 0). Figures 5.23(b) and 5.24(b) show the distribution of the tractions at the section z = H/4. These figures show the effect of the total number of terms |M| taken in the Fourier series representation. Of course, the number of terms needed depends upon the half-width of the load, the axial distance of the observation point from the load, and the frequency.

5.10 Forced Motion of Finite-Width Plate In this section, we discuss the forced motion of a cylinder of rectangular cross section. The geometry of the cylinder with the forces acting on the top surface of the plate is shown in Fig. 5.25. The equations governing the motion of the plate were derived in Section 5.7 (see equation (5.7.24)) using the SAFE method. Taking the Fourier transform of equation (5.7.24) with respect to X, we get ˆ = ω 2[M]U ˆ +F  k 2 K1 + ikK 2 + K 3  U



(5.10.1)

Observation point

10H f (t )

H

Z X

Y

B

FIGURE  5.25  Schematic diagram of the plate with the line excitation force in the negative Z-direction acting along the line X = 0, Z = H/2. Only one quadrant is shown. (Reprinted with permission from Mukdadi and Datta 2003, Fig. 1.)

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Elastic Waves in Composite Media and Structures

where ˆ= U

∞

∫ U(X ,ω )e

− ikX

dX

(5.10.2)

−∞

As in the previous section (see also Chapter 4, Section 4.7), equation (5.10.1) can be rewritten as [A − kB]V = P



(5.10.3)

where  0 K3 − ω 2M  , A=  K − ω 2 M iK  2   3



K 3 − ω 2M 0  B=  0  − K 1 

(5.10.4a)

and ˆ  U V =  ,  kU ˆ  



0  P=  F 

(5.10.4b)

As noted before, the homogeneous equation associated with equation (5.10.3) defines an eigenvalue problem. For each eigenvalue km, there are right and left eigenvectors, Φm and Ψm, respectively. They satisfy the equations [A − km B]Φm = 0, [A T − km BT ]ψ m = 0



(5.10.5)



The dual sets of left and right eigenvectors also satisfy the bi-orthogonality relations

ψ mT BΦ p = diag(Bm ), ψ mT AΦ p = diag(km Bm )



(5.10.6)

The right and left eigenvectors can be partitioned into upper and lower halves as follows:



 Φmu   Φmu  , = Φm =  Φ  k Φ   ml   m mu 

ψ mu  ψ mu   = ψm =  ψ   k ψ   ml   m mu 

(5.10.7)

Substituting equation (5.10.7) into the bi-orthogonality relations in equation (5.10.6) leads to explicit forms of these relations as

ψ Tpu (K 3 − ω 2 M)Φmu − km k pψ Tpu K1 Φmu = δ pm Bm

(5.10.8a)

(km + k p )ψ Tpu (K 3 − ω 2 M)Φmu + ik p kmψ Tpu K 2 Φmu = δ pm km Bm

(5.10.8b)

The solution V can be represented in a series involving the right eigenvectors, and we can write 2M

V=

∑V Φ m

m =1

m



(5.10.9)

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Wave Propagation in Composite Cylinders

The coefficients Vm can be determined by substituting V from equation (5.10.9) and using the bi-orthogonality relations. This gives

ψ mT P (km − k )Bm

Vm =



(5.10.10)



Thus, V is found to be 2M

V=

ψ mT P Φm m − k )Bm

∑ (k m =1

(5.10.11)



In terms of the right eigenvectors, equation (5.10.11) can be expressed as ˆ= U

2M

∑ m =1

kmψ mTu F Φ (km − k )Bm mu

(5.10.12)



The displacement at any point in the rectangular elastic cylinder can be obtained by taking the inverse Fourier transform of equation (5.10.12), thus obtaining



1 U( X , ω ) = 2π

2M ∞

kmψ mTu F Φmu e ikX dk m − k )Bm

∑ ∫ (k m =1 −∞

(5.10.13)

The integral appearing in equation (5.10.13) can be evaluated by the method of residues to give M

U ( X , ω ) = −i

∑ m=0

T F kmψ mu Φ mu e ikm X Bm

(5.10.14)



Equation (5.10.14) is for the waves propagating in the positive X-direction. Note that this is the time-harmonic response under the assumption that the force is time harmonic of the form F(w)e−iwt. The transient response for particular time-dependent force is obtained as U( X , t ) =

1 2π

∞

∫ U� (X ,ω )F(ω )e

−∞

− iωt

dω

(5.10.15)

� (X,w) is the expression on the right-hand side of equation (5.10.14) when F = 1. where U Note that this is the fundamental response of the structure when the time dependence of the force is d(t). The integral in equation (5.10.15) has to be evaluated numerically for particular forms of the Fourier transform F(w) of the force f(t). For dispersive modes, difficulties of such

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Elastic Waves in Composite Media and Structures 2

2

Symmetric

1.5

1

1

0.5

0.5

C g/Cs

1.5

0

0

2

4

6

8

10

0

Anti-symmetric

0

2

4

6

8

10

Ω

Ω

FIGURE 5.26  Group-velocity dispersion curves for longitudinal and flexural waves in an infinite Ni plate. (Reprinted with permission from Mukdadi and Datta 2003, Fig. 6.)

integration may occur at the cutoff frequencies or near w = 0. These can be circumvented by adding a small imaginary part to the frequency and writing equation (5.10.15) as U( X , t ) =

e −ηt 2π

2

∞

∫ U� (X ,ω − iη)F(ω − iη)e

− iωt

dω

(5.10.16)

−∞

Extensional

Torsional

Cg /Cs

1.5 1 0.5 0

0

1

2

3

4

2

5

6

0

1

2

3

4

Flexural - Y

5

6

Flexural - Z

Cg /Cs

1.5 1 0.5 0

0

1

2

3

4

Ω

5

6

0

1

2

3

4

5

6

Ω

FIGURE 5.27  Group-velocity dispersion curves for a finite-width Ni plate (B/H = 2). (Reprinted with permission from Mukdadi and Datta 2003, Fig. 7.)

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Wave Propagation in Composite Cylinders

Some numerical results are shown below for a homogeneous isotropic plate of rectangular cross section. The material of the plate is taken to be nickel, and its properties are listed in Table 4.4. Three geometries are considered for comparison purposes: (a) an infinite plate with B/H → ∞, (b) a finite-width plate having an aspect ratio B/H = 2, and (c) when B/H = 4. A schematic diagram of the plate with a distributed vertical force acting on the surface Z = 0 along the half-width of the plate at X = 0 is shown in Fig. 5.25. As mentioned before, the guided waves in the homogeneous plate can be classified according to the symmetry and antisymmetry of the particle motion about the Y- and Z-axes. Group velocities of the different modes in the three plates are shown in Figs. 5.26–5.28. Here, the nondimensional frequency Ω = wH/C s. It is noticed that the cutoff frequencies of the extensional (L) and flexural (about the Y-axis) (BY) modes in finite-width plates are lower than those in the infinite plate. Furthermore, the cutoff frequencies of these modes decrease as B/H increases. The effect of the finite width of the plate on the group-velocity dispersion is seen to be more pronounced on the L and BZ modes than on the T (torsion) and BY modes. As the frequency increases, however, all of the high modes are quite dispersive. The L 0 and BY modes have similar behav0 ior as the S 0 and A0 modes of the infinite plate. In fact, the width does not affect the

Torsional

Cg /Cs

Extensional

0

1

2

3

4

5

6

0

1

2

3

5

6

Flexural - Z

Cg /Cs

Flexural - Y

4

0

1

2

3 Ω

4

5

6

0

1

2

3 Ω

4

5

6

FIGURE 5.28  Group-velocity dispersion curves for a finite-width Ni plate (B/H = 4). (Reprinted with permission from Mukdadi and Datta 2003, Fig. 8.)

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Elastic Waves in Composite Media and Structures

BY0 mode very much. The extensional modes share the common feature that they are not very dispersive at low and high frequencies. At low frequencies, the L 0 mode propagates at the rod velocity, which is lower than the plate velocity approached by the S 0 mode. At high frequencies, these modes travel at the Rayleigh wave speed. Time dependence of the pulse is taken to be f (t ) =

 (t − t0 )2  exp  − sin(ω c t ) 2σ 2   σ 2π 2

(5.10.17)

Symmetric Ni (H = 100 µm)

0.04

B/H = ∞ B/H = 4

G33

0.02 0 –0.02 –0.04 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.5 t (µs)

0.6

0.7

0.8

0.9

1

0.1

G13

0.05

0

–0.05

–0.1

FIGURE 5.29  Comparison of the transient extensional mode response of an infinite plate and a finite-width plate (B/H = 4). (Reprinted with permission from Mukdadi and Datta 2003, Fig. 9.)

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Wave Propagation in Composite Cylinders

Because of the complex nature of the dispersion of propagating modes, the transient wave will also show interaction of several modes in a finite-width plate, and it will not afford an easy interpretation. For this reason, w c is taken as 3.1 Cs/H. Also, s and t0 are chosen to be 1.2 and 3.0 H/Cs, respectively. This gives the pulse duration about 0.2 µs, and in the frequency domain, the width is 2 < Ω < 4. Examination of Fig. 5.29 shows that primarily S 0 and A0 modes are excited in an infinite plate (although some contribution will be from A1 mode). Here, results are presented for symmetric mode excitation by equal and opposite line vertical loads applied to the top and bottom surfaces of the plate (along lines parallel to the Y-axis) as well as antisymmetric mode excitation by equal vertical forces acting in the same direction. Figures 5.29 and 5.30 show the comparison 0.2

Symmetric Ni (H = 100 µm)

B/H = ∞ B/H = 4

G33

0.1

0

–0.1

–0.2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.5 t (µs)

0.6

0.7

0.8

0.9

1

0.1

G13

0.05

0

–0.05

–0.1

FIGURE 5.30  Comparison of the transient flexural mode response of an infinite plate and a finite-width plate (B/H = 4). (Reprinted with permission from Mukdadi and Datta 2003, Fig. 10.)

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between the surface displacement amplitudes in the two cases for an infinite plate and a plate with B/H = 4. The observation point is at (10H, 0, H/2). The variable Gij represents displacement in the i-direction for a force applied in the j-direction. It is seen from Fig. 5.29 that in both cases the wave is extensional in the early period followed by some vertical motion, particularly in the finite-width plate. In the finite-width plate, initial extensional motion is almost the same as in an infinite plate, with much larger extensional motion at later times (caused by reflections from the sides). Figure 5.30 shows the remarkable similarity (with some differences) in the flexural motion of the two plates. Figures 5.31 and 5.32 show the comparison between plates having widths B/H = 2 and 4. 0.06

B/H = 2 B/H = 4

G33

0.03

0

–0.03

–0.06

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.5 t (µs)

0.6

0.7

0.8

0.9

1

0.1

G13

0.05

0

–0.05

–0.1

FIGURE  5.31  Comparison of the transient extensional mode response of finite-width plates with B/H = 2 and 4. (Reprinted with permission from Mukdadi and Datta 2003, Fig. 11.)

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Flexural - Y Ni

G33

0.1

B/H = 2 B/H = 4

0

–0.1 0

0.1

0.2

0.3

0.4

0.5 t (µs)

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.5 t (µs)

0.6

0.7

0.8

0.9

1

G13

0.1

0

–0.1

FIGURE 5.32  Comparison of the transient flexural mode response of finite-width plates with B/H = 2 and 4. (Reprinted with permission from Mukdadi and Datta 2003, Fig. 12.)

At early times, the extensional motion (Fig. 5.31) in the wider plate is found to be much larger than in the narrower plate. However, the opposite is seen at a later time. Again, the flexural motion (Fig. 5.32) in both cases is almost the same.

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6 Scattering of Guided Waves in Plates and Cylinders 6.1 Introduction ............................................................................ 223 6.2 Scattering in a Plate................................................................ 224 Scattering by Symmetric Surface-Breaking Cracks in a Composite Plate • Transient Scattering by a SurfaceBreaking Crack in a Plate • Transient Scattering by a Delamination in a Composite Plate • Reflection of Plane-Strain Waves at the Free Edge of a Semi-Infinite Composite Plate 6.3 Scattering in a Pipe ................................................................ 257 Scattering by Circumferential Cracks in a Pipe • Scattering by Cracks in a Welded Steel Pipe • End Reflection of Guided Waves in a Circular Cylinder

6.1 Introduction In the previous chapters, discussion has been focused on characteristics of ultrasonic guided waves in, and the forced response of, laminated and homogeneous plates and cylinders without any defects. Guided waves provide an efficient means of determining material properties of the structures. These are important for designing and manufacturing of structures or their components. Comparison of model predictions and experimental observations of dispersion characteristics has been used to inversely determine the anisotropic elastic constants, damping parameters, and dimensional properties like thickness. Ultrasonic guided waves have been used to assess deterioration in these properties due to the service environment. These have also proven to be effective in characterizing isolated critical defects like cracks, cavities, welds, and joints. Guided waves can propagate long distances and can be tailored for use in monitoring structural health. Important advances have been made during the last two decades in the development of ultrasonic nondestructive evaluation techniques. Modeling of scattering of ultrasonic guided waves by cracks, delamination, and welds is the subject of this 223

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chapter. In the following discussion, scattering in a plate is studied first and is followed by an investigation of scattering in a cylindrical shell.

6.2 Scattering in a Plate Two-dimensional scattering of guided waves by surface-breaking and subsurface cracks in a semi-infinite space has received considerable attention. Research in this arena includes work by: Achenbach et al. (1980a), who used the elastodynamic ray theory; Kundu and Mal (1981), who used an asymptotic diffraction theory to study a surface crack in a plate; Tuan and Li (1974) and Simons (1978), who used perturbation methods; and Mendelsohn et al. (1980) and Achenbach et al. (1980b), who expressed the solution in terms of singular integral equations. Integral equations were also used to study scattering of longitudinal, transverse, and surface waves by subsurface cracks by Brind and Achenbach (1981), Achenbach and Brind (1981a, 1981b), and Keer et al. (1984). Scattering of SH waves by surface-breaking cracks was studied by Stone et al. (1980) using an integral equation formulation. Karim and Kundu (1988, 1991) also used integral equations to solve dynamic response problems in an anisotropic half-space. Huang (1996) studied scattering by a surface-breaking crack using the integral formulation and Galerkin’s method. Propagation of plane-strain waves in an elastic half-space containing an embedded cylindrical cavity was studied by Thiruvenkatachar and Viswanathan (1965) using a successive-reflection technique. This problem was also studied by Gregory (1970), who used a multipolar expansion. These methods are restricted to simple shapes in an isotropic medium. Datta (1979) and Datta et al. (1982) studied scattering of SH waves by surface-breaking cracks using a matched asymptotic expansion (MAE) technique as well as a hybrid method combining finite-element (FE) representation of the near field and wave function expansion in the far field. MAE was used by Datta and El-Akily (1978a, 1978b) to study scattering by a cylindrical cavity in a half-space. Datta and Shah (1982) used the MAE and hybrid methods to solve scattering of SH waves by embedded cavities and cracks. These methods are rather general and have been applied to more complexly shaped scatterers. Shah et al. (1982, 1985, 1986) studied diffraction of SH waves by arbitrarily shaped cavities, and of body and surface waves by cracks, in a half-space by using the hybrid method. Bouden et al. (1991) used a hybrid method combining finite element representation of the near field and a boundary integral representation of the far field (see also Cho and Rose 2000). Scheidl and Ziegler (1978) and Höllinger and Ziegler (1979) developed an approximate method suitable for analyzing scattering by a circular cavity in a semi-infinite elastic medium. Other numerical methods that have been used to investigate scattering by surfacebreaking or near-surface cracks and other surface irregularities include the T-matrix method (Boström 1980; Karlsson 1984; Boström and Karlsson 1985), the finite-difference method (FDM) (Fuyuki and Matsumoto 1980; Hirao et al. 1982; Harker 1984; Scandrett and Achenbach 1987; Saffari and Bond 1987; Wu et al. 1995; Masserey and Mazza 2005), boundary-element method (BEM) (Zhang and Achenbach 1988; Hévin et al. 1998; see also the review by Beskos 1997), and the finite-element method (Hassan and Veronesi 2003). There have also been some investigations of three-dimensional defects in a half-space. Among these are: Boström and Kristensen (1980), Angel and Achenbach (1984), Lin and Keer (1986), Bövik and Boström (1997), and Boström et al. (2003).

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Scattering of Guided Waves in Plates and Cylinders

225

Guided-wave techniques for ultrasonic characterization (location, size, shape, etc.) of defects in a plate have been well developed over the last 20 years or so. References to earlier works on problems of wave diffraction can be found in the review article by Miklowitz (1960). Later developments in various techniques and experiments can be found in Datta (1978), Varadan and Varadan (1986), and Datta et al. (1990a). Early works dealing with SH-wave scattering in plates were by Lysmer and Waas (1972), who used finite elements coupled with semi-infinite elements for dynamic response of foundations. Egorov and Kharitonov (1979) treated scattering by a normal surface crack as a Riemann–Hilbert problem. Scattering of Lamb waves by an infinitely thin crack parallel to the surface of an isotropic plate was studied by Rokhlin (1980) using the Wiener–Hopf technique and by the method of multiple diffractions (Rokhlin 1981). Tan and Auld (1980) investigated the scattering by a normal surface-breaking crack in an isotropic plate by a normalmode variational technique. Fortunko et al. (1982) used the variational method as described by Tan and Auld (1980) to study SH-wave scattering by surface-breaking and buried cracks in a plate. Abduljabbar et al. (1983) presented a detailed solution of scattering of SH waves by surface-breaking cracks in a plate using the hybrid method, which combined the finite element representation of the region containing the defects bounded by two vertical boundaries and the exact guided-wave function expansion in the semiinfinite external regions. They established the reciprocity relations that hold between the reflected and transmitted modes and used the energy-balance criteria to check the accuracy of the reflection and transmission coefficients. Rokhlin and Bendec (1983) used the results from the earlier works (Rokhlin 1980, 1981) to analyze and experimentally measure the transmission of Lamb waves through an aperture in a plate. Sabbagh and Krile (1973) used finite elements to investigate scattering of SH waves by a crack in a plate. They considered only the first symmetric mode propagating. Koshiba et al. (1981) solved the general problem of SH-wave scattering in a plate using a very similar method that was presented by Abduljabbar et al. (1983). Koshiba et al. (1984) extended the approach to Lamb wave scattering. They considered only the first symmetric (S 0) propagating mode. Koshiba et al. (1987) generalized the work by Koshiba et al. (1981) to include all propagating as well as several nonpropagating modes. Al-Nassar et al. (1991) and Datta et al. (1991) used the hybrid method to analyze the general Lamb wave scattering by a strip weldment and by a surface-breaking crack, respectively, in an isotropic plate. In these, they included the propagating and nonpropagating modes to achieve the necessary convergence. Karunasena et al. (1991c) generalized the hybrid method to the case of scattering by symmetric surface-breaking cracks in laminated composite plates. For other methods, the reader is referred to Alleyne and Cawley (1992) and Gunawan and Hirose (2004) for example. In the following, the hybrid method is presented together with numerical results for reflection and transmission coefficients of scattered modes.

6.2.1 Scattering by Symmetric Surface-Breaking Cracks in a Composite Plate Figure 6.1 shows the geometry of the composite plate and the interior and exterior regions. The interior region R I is bounded by the artificial vertical boundaries B + and B -. The exterior right and left domains are respectively denoted by R+ and R −.

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226

Elastic Waves in Composite Media and Structures Interface number 1

Free surface x y z Layer

Zi Zi+1

I I+1

i -th sublayer layer

2H

N+1

Free surface (a) x–

x a

B– R–

x+

B+ z Interior region R

Crack

a

R+

Incident wave

(b)

FIGURE 6.1  Geometry of a composite plate with symmetrical surface-breaking cracks: (a) laminated plate, (b) symmetrical surface cracks. (Reprinted with permission from Karunasena et al. 1991c, Fig. 1.)

The symmetric surface-breaking cracks are assumed to be located at x = 0, and they are infinite in length along the y-axis. The plate is composed of perfectly bonded layers of orthotropic material symmetry having the same symmetry planes coinciding with the xy-, yz-, and zx-planes. We will consider plane-strain waves in the plate in the zx-plane. As discussed in Chapter 4, any dynamic disturbance in the plate can be expressed as the sum of an infinite number of plate modes, not all of which are propagating. A timeharmonic plane propagating mode that is generated at x = +∞ and propagating in the negative x-direction is assumed to be incident on the cracks. The surfaces of the plate at z = 0 and 2H and the crack faces at x = ±0 are assumed to be traction free. The incident wave is scattered by the crack into multiple modes propagating in the ± x directions. Because of the symmetry of the problem, an incident symmetric mode will be scattered into symmetric modes only. Similarly, an antisymmetric incident mode will be scattered into antisymmetric modes only. The dynamic field at any point of the plate will be composed of the incident field and the scattered fields. The problem of finding the amplitudes of the scattered wave modes in the exterior regions R+ and R− is solved by using a hybrid method, in which the total field in the interior region is modeled by finite elements, and that in the exterior regions is modeled as a sum of the plate modes and the incident mode. The unknown coefficients

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Scattering of Guided Waves in Plates and Cylinders

of the modal sum and the displacements at the interior finite elements are found by matching the displacements and tractions at the nodes on the fictitious boundaries B + and B −. 6.2.1.1 Wave Functions for Exterior Regions Details of the methodology used to obtain wave functions for the laminated plate are given in Section 4.2.1. As discussed there, each lamina is divided into several sublayers. The total number of sublayers through the thickness of the plate is N. Let u(x, z, t) and w(x, z, t) be the displacement components in the x- and z-directions, respectively. Similar to equation 4.2.17, these have the forms u( x , z , t ) = U (z )exp[i(kx - ω t )]

(6.2.1)

w( x , z , t ) = W (z )exp[i(kx - ω t )]

Denoting the displacement–stress vector at the top interface (z = zi) of the ith sublayer as Qi = 〈ui wi σ zzi σ zxi 〉T



(6.2.2)



we obtain the relation Qi +1 = Pi Qi



(6.2.3)



where Pi is the propagator matrix for the ith sublayer. Applying equation (6.2.3) to successive sublayers, we obtain (see equation (4.2.3))

Q N +1 = PQ1 , P = PN …P1

(6.2.4)



The exact dispersion relation governing k andw in equation (6.2.1) is obtained from the equation P31 P32 P41 P42



=0

(6.2.5)

For a given frequency, equation (6.2.5) is solved to find the wave numbers km (m = 1, 2, …). Then, displacement–stress vectors at the interfaces of the sublayers are obtained by using equation (6.2.3). Consider the case in which the incident wave is the pth propagating mode corresponding to the wave number kp. After encountering the crack at x = 0, it generates a scattered wave field. Using the wave-function expansion, the displacement vector of the + scattered field, q s+ x , in region R at arbitrary x can be written as M



q s+ x =

∑A q + m

m =1

m

exp(ikm x ), x ≥ x +



(6.2.6)

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Elastic Waves in Composite Media and Structures

Here, Am+ is the coefficient of the mth scattered mode in R+, and qm is the mth modal displacement vector evaluated at the N + 1 surfaces of the sublayers. Thus, q Tm =



1 〈u w … u( N +1)m w( N +1)m 〉 g m 1m 1m

(6.2.7)

N +1

∑ (| u | + | w

gm =

im

2



(6.2.8)

|2 )

im

i =1



Equation (6.2.6) is evaluated at the nodes on the boundary B + to yield q Bs+ = G + D +



(6.2.9)



where G + = [q1 q 2 … q M ]



D + = D1+ D2+ … DM+



)

(

Dm+ = Am+ exp ikm x + ,



(6.2.10)

T

(6.2.11)



m = 1, 2, …, M

(6.2.12)



Note that G + is a matrix of size 2(N + 1) × M. The consistent nodal force vector due to the scattered field at the nodes can be formed (Karunasena et al. 1991c) as PBs+ = F + D +



(6.2.13)



where F + = [F1 F2 … FM ] FmT = F1xm F1zm F2xm F2zm … F(xN +1)m F(zN +1)m



F1xm =

h1 (- ) 2σ xx1m + σ xx 2m 6

F1zm =

h1 (2σ zx1m + σ zx 2m ) 6

Fimx =

h hi -1 ( + ) ( - ) + i 2σ ( + ) + σ ( - ) σ xx (i -1)m + 2σ xxim xxim xx (i +1)m 6 6

Fimz =

hi -1 h (σ zx (i -1)m + 2σ zxim ) + i (2σ zxim + σ zx (i +1)m ) for 2 ≤ i ≤ N 6 6

(

)

) (

(

F(xN +1)m =

hN ( + ) + 2σ xx ( N +1)m σ 6 xxNm

F(zN +1)m =

hN (σ + 2σ zx ( N +1)m ) 6 zxNm

(

)

(6.2.14)

fo or 2 ≤ i ≤ N

)

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Scattering of Guided Waves in Plates and Cylinders

( + ) and σ ( - ) (2 ≤ i ≤ N) denote the normal stresses in the xIn equation (6.2.14), σ xxim xxim direction associated with the mth mode, just above and below the ith interface, respectively. Following a similar procedure at the boundary B − , we obtain the displacement and force vectors due to the scattered field as



q Bs- = G - D -



(6.2.15)



PBs- = F - D -



(6.2.16)

where G − is obtained from G + after replacing each x-direction displacement component by its negative value. Similarly, F− is obtained from F+ after replacing each x-direction force component by its negative value. D − is given by D -T = 〈D1- D2- … DM- 〉

where

)

(

(6.2.17)



Dm- = Am- exp -ikm x - , m = 1, 2, …, M



(6.2.18)

and Am- is the coefficient of the mth scattered mode in R−. In a similar manner, the boundary displacement and force vectors due to the incident wave can be constructed as

( ) = A G exp ( -ik x ) = - A F exp ( -ik x ) = A F exp ( -ik x )

q inB + = Ainp G -p exp -ik p x + q inB P

in + B



PBin-

in p

p

in p p

in p p

-

p

p

p

(6.2.19)

+

-

Here, G -p and Fp- are the pth columns of G − and F− matrices, respectively, and Ainp is the amplitude of the incident mode. 6.2.1.2 Finite Element Model of Interior Region The interior region R is modeled by finite elements. The crack-tip singularity is modeled using six-node, quarter-point, triangular crack-tip elements (Barsoum 1976). The cracktip elements are surrounded by a layer of transition elements followed by conventional four-node elements. The layers adjacent to the vertical boundaries B + and B − consist of five-node elements, with the midside node lying on the boundaries. By following the conventional finite-element discretization process (Zienkiewicz 1977), we obtain

δ q T Sq - δ q TB PB = 0



(6.2.20)

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Elastic Waves in Composite Media and Structures

where



 SII SIB   S = K I - ω 2 MI =   SBI SBB    q T = q TI q TB



(6.2.21) (6.2.22)



Here, K I and MI are the global stiffness and mass matrices of the interior region, respectively; qI is the nodal displacement at the interior nodes; qB is the nodal displacement vector at the boundary nodes; and PB is the interaction force vector at the boundary nodes. The variable d denotes the first variation. 6.2.1.3 Global Solution The modal expansion coefficients for (a) the scattered field in the exterior regions and (b) the nodal displacement vectors in the interior region are obtained by imposing the following continuity conditions on displacements and tractions on the boundaries B + and B −:

q B = q inB + q Bs



(6.2.23)



PB = P + P



(6.2.24)

in B

s B

where q inB T = q inB -T q inB +T q BsT = q Bs-T q Bs+T

(6.2.25)

PBinT = PBin-T PBin+T PBsT = PBs-T PBs+T





Using equations (6.2.9) and (6.2.15) in equation (6.2.23) and, in turn, in equation (6.2.20), one obtains

SII q I + SIB q B = 0



GT (SBI q I + SBB q B ) = GT PB

(6.2.26)



(6.2.27)

where



G - 0   G=  0 G + 

(6.2.28)

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Scattering of Guided Waves in Plates and Cylinders

231

Using equation (6.2.26) in (6.2.27), it is found by virtue of equations (6.2.23) and (6.2.24) that

(

ΛD = GT PBin - S*BB q inB

)

(6.2.29)

Here, S*BB = SBB - SBI SII-1 SIB Λ = GT (S*BB G - F) (6.2.30)

DT = 〈D -T D +T 〉



F- 0   F=  0 F + 



and Λ is a matrix of size 2M × 2M. D − and D+ are obtained by solving equation (6.2.29). Coefficients Am+ and Am- are obtained from equations (6.2.12) and (6.2.18), respectively. The vector qB is computed from equation (6.2.23) by making use of equations (6.2.9) and (6.2.15). Then, qI is found from equation (6.2.26). The reflection coefficient Rpm of the mth reflected mode and the transmission coefficient Tpm of the mth transmitted mode due to the pth incident mode will be defined as



R pm =

Am+ Apin

(6.2.31)

AmTpm = in , m ≠ p Ap Tpm = 1 +

(6.2.32)

Am, m= p Apin

(6.2.33)

6.2.1.4 Energy Conservation and Reciprocity Relations Reflected and transmitted energies are carried only by the propagating modes. The time-averaged value of the energy flux associated with the nth reflected propagating mode through the plate cross section due to the pth incident mode is given by 2



+ = ω Ain |R |2 β I pn p pn n



2



- = ω Ain |T |2 β I pn p pn n 2



I pin = ω Apin β p





(6.2.34) (6.2.35) (6.2.36)

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Elastic Waves in Composite Media and Structures

Here,

βn = Im(Fn+T . q n ), 1 ≤ n ≤ N pr



(6.2.37)

with Npr being the number of propagating modes at the frequency considered and Im standing for imaginary part. Let e be the percentage error in energy balance given by

ε=

 100  in I I pin  p 

 -  + I pn  

N pr

∑(I

)

+ pn

n =1

(6.2.38)

Application of the principle of conservation of energy to the closed region R bounded by x = x+, x = x − , and the top and bottom surfaces of the plate shows that e should be zero. This condition is used to check the accuracy of the numerical computations. Application of the real elastodynamic reciprocity relations (Auld 1990; Tan and Auld 1980; Achenbach 2003) to the region R results in R pnα n = Rnpα p



Tpnα n = Tnpα p



(6.2.39)



(6.2.40)



where

α n = -2Fn-T . q n , 1 ≤ N ≤ N pr



(6.2.41)

A close examination of the elements of the matrices PB, G +, G − , F+, and F− reveals that a n is related to b n by

α n = 2iβn



(6.2.42)



+ be the proportion of the incident energy transferred to the nth reflected mode. Let E pn Then



+ = E pn

+ I pn

I pin

=|R pn|2

βn βp



(6.2.43)

Similarly,



+ = Enp

+ Inp

In

= |Rnp|2

βp βn



(6.2.44)

It follows from equations (6.2.39) and (6.2.41)–(6.2.43) that

+ = E+ E pn np



(6.2.45)

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233

In a similar manner, it can be shown that for the transmitted modes the following relation holds.

- = EE pn np



(6.2.46)

Reciprocity relations in equations (6.2.39), (6.2.40), (6.2.44), and (6.2.45), as well as the energy conservation (indicated by e ≈ 0), are important checks for the accuracy of the numerical results. 6.2.1.5 Numerical Results In this section, some representative numerical results are presented for reflection and transmission coefficients when the first symmetric mode, S 0, is incident on the symmetric surface-breaking cracks shown in Fig. 6.1. Also shown are results for a single surface-breaking crack. First consider a homogeneous isotropic plate with Poisson ratio ν = 0.31. For symmetric surface-breaking cracks, the problem is either symmetric or antisymmetric. So, only half of the plate needs to be modeled. The total number of sublayers, N, used to compute the wave functions for the exterior region plays an important role in the accuracy of the numerical results. To select a suitable value for N, the quantity b n, defined in equation (6.2.37), was computed by increasing N to a few selected low, intermediate, and high frequencies until convergent values for b n were obtained within the frequency range considered. It was found that for Ω (=wH/ µ/ρ ) ≤4, a reasonable value for N through the half-thickness was 40. Here, m and r are the shear modulus and density, respectively. The finite-element mesh was automatically generated with arbitrary normalized crack length, a/H, with an increment of 1/20 in steps. The finite elements were taken to be square (except near the crack tip). Overall, the normalized width of the interior region was taken to be 0.4 (i.e., x − /H = −0.2, x+/H = 0.2), and there were eight columns of finite elements symmetrically about the z-axis. The mesh contained 20 rows. Three checks were made to validate the numerical results: (a) for a plate without the crack, the scattered field should be zero; (b) the percentage error e in energy should be nearly zero; and (c) the reciprocity relations should be satisfied. These were found to be satisfied with negligible errors. For further verification, a comparison of the results obtained here for R11 with those given by Koshiba et al. (1984) is shown in Fig. 6.2 when Ω was taken to be p/2. In this case, Npr is 1. A total number of 21 modes was used in the calculations here, and |e| was less than 0.005%. The comparison is found to be excellent. Next, we consider a homogeneous graphite-fiber-reinforced epoxy composite plate with fibers aligned along the x-axis. Elastic properties of the composite material are given in Table 4.1. The number of sublayers through the half-thickness was determined to be 40 to give reasonable accuracy within the same frequency range as in the previous example. Figure 6.3 shows the variation of R1n (n = 1, 2, 3) for the normalized frequency Ω (=wH/ (c /ρ ) ) having the value 4.0. 55 0� There are three propagating symmetric modes at this frequency (see Fig. 4.2). As in the previous case, 20 modes were found to give reasonable accuracy (|e| = 0.35%). It is

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|R11|

0.8 0.6 0.4 0.2 0.0 180

Arg(R11) Deg.

160 140 120 100 80 0.0

0.2

0.4

0.6

0.8

1.0

a/H

FIGURE 6.2  R11 vs. normalized crack length at Ω = p/2 for homogeneous isotropic plate: (ν = 0.31) for incident first symmetric plate mode (S 0). Present results: o o o (Koshiba et al. 1984). (Reprinted with permission from Karunasena et al. 1991c, Fig. 3.)

seen that the variation of R11 is quite different when there is more than one propagating mode, because the energy is now shared among three modes. Also interesting to note is that |R12| goes to zero as the crack approaches the thickness of the plate. Thus, the total energy is shared between the first and third symmetric modes. Figure 6.4 shows the variation of R1n (n = 1, 2) for a 35-layer (90°/0°/…/0°/90°) graphite–epoxy cross-ply composite plate. For this case, N was taken to be 70 to obtain satisfactory accuracy. The frequency Ω is 2.5, and there are two propagating modes. As discussed in Chapter 4, the plate can be modeled as an effective homogeneous anisotropic plate, with the elastic constants given by c11 = 87.32 GPa, c33 = 13.92 GPa, c13 = 6.68 GPa, and c55 = 4.68 GPa. Reflection coefficients for this effective homogeneous plate are also shown in this figure. Results for the homogenized plate are seen

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n=1 n=2 n=3

|R1n|

0.8 0.6 0.4 0.2 0.0 360

Arg(Rin) Deg.

300 240 180 120 60 0 0.0

0.2

0.4

0.6

0.8

1.0

a/H

FIGURE 6.3  R1n vs. normalized crack length at Ω = 4.0 for incident S 0 mode in a homogeneous graphite–epoxy plate. (Reprinted with permission from Karunasena et al. 1991c, Fig. 5.)

to agree well with those for the cross-ply plate. As in the case of a uniaxial graphite– epoxy plate, |R12| approaches zero as the length of the crack becomes the same as the thickness of the plate. The method discussed above for symmetric surface-breaking cracks can also be used for a single surface-breaking crack. This was discussed by Bratton (1990) using the hybrid method. Results for a uniaxial graphite–epoxy plate of thickness H when the symmetric S 0 mode is incident on a surface-breaking crack are shown in Fig. 6.5. The frequency Ω is chosen to be 0.6p. There are three propagating modes (S 0, A0, and A1) at this frequency. Thus, for the incident S 0 mode, there are reflected and transmitted S 0, A0, and A1 modes. Reciprocity relations are seen to be satisfied by the reflected and transmitted modes. It is to be noted that the problem of scattering by a single surface-breaking crack in isotropic plates has been considered by Flores-Lopez and Gregory (2006) and Castaings et al. (2002). Their results are qualitatively similar to the ones presented here.

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Results using layered model Homogeneous model

|R1n|

0.8 0.6 0.4 0.2 0.0 360

n=1 n=2

Arg(Rin) Deg.

300 240 180 120 60 0

0.0

0.2

0.4

0.6

0.8

1.0

a/H

FIGURE 6.4  R1n vs. normalized crack length at Ω = 1.2 for incident S 0 mode in a 35-layer (90°/0°/…/0°/90°) graphite–epoxy cross-ply plate. (Reprinted with permission from Karunasena et al. 1991c, Fig. 7.)

The hybrid method combining FE and modal expansion (as discussed above) has also been used to model reflection and transmission of propagating modes incident upon a welded or jointed plate by Al-Nassar et al. (1991) and Karunasena et al. (1994, 1995).

6.2.2 Transient Scattering by a Surface-Breaking Crack in a Plate In the previous section, a hybrid method combining finite elements and a modal expansion technique was used to find the reflection and transmission coefficients for the scattered field. This requires that the cracks (or other discontinuities) be bounded within a region that is bounded by two vertical boundaries that span the entire thickness of

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0.8 |T| & |R|

S0

S0

0.6 A1 0.4

A0

0.2 0.0 0.0

Trans. Reflc. 0.2

0.4

0.6

0.8

1.0

d/H

FIGURE 6.5  Magnitudes of the reflection and transmission coefficients for the incident S0 mode and the converted A0 and A1 modes for a surface-breaking crack in a homogeneous graphite–epoxy plate at the normalized frequency of 0.6p. (Reprinted with permission from Bratton 1990, Fig. 3.11.)

the plate. Thus, the method is limited to low and medium frequencies. A method that bypasses this is one that combines the finite element method with a boundary integral representation. This hybrid method is very versatile and has quite general applications (see Ju et al. 1990; Bouden et al. 1991; S. Liu et al. 1991a, 1991b, 1996, 1997, 2002; Datta et al. 1992b; Ju and Datta 1992b; Liu and Datta 1991, 1993; Liu and Huang 2003). Recently, Galán and Abascal (2003, 2005) (see the many references in these papers for applications of the hybrid method) have also used a combined FE and BE method to study scattering in an infinite plate. Alleyne and Cawley (1992) used a finite element to study scattering by defects in a plate. One of the attractive features of the combined FE and BE method, as with the one discussed in the previous section, is that scatterers of arbitrary shapes and properties can be considered. The other advantage is that the finite elements are used only in a small interior region containing the defects. The field in the exterior region is represented by boundary integrals involving the Green’s functions for the plate (which can be layered and anisotropic). Since the computation of the Green’s functions is independent of the scatterers, these have to be evaluated only once for the plate. 6.2.2.1 Exterior Region R E and Boundary Integral Representation The problem considered here is illustrated in Fig. 6.6, which shows the two-dimensional configuration of a surface-breaking crack in a plate. The crack is assumed to be inside the region bounded by the contour C. The interior region RI is defined to be the region bounded by the contour B, and the exterior region R E is bounded inside by C. The region between C and B is shared by both regions. The field inside RI is represented by finite elements, and that in R E is expressed as a boundary integral over C.

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Impact force B

C

x

Crack

R1

RE Plate

z

FIGURE 6.6  Geometry of the plate showing the surface-breaking crack and the contours B and C for boundary integral representation. (Reprinted with permission from S. Liu et al. 1991b, Fig. 1.)

In RE, the displacement field u is composed of two parts: the field generated by external sources u(f) in the absence of any defects and the scattered field u(s) due to the crack. Thus, (6.2.47)

u = u( f ) + u( s )



The free-field displacement is given by the solution to the equation of motion (2.1.31) that satisfies the traction-free boundary conditions on the free surface of the plate and the continuity conditions at the interfaces between the layers comprising the plate. Note that this can be a propagating mode supported by the plate. A time dependence of the form e −iωt will be assumed. To derive the boundary integral representation for u(s), we apply the following elastodynamic reciprocity theorem involving two elastodynamic states, (u, t, f) and (v, s, g), in R E.



∫∫ (f .v - g .u)dxdz = �∫ (u.s - v .t )dC RE

C

(6.2.48)

Here, u, t represent the displacement and surface tractions caused by body force f, and v, s that due to force g. Taking the scattered field as the first state and the Green’s functions for the plate as the second state, equation (6.2.48) gives -ui( s ) ( x ′ , z ′ ) =

�∫ (u C

(s) j

)

Σ ijk - Gijσ (jks ) (-nk )dC , ( x ′ , z ′ ) ∈ B

(6.2.49)

where the integration along C is clockwise, and n is the outward normal to C. The variables Σijk and Gij satisfy the equation

Σ ijk ,k + ρω 2Gij = -δ ijδ ( x - x ′ )δ (z - z ′ ),

( x , z ) ∈ RE



(6.2.50)

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Scattering of Guided Waves in Plates and Cylinders

Next, equation (6.2.48) is applied to the region including the scatterer and the contour C, with the two states being u(f) and G. This gives

�∫ (u

0=

(f ) j

)

Σ ijk - Gijσ (jkf ) nk dC

C

(6.2.51)

The integral is performed in the counterclockwise direction. Combining equations (6.2.49) and (6.2.51), it is found that ui ( x ′ , z ′ ) = ui( f ) ( x ′ , z ′ ) +

�∫ (u Σ j

ijk

- Gijσ jk )nk dC

(6.2.52)

C

Equation (6.2.52) is the integral representation of the total field at any point in R E. 6.2.2.2 Interior Region R I and the Finite-Element Formulation This region contains all the defects and is discretized into finite elements. The equation of motion for each element can be written in the form K e u e - ω 2 Me u e = re



(6.2.53)



where Ke and Me are the stiffness and mass matrices, respectively, re is the nodal force, and ue is the nodal displacement. Equation (6.2.53) may be written in an alternative form as Se u e = re



(6.2.54a)



where Se = Ke −w 2Me. Assembling all the elements, we obtain the global equation in RI ∪ B as Su = r



(6.2.54b)

The nodal displacements uB can be separated from the interior nodal displacements u I, and equation (6.2.54) can be written in the partitioned form



 SBB SBI  u B  rB     =    SIB SII  u I  0     

(6.2.55)

where rB is the interaction force at B. 6.2.2.3 Solution for the Nodal Displacements in R I ∪ B Now, evaluating the integrals at all the nodes NB on B and separating u B from u I, equation (6.2.52) gives

u B = u(Bf ) + A BI u I + A BB u B



(6.2.56)

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Elastic Waves in Composite Media and Structures

A BI and A BB are complex matrices. Combining the second equation in (6.2.55) and equation (6.2.56), the system of complex equations for determining the nodal displacements uI and uB are obtained as



I - A BB - A BI  u B  u(Bf )     =    SIB  u I   0  S II      

(6.2.57)

Thus, the total displacements at the nodes in RI ∪ B are obtained in terms of the freefield nodal displacements on B by solving equation (6.2.57). These equations form a fairly large, complex, and sparse unsymmetric system. Thus, the choice of a fast and accurate method of solution involves minimization of some function of both computational time and required storage. The biconjugate gradient method (Sarkar et al. 1981; Langtangen and Tveito 1988) and a compacted data structure (Nour-Omid and Taylor 1984) are used to solve this system of equations. The error criterion used to terminate the iteration involved to solve the system Ax = b is

ε = ||Ax k - b|| / ||b|| < 10-6

(6.2.58)



The Green’s function for the plate can be written (see Chapter 4, Section 4.5) in the Fourier integral form as ∞

∫

G( x , z ; x ′ , z ′ ; ω ) = F(k , z , z ′ ; ω )e ik( x - x ′ )dk

0

(6.2.59)

where F has simple poles at the roots kn of the dispersion equation for a given w. The integration is performed by using the adaptive integration scheme, as discussed in Chapter 4 (Section 4.7.1.3). 6.2.2.4 Numerical Results and Comparison with Experiment Numerical results are presented for the surface response of an infinite plate of thickness H = 25.4 mm due to a vertical force acting on the top surface of the free–free plate (see Fig. 6.6). The surface-breaking crack is a vertical cut 6 mm deep and 0.5 mm wide. This problem was studied experimentally by Paffenholz et al. (1990). The material properties of the glass plate are: vL = 5.64 mm/ms, vS = 3.35 mm/ms, and density is 2.61 g/cm3. In the experiment, the plate was excited by impacting it with a steel ball (1.6-mm diam.) dropped from a height on the top surface of the plate. Thus, the location and orientation of the impact was known. However, the time history of the excitation was not known a priori, so it had to be determined experimentally. The source function is extracted from the signal recorded at a distance 5H from the point of impact in the absence of the crack. Fig. 6.7(a) shows the time history of the signal, and Fig. 6.7(b) shows its frequency spectrum (R(f)). Note that R(f) is given by S(f).D(f), where S(f) is the source spectrum and D(f) is the frequency spectrum of the vertical displacement of the receiver point on the top surface due to an impulsive (delta) time-dependent vertical force.

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Displacement, Uz

1000

0

1000

2000

0

1.0*10–4

2.0*10–4 3.0*10–4 Time (sec)

4.0*10–4

5.0*10–4

5*105

6*105

(a)

Amplitude Spectrum, Uz

0.0600

0.0400

0.0200

0.0000

0

2*105

3*105 4*105 Frequency (Hz) (b)

FIGURE 6.7  (a) Experimental signal at 5H from the source generated by a steel ball impact when there is no crack in the plate, and (b) corresponding frequency spectrum. (Reprinted with permission from S. Liu et al. 1991b, Fig. 2.)

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Elastic Waves in Composite Media and Structures

D(f) can be determined analytically by computing the Green’s function. Then S(f) is obtained as R(f)/D(f). As seen from Fig. 6.7(b), the frequency content of R(f) is confined to the range 0–0.5 MHz. So, to obtain sufficiently accurate representation of the Green’s displacement and the scattered field due to the crack, the maximum frequency is chosen to be 0.625 MHz, and 257 sampling points are taken. Next, the zero-scatter test (without crack) is performed to determine the largest finite element. The finite-element mesh for the surfacebreaking crack consists of 173 elements and 486 nodes. The area between C and B is discretized by one layer of 26 seven-node elements, with two nodes on B. The rest of the elements are either six-node triangular, or eight-node quadrilateral elements. No special singular crack-tip elements are used. Details of the FE mesh can be found in Liu (1991). Figures 6.8(a) and (b) show the experimentally measured vertical surface displacements in the near vicinity of the origin with and without the crack, respectively. The receiver was positioned at x = −5 mm to x = 5 mm in 1-mm intervals. The source was at x = −5H. The corresponding numerical simulations are shown in Figs. 6.9(a) and (b). Frequency spectra of the time history are shown in Figs. 6.10(a) and (b). Comparing Figs. 6.8 and 6.9, it is found that the numerical simulations and the experimental results are in very good agreement. The surface displacement changes only very little over this region if no crack is present. In the presence of the crack, the displacements have sharper and larger maximum amplitudes. Comparing the spectra without and with the crack, it is found that the introduction of the crack causes a remarkable change in the spectra at around 0.1 MHz. Note that there is a peak appearing in Fig. 6.10(b) at the first antisymmetric cutoff frequency (0.0659 MHz). Figure 6.10(a) shows a sharp peak at 0.11 MHz that corresponds to the cutoff frequency of the first symmetric thickness stretch mode.

6.2.3 Transient Scattering by a Delamination in a Composite Plate 6.2.3.1 Hybrid Method As mentioned above, the hybrid method described in Section 6.2.2 can be used to find a solution for the scattered field due to any arbitrarily shaped crack or inhomogeneity. In particular, it can be used to model scattering by a delamination in a fiber-reinforced composite plate. This was done by Datta et al. (1992b). In the following discussion, the solution to the problem of transient scattering by a delamination (crack) parallel to the free surfaces of a plate is presented. The geometry of the plate with a delamination and its dimensions are shown in Fig. 6.11. Also shown are the contours C and B used to develop the equations governing the displacements at the nodes in RI ∪ B. These are given by equation (6.2.57). As in the previous section, an adaptive integration scheme is used to obtain the Green’s functions in the frequency domain. The system of equations (6.2.57) is solved by a biconjugate gradient method. Details are given in Datta et al. (1992b) (see the previous section). The plate is considered to be transversely isotropic with the x-axis as the symmetry axis. The plate specimen for which the numerical results are shown in the following is made of graphite fiber/epoxy with elastic constants, c11 = 160.7, c33 = 13.96, c55 = 7.07, and

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4 No crack 7

Receiver at Location

Normal Surface Displacement 10 mm

1

10 0.00

0.04

0.08

0.12

0.16 0.20 0.24 Time (ms) (a)

0.28

0.32

0.36

0.40

4 Crack 7

Receiver at Location

Normal Surface Displacement 10 mm

1

10 0.00

0.04

0.08

0.12

0.16 0.20 0.24 Time (ms) (b)

0.28

0.32

0.36

0.40

FIGURE 6.8  Experimental normal surface displacements as a function of time (a) without and (b) with a crack in the near field studied by Paffenholz et al. (1990). (Reprinted with permission from S. Liu et al. 1991b, Fig. 3.)

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Elastic Waves in Composite Media and Structures 10000

Z-Displacement

5000

0

–5000

–10000

0

2.00*10–4 Time (sec)

4.00*10–4

(a) 10000

Z-Displacement

5000

0

–5000

–10000

0

2.00*10–4 Time (sec)

4.00*10–4

(b)

FIGURE 6.9  Numerical simulation corresponding to Figure 6.8: (a) without the crack and (b) with the crack. (Reprinted with permission from S. Liu et al. 1991b, Fig. 4.)

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Z-Amplitude

0.1

0.0

–0.1

–0.2 0.00

0.10

0.20 0.30 Frequency (MHz)

0.40

0.50

0.40

0.50

(a) 0.2

Z-Amplitude

0.1

0.0

–0.1

–0.2 0.00

0.10

0.20 0.30 Frequency (MHz) (b)

FIGURE 6.10  Frequency spectra corresponding to Figure 6.9: (a) without the crack and (b) with the crack. (Reprinted with permission from S. Liu et al. 1991b, Fig. 5.)

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Elastic Waves in Composite Media and Structures

ct pa

ce

for

Im

H 0.75 H H B

Composite plate

H/8

x

1.25 H C

Ri

Crack

H

RE

y z

FIGURE 6.11  Composite plate with delamination. (Reprinted with permission from Datta et al. 1992b, Fig. 1.)

c13 = 6.44, all in units of GPa. The density is 1.8 g/cm3. Thus, the longitudinal and shear wave speeds along the fiber direction are 9.4 mm/ms and 1.2 mm/ms, respectively. The plate thickness (H) is 5.08 mm, and the crack of length 6.4 mm is located at a depth of 0.635 mm from the top surface. The line load is applied in the z-direction on the top surface at x = −1.75H. The hybrid method used here was tested for its accuracy for different finite-element meshes by performing the zero-scattering test (when there is no crack). This was done by giving the crack the same material properties as those of the surrounding medium. The relative error of calculated total displacement and the incident displacement field was kept within 5% by adjusting the number of finite elements. Generally, 10 elements per wavelength were enough to obtain the accuracy desired. There were 400 elements and 1274 nodes used for the computation. The number of nodes on the boundary B was 49. The number of iterations used varied from 1717 to 2558, depending upon the range of frequencies considered here. Frequency spectra of vertical surface displacements at x = 0 and x = 6.5H due to an impulsive load are shown in Figs. 6.12 and 6.13. These figures show some resonance peaks at values of k2H (=wH/cs), where cs is the shear wave speed in the x-direction, that are related to some cutoff frequencies of the plate and the portion of the plate between the delamination and the bottom surface. The sharp peak at k2H = 1.2 in Fig. 6.12 is due to the resonance of the finite plate above the delamination. Keer et al. (1984) and Cawley and Theodorakopoulos (1989) studied this effect. They proposed that resonance frequency for a horizontal crack could be predicted by using plate theory, with the plate having the length and thickness equal respectively to the crack length and its depth below the free surface. The natural frequencies of this plate are k2H = 1.13 and k2H = 2.56 for simple and clamped supports, respectively. These provide the lower and upper bounds for the peak seen in Fig. 6.12. Other pronounced peaks seen in Figs. 6.12 and 6.13

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With crack Without crack

Z - Amplitude

2.0

1.5

1.0

0.5

0.0

0

5 A1 S1

10 15 S2 A2 A3 S3 S4 Normalized Frequency (K2H)

A4

20

FIGURE 6.12  Frequency spectrum of surface displacement of a graphite–epoxy plate with and without delamination due to an impulsive load. Receiver is at the origin. (Reprinted with permission from Datta et al. 1992b, Fig. 6.)

correspond to the cutoff frequencies of S1, A2, and S3 modes, as noted in these figures. These also show significant differences between the spectra with and without the crack. 6.2.3.2 Boundary Element Method As mentioned before, various boundary-element methods (BEM) have been used to study scattering problems. This section presents a brief account of the BEM that was used by Zhu et al. (1998) to model scattering by a horizontal crack (as discussed above). A multidomain technique (Blanford et al. 1981) was used to overcome the difficulties encountered in evaluating the hypersingular integrals that arise in dealing with problems involving cracks via integral equations. Details of the technique to evaluate the Cauchy Principal Value integrals by introducing the fictitious-source concept can be found in Zhu et al. (1996). The Green’s function in the frequency domain is obtained by using the modal summation technique described in Zhu et al. (1995a, 1995b) and is given by equation (4.7.37). The total displacement at any point x due to a force acting at x is obtained from equation (6.2.52) as

∫

ciju j (ξ ) = [Gij ( x ,ξ )t j ( x ) - Hij ( x ,ξ )]dS + uif (ξ )

S

(6.2.60)

where Gij and Hij are the Green’s displacement and stress tensors due to a unit harmonic line load applied at x. The variable cij is the shape factor associated with the boundary.

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With crack Without crack

Z - Amplitude

2.0

1.5

1.0

0.5

0.0

0

5

10 15 Normalized Frequency (K2H)

20

FIGURE 6.13  Frequency spectrum of surface displacement of a graphite–epoxy plate with and without delamination due to an impulsive load. Receiver is at 6.5H from source on the top surface. (Reprinted with permission from Datta et al. 1992b, Fig. 7.)

Equation (6.2.60) cannot, in general, be solved analytically, and numerical methods must be used instead. For this purpose, the boundary is represented by a series of elements connected to boundary nodes. With the spatial discretization, equation (6.2.60) can be written in the discrete form as

Hu = Gt + b

(6.2.61)

where matrix H contains the cij tensor and the Cauchy Principal Value of the traction kernel integrals. Matrix G contains weakly singular displacement kernel integrals. These are evaluated using the method described by Zhu et al. (1996). The evaluation of the integrals around the corner points of the boundary is complicated. A double-nodes technique (Brebbia et al. 1984) is incorporated here. See Zhu et al. (1998) for details. Numerical results are presented here for a horizontal crack in a uniaxial graphite– epoxy plate having the same properties as considered in Section 6.2.1. The geometry of the plate with the crack and the fictitious boundary (dotted line) are shown in Fig. 6.14. Thus, the plate is divided into two subdomains, as shown. In the first step, the plate is divided into 16 sublayers to compute the eigenvalues and eigenvectors for calculating the Green’s displacement tensors using equation (4.7.37). The associated stress tensors are then computed by using the constitutive equation. To overcome the difficulties encountered in the evaluation of the hypersingular integrals due to the delamination, the multidomain (Blanford et al. 1981) discretization and the boundary integral equations for each domain are used. The displacement continuity and

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O

x 2.52H

H/4

2.8H 2H Domain I

z

Domain II

FIGURE 6.14  Configuration of a composite plate with delamination and the domain decomposition for boundary element formulation. (Reprinted with permission from Zhu et al. 1998, Fig. 3.)

stress equilibrium conditions are employed on the fictitious boundaries. The delamination is assumed to be open with stress-free surfaces. The boundary element mesh has 124 quadratic elements and 204 nodes. Eight crack-tip singular elements (Blanford et al. 1981) are used. The problem was also solved by the hybrid method combining FE and modal expansion techniques (Zhu et al. 1995b). Results obtained by BE and the hybrid methods were found to agree (not shown) well (Zhu et al. 1998). Vertical displacement amplitudes for points at the top surface of the plate between x = −1.5H and x = 1.5H are shown in Fig. 6.15 for the nondimensional frequency Ω = 1.57. Here, Ω is defined as wH/ c55 /ρ . It is seen that the surface response in the presence of the crack shows pronounced maxima almost symmetrically

Z - Displacement

1.5

With crack Without crack

1

0.5

0 –1.5

–1

–0.5

0 x/H

0.5

1

1.5

FIGURE 6.15  Top-surface response at Ω = 1.57. (Reprinted with permission from Zhu et al. 1998, Fig. 6.)

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With crack Without crack

Z - Displacement

0.1

0

–0.1

–0.2

5

7.5

10 Cst/H

12.5

1.5

FIGURE 6.16  Transient response of point (1.26H, 0). (Reprinted with permission from Zhu et al. 1998, Fig. 9.)

located about the origin (above the center of the crack). Each peak is found to be located at L/4 distance from the origin. This can be used to estimate the crack length. Transient response of the plate surface at the point above the right edge of the crack (x = 1.26H, z = 0) is shown in Fig. 6.16. The time dependence of the load is given by

2 /2

f (τ ) = 2 /π e -(τ -τ 0 )

sin(Ω c τ )



(6.2.62)

where the normalized time t = ( c55 /ρ )t/H. The time delay t 0 is taken as 3.0. The normalized central frequency Ωc is chosen to be 3.14. There is a significant difference between the signals for the plate with and without delamination.

6.2.4 Reflection of Plane-Strain Waves at the Free Edge of a Semi-Infinite Composite Plate Reflection and transmission of an incident propagating wave mode after scattering from symmetric surface-breaking cracks in a composite plate are analyzed in Section 6.2.1. In this section, reflection of such an incident wave from the free edge (i.e., when the cracks go through the thickness of the plate) of the composite plate is considered. Reflection from the free end of a semi-infinite homogeneous isotropic plate has been considered by several investigators over the last 30 years. Among these are the studies by Torvik (1967), Wu and Plunkett (1967), Auld and Tsao (1977), Koshiba et al. (1983), and Gregory and Gladwell (1983). In the last publication, the authors also considered reflection from the fixed edge. Recently, Cho and Rose (1996) have provided further insight into this reflection problem. Reflection from the free edge of a composite plate was considered by Karunasena et al. (1991d).

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x y

Interface number 1

Free surface

z Layer

Free edge Zi

Z i+1

i i+1

I -th sublayer layer

Incident wave

N+1

H Free surface

FIGURE 6.17  Geometry of the laminated plate with free edge. (Reprinted with permission from Karunasena et al. 1991d, Fig. 1.)

Consider the laminated composite plate as described in Section 6.2.1 with a free edge at x = 0 (see Fig. 6.17). A train of plate waves traveling in the negative x-direction is assumed to be incident on the edge. The reflected field consists of a finite number of propagating and an infinite number of nonpropagating modes. Wave functions required for the reflection analysis are obtained by considering the plane-strain wave propagation in the corresponding infinite plate. This is discussed in Section 6.2.1. It is assumed that each lamina is orthotropic, with the symmetry axes coinciding with the x-, y-, and z-axes. The dispersion equation is given by equation (6.2.5). Traction-free conditions at interface 1 and equation (6.2.5) give the components of the mth eigenvector at interface 1 as Q1Tm = 〈1 - P31 /P32 0 0〉



(6.2.63)



Then, applying equation (6.2.3) at successive interfaces, the mth mode eigenvector is obtained as

QTm = Q1Tm QT2m … QT( N +1)m

(6.2.64)



where

QTim = 〈uim wim σ zzim σ zxim 〉, i = 1,…, N + 1, m = 1,…, M



(6.2.65)

The eigenvector Qm is normalized by dividing by a factor gm given by N +1

gm =

∑ (| u | + | w im

i =1

2

|2 )

(6.2.66)

im



Now, let the incident wave be the pth mode corresponding to the wave number kp. After this wave strikes the edge, a reflected wave field is generated. The displacement

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vector corresponding to this reflected field can be approximated by the modal sum of a finite number of modes M in the form (see equation [6.2.6]) M

qr =

∑B q e m

m

ikm x

, x >0

(6.2.67)



m =1

Here,

q Tm = 〈u1m w1m … u( N +1)m w( N +1)m 〉

(6.2.68)



Then the reflected wave field at x = 0 is given by q r0 = GB

with

(6.2.69)



G = [q1 q 2 … q M ]



(6.2.70)



B = 〈 B1 B2 … BM 〉



(6.2.71)

Using the constitutive equation and the solution of the wave equation, the stress component s xx can be written as



σ xx =

c55 [c (de - cf )u + ik( f - be )σ zz ] ∆1 55

(6.2.72)



The constants c, d, e, f, and ∆1 are given by c = (1 - δ )k 2 - β r12 a, d = (1 - δ )k 2b - β r22

(6.2.73)

e = (1 - δ )ar12 - λ k 2 , f = (1 - δ )r22 - λ k 2b

 r2  ∆1 = c55ikβ r22  12 ab - 1  r2 



where l, d, and b are defined in equation (4.2.12), and r1 and r2 are given by equation (4.2.18). The variables a and b replace A and B, respectively, appearing in equation (4.2.16). Since uim and s zzim for the mth mode are known from equation (6.2.64), equation (6.2.72) can be used to calculate s xxim. It should be noted that s xxim is discontinuous at the interfaces between layers having different material properties. The force vector at the edge due to the reflected field is now formed as

(6.2.74)

R r = -FB

where F is the same as F+ given in equation (6.2.14). The edge force vector due to the incident field can be written as

R in = Apin Fp-



(6.2.75)

The vector Fp- is obtained from the pth column of F after replacing each x-direction force component by its negative value.

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253

The traction-free edge condition requires that R = R r + R in = -FB + ApinFp- = 0





(6.2.76)

A variational solution to the problem can be obtained by applying the principle of virtual displacement, as in Wu and Plunkett (1967). This gives

δ qT R = 0



(6.2.77)



where d denotes variation. It is noted that the total displacement field at x = 0 is given by q = q r0 + q in0



(6.2.78)



and

δ q = δ q r0



(6.2.79)



Substituting equations (6.2.76) and (6.2.79) in equation (6.2.77), and making use of equation (6.2.69), it is found that B = Apin GT F -1GT Fp-



(6.2.80)



The normalized amplitude is defined as Am = Bm /Apin



(6.2.81)



To assess the accuracy of the numerical results, the energy conservation principle is used. The reflected energy is carried only by the propagating modes. The time-averaged value of the energy flux associated with the jth reflected propagating mode through the plate cross section is given by I rj = ω | B j |2 J j ,



j ≤ N pr



(6.2.82)

where J j = Im FjT . q j 





(6.2.83)

Let Ej be the proportion of incident energy transferred into the jth reflected mode. Then E j = I rj /I pin





(6.2.84)

The percentage error in energy balance, e, is given by



 ε =  I pin  

N pr

∑ j =1

 I rj 100/I inp  

(6.2.85)

The principle of energy conservation requires that e should be equal to zero. The technique outlined above was used to investigate reflection of the symmetric mode S 0 from the free edge of a homogeneous isotropic plate with Poisson ratio ν = 0.25.

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Results obtained here are compared with those given by Gregory and Gladwell (1983). Since the problem is symmetric, only the half-thickness of the plate is considered in the analysis. The total number of sublayers N, used to compute the number of modes M needed for the modal expansion, plays an important role in the accuracy of the results. To select a suitable value for N, the quantity Jj given by equation (6.2.83) was computed at a few selected low, intermediate, and high frequencies until convergent values were obtained. In this way, a reasonable value for N through the half-thickness was found to be 50. Thereafter, the reflection problem was solved at the selected frequencies by the variational method by increasing the number of modes. Figure 6.18 shows the comparison of the partition of energy between various modes predicted by the presented analysis and those reported by Gregory and Gladwell (1983). 1.2

Present Mode 1 Mode 2 Mode 3 Mode 4

1.0

Ej

0.8

Gregory and Gladwell (1983)

0.6 0.4 0.2 0.0 1.48

1.50

1.52

1.54 Ω

1.56

1.2

1.58

1.60

Gregory and Gladwell

1.0

Ej

0.8 0.6 0.4 0.2 0.0

1.6

2.0

2.5

3.0

Ω

3.5

4.0

4.5

5.0

FIGURE 6.18  The partition of energy Ej between reflected modes in a homogeneous isotropic plate due to the first symmetric (S 0 ) incident mode. (Reprinted with permission from Karunasena et al. 1991d, Fig. 3.)

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Scattering of Guided Waves in Plates and Cylinders 1.2 Mode 1

1.0

Mode 2 Mode 3

Ej

0.8 0.6 0.4 0.2 0.0 2.196 2.197 2.198

2.199 2.200

2.201 Ω (a)

2.202

2.203

2.204

2.205

5.0

5.5

6.0

6.5

1.2 1.0

Ej

0.8 0.6 0.4 0.2 0.0 2.0

2.5

3.0

3.5

4.0

Ω (b)

4.5

FIGURE 6.19  The partition of energy Ej between the modes for the homogeneous graphite– epoxy plate due to the incident S 0 mode. (Reprinted with permission from Karunasena et al. 1991d, Fig. 4.)

The modal expansion consisted of 21 modes. For the range of Ω considered here, |e| was found to be less than 0.18%. The comparison of the results is seen to be excellent. Results for the reflection coefficient |A1n| (not shown) were found to also agree well with those of Gregory and Gladwell (1983). The energy distribution among various modes for edge reflection in a uniaxial graphite–epoxy plate is shown in Fig. 6.19. The modal expansion consisted of 20 modes. For the range of frequency 0 ≤ Ω ≤ 2.196, not shown in Fig. 6.19, |e| was less than 0.05%. For the range of frequency in which results are presented here, |e| was less than 0.88%.

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Elastic Waves in Composite Media and Structures 1.2 1.0

|A1|

0.8 0.6 0.4 0.2 0.0 0.0 0.5

1.0 1.5

2.0

2.5 3.0

3.5 Ω (a)

4.0

4.5

5.0

5.5

6.0

6.5

40 35 30

|A2|

25 20 15 10 5 0 2.1505

2.1510

2.1515

2.1520 Ω (b)

2.1525

2.1530

2.1535

FIGURE 6.20  The amplitude |Aj| for the homogeneous graphite–epoxy plate due to the incident S 0 mode: (a) amplitude of |A1|, (b) amplitude of |A2|. (Reprinted with permission from Karunasena et al. 1991d, Fig. 5.)

Note that the range 2.197 < Ω ≤ 2.2041 is the backward phase-velocity region for the third mode. It is observed that at the first cutoff frequency, Ω = 2.2041, only the second mode carries energy. At the second cutoff frequency, Ω = 3.142, only the first mode carries energy. Then, at the third cutoff, Ω = 6.283, all three modes carry energy. It is found that in the range 2.4 < Ω < 5.9, the first and third modes share almost the entire reflected energy. In Fig. 6.20(a), the variation of amplitude |A1| with Ω is shown. After a careful search, it was found that edge resonance occurred in the second mode near Ω = 2.1520.

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257

The variation of amplitude |A2| near the resonant frequency is shown in Fig. 6.20(b). At Ω = 2.1520, by increasing M from 20 to 30, e changed from 0.14% to 0.01%, and only a 0.11% increase in |A2| was observed.

6.3 Scattering in a Pipe Circular tubes and pipelines are widely used in chemical, transportation, and energy industries. In many applications, these span long distances and are subject to harsh environments. Because of the criticality of these structures, nondestructive evaluation of damage has been receiving attention from many investigators. Among the various techniques used to characterize material property changes and defects, ultrasonic techniques are the most widely used. Significant advances have been made in recent years in the analysis and application of ultrasonic inspection systems. References are made to the works by Rose (1999, 2002) and Cawley et al. (2002, 2003). Guided waves in cylinders are similar in nature to those in plates. However, as discussed in Chapter 5 (Sections 5.3–5.6), many more modes are excited in cylinders than in plates. This makes appropriate mode selection critical for the success of an ultrasonic technique for pipeline inspection. Dispersion of guided waves in cylinders has been analyzed in Chapter 5. A semianalytic finite element (SAFE) method has been found to be very effective for analyzing dispersive guided-wave propagation and scattering in plates and cylinders. Studies of wave propagation and reflection in a semi-infinite cylinder and wave propagation in a cylinder with a region of material inhomogeneity have been reported by Kohl et al. (1992a), Rattanawangcharoen (1993), and Rattanawangcharoen et al. (1994, 1997). These studies used superposition of modes, propagating and nonpropagating, to satisfy the conditions at the boundaries. In the last-mentioned investigation and in Zhuang et al. (1997), a hybrid technique—similar to that discussed in Section 6.2.1 for plates, combining finite elements in the interior region containing the inhomogeneities and defects, and wave function expansions in the exterior regions—was used. Olsson (1994) used the T-matrix method to analyze reflection and transmission of guided waves in a solid cylinder containing a spherical inclusion. Bai et al. (2001) studied scattering by circumferential cracks in an isotropic homogeneous pipe using finite element representation of the cross section of the pipe containing the cracks and modal expansion outside of this region. Alleyne et al. (1998) and Lowe et al. (1998) reported experimental and modeling results for the problem of scattering by circumferential notches in a steel pipe. They used a finite-element method to model the scattering problem. In the following section, analysis of scattering by circumferential cracks and comparison with experiments are presented.

6.3.1 Scattering by Circumferential Cracks in a Pipe Scattering of guided propagating waves in a homogeneous isotropic hollow cylinder by a circumferential crack is considered here. The crack is assumed to be located at z = 0

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y

d R0

R1 θ0

x

z

FIGURE 6.21  Geometry of a steel pipe. R i, inner radius; Ro, outer radius; d, crack depth; q o, halfcircumferential crack angle. (Reprinted with permission from Bai et al. 2001, Fig. 1.)

(see Fig. 6.21). The half-circumferential crack angle is taken to be q 0, and the depth of the crack is d. The pipe has the outer and inner radii, Ro and R i, respectively, and the thickness H (= Ro − R i). To calculate the wave functions, the pipe is discretized into N concentric cylindrical layers (see Chapter 5, Section 5.2). 6.3.1.1 Wave Modes Time-harmonic wave propagation is considered. The displacement components in cylindrical polar coordinates (r, q, z) can be obtained from the general results for a transversely isotropic cylinder, as seen in equation (5.3.8). These are ∞

∞

ur =

∑

urme imθ e i (ξz -ωt ) ,

m =-∞

uθ =

∑

∞

uθme imθ e i (ξz -ωt ) ,

and uz =

m =-∞

∑u

zm

e imθ e i (ξz -ωt )

m =-∞



(6.3.1)

The circumferential modal components urm, uqm, and uzm are given by urm = fm′ (r ) - ξh1m (r ) +

im h (r ) r 2m

m  uθm = i  fm + ξh1m - h2′m  r 



  m +1 uzm = i ξ fm h1m - h1′m  r  

(6.3.2)



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where fm = A1m Hm(1) (α r ) + B1m Hm( 2 ) (α r ) h1m = A2m Hm(1+) 1 (β r ) + B2m Hm( 2+)1 (β r )

(6.3.3)

h2m = A3m Hm(1) (β r ) + B3m Hm( 2)) (β r )

α 2 = ω 2 /c12 - ξ 2 , β 2 = ω 2 /c22 - ξ 2





A prime sign (′) denotes differentiation with respect to r, m is the circumferential wave number, and x is the axial wave number. Hm(1) and Hm( 2 ) are the Hankel functions of the first and second kinds, respectively. The variables c1 and c2 are the longitudinal and shear wave speeds in the medium. The stress components Σrrm, Σrqm, and Σrzm can now be calculated, as described by equation (5.3.10). Then, the displacement–traction vector at the kth interface can be related to that at the (k+1)th interface (using continuity at the interfaces) as (see equation [5.3.18]) U km  U km+1  m = P  k    Skm  Skm+1 



(6.3.4a)

where Pkm is the transfer matrix for the kth sublayer. Repeated application of equation (6.3.4) to every sublayer of the composite cylinder having N sublayers gives U Nm+1  U1m  = P   m m S1  SN +1 



(6.3.4b)

The dispersion equation for the cylinder is obtained by setting the traction-free boundary conditions at R = R i and R = Ro. This equation relates the axial wave number x with the frequency w for a circumferential wave number m. These will be denoted by xmn for a givenw. For a solid circular cylinder (ri = 0), equation (6.3.4a) will be modified, as discussed in Chapter 5 (Section 5.3). 6.3.1.2 Modal Expansion Assume that an incident propagating mode traveling in the negative z-direction from z = + ∞ strikes the crack at z = 0. There will be reflected and transmitted waves moving in the ±z-directions. These wave fields can be expanded in a finite number of propagating modes and an infinite number of nonpropagating modes. Due to the symmetry of the crack, only the reflected waves need be considered. The displacements and stresses associated with the reflected field will be written in the form of modal sums as M



ur =

Nm

∑∑

M

urmn aklmn e iξmnz +imθ -iωt =

m =- M n =1

∑U

m =- M

rm

(r )Em (z )aklm e imθ -iωt

(6.3.5a)

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M

uθ =

∑∑

M

uθmn aklmn e iξmnz +imθ -iωt =

m =- M n =1 M



Nm

uz =

Nm

∑U

θm (r )Em (z )aklm e

∑ ∑u

(6.3.5b)

imθ -iωt

m =- M



M

a

zmn klmn

e iξmnz +imθ -iωt =

m =- M n = 0

∑U

zm

(r )Em (z )aklm e imθ -iωt

m =- M

(6.3.5c)

and M



∑S

σ zr =

zrm

(r )Em (z )aklm e imθ -iωt

m =- M

(6.3.6a)

M



∑S

σ zθ =

zθm

(r )Em (z )aklm e imθ -iωt

(6.3.6b)

m =- M M



σ zz =

∑S

zzm

(r )Em (z )aklm e imθ -iωt

m =- M

(6.3.6c)



Here, U rm (r ) = [urm1 (r ) urm 2 (r ) … urmNm (r )]

(6.3.7)

Uθm (r ) = [uθm1 (r ) uθm 2 (r ) … uθmNm (r )]

U zm (r ) = [uzm1 (r ) uzm 2 (r ) … uzmNm (r )] Szrm (r ) = [σ zrm1 σ zrm 2 … σ zrmNm ]

(6.3.8)

Szθm (r ) = [σ zθm1 σ zθm 2 … σ zθmNm ]

Szzm (r ) = [σ zzm1 σ zzm 2 … σ zzmNm ]





aklm = 〈aklm1 aklm 2 … aklmNm 〉

T



Em (z ) = diag[e iξm1z e iξm 2z … e

iξmN z m

]

(6.3.9)

The constants aklmn are as-yet-undetermined complex numbers. The first two subscripts denote, respectively, the axial and circumferential wave numbers associated with the incident field, and the last two correspond to those associated with the reflected field. For computational purposes, the continuous functions in the expansions in equations (6.3.5) and (6.3.6) are evaluated at discrete points on the cross section. The cross section is divided into six-node elements with uniform distribution in both circumferential and radial directions (see Fig. 6.22). The number of divisions in the radial and circumferential directions are p and q, respectively. To represent curvature in the circumferential direction more accurately, three nodes are taken on the circle, whereas two nodes are used in the radial direction. Consistent forces at the nodes are used (Bathe 1982). The expansions in equations (6.3.5) and (6.3.6) satisfy the traction-free conditions on the outer and inner surfaces of the pipe. So, only the boundary conditions at z = 0 have

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x

FIGURE 6.22  A typical mesh in the cross section. The shadow region represents the crack. (Reprinted with permission from Bai et al. 2001, Fig. 2.)

to be satisfied. Because of the occurrence of the crack on this face, care must be taken on this part of the cross section when dealing with boundary conditions. Traction-free boundary conditions are imposed on the crack face for both symmetric and antisymmetric cases. Therefore, the boundary conditions for the symmetric case are SCI  SCR  S = S + S =   +   = 0,, SNI  SNR  I



(6.3.10)

at z = 0

R



and for the antisymmetric case, these are



 ACI   ACR  A = AI + AR =   +   = 0,,  ANI   ANR 

at z = 0

(6.3.11)

Here,  f rΧ      SNΧ =  fθΧ  ,   uzΧ 

Χ = I or R



 f rΧ      SCΧ =  fθΧ  ,    f zΧ 

u      ANΧ = uθΧ  ,    f zΧ 

Χ = I or R



f      ACΧ =  fθX  ,    f zX  X r

(6.3.12)

Χ r

(6.3.13)

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where fr, fq , and fz are the consistent force components in the r-, q-, and z-directions, respectively, at the z = 0 boundary. The superscripts I and R refer to the incident and reflected fields, respectively. The subscripts C and N represent whether the point considered is located in the cracked or noncracked regions, respectively. Without loss of generality, we assume that the nodal sequence is arranged such that the first PC points are located in the cracked region and all the rest P N (= P – PC) points are located in the noncracked region. Let us consider the symmetric case first. It may be noted that when the mixed displacement and force components fr, fq , and uz are known, the dual components ur, uq , and fz are unknown at a point in the noncracked region, and vice versa. For the cracked region, all the force components are known, and the corresponding displacement components are unknown. The vectors SCR and SNR appearing in equation (6.3.12) may be written as

SCR = GCR a,

SNR = GNR a

(6.3.14)



where GCR = GCR,- M GCR,- M +1 … GCR, M  (6.3.15)

GNR = GNR ,- M GNR ,- M +1 … GNR , M 

a = 〈akl ,- M akl ,- M +1 … akl , M 〉T



Matrices GCR,m and GNR ,m are of sizes 3PC × Nm and 3P N × Nm, respectively. They are given by

GCR,m

 FCR,rm   FNR,rm      =  FCR,θm  , GNR ,m =  FNR,θm       R   R   FC ,zm  U N ,zm 

(6.3.16)

The matrices FCR,rm , FCR,θm , and FCR,zm are PC × Nm matrices of the force components; FNR,rm , FNR,θm , and U NR ,zm are P N × Nm matrices of the appropriate force and displacement components. These are given in Bai et al. (2001). The total number of wave modes, NT, considered in the wave function expansion is m

NT =



∑N

N =- m

m

(6.3.17)



and P is the total number of nodes in the cross section z = 0, PC is the total number of nodes in the cracked region, and P N (= P − PC) is the number of nodes in the noncracked region. In a similar manner, the incident wave field SI can be constructed as

SCI = aklI g CI ,kl ,

SNI = aklI g NI ,kl



(6.3.18)

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Here, the vector g CI ,kl is obtained from the lth column of the matrix GCR,k after replacing m each r- and q-component force by its negative value. The vector g N, kl is obtained from R the lth column of the matrix GN ,k after replacing each z-direction force component by its negative value. The variable aklI is the amplitude of the incident wave. Equation (6.3.10) (see also equation (6.3.14)) is solved by the principle of virtual work. It is found that



 H CR    H R   N

*T

GCR   H CR    a = -aklI   G R  H R   N  N

*T

 g CI ,kl      I  g N ,kl 

(6.3.19)

where the superscript asterisk (*) denotes a complex conjugate. Here, H CR =  H CR,- M H CR,- M +1 … H CR, M  H NR =  H NR ,- M H NR ,- M +1 … H NR , M 



U CR,rm    H CR,m = U CR,θm  ,    R  U  C ,zm 

U NR ,rm    H NR ,m = U NR ,θm     R  F  N ,zm 

(6.3.20)



The block matrices appearing in the expressions for H CR,m and H NR ,m are given in Bai et al. (2001). It is noted that equation (6.3.19) is derived by using pointwise conditions. If we consider the idea of a transfer matrix from radius to radius, instead of point to point, calculations can be performed more efficiently. This idea is based upon the circular symmetry and the symmetry of the problem. Results presented in the following section were obtained by making this simplification (Bai et al. 2001). Equation (6.3.19) was derived for the symmetric problem. A similar equation can easily be derived for the antisymmetric case. Once these two are solved, the reflection and transmission coefficients, Rkl,mn and Tkl,mn, are obtained as



Rkl ,mn =

S A aklmn + aklmn , I 2akl

Tkl ,mn =

S aklmn - akAlmn 2aklI



(6.3.21)

The numerical accuracy of the coefficients is checked by the principle of energy conservation. 6.3.1.3 Numerical Results The method described above has been used to study scattered reflected fields due to a circumferential crack with different half-angles (q 0) and radial depths (d). It is assumed here that the crack has broken the outer surface. However, the method can be used for buried cracks as well. The problem studied here was considered by Alleyene et al. (1998)

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Elastic Waves in Composite Media and Structures 20

F(1, 8)

Phase Velocity (mm/µs)

18 16 14

F(1, 5)

12

L(0, 3)

10 8

F(1, 3)

6

0

L(0, 4)

L(0, 5) F(1, 7)

F(1, 4)

F(1, 2) L(0, 2)

4 2

F(1, 6)

L(0, 1) F(1, 1) 0

50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 Frequency (kHz)

FIGURE 6.23  Phase velocity vs. frequency for a steel pipe. H/R = 0.135, ν = 0.287. (Reprinted with permission from Bai et al. 2001, Fig. 4.)

and Lowe et al. (1998) numerically (by using a finite-element method) and experimentally. The results presented here use the same geometric and material properties so that comparisons can be made between the model predictions and the experimental results. The material constants for all the examples discussed below are for steel with longitudinal and shear wave velocities c l = 5.96 km/s and cs = 3.26 km/s. The Young’s modulus is 216.9 GPa. The inner radius, R i, and thickness, H, of the pipe are taken to be 38 mm and 5.5 mm, respectively. This was one of the cases reported in Alleyene et al. (1998). Note that the Poisson ratio is 0.287 and H/R = 0.135. Figure 6.23 shows phase velocity vs. frequency for different propagating modes in this pipe. Note that there are also other modes in the frequency range considered here, but these are not shown. Alleyene et al. (1998) presented results for reflection of the incident mode L(0,2) in the frequency range 60–85 kHz. It is seen that this mode is nearly nondispersive in the frequency range 60–300 kHz. The phase velocity of the F(1,3) mode approaches that of L(0,2) at frequencies higher than about 75 kHz, and the phase velocity of the F(1,2) mode approaches the shear wave velocity at high frequencies. In the following discussion, results are presented when two different modes (L[0,2] and F[1,3]) are incident on the crack. Convergence of the numerical procedure was tested with different mesh sizes, and results were compared for several scattering problems. Different numbers of circumferential and axial modes were considered for different scattering geometries to test the convergence of the modal analysis. The incident wave was chosen as L(0,2). It was found that the results converged with the choice of mesh sizes such that p and q were 100 and 10,000, respectively. For best results, the number of modes was found to be M = 11, Nm = 51 for m = 0, and Nm = 40 for m = ±1, ±2, …, ±M. The convergence was tested by using the criterion of energy conservation. Reflection coefficients were calculated for two different crack depths, namely 0.5H and 0.55H. It was found that results for d = 0.55H agreed better with experiments, and

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Scattering of Guided Waves in Plates and Cylinders 0.40 0.35 Experimental Numerical 100% 50% 10%

R02,02

0.30 0.25 0.20 0.15 0.10 0.05 0.00 0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

Ω

FIGURE 6.24  Reflection coefficients for three different crack extensions: ——, crack depth = 0.5H; -----, crack depth = 0.55H. (Reprinted with permission from Bai et al. 2001, Fig. 5.)

the relative error was less than 6% for the full circumferential crack (q 0 = 180°). Using the same mesh and crack depth, the relative error for a 50% circumferential crack length was about 10%. Reflection coefficients for three different lengths and two different crack depths as functions of frequency are shown in Fig. 6.24. Here, Ω =wH/cs. Experimental results shown in this figure are found to agree reasonably well with numerical results. Reflection and transmission coefficients for mode conversions as functions of frequency when the crack length is 50% are shown in Fig. 6.25. The largest magnitude was found for |R02,01|, closely followed by |R02,02| and |R02,11|. These are seen to be not very sensitive to frequency in the range 0.64 ≤ Ω ≤ 0.90. For the transmission coefficients, |T02,02| is much higher in magnitude than the other converted mode coefficients. Finally, variations of the reflection and transmission coefficients with crack length at 70-kHz frequency are shown in Fig. 6.26. This figure shows that both |R02,01| and |R02,02| increase with increasing crack length, the former being higher in magnitude. Experimental results for the latter are seen to agree well with the model results. |T02,02| decreases with increasing crack length, whereas |T02,01| increases with increasing crack length. The former has the highest value for all crack lengths. Also interesting to note is that these coefficients are linear in L. This is corroborated by the experimental observations by Alleyene et al. (1998). Results for the incident F(1,3) mode were also computed. Those can be found in Bai et al. (2001). The method described here can also be used to study scattering by circumferential cracks in composite cylinders (Bai et al. 2002). Although this is well suited for the study of scattering by planar cracks located in a cross section of a cylinder, it is not applicable to problems of inhomogeneities and cracks that have both circumferential and axial extents. For these problems, the hybrid method combining finite elements for an inner region and modal expansion in the outer region is very effective

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Elastic Waves in Composite Media and Structures 0.30 0.25

R02,mn

0.20 0.15 0.10 0.05 0.00 0.60

0.65

0.70

0.75

Ω (a)

0.80

0.85

0.90

0.95

0.90

0.95

1.00 0.90 0.80

T02,mn

0.70

m = 0, n = 1 m = 0, n = 2

0.60

m = 0, n = 3 m = 0, n = 2

m = 0, n = 1

0.50 0.40 0.30 0.20 0.10 0.00 0.60

0.65

0.70

0.75

Ω (b)

0.80

0.85

FIGURE 6.25  Normalized reflection and transmission coefficients in a steel pipe. H/R = 0.135, ν = 0.287, crack length is 50% of the circumference, and crack depth = 0.55H: (a) reflection coefficient, (b) transmission coefficient. (Reprinted with permission from Bai et al. 2001, Fig. 8.)

(Rattanawangcharoen et al. 1997; Zhuang et al. 1997; Mahmoud et al. 2004). In the following subsection, a solution for scattering by cracks in a welded pipe using the hybrid method is presented.

6.3.2 Scattering by Cracks in a Welded Steel Pipe Guided-wave scattering by welds and cracks in plates was studied by Al-Nassar (1990) and Al-Nassar et al. (1991) using the hybrid FE and modal expansion technique developed in Section 6.2.1. Here, the same technique is used to model scattering by cracks in welded pipes. The problem of two identical semi-infinite steel pipes coaxially welded

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R02,mn

0.40 0.30 0.20

m = 0, n = 2 experimental data

0.10 0.00 0.00 0.10

0.20

0.30

0.40

0.50 (a)

0.60

0.70

0.80

0.90

1.00

0.40

0.50 (b)

0.60

0.70

0.80

0.90

1.00

1.00 0.90 0.80

T02,mn

0.70 0.60 0.50 0.40 0.30

m = 0, n = 1 m = 0, n = 2 m = 1, n = 1 m = 1, n = 2 m = 1, n = 3

0.20 0.10 0.00 0.00 0.10

0.20

0.30

FIGURE 6.26  Normalized reflection and transmission coefficients in a steel pipe as functions of the crack length at f = 70 kHz. H/R = 0.135, ν = 0.287, crack depth = 0.55H: (a) reflection coefficient; (b) transmission coefficient. (Reprinted with permission from Bai et al. 2001, Fig. 10.)

together by a V-shaped weld at z = 0 is considered. The circumferential crack is assumed to be at the interface between the weld region and the pipe. The welded region and the crack are contained in the interior region, bounded by two artificial annular boundaries B + and B − located at z+ and z − , respectively. Figure 6.27 shows half of the pipe in a plane (vertical) containing the z-axis. The wave numbers and corresponding modes of propagation in the homogeneous pipe for a given frequency are calculated by using the stiffness method II, which has been described in Chapter 5. For this purpose, the pipe was divided into a number of coaxial circular cylindrical sublayers, and the radial dependence of displacement was approximated by quadratic interpolation functions in the radial coordinate (see equation (5.5.3)). The weld region and the crack are assumed to be axisymmetric, so that the displacement is independent of q. Denoting by {Q� } the assembled nodal displacement components, the equation for

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Elastic Waves in Composite Media and Structures Interior region r B–

H

B+

R–

R+

H/16 Z–

Z+ z

o 3H/32

FIGURE 6.27  Geometry of the weld region of a welded steel pipe and the annular region for finite-element discretization. (Reprinted with permission from Zhuang et al. 1997, Fig. 2.)

� is given by equation (5.5.8). The solution to this equation is assumed as {Q} � = {Q } {Q} 0 ei(xz−wt). Then, {Q 0} satisfies the equation

(-ξ 2[K1 ] + iξ[K 2 ] - [K 3 ] + ω 2[ M ]){Q0 } = {0}



(6.3.22)

The equation obtained by setting the determinant of the matrix coefficient of {Q 0} to zero is the dispersion equation that is solved to obtain the eigenvalues (wave numbers) for fixed w. The eigenvectors composed of the displacement components at the discrete nodal points are obtained using the Rayleigh–Ritz-type procedure. The incident wave is assumed to be a propagating mode moving in the negative zdirection. Thus, it is given by {qin } = Ain {qi- }e -i (ξi z +ωt )





(6.3.23)

z ≥ z+

(6.3.24)

6.3.2.1 Scattered Field The scattered field in R+ is given by J

{q s } =

∑ A {q }e + j

+ j

i (ξ j z -ωt )

,

j =0



where

{q +j } = 〈u1 j w1 j … uNj w Nj 〉T



(6.3.25)

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Here, N is the number of nodes in the radial direction. Equation (6.3.24) can be written in the matrix form, after suppressing the factor e −iωt, as {q s + } = [G + ]{D + }



B

(6.3.26)



where

[G + ] = [{q1+ } {q2+ } … {q J+ }] D +j = A+j e



ik j z +

(6.3.27a)



(6.3.27b)



The stress components, s rz and s zz, at discrete points of the kth sublayer at the boundary B +, are obtained by using the stress–strain and strain–displacement relations. The stress vector associated with the reflected field at the boundary B + is obtained as {σ s + } = [σ ]{D + }



B

(6.3.28)



where [σ ] = [{S1 } {S2 } … {SJ }]

{S j }T = 〈σ rz 1 j σ zz 1 j … σ rzNj σ zzNj 〉T

(6.3.29)

The nodal force vectors are then constructed in the usual manner (see Section 6.2.1) as {P s+ } = [F + ]{D + }



B

(6.3.30)



Following the same procedure, the scattered displacement and force vectors at the boundary B − are obtained as {q s - } = [G - ]{D - }



B

{P s- } = [F - ]{D - }



B

(6.3.31)



(6.3.32)



Here, D -j = A -j e



- iξ j z -

(6.3.33)



The boundary displacement and force vectors for the incident wave are constructed in a similar manner. These are

+

{qin- } = Alin {ql- }e -iξl z

-

{P in+ } = - Alin {Fl - }e -iξl z

+



B

{P in- } = Alin {Fl - }e -iξl z B

(6.3.34b)



B



(6.3.34a)



B



{qin+ } = Alin {ql- }e -iξl z

-



(6.3.34c) (6.3.34d)

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The interior region between the boundaries B + and B − containing the weldment and the crack is modeled by conventional finite elements. This yields

δ {q }T [S]{q} - δ {qB }T {PB } = 0



(6.3.35)



where [SII ] [SIB ]   [S] = [K I ] - ω 2[ M I ] =  [SBI ] [SBB ]  



{q} = {qI } {qB } T



T

T

(6.3.36a) (6.3.36b)



and [KI] and [MI] are the global stiffness and mass matrices of the interior region, respectively. The variables {qI} and {qB} are the interior and boundary nodal displacements, respectively, and {P B} is the interaction force vector at the nodes on B. As in Section 6.2.1, imposing the continuity of displacement and force vectors at B + and B − , the following equations are obtained [SII ]{qI } + [SIB ]{qB } = 0

[G]T ([SBI ]{qI } + [SBB ]{qB }) = [G]T {PB }



(6.3.37a) (6.3.37b)



where



[G + ] [0]  [G] =  -   [0] [G ]

(6.3.38)

{ } { } {P } = { P } + { P }

(6.3.39)

and {qB } = qBin + qBs

in B

B

s B

Solving equation (6.3.37a) for {qI}, we find [qI ] = -[SII ]-1[SIB ][qB ]



(6.3.40)



Using equation (6.3.40) in (6.3.37b) and making use of equation (6.3.39), it is found that

(

{ } { }) = G  ({P } - S� {q }) T

[G]T  S�BB  qBs - PBs

in B

BB

in B

(6.3.41)

Finally, the equations for the modal coefficients {D+} and {D −} are obtained by using equations (6.3.26), (6.3.30)–(6.3.33), and (6.3.34) in equation (6.3.41). The reflection and transmission coefficients, Rlj and Tlj, for the jth mode when the lth mode is incident are defined as



Rlj =

A +j Alin



(6.3.42a)

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 A-j  in , A Tlj =  l  Aj 1 + in ,  Al



l≠ j (6.3.42b) l= j

The numerical accuracy of these coefficients are checked by energy conservation.

6.3.2.2 Numerical Results The hybrid numerical technique outlined above was used to study scattering by the welded region in a pipe with and without a crack. As shown in Fig. 6.27, a surface-breaking crack was assumed to be located at the interface between the weld and the pipe in the positive z-direction. The material constants for the weld material and the steel pipe are given in Table 6.1. Total thickness of the pipe was taken as H = 5.08 mm. The inclination of the weld boundary to the z = 0 plane was taken as q, which was assumed to take the values of 0°, 30°, and 37.5°. Also, the length of the crack was varied. For the computation of the guided mode wave numbers and mode shapes, the pipe was divided into 16 coaxial cylindrical sublayers. Nondimensional frequency and wave number are defined as Ω =wH/c s and g = xH, where Ω was varied from 0 to 5. The first three cutoff frequencies of axisymmetric modes of the pipe are: 0.170, 3.150, and 5.440. So, there are three propagating modes within the range of frequency considered. Numerical results for the amplitude of the reflection coefficient |R11| are shown in Fig. 6.28 for different shapes of the welded region with and without cracks. The reflection coefficients for different shapes of the weld are seen (Fig. 6.28(a)) to have sharp resonant peaks at the first cutoff frequency. These are absent (Fig. 6.28(b)) when there is a crack. Results for normal slope differ significantly from those for the V-shaped welds. It is seen that, for V-shaped welds without cracks, the reflection coefficients have pronounced maxima at frequencies that are higher than the first cutoff. The locations of the maxima depend on the slopes. In the presence of cracks, the reflection coefficients change considerably. For the V-shaped welds, these reach sharp peaks at the second cutoff frequency. It is interesting to note that, for V-shaped welds with the crack, the reflection coefficients have pronounced maxima after this second cutoff

TaBLE 6.1  Material Properties of Steel and Weld Elastic Stiffness (GPa)

Density, r

Region

c11

c13

c33

c55

c66

(g/cm3)

Steel Weld

278.00 229.00

112.00 160.00

278.00 262.00

81.00 82.00

81.00 81.00

7.80 7.80

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Elastic Waves in Composite Media and Structures 6

Vertical weldment

R11 (×10–2)

30° V-shaped weldment 37.5° V-shaped weldment 4

2

0

0

1

2

3 Ω (a) Without Cracks

4

5

8

R11 (×10–1)

6

4

2

0

H/4 vertical weldment

0

1

2

H/4 30° V-shaped weldment H/4 37.5° V-shaped weldment 3 4 5

Ω (b) With Cracks

FIGURE 6.28  Reflection coefficients vs. frequency for a V-shaped weldment with different slopes. (Reprinted with permission from Zhuang et al. 1997, Fig. 5.)

frequency. It is found that the reflection coefficient for a weld having the boundary slope 0 (q = 0°) increases with frequency, and it has a small peak at the second cutoff frequency and has a maximum afterward. Figure 6.29 shows the results for different crack lengths and welds having different boundary slopes. In the Figures 6.29(a) and (b) the vertical axes are |R11| (× 10 -1). Although the results for different weld shapes (with cracks) are generally similar, there are some significant differences in details. Figure 6.29(a) shows that, for a vertical crack (and normal weld), the reflection coefficient reaches sharp peaks at lower frequencies as the crack length increases. On the other hand, for an inclined crack (and V-shaped weld), the peak in the reflection coefficient becomes broader and occurs at lower frequencies with increasing crack length.

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Scattering of Guided Waves in Plates and Cylinders 8

H/8 vertical crack H/4 vertical crack H/2 vertical crack

6

4

2

0

0

10

1

2

3 Ω (a) Vertical Weldment

4

5

3 Ω (b) V-shaped Weldment

4

5

H/8 30° crack H/4 30° crack H/2 30° crack

8 6 4 2 0

0

1

2

FIGURE 6.29  Reflection coefficients vs. frequency for various crack lengths. (Reprinted with permission from Zhuang et al. 1997, Fig. 6.)

In conclusion, these results show that the reflection coefficients are sensitive to the weld geometry as well as to the crack slope. The hybrid method used here is found to be equally suited for the analysis of crack scattering in plates and pipes.

6.3.3 End Reflection of Guided Waves in a Circular Cylinder Reflection of guided waves from the free end of a semi-infinite plate has been analyzed in Section 6.2.4. The corresponding problem for a semi-infinite circular cylinder is analyzed in this section. The free-end reflection of axisymmetric waves in a homogeneous isotropic elastic solid rod was investigated experimentally by Oliver (1957). McNiven

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(1961) used a three-mode approximate theory to investigate the free-end reflection problem studied by Oliver (1957). Later, Zemanek (1972) presented detailed analytical solutions for dispersion of guided waves in circular cylinders and used five-, seven-, and nine-mode approximations for the end-reflection problem. These authors made careful investigations of the occurrence of the end-resonance when the first axisymmetric extensional mode L(0,1) was incident on the free end. The end resonant frequency was found to be just below the cutoff frequency of the second axisymmetric mode, L(0,2). More recently, Kim and Steele (1989) used a stiffness matrix, based on analytical solutions, to analyze guided-wave behavior for different end conditions. Gregory and Gladwell (1989) studied problems of generation and reflection of axisymmetric waves in a semi-infinite rod analytically. They also made a detailed study of the end-resonance frequency and its dependence on the Poisson ratio of the material. The free-end reflection of the extensional waves in homogeneous, isotropic, hollow cylinders was studied by McNiven and Shah (1967). The approximate three-mode theory was employed to predict the resonant frequency. Reflection of elastic waves from the free end of a semi-infinite circular cylinder is considered here. It is assumed that the cylinder is composed of composite layers perfectly bonded together. The reflected wave field is represented by the modal sum of a finite number of guided-wave functions. To obtain these eigenfunctions at discrete points through the thickness when the layers are isotropic, the propagator matrix approach is used (see Section 6.3.1). Also, for general applicability, stiffness method II is used to obtain the wave numbers for a given frequency, w, and the circumferential wave number, m (see Chapter 5, Section 5.5). The reflected field for z ≥ 0 can be written as J

{ } ∑A qmre =



jm

{q jm }e

i (ξ jm z -ωt )+imθ

z ≥0

,

j =0

(6.3.43)

where

{q jm }T = 〈u1 jm υ1 jm w1 jm … uNjm υ Njm w Njm 〉



(6.3.44)

N is the number of nodal points, and the superscript “re” stands for “reflected field.” The complex coefficients Ajm are to be determined so that the associated stress field together with the stress field associated with the incident wave satisfy the stress-free boundary condition at z = 0. The factor e −iωt+imq will be suppressed in the following discussion. The reflected wave field at z = 0 can then be written as

{q }

re m 0



= [G]{ A}

(6.3.45)



where

[G] = [{q1m } {q2m } … {q Jm }]



{ A}T = 〈 A1m A2m … AJm 〉





(6.3.46a) (6.3.46b)

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The stress components (s rz, s qz, s zz) at discrete points on the surface z = constant on the kth sublayer can be obtained by using the strain–displacement and stress–strain relations. The stress vector containing these stress components due to the reflected field at the free end can be written in the form

{R }

re m 0



= [F ]{ A}

(6.3.47)



The matrix [F] is composed of the traction components at the layers and is formed as [F ] = [{S1m } {S2m } … {SJm }]



(6.3.48)



with {S jm }T = 〈σ 1rzjm σ 1θ zjm σ 1zzjm σ 2rzjm σ 2θ zjm σ 2zzjm … σ Nrzjm σ Nθ zjm σ Nzzjm 〉 (6.3.49)



Now, for the incident field, consider one of the propagating modes traveling in the negative z-direction. Thus, in {q }e - i ( km z +ωt ) {qin } = Aim im



(6.3.50)



where the superscript “in” stands for the incident field. Then, the stress vector at the free end due to the incident field is

{R }

in m 0



{ }

in S in = Aim im

(6.3.51)

The traction-free condition at the free end requires that

{ } - {R }

{Rm } = Rmre

in m 0

0

= {0}



(6.3.52)

By minimizing the sum of squares of the residuals of {R}, the least-squares solution for the complex amplitude {A} is obtained as



 ro  in { A} = Aim  [F ]T [F ]rdr     ri 

∫

-1

 ro   [F ]T S in rdr  im    ri 

∫ { }

(6.3.53)

where the overbar denotes a complex conjugate. An alternative approach to determining the complex amplitude {A} is to employ the variational principle (Wu and Plunkett 1967). Using the principle of virtual displacement, one obtains

δ {qm }T0 {Rm } = 0



(6.3.54)

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where d denotes the first variation. The total displacement at the free end is

{ } + {q }

{qm }0 = qmre



in m 0

0

(6.3.55)



So, d{qm}0 = d{ qmre }0. The solution is obtained by using equations (6.3.45) and (6.3.54) in equation (6.3.52) as



 ro  { A} = Amin  [G]T [F ]rdr     ri 

-1

∫

 ro   [G]T S in rdr  im    ri 

∫ { }

(6.3.56)

The normalized reflection coefficient is defined by B(jm ) =



A jm in Aim

(6.3.57)



where B(jm ) gives the reflection coefficient for the jth mode due to the incident ith mode when the circumferential wave number is m. One of the physical quantities of interest is the reflected energy per unit incident energy. The reflected energy is carried by the propagating modes only. The timeaveraged value of the energy flux associated with the jth reflected propagating mode through the cross section of the cylinder is given by dropping the index m,  ro  E j = ω |A j|2 Im  {Fj }T .{q j }rdr      ri

∫



(6.3.58)

for j ≤ Npr, Npr being the number of propagating modes. The energy flux due to the incident field can be written as E =ω A in i

in i



2

  ro Im   Siin . qiin  rdr      ri

∫ { }{ }

(6.3.59)

Thus, the proportion of the incident energy transferred to the jth reflected mode is given by Iij =



Ej Eiin

(6.3.60)

Since there is no energy dissipation at the free end, the percentage difference in energies carried by the incident and reflected fields should be zero, i.e.,



 ε =  Eiin  

N pr

∑ j =1

 E j  100/Eiin = 0  

(6.3.61)

The accuracy of the numerical results is checked by this test.

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6.3.3.1 Numerical Results The following three examples are considered:

1. A homogeneous isotropic elastic rod 2. A two-layered isotropic hollow cylinder 3. A four-ply (+15°/−15°/+15°/−15°) graphite–epoxy hollow cylinder

In all examples, H and R are the total thickness and mean radius of the cylinder, respectively. The frequency and wave number are normalized as Ω=

ω , ω ref

γ=

ξ ξref



(6.3.62)

where w ref and xref are the reference frequency and wave number, respectively. The total number of sublayers N used to compute the discrete eigenvectors and the number of modes J used in the modal expansion are very important factors for the accuracy of the results. Values of these are chosen so that the reflection coefficient {A} in equation (6.3.53) or (6.3.56) and the proportion of energy carried by each reflected mode in equation (6.3.58) converge. Example 1 Results for the reflection of the first symmetric mode in an elastic isotropic solid rod from its free end are shown in Fig. 6.30. The Poisson ratio of the solid is ν = 0.25. The reference frequency and wave number are, respectively, taken asw ref = c l/H and xref = 1/H. A full discussion of the frequency spectra for this case is given by Onoe et al. (1962). See also Meitzler (1965) for a detailed discussion of the first three branches of extensional motion in plates and cylinders. The first three cutoff frequencies are Ω = 1.931, 2.069, and 2.212. The propagator matrix approach with 20 sublayers (corresponding to 42 degrees of freedom) and 21 modes was employed in this example. Figures 6.30(a) and (b) show, respectively, the normalized amplitude Bj and the proportion of energy Ij for each reflected mode. Comparison of the energy partition among the propagating modes above the first cutoff frequency with the results presented by Gregory and Gladwell (1989) shows excellent agreement. For the range of frequency considered in Fig. 6.30, |e| was less than 0.5%. From the numerical experimentation, it was observed that both the least-squares and variational methods gave the same results. The end resonant frequency was found to be 1.644, which agreed with that obtained by Gregory and Gladwell (1989). Example 2 Reflection of the first propagating mode with m = 1 in a two-layered isotropic hollow cylinder is considered next. The properties of the two layers are (Armenàkas 1971): ν1 = ν2 = 0.3, m1/m2 = 1, r1/r2 = 2, and h1/h2 = 1. The ratio of the thickness to the mean radius of the outer layer, h2/R 2, is 0.20. The reference frequency and wave

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Elastic Waves in Composite Media and Structures 1.5

Bj

1.0

0.5

0.0 1.8

1.5

2.0

Present

2.2

2.4

Gregory and Gladwell (1989) Mode 1 Mode 2 Mode 3

Ij

1.0

Ω (a)

0.5

0.0 1.8

2.0

Ω (b)

2.2

2.4

FIGURE 6.30  Reflection of the first symmetric mode (Mode 1) from the free end of an isotropic elastic solid cylinder: (a) the normalized amplitude |Bj|, (b) the partition of energy |Ij|. (Reprinted with permission from Rattanawangcharoen et al. 1994, Fig. 2.)

number are, respectively, w ref = 2pcs2/H and xref = 2p/H. The frequency spectrum for guided waves in this cylinder was presented by Armenàkas (1971) and was also investigated by Rattanawangcharoen (1993) using the stiffness method. The complex and real wave numbers were also presented by Rattanawangcharoen (1993). The first four cutoff frequencies are: 0.059, 0.145, 0.428, and 0.438. The propagator matrix approach using the exact solution and the stiffness method II are used to obtain the wave numbers and associated displacement and stress vectors. It was

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Scattering of Guided Waves in Plates and Cylinders Rayleigh-Ritz Mode 1 Mode 2 Mode 3 Mode 4 Mode 5

Bj

2.0

Propagator matrix

0.80

1.5 0.61

1.0

0.65

0.5 0.0 0.0

0.1

0.2

0.3

Ω (a)

0.4

0.5

0.6

0.7

0.65 1.5

1.0 Ijl

0.61

0.65

0.5

0.0 0.0

0.1

0.2

0.3

Ω (b)

0.4

0.5

0.6

0.7

FIGURE 6.31  Reflection in a two-layered isotropic elastic solid cylinder: (a) the normalized amplitude |Bj|, (b) the partition of energy Ijl . (Reprinted with permission from Rattanawangcharoen et al. 1994, Fig. 3.)

found that 99 degrees of freedom (corresponding to 32 nodes and 16 sublayers, respectively, in the propagator matrix formulation and the stiffness method II) and 30 modes gave convergent results. Reflection coefficients |Bj| and the proportion of energy |Ij| are illustrated in Figs. 6.31(a) and (b), respectively. Results obtained by the two methods are seen to have excellent agreement. For a very short range of frequency after the first cutoff,

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Elastic Waves in Composite Media and Structures 2.0

Mode 1 Mode 2 Mode 3 Mode 4 Mode 5

Bj

1.5

1.0

0.5

0.0 0.0

0.2

0.4

Ω (a)

0.6

0.8

1.0

0.6

0.8

1.0

1.5

Ijl

1.0

0.5

0.0 0.0

0.2

0.4

Ω (b)

FIGURE 6.32  Reflection in the four-ply (+15°/−15°/+15°/−15°) graphite–epoxy cylinder: (a) the normalized amplitude |Bj|, (b) the partition of energy Ijl . (Reprinted with permission from Rattanawangcharoen et al. 1994, Fig. 5.)

the second mode is predominant. However, the amplitude and the proportion of energy for this mode drop rapidly away from the cutoff, and the first mode becomes predominant. This continues past the second cutoff. Between the third and fourth cutoff frequencies, the fourth mode has very large magnitude. The fifth mode dominates after the fourth cutoff, except around Ω = 0.62, when its amplitude reaches a maximum and those of the first three modes reach local maxima. An investigation in this region reveals that the first mode is a breathing mode, the second and fourth are torsional modes, while the third and fifth modes are coupled longitudinal.

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For the range of frequency considered here, |e| is less than 0.5%. As in the first example, results obtained using the least-squares and variational methods showed little discrepancy. Example 3 The reflection of the first incoming mode when m = 1 was investigated for a fourply (+15°/−15°/+15°/−15°) hollow composite cylinder. The elastic properties for each ply relative to their symmetry axes (see Chapter 5, Section 5.6) were taken as EL = 139.274 GPa, ET = 15.169 GPa, GLT = G TL = 5.861 GPa, and νLT = νTL = 0.21. The reference frequency and wave number are taken asw ref = c L/H, xref = p/H, c L = EL /ρ . The ratio H/R was taken to be 0.667. In this example, the wave numbers for waves traveling in the positive and negative z-directions are different. The frequency spectrum for this case was given by Fig. 3.4 in Rattanawangcharoen (1993). The first four cutoff frequencies are Ω = 0.215, 0.295, 0.696, and 0.760. This case was studied using the stiffness method II (since no analytical methods are available). The cylinder was discretized into 16 sublayers (corresponding to 99 degrees of freedom) and 30 modes were used. Figures 6.32(a) and (b) show |Bj| and |Ij| for the reflected modes. As mentioned above, the propagating modes in the positive and negative z-directions have different frequency spectra. Thus, the reflection coefficient of the first mode propagating in the positive z-direction and that for the incident first mode propagating in the negative z-direction need not be unity, as seen in Fig. 6.32(a). However, the conservation of energy holds, and the energy carried by the incident mode is transferred to the reflected mode, as seen in Fig. 6.32(b). It is noted that, after the first cutoff frequency, the second mode becomes predominant, but its amplitude drops rapidly as the frequency increases and that of the first mode increases. Between the second and third cutoffs, the first mode is dominant, although the other two modes also participate in carrying the energy. After the third cutoff, the energies carried by different modes are of the same order. In this example, it was found that the numerical results obtained by the least-squares method had large errors (|e| ≈ 15%). On the other hand, the variational method gave results with |e| ≤ 0.5% within the range of frequency considered. A careful search was made for end resonance frequencies for Examples 2 and 3, but none were found.

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Appendix A: Computer Programs A.1 Programs.................................................................................. 283 Guided Waves • Material Properties and Bulk Waves A.2 Executing Programs............................................................... 286 Table of Contents • Notes on Input/Output for Computer Programs

A.1 Programs There are 17 programs contained on the accompanying CD-ROM: 15 are for guided Lamb waves in laminated composite structures, while the other two are miscellaneous. Refer to Fig. A.1 for the list of programs.

A.1.1 Guided Waves The problems considered for guided waves are: Antiplane waves in plates Plane-strain waves in plates Three-dimensional waves in plates Waves in composite cylinders Waves in piezoelectric cylinders* For each problem, programs are categorized into three types:

1. Calculation of frequency spectra 2. Determination of phase and group velocities 3. Evaluation of Green’s function

* While not discussed in the main text, “waves in piezoelectric cylinders” is included here for the reader to compare and contrast waves in piezoelectric cylinders to those occurring in elastic cylinders. The interested reader is encouraged to consult material contained in the literature for further details. Selected references are provided at the end of this appendix to provide a starting point for further study.

283

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Figure A.1  List of programs.

The plates and cylinders considered here are composed of perfectly bonded fiberreinforced layers (laminae) of possibly distinct thicknesses. To attain greater numerical accuracy, each layer can be further divided into sublayers. A numerical procedure, the stiffness method II described in the text, is used to program these problems. Quadratic interpolation functions having the nodes at top, middle, and bottom of each sublayer are chosen here. The problem type is symmetric or antisymmetric if the plate is materially and geometrically symmetric about the mid-plane; otherwise it is asymmetric. Plates are divided into N sublayers through the thickness for all problem types. However to reduce the numerical computation to half, N must be an even integer for symmetric or antisymmetric problem types. Solutions to the asymmetric problem contain the results of both the symmetric and antisymmetric problems. Sublayer numbering must be done from the top of the plate for all problem types. The total thickness of the plate is 2H, which for the sample data is taken

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to be 10 mm. The number of degrees of freedom (DOF) for antiplane problems is 1; for plane-strain problems, it is 2; and for three-dimensional problems, it is 3. Cylinders are divided into N sublayers. Sublayer numbering must be done from the innermost sublayer. The total thickness of the cylinder is H and the thickness-to-mean ratio is H/R, which for the sample data are taken to be 10 mm and 0.1, respectively. For composite cylinders, the DOF is 1 for torsional waves, 2 for longitudinal waves, and 3 for coupled waves. For piezoelectric cylinders, DOF is 4, three displacements and one electric potential. The stiffness method II, described in the text, is used to program these problems. This method involves numerically evaluating the stiffness and mass matrices. The stiffness matrices may be complex. The total size of these matrices will be (2N + 1)*DOF. The matrix sizes are then doubled in the solution process for the phase and group velocities and Green’s function programs. A plane-wave superposition technique is used in the programs to evaluate the Green function for three-dimensional waves in plates. The increment of the plane wave’s propagation direction is taken as 2°, for 0 < θ ≤ 180°, resulting in the superposition of 90 waves. For the Green function for cylinder programs, 121 circumferential waves are superposed, −60 ≤ M ≤ 60, where M is the circumferential wave number. Thus, it can be seen that the Green function programs are very computationally intensive. It is suggested that a maximum of 15 sublayers be used. Ten sublayers are taken in the sample data. In all the programs, the following normalized values are used for input and output. Designating the shear wave speed (length/time) in the first sublayer as Cs, Cphase as phase velocity (length/time), Cgroup as group velocity (length/time), frequency (radian/time) as w, wave number as k (1/length), and H as the half thickness of the plate and the total thickness of the cylinder, respectively, the normalized quantities are normalized frequency (omega), Ω =

ω×H Cs

normalized wave number, γ = kH



normalized phase velocity, CPHV = normalized group velocity, CGRV =



Cphase Cs Cgroup Cs

A.1.2 Material Properties and Bulk Waves The book essentially deals with transversely isotropic materials. For a transversely isotropic material, with the axis of symmetry aligned with the material (local) axis:

1. The program Transformation of Material Properties calculates the material properties in the principal (global) coordinates from those given in the local coordinates. 2. The program Velocity and Slowness Surfaces and Skew Angle evaluates velocity and slowness surfaces, group velocities, and skew angle.

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Figure A.2  Phase and group velocities-plane-strain waves.

A.2 Executing Programs A.2.1 Table of Contents the Table of Contents icon, from the Start Menu, will open the screen as shown in Fig. A.1. Manual buttons are next to the corresponding program buttons. on one of the program buttons, say Phase and Group Velocities—Planestrain Waves, for example, will cause a screen similar to that shown in Fig. A.2 to appear. Sample data can be entered by the Load Sample Data button. The manual for the program, in PDF,* can be seen by the corresponding manual button in the Table of Contents. The manual will guide the user through the entire process of executing the program. Each manual is short. It is recommended that the user should read the applicable manual and experiment with the sample data to get familiar with the programs. An electronic copy of Appendix A will open by the button Appendix A, at the top left corner of the Table of Contents screen. * Note that a Portable Document Format (PDF) viewer, such as Adobe Reader®, is required to view these documents. The free Adobe Reader is available at http://www.adobe.com/products/acrobat/readstep2. html.

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287

A.2.2 Notes on Input/Output for Computer Programs Material and geometric properties need not be in SI (metric) units as long as they are in consistent units. Angles ALFA, BETA, and PHI are in degrees. Sample data presented in the programs use metric units for material and geometric properties as follows: • • • • •

Elastic constants Cij or Dij are in units of Pa. Mass density ρ is in units of kg/m3. Thickness h is in units of m. Piezoelectric constants EPij are in units of C/m2. Dielectric constants EDij are in units of F/m.

Material and geometric properties as well as angles are real numbers; they require that a decimal point be included in the input. Only ten modes are presented in the tables and plots (graphs) of the relevant program output. Stiffness method II, used in the programs, is a variant of a finite-element procedure. It has the same inherent numerical accuracy problems associated with the finite-element method. The program may be unreliable near the cutoff frequencies.

References Tiersten, H. F. 1969. Linear piezoelectric plate vibrations. New York: Plenum Press. Heyliger, P., and S. Brooks. 1995. Free vibration of piezoelectric laminates in cylindrical bending. Int. J. Solids Structures 32: 2945–2960. Siao, J. C.-T., S. B. Dong, and J. Song. 1995. Frequency spectra of laminated piezoelectric cylinders. J. Vibr. Acoust. 116: 364–370. Bai, H., E. Taciroglu, S. B. Dong, and A. H. Shah. 2004a. Elastodynamic Green’s functions for a laminated piezoelectric cylinder. Int. J. Solids Structures 41: 6335–6350.

Disclaimer The programs contained here are intended for educational use. While every reasonable effort has been made to ensure that the programs are accurate and error free, use of the programs is at your own risk. Neither the authors nor publisher assumes any responsibility for errors or omissions in these materials. These materials are provided “as is” without warranty of any kind, either express or implied, including but not limited to, the implied warranties of merchantability or fitness for a particular purpose. The authors, publishers, and other parties involved in creating and delivering these materials shall not be liable for any special, indirect, incidental, or consequential damages, including without limitation, lost revenues or lost profits, or any other damages that may result from the use of these materials.

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References Aalami, B. 1973. Waves in prismatic guides of arbitrary cross section. ASME J. Appl. Mech. 40: 1067–1072. Abduljabbar, Z., S. K. Datta, and A. H. Shah. 1983. Diffraction of horizontally polarized shear waves by normal edge cracks in plates. J. Appl. Phys. 54: 461–472. Achenbach, J. D. 1973. Wave propagation in elastic solids. Amsterdam: North-Holland. Achenbach, J. D. 1976. Generalized continuum theories for directionally reinforced solids. Arch. Mech. 28: 257–278. Achenbach, J. D. 2003. Reciprocity in elastodynamics. London: Cambridge University Press. Achenbach, J. D. 2005. The thermoelasticity of laser-based ultrasonics. J. Thermal Stresses 28: 713–727. Achenbach, J. D., and R. J. Brind. 1981a. Elastodynamic stress-intensity factors for a crack near a free surface. ASME J. Appl. Mech. 48: 539–542. Achenbach, J. D., and R. J. Brind. 1981b. Scattering of surface waves by a sub-surface crack. J. Sound Vib. 76: 43–56. Achenbach, J. D., A. K. Gautesen, and D. A. Mendelsohn. 1980a. Ray analysis of surface wave-interaction with an edge crack. IEEE Trans. Sonics Ultrasonics 27: 124–129. Achenbach, J. D., L. M. Keer, and D. A. Mendelsohn. 1980b. Elastodynamic analysis of an edge crack. ASME J. Appl. Mech. 47: 551–556. Achenbach, J. D., and C. T. Sun. 1972. The directionally reinforced composite as a homogeneous continuum with microstructure. In Dynamics of composite media, ed. E. H. Lee, 48–69. New York: American Society of Mechanical Engineers. Achenbach, J. D., C. T. Sun, and G. Herrmann. 1968. On the vibrations of a laminated body. ASME J. Appl. Mech. 35: 689–696. Aggelis, D. G., S. V. Tsinopoulos, and D. Polyzos. 2004. An iterative effective medium approximation (IEMA) for wave dispersion and attenuation predictions in particulate composites, suspensions and emulsions. J. Acoust. Soc. Am. 116: 3443–3452. Ahmad, F., and A. Rahman. 2000. Acoustic scattering by transversely isotropic cylinders. Int. J. Eng. Sci. 38: 325–335. Ahmad, F. 2001. Guided waves in a transversely isotropic cylinder immersed in a fluid. J. Acoust. Soc. Am. 109: 886–890. Alleyne, D. N., and P. Cawley. 1992. The interaction of Lamb waves with defects. IEEE Trans. Ultrasonics, Ferroelectrics, Freq. Control 39: 381–397. 289

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Alleyne, D. N., M. J. S. Lowe, and P. Cawley. 1998. The reflection of guided waves from circumferential notches in pipes. ASME J. Appl. Mech. 65: 635–641. Al-Nassar, Y. 1990. Scattering of Lamb waves in an elastic plate for ultrasonic nondestructive evaluation of weldments and cracks. Ph.D. thesis, Department of Mechanical Engineering, University of Colorado, Boulder. Al-Nassar, Y., S. K. Datta, and A. H. Shah. 1991. Scattering of Lamb waves by a normal rectangular strip weldment. Ultrasonics 29: 125–132. Al-Qahtani, H. M., and S. K. Datta. 2004. Thermoelastic waves in an anisotropic infinite plate. J. Appl. Phys. 96: 3645–3658. Al-Qahtani, H. M., S. K. Datta, and O. M. Mukdadi. 2005. Laser-generated thermoelastic waves in an anisotropic infinite plate: FEM analysis. J. Thermal Stresses 28: 1099–1122. Al-Qahtani, H., S. Mukdadi, and S. K. Datta. 2008. Laser generated thermoelastic waves in an infinite anisotropic plate. Unpublished. Al-Qahtani, H. M. and S. K. Datta. 2008. Laser-generated thermoelastic waves in an anisotropic infinite plate. J. Thermal Stresses. 31:569–583. Ament, W. S. 1953. Sound propagation in gross mixtures. J. Acoust. Soc. Am. 25: 638–641. Angel, Y. C., and J. D. Achenbach. 1984. Reflection and transmission of obliquely incident Rayleigh waves by a surface-breaking crack. J. Acoust. Soc. Am. 75: 313–319. Arias, I., and J. D. Achenbach. 2003. Thermoelastic generation of ultrasound by linefocused laser irradiation. Int. J. Solids Structures 40: 6917–6935. Armenàkas, A. 1967. Propagation of harmonic waves in composite circular-cylindrical shells, I: Theoretical investigation. AIAA J. 5: 740–744. Armenàkas, A. E. 1970. Propagation of harmonic waves in composite circular-cylindrical rods. J. Acoust. Soc. Am. 47: 822–837. Armenàkas, A. 1971. Propagation of harmonic waves in composite circular cylindrical shells, II: Numerical analysis. AIAA J. 9: 599–605. Armenàkas, A., D. C. Gazis, and G. Herrmann. 1969. Free vibration of circular cylindrical shells. Oxford: Pergamon. Armenàkas, A., and E. S. Reitz. 1973. Propagation of harmonic waves in orthotropic circular cylindrical shells. ASME J. Appl. Mech. 30: 168–174. Auld, B. A. 1990. Acoustic fields and waves in solids. Vol. II. Malabar, FL: Krieger. Auld, B. A., and E. M. Tsao. 1977. Variational analysis of edge resonance in a semi-infinite plate. IEEE Trans. Sonics Ultrasonics 24: 317–326. Bai, H., A. H. Shah, N. Popplewell, and S. K. Datta. 2001. Scattering of guided waves by circumferential cracks in steel pipes. ASME J. Appl. Mech. 68: 619–631. Bai, H., A. H. Shah, N. Popplewell, and S. K. Datta. 2002. Scattering of guided waves by circumferential cracks in composite cylinders. Int. J. Solids Structures 39: 4583–4603. Bai, H., E. Taciroglu, S. B. Dong, and A. H. Shah. 2004a. Elastodynamic Green’s functions for a laminated piezoelectric cylinder. Int. J. Solids Structures 41: 6335–6350. Bai, H., J. Zhu, A. H. Shah, and N. Popplewell. 2004b. Three-dimensional steady-state Green function for a layered isotropic plate. J. Sound Vib. 269: 251–271. Baik, J. M., and R. B. Thompson. 1984. Ultrasonic scattering from imperfect interfaces: A quasi-static model. J. Nondestr. Eval. 4: 177–196. Barbero, E. J., J. N. Reddy, and J. L. Teply. 1990. General two-dimensional theory of laminated cylindrical shells. AIAA J. 28: 544–553.

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