Guided Waves in Structures for SHM

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1 229 x mm RED BOX RULES ARE FOR PROOF STAGE ONLY. DE:ETE BEFORE FINAL PRINTING. Ostachowicz Guided Waves in Structures for SHM The Time-Domain Spectral Element Method Wieslaw Ostachowicz Pawel Kudela Marek Krawczuk Arkadiusz Zak Kudela Krawczuk Zak Polish Academy of Sciences Guided Waves in Structures for SHM reports on the simulation, analysis and experimental investigation related propagation of elastic waves in isotropic or laminated structures. The full spectrum of theoretical and practical issues associated with propagation of elastic waves is presented and discussed in this one study. Key features: Covers both numerical and experimental aspects of modeling, analysis and measurement of elastic wave propagation in structural elements formed from isotropic or composite materials Comprehensively discusses the application of the Spectral Finite Element Method for modelling and analysing elastic wave propagation in diverse structural elements Presents results of experimental measurements employing advanced laser technologies, validating the quality and correctness of the developed numerical models Accompanying website ( contains demonstration versions of commercial software developed by the authors for modelling and analyzing elastic wave propagation using the Spectral Finite Element Method Guided Waves in Structures for SHM provides a state of the art resource for researchers and graduate students in structural health monitoring, signal processing and structural dynamics. This book should also provide a useful reference for practising engineers within structural health monitoring and non-destructive testing. Guided Waves in Structures for SHM Understanding and analysing the complex phenomena related to elastic wave propagation has been the subject of intense research for many years and has enabled application in numerous fields of technology, including structural health monitoring (SHM). In the course of the rapid advancement of diagnostic methods utilising elastic wave propagation, it has become clear that existing methods of elastic wave modeling and analysis are not always very useful; developing numerical methods aimed at modelling and analysing these phenomena has become a necessity. Furthermore, any methods developed need to be verified experimentally, which has become achievable with the advancement of measurement methods utilising laser vibrometry. Guided Waves in Structures for SHM The Time-Domain Spectral Element Method Wieslaw Ostachowicz Pawel Kudela Marek Krawczuk Arkadiusz Zak Cover design: Gary Thompson

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5 Guided Waves in Structures for SHM The Time-Domain Spectral Element Method Wieslaw Ostachowicz Pawel Kudela Marek Krawczuk Arkadiusz Zak Polish Academy of Sciences, Institute of Fluid Flow Machinery A John Wiley & Sons, Ltd., Publication

6 This edition first published John Wiley & Sons Ltd. Registered Office John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at The right of the author to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The publisher is not associated with any product or vendor mentioned in this book. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought. MATLAB R is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book s use or discussion of MATLAB R software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB R software. Mathematica R is a registered trademark of Wolfram Research, Inc. and is used with permission. Wolfram Research, Inc. does not warrant the accuracy of the text or exercises in this book. The book s use or discussion of Mathematica R or related products does not constitute endorsement or sponsorship by Wolfram Research, Inc. nor is Wolfram Research, Inc. directly involved in this book s development or creation. Library of Congress Cataloging-in-Publication Data: Guided waves in structures for SHM : the time-domain spectral element method / [edited by] Wieslaw Ostachowicz...[et al.]. p. cm. Includes bibliographical references and index. ISBN (hardback) 1. Elastic analysis (Engineering) 2. Elastic wave propagation Mathematical models. 3. Composite materials Analysis. 4. Finite element method. I. Ostachowicz, W. M. (Wieslaw M.) TA653.G dc A catalogue record for this book is available from the British Library. Print ISBN: ; obook ; epdf ; epub ; Mobi Set in 10/12.5pt Palatino by Aptara Inc., New Delhi, India

7 Contents Preface ix 1 Introduction to the Theory of Elastic Waves Elastic Waves Longitudinal Waves (Compressional/Pressure/Primary/P Waves) Shear Waves (Transverse/Secondary/S Waves) Rayleigh Waves Love Waves Lamb Waves Basic Definitions Bulk Waves in Three-Dimensional Media Isotropic Media Christoffel Equations for Anisotropic Media Potential Method Plane Waves Surface Waves Derivation of Lamb Wave Equations Numerical Solution of Rayleigh Lamb Frequency Equations Distribution of Displacements and Stresses for Various Frequencies of Lamb Waves Shear Horizontal Waves Wave Propagation in One-Dimensional Bodies of Circular Cross-Section Equations of Motion Longitudinal Waves 36

8 vi Contents Solution of Pochhammer Frequency Equation Torsional Waves Flexural Waves 43 References 45 2 Spectral Finite Element Method Shape Functions in the Spectral Finite Element Method Lobatto Polynomials Chebyshev Polynomials Laguerre Polynomials Approximating Displacement, Strain and Stress Fields Equations of Motion of a Body Discretised Using Spectral Finite Elements Computing Characteristic Matrices of Spectral Finite Elements Lobatto Quadrature Gauss Quadrature Gauss Laguerre Quadrature Solving Equations of Motion of a Body Discretised Using Spectral Finite Elements Forcing with an Harmonic Signal Forcing with a Periodic Signal Forcing with a Nonperiodic Signal 84 References 92 3 Three-Dimensional Laser Vibrometry Review of Elastic Wave Generation Methods Force Impulse Methods Ultrasonic Methods Methods Based on the Electromagnetic Effect Methods Based on the Piezoelectric Effect Methods Based on the Magnetostrictive Effect Photothermal Methods Review of Elastic Wave Registration Methods Optical Methods Laser Vibrometry Analysis of Methods of Elastic Wave Generation and Registration Exemplary Results of Research on Elastic Wave Propagation Using 3D Laser Scanning Vibrometry 116 References 121

9 Contents vii 4 One-Dimensional Structural Elements Theories of Rods Displacement Fields of Structural Rod Elements Theories of Beams Displacement Fields of Structural Beam Elements Dispersion Curves Certain Numerical Considerations Natural Frequencies Wave Propagation Examples of Numerical Calculations Propagation of Longitudinal Elastic Waves in a Cracked Rod Propagation of Flexural Elastic Waves in a Rod Propagation of Coupled Longitudinal and Flexural Elastic Waves in a Rod 162 References Two-Dimensional Structural Elements Theories of Membranes, Plates and Shells Displacement Fields of Structural Membrane Elements Displacement Fields of Structural Plate Elements Displacement Fields of Structural Shell Elements Certain Numerical Considerations Examples of Numerical Calculations Propagation of Elastic Waves in an Angle Bar Propagation of Elastic Waves in a Half-Pipe Aluminium Shell Propagation of Elastic Waves in an Aluminium Plate 195 References Three-Dimensional Structural Elements Solid Spectral Elements Displacement Fields of Solid Structural Elements Six-Mode Theory Nine-Mode Theory Certain Numerical Considerations Modelling Electromechanical Coupling Assumptions Linear Constitutive Equations Basic Equations of Motion 214

10 viii Contents Static Condensation Inducing Waves Recording Waves Electrical Boundary Conditions Examples of Numerical Calculations Propagation of Elastic Waves in a Half-Pipe Aluminium Shell Propagation of Elastic Waves in an Isotropic Plate Experimental Verification Modelling the Bonding Layer 227 References Detection, Localisation and Identification of Damage by Elastic Wave Propagation Elastic Waves in Structural Health Monitoring Methods of Damage Detection, Localisation and Identification Energy Addition Method Phased Array Method Methods Employing Continuous Registration of Elastic Waves within the Analysed Area Damage Identification Algorithms Examples of Damage Localisation Methods Localisation Algorithms Employing Sensor Networks Algorithms Based on Full Field Measurements of Elastic Wave Propagation 275 References 288 Appendix: EWavePro Software 295 A.1 Introduction 295 A.2 Theoretical Background and Scope of Applicability (Computation Module) 296 A.3 Functional Structure and Software Environment (Pre- and Post-Processors) 298 A.4 Elastic Wave Propagation in a Wing Skin of an Unmanned Plane (UAV) 312 A.5 Elastic Wave Propagation in a Composite Panel 320 References 333 Index 335

11 Preface This book is aimed at professionals whose scientific interests are directly associated with propagation of elastic waves in structural elements. This book may be useful not only for students of technical universities but also for researchers and engineers who solve practical problems involving propagation of elastic waves in structural elements made of isotropic materials or laminated composites. Waves propagating in elastic media have been known for many centuries and have been the subject of scientific research of many scholars. Elastic waves result from stresses acting within the media and are associated with volume (compression and tension) and shape (shear) deformations. Better recognition and understanding of the complex phenomena behind the propagation of elastic waves in structural elements have promoted various novel and practical applications in many fields of technology. One such field is diagnostics of structural elements, where the use of elastic waves increases rapidly each year. Local methods employing elastic waves have been employed successfully for many years, but attempts to apply elastic waves in a global sense for diagnosing structural elements are still at an early stage of development. The measure of success in these attempts comes from various achievements made in parallel in several different fields. The first of them is the development of numerical simulation methods and tools aimed at modelling and analysing the phenomena associated with propagation of elastic waves in structural elements. The second, independent, one is the development of appropriate experimental methods and techniques allowing verification and validation results of numerical simulations. Recently these goals have become achievable in practice thanks to employing the most advanced measuring techniques based on three-dimensional (3D) laser scanning vibrometry.

12 x Preface This book is intended to report on the challenges associated with numerical simulation methods, analyses and experimental investigations related to the propagation of elastic waves in structural elements made of isotropic materials or composite laminates. For the first time the full spectrum of theoretical and practical issues associated with the propagation of elastic waves are presented and discussed in one study. The first part of the book, devoted to various modelling and analysis issues associated with propagation of elastic waves, is focused on the Spectral Finite Element Method, which in the authors opinion is the most suitable modelling technique out of a variety of numerical methods used nowadays to solve wave propagation-related problems. This part of the book gives a broad overview of the existing state of the art and knowledge concerning modelling of elastic wave propagation in structural elements, while emphasising the problems associated with developing efficient numerical methods and tools and verifying them. Original solutions developed by the authors, suitable for constructing appropriate numerical models for simulating propagation of elastic waves in 1D, 2D and 3D structural elements made of isotropic and laminated composites are presented and discussed. Based on the developed spectral finite elements, a range of numerical tests has been carried out in order to verify the accuracy of the models, beginning from wave propagation in simple rods, beams, membranes and plates, and ending with shells or 3D structures. The second part of the book, devoted to experimental measurements, presents the application of 3D laser scanning vibrometry for measuring, investigating and visualising the propagation of elastic waves in real-life structural elements. This part of the book naturally complements the theoretical and numerical investigations of the earlier part. Numerous scenarios and results of experimental measurements carried out on 1D, 2D and 3D structures are presented and discussed. The last part of the book is concerned with various practical applications associated with wave propagation phenomena in structural elements. Problems of damage detection and location are discussed and investigated here. These problems are a part of a wide multidisciplinary research subject known as Structural Health Monitoring. Several damage detection methods developed or/and implemented by the authors and their practical applications in the context of Structural Health Monitoring are described in great detail, based on the results of either numerical or experimental investigations. The results of experimental studies included in this book make use of excitation and registration of elastic waves within structural elements using

13 Preface xi piezoelectric transducers. Additionally, and in parallel, independent registration of propagating elastic waves employs advanced 3D laser scanning vibrometry. These two techniques have been applied and investigated in order to qualitatively and quantitatively characterise the wave propagation phenomena. The authors would like to underline the unique character of this book resulting from its complex and multidisciplinary character. Various acclaimed books dedicated to wave propagation phenomena in elastic media are usually theoretical in nature, while the question of appropriate verification of the developed numerical methods is addressed in a very limited manner. The authors of the studies mentioned often use analytical models of the wave propagation phenomena and/or apply different numerical methods based on either the finite element method or spectral methods in the frequency domain. The intention of the authors of this book is to present for the first time in one place new models of spectral finite elements defined in the time domain developed to facilitate analysis of propagation of elastic waves in structural elements. Originality of the material presented in this book comes from the attempt to connect together the results of both numerical and experimental investigations, as well as to indicate their practical implications. Until now, the original numerical models discussed in the book as well as the results of experimental studies using 3D laser scanning vibrometry have had no equivalents in published books dedicated to this field. Therefore the level of scientific research of this book, in the opinion of the authors, closely follows the latest trends in this area. It is worth noting that this book is accompanied by a demonstration version of software employing methods of analysing elastic wave propagation in structural elements using spectral finite elements. It should be emphasised that this software has been developed by the authors of this book. Guided Waves in Structures for SHM: The Time-Domain Spectral Element Method is accompanied by a website ( The website contains and describes the EWavePro (Elastic Wave Propagation) software, which can be used for analysing phenomena of propagation of longitudinal, shear and flexural waves in two and three dimensional thin walled structures composed of isotropic materials or composite laminates. The abbreviation EWavePro is used here to distinguish the software developed by the authors. The software is developed in order to facilitate better understanding of elastic wave propagation phenomena and to be used as a tool in designing structural health monitoring systems based on changes in the elastic wave propagation patterns.

14 xii Preface The authors want to thank their colleagues from the Department of Mechanics of Intelligent Structures: Dr P. Malinowski, Dr M. Radzienski and Dr T. Wandowski for assistance with writing Chapters 3 and 7, as well as Dr L. Murawski for involvement in writing the Appendix. Their efforts contributed to the development of the mentioned parts of this book cannot be overstated.

15 1 Introduction to the Theory of Elastic Waves 1.1 Elastic waves Elastic waves are mechanical waves propagating in an elastic medium as an effect of forces associated with volume deformation (compression and extension) and shape deformation (shear) of medium elements. External bodies causing these deformations are called wave sources. Elastic wave propagation involves exciting the movement of medium particles increasingly distant from the wave source. The main factor differentiating elastic waves from any other ordered motion of medium particles is that for small disturbances (linear approximation) elastic wave propagation does not result in matter transport. Depending on restrictions imposed on the elastic medium, wave propagation may vary in character. Bulk waves propagate in infinite media. Within the class of bulk waves one can distinguish longitudinal waves (compressional waves) and shear waves. A three-dimensional medium bounded by one surface allows for propagation of surface waves (Rayleigh waves and Love waves). Propagation of bulk waves and surface waves is used for describing seismic wave phenomena. Bounding the elastic medium with two equidistant surfaces causes compressional waves and shear waves to interact, which results in the generation of Lamb waves. One can say that a free boundary restricting an elastic body guides and drives waves; therefore the term Guided Waves in Structures for SHM: The Time-Domain Spectral Element Method, First Edition. Wieslaw Ostachowicz, Pawel Kudela, Marek Krawczuk and Arkadiusz Zak. C 2012 John Wiley & Sons, Ltd. Published 2012 by John Wiley & Sons, Ltd.

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