ABSTRACT. This dissertation presents a derivation for the transient wave response of an infinite

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1 ABSTRACT BOGERT, PHILIP B. Transient Waves fro Acoustic Eission Sources in Isotropic Plates Using a Higher Order Extensional and Bending Theory. (Under the direction of Fuh-Gwo Yuan). This dissertation presents a derivation for the transient wave response of an infinite isotropic plate to a general acoustic eission (AE) point source discontinuity loading, based on third-order plate theory. The calculation of the wave response is facilitated by eploying the concept of a seisic oent tensor (or derived equivalent body-forces) to describe the loading fro highly localized displaceent discontinuities on a fracture surface. Further, the body forces fro 3-D elasticity are converted to plate loadings for use in the plate theory wave equations of otion. The transient wave response can be detected as AE signals using piezoelectric sensors. In particular, tie-dependent surface strains can be readily obtained experientally. Therefore the results ephasize the calculation of the surface strains for potential coparison with future experients. The calculated transient response, which represents waves propagating fro a general AE point source in the plate, is expressed in an explicit integral for. It is shown that the transient response, which is given by double inverse Fourier transfors, can be siplified into a finite series involving inverse Hankel transfors which only require onediensional inversions for an isotropic plate. Thus nuerical evaluation of the transient wave is ore robust and accurate than that generated using two-diensional inverse transfors and also, asyptotic solutions can be readily obtained. Nine types of AE sources representing different icro-daage echaniss and their corresponding plate loads are discussed. Nuerical results for four types of AE point sources with a Heaviside tie history loading are presented.

2 The long-ter goal of the developent, having established a relationship between disturbance and response, is to onitor responses in a structure and be able to deterine the source, i.e. daage, type and location by solving the inverse proble in real tie. What is new and different fro previous work upon which this is building is that the extensional forulation is evaluated for general AE loading, and a higher order bending theory is developed and evaluated. Additionally, the polar conversion reduction to a single variable spatial integration is ipleented for both theories.

3 Transient Waves fro Acoustic Eission Sources in Isotropic Plates Using a Higher Order Extensional and Bending Theory by Philip B. Bogert A dissertation subitted to the Graduate Faculty of North Carolina State University in partial fulfillent of the requireents for the degree of Doctor of Philosophy Aerospace Engineering Raleigh, North Carolina APPROVED BY: Fuh-Gwo Yuan Coittee Chair Eric Klang Kara Peters Yong Zhu

4 BIOGRAPHY Mr. Bogert is an aerospace engineer in the Structural Mechanics and Concepts Branch at the NASA Langley Research Center (LaRC). He has nearly years of experience with NASA developing, designing and analyzing structures for the aerospace industry and anaging aerospace research progras. He chaired the Loads and Dynaics, Microgravity and Orbital Debris Penetration Working Groups in his role as Loads and Dynaics Manager for the Space Station Progra in the early 99s. In the id 99s he was the Headquarters Manager of the Advanced Subsonic Technology Progra s Advanced Coposites Technology and Aging Aircraft Eleents. In the late 99s he anaged the High Speed Research Progra s Coposite Fuselage Eleent. He is currently the Associate Principal Investigator for the Lightweight Durable Airfraes portion of the Supersonics Eleent of NASA s Fundaental Aeronautics Progra. Before working at NASA, Mr. Bogert worked for 5 years in private industry. He anaged a structural dynaics division for an engineering copany where, in addition to anageent duties, he perfored underwater explosion qualification analysis for all recent classes of US Navy subarine propulsion plants. Mr. Bogert started his career in the Nuclear Power industry where he perfored seisic analysis for nuclear power plant structures and later anaged a structural analysis and pipe rupture itigation group. Before coencing his current research at the North Carolina State University, Mr. Bogert earned a B.S. degree in Civil Engineering fro Lafayette College, a M.S degree in Structures fro Colubia University and a M.S. degree in Solid Mechanics fro the George ii

5 Washington University. He has been arried for 3 years and has three children and three grandchildren. In his non working tie he enjoys, church, golf, running, bird watching and hunting with faily and friends. iii

6 ACKNOWLEDGMENTS I would like y express gratitude and appreciation to y advisor and friend, Dr. Fuh- Gwo Yuan for his technical guidance and great patience throughout this long process which included any interruptions due to y work coitents at NASA. I would not have copleted this effort without his constant encourageent, guidance and understanding over the years. I a also very grateful to Dr. Shaorui Yang who patiently answered y any and detailed technical questions throughout this tie drawing on his broad and deep knowledge of atheatics and solid echanics. I a grateful to y other coittee ebers, Dr. Eric Klang, Dr. Kara Peters and Dr. Yong Zhu whose tie coitent and encourageent were never in doubt, and also to Dr. Bob Nagel who kept e on track procedurally when I ost needed it. I a greatly indebted to Dr. Olaf Weckner of the Boeing Copany who unselfishly helped e navigate through the learning curve of the MATHEMATICA code in addition to supplying keen technical insights into the physics of the proble often giving up evening and weekend tie in his support. I a also quite grateful to several individuals who ade a big ipression during y tie with NASA, for their support but, unfortunately, are not here to witness the copletion of this journey. The first was the late Dr. Jaes H. Starnes, Jr. who encouraged e to ebark on this quest late in y career, along with y first Advisor, the late Dr. Harold Liebowitz fro the George Washington University. Secondly I a grateful to y forer Branch Head, Dr. Daodar Abur who supported e and created an environent where I iv

7 could fulfill the requireents of university residency with uch tie away fro Langley. My forer and current Branch Heads, Mr. Kevin Rivers, Dr. Stephen Scotti and Mr. David Brewer, encouraged e greatly and shielded e as uch as possible fro additional work coitents so that I could focus on copleting y research, especially towards the end. I a very grateful to y Progra Lead, Mr. Peter Coen, who supported y research tie even when it eant being less dedicated to y day to day progra tasks, and also to y coworker and friend Mr. Willia T. Freean who filled the gaps in y issing efforts in those day to day tasks. Finally, and perhaps ost of all, I want to thank y wife, Mary Ellen, for her treendous patience and support and for giving up uch tie when we otherwise could have been together, so that I could coplete long hours or study and frequent travel fro Hapton Roads to Raleigh. With her love I think I could do just about anything. v

8 TABLE OF CONTENTS LIST OF FIGURES... viii CHAPTER. INTRODUCTION.... Preliinaries.... Relevant Literature....3 Research Objectives....4 Organization... CHAPTER. - EQUIVALENT BODY-FORCES FOR DISPLACEMENT DISCONTINUITIES.... Moent Density Tensor.... Equivalent Body Force... 3 CHAPTER 3. - EQUIVALENT PLATE LOADS FOR A POINT SOURCE IN A PLATE Plate Equations of Motion AE Loading-Replaceent of Tractions with Body Forces Diensionless Plate Loads Nine Types of AE Sources and Their Moent Tensor Coponents... 6 CHAPTER 4. - VECTORIZED EQUATIONS OF MOTION/ HOMOGENEOUS INTEGRAL TRANSFORM SOLUTIONS Copact Equation of Motion-Extensional Wave Diensionless Dispersion Relation-Extension Copact Equation of Motion-Flexural Wave Diensionless Dispersion Relation-Flexure CHAPTER 5. - NON HOMOGENEOUS INTEGRAL TRANSFORM SOLUTIONS Extensional Wave Transfored Solution Extensional Transfored Plate Loads Flexural Wave Transfored Solution Flexural Transfored Plate Loads Transient Solution by Application of Inverse Fourier Transfor vi

9 5.6 Strain Forulation Cartesian to Polar Transforation Reduction to D Hankel Transfors Outgoing AE waves and Asyptotic Solutions... 6 CHAPTER 6. - NUMERICAL RESULTS Results Overview Results-General Vertical Tensile Crack Horizontal Tensile Crack Shear crack I Inclined Shear Crack I Coparison with Elasticity Theory... 7 CHAPTER 7. - DISCUSSION AND CONCLUSIONS REFERENCES APPENDIX... 9 APPENDIX A - FORMULATION OF THE HIGHER-ORDER PLATE THEORY CONSTITUTIVE EQUATION APPENDIX B SAMPLE PLATE LOAD DERIVATION.... APPENDIX C PLATE LOAD NORMALIZATION EXAMPLE...4 APPENDIX D EXAMPLE CALCULATION OF MOMENT TENSOR FOR VERTICAL TENSILE CRACK...5 APPENDIX E -TRANSFORMING OF NORMALIZED PLATE FORCES EXAMPLE...6 APPENDIX F- SHAPE CORRECTION FACTORS...7 APPENDIX G - STRAINS FROM DISPLACEMENTS FORMULATION...8 APPENDIX H HEAVYSIDE TEMPORAL LOADING...3 APPENDIX I - SUMMARY OF MINDLIN (FIRST ORDER) THEORY...5 APPENDIX J - PLANE STRESS FORMULATION...3 Appendix J.A Calculation of Coefficients U n for the u Displaceent Mode Contribution...43 Appendix J.B- Derivation of Eq. (J.4.4)...45 vii

10 LIST OF FIGURES FIGURE Page Figure. Typical AE paraeters... 3 Figure. Moent Tensor Coponents M ij... 6 Figure. Moent tensor coponent corresponding dipoles... 6 Figure.3 A point source with oent tensor M ij and inclined surface at (,, z ) in a plate... 7 Figure 3. Lower order plate forces... Figure 3. AE source orientations... 7 Figure 3.3 Vertical tensile crack... 7 Figure 3.4 Horizontal tensile crack (axi-syetric w.r.t. x 3 -axis)... 8 Figure 3.5 Horizontal shear crack... 9 Figure 4. Mode shapes for extension bending and transverse shear Figure 4. Higher order theory extensional dispersion relationships - frequency vs. wave-nuber Figure 4.3 Higher order theory extensional theory dispersion group velocity vs. frequency Figure 4.4 Higher order theory bending dispersion relationships frequency vs. wave nuber... 4 Figure 4.5 Higher order theory bending dispersion relationships group velocity vs. frequency... 4 Figure 4.6 First order theory dispersion relationships (a) frequency vs. wave nuber group velocity vs. frequency Figure 4.7 Dispersion relation of lower order odes for an isotropic plate with =.33 fro 3-D elasticity viii

11 Figure 6. Surface strain transient response to source: Vertical Tensile Crack =. Third Order Theory The arrows indicate the corresponding odes near arrival. r=, z =h/ Figure 6. Surface strain transient response to source: Vertical Tensile Crack =. First Order Theory. The arrows indicate the corresponding odes near r=, z =h/ Figure 6.3 Surface strain transient response to source: Vertical Tensile Crack = 45. Third Order Theory. The arrows indicate the corresponding ode near arrival r=, z =h/ Figure 6.4 Surface strain transient response to source: Vertical Tensile Crack = 45. First Order Theory. The arrows indicate the corresponding odes near arrival. r=, z =h/ Figure 6.5 Surface displaceent transient response to source: Vertical Tensile Crack = and = 45 for A flexural ode. Third Order Theory. The arrows indicate the corresponding odes near arrival. r=, z =h/ Figure 6.6 Surface strain transient response to source: Horizontal Tensile Crack (axi-syetric). Third Order Theory. The arrows indicate the corresponding odes near arrival. r=, z =h/ Figure 6.7 Surface strain transient response to source: Horizontal Tensile Crack (axi-syetric). First Order Theory. The arrows indicate the corresponding odes near arrival. r=, z =h/ Figure 6.8 Surface strain transient response to source: Horizontal Shear Crack I θ=. Third Order Theory. The arrows indicate the corresponding odes near arrival. r=, z =h/ Figure 6.9 Surface strain transient response to source: Horizontal Shear Crack I θ =. First Order Theory The arrows indicate the corresponding odes near arrival. r=, z =h/ Figure 6. Surface strain transient response to source: Inclined Shear Crack I β =3,θ =. Third Order Theory. The arrows indicate the corresponding odes near arrival. r=, z =h/ ix

12 Figure 6. Surface strain transient response to source: Inclined Shear Crack I β =3,θ =. First Order Theory. The arrows indicate the corresponding odes near arrival. r=, z =h/ Figure 6. Surface strain wavefor odeled by Theory of Elasticity Third Order Theory fro AE source: Horizontal Tensile Crack at z = h/4, wavefor observed at r = Figure J. Coparison of closed for and nuerical solutions for the ode u displaceent for a vertical tensile crack. r=5, θ= Figure J. Coparison of closed for and nuerical solutions for the ode u displaceent for a vertical tensile crack. r=5, θ=45 J. Coparison of closed for and nuerical solutions for the ode u displaceent for a vertical tensile crack. r=5, θ= Figure J.3 MATHEMATICA aniation of u displaceent for SH ode as a function of r,θ, at four snapshots in tie...4 Figure J.4 MATHEMATICA aniation of u displaceent for S ode as a function of r,θ, at four snapshots in tie...4 x

13 CHAPTER. INTRODUCTION. Preliinaries Acoustic eission (AE), a passive onitoring technique, involves instruenting the chosen structures and listening for characteristic signals fro the structures under stress (Bray and McBride, 99 [6]). Physically, acoustic eission is the ter used to describe elastic waves eitted by rapid release of strain energy fro sudden localized changes within a structure under stress. The sudden local icrostructural changes fro, for exaple, the foration of daage, plasticity, or phase transforations, are often referred to as localized sources. Such localized sources act as a center of radiation over a sall or finite area for the duration of bursts of elastic waves in the khz to several MHz range that propagate throughout the structure. Unlike ost of the various non-destructive evaluation (NDE) techniques, the AE onitoring technique detects, locates, and then identifies the daage under stress the oent the daage is created. In contrast, ultrasonic testing detects and characterizes daage after it has been created. Provided that the elastic waves are of sufficient agnitude to be detected by sensors as AE signals, they can be used to estiate localized changes in assessing the health of the structure. The potential of the AE onitoring technique as a ethod for reotely detecting and locating daage in a wide range of loaded structures has been long recognized, with initial work developed in the early 96s (e.g., Hastad, [,3]). One of the distinct advantages of the technique for future structural health onitoring systes, copared to

14 other NDE techniques for aerospace structures, is the continual in-situ onitoring capability during the entire load history in flight. As elastic waves are eitted fro icro-daage forations, and propagate through the structure, the wavefor shape transitted by AE contains inforation about the location and nature of the source and the host aterial itself; therefore, accurate sensing of the transitted transient wave response is necessary to fully evaluate AE wavefors. High-fidelity sensors yield undistorted wavefors that allow distinction between the arrival ties of different wave odes as well as subtle distinctions about the source kineatics. In coposite structures, localized sources ay include various icro-daage echaniss such as fiber breakage, atrix cracking, and delaination due to foreign object ipact, fatigue loading, or even anufacturing defects. In this paper, the ters, icro-daage and AE source are used interchangeably.. Relevant Literature In recent decades two ain approaches were developed to characterize the AE signals (Aki and Richards [4]): () Signal-based approach which characterizes a wavefor by siple event counting and statistical correlations between daage echaniss and experiental data and using these for future event description; () Wavefor-based approach which ais at reconstructing the daage location and daage echaniss using transient wave theory.

15 Signal based Approaches - In the conventional signal-based approach, three types of signal analysis have been explored: AE paraeter analysis, AE activity analysis, and AE frequency analysis. Figure. Typical AE paraeters AE paraeter analysis - Here a nuber of paraeters are extracted fro individual AE signals. Soe of the paraeters are defined with reference to Fig., including nuber of counts, peak aplitude, duration, and rise tie. Other paraeters can be defined, in ters of energy. Apparently, the nuber of counts is affected by the threshold setting. These AE paraeters for each AE event are usually plotted against test paraeters such as load, strain, and teperature. After repeated tests on identical speciens, epirical inferences are ade concerning the severity of daage or the discriination between different daage echaniss. A nuber of studies have deonstrated the success of this approach in laboratory settings (Favre and Laizet, 989, Dzenis and Qian, [,]), but few practical applications of this technique have eerged. Overall, this AE analysis ethod is capable of providing useful inforation on daage developent in coposites. However, 3

16 values obtained fro conventional paraeters are in no way unique to a given source. Derived paraeters do not copletely describe the signal, and are not generally reproducible with different structures. Further, the discriination of daage echaniss is difficult to achieve due to the overlap of AE paraeters caused by coplex daage and wave propagation processes in coposites. AE activity analysis This ethod focuses on easuring the aount of AE signals produced by a specien or a structure. It priarily results in inforation about the initiation and the evolution of daage throughout a test or during the service life of a coponent. A good exaple is the well known Felicity ratio analysis which is nowadays being used for the onitoring of fiberglass tanks, pressure vessels, and pipes (Fowler et al., 989 and Downs and Hastad, 998 [3.]). This type of analysis has been standardized in a nuber of CARP and ASME codes and constitutes one of the ain fields where AE has found successful application. AE frequency analysis This technique uses Fast Fourier Transfor (FFT) techniques to exaine the frequency spectru of AE waves. This approach is priarily used for discriination purposes, based on the assuption that different daage phenoena will produce signals with different frequency contents. A liited nuber of papers have been published on this subject and it has not yet found any widespread practical application (Ghaffari and Awerbuch, 99; and de Groot et al., 995 [4,8]). The ain deficiency of this analysis is that the wave propagation through the structure exhibits several dispersive wave odes. Due to the characteristics of propagation of these wave odes, there are 4

17 certain characteristic frequency ranges that propagate with sufficient agnitude to be sensed by the AE sensors. These frequency ranges vary as a function of structural properties and geoetry even when the daage echanis is identical. Thus, it is not reasonable to directly associate certain AE signal frequency ranges with certain daage echaniss in a way that is independent of the aterial and geoetry of the structure. None of these AE signal analysis techniques used in laboratory studies has proven to be capable of consistently dealing with the difficulties encountered in large structure, naely, large aounts of data, the eliination of noise sources, aterial anisotropy, and the influence of wave propagating effects (attenuation and dispersion). These analyses often gave controversial results because they lack a physical justification (based on the theory of AE) for the signal features used to sort the experiental signals into different source echaniss. Suarizing these observations one can state that the conventional AE approaches have reained ainly qualitative techniques. Wavefor Based Approach - For AE to onitoring be accepted in practical use as a viable technique for assessing the state of structural health, a quantitative approach is required (Scruby, 985[35]). Such an approach is based on the recognition that the wavefor (signature) of AE signals is a characteristic of the process which produces it. Hence the AE signals contain inforation about (a) the AE source, which includes location, agnitude, and daage echaniss; (b) the structure through which the wave propagates in the for of transient stress waves; and (c) the onitoring syste including the sensor (i.e., piezoelectric sensors) either ounted on or ebedded in the structure and the associated signal processing electronics. With the ever increasing power of data 5

18 acquisition systes and the increased sensitivity of new sensors, recording the entire wavefor with high fidelity becoes feasible. This wavefor approach characterizes AE signals by eploying transient wave theory to predict the signals generated by different types of daage echaniss. This, in turn, enables experientally easured acoustic eission data to be interpreted in a physically eaningful anner. A clear understanding of the quantitative relationship between the waves and their sources is essential in developing algoriths for detecting and characterizing the daage. Matheatically, identification of an AE source is an inverse proble. If the area of the source foration (sudden icro-daage growth) is sall copared to the easured doinant wavelength of the AE signal and source-sensor distance, the AE source can effectively be considered as a point source. The localized sources over a finite area, however, can be odeled as the suation (or integration in the liit) of point sources (Chang and Sachse, 985 [8]), each of which accounts for the evolutionary icro-daage over a discretized sub-area eitted at different ties. Therefore, accurately characterizing the point source is a key in understanding the nature of the icro-daage echaniss. Micro-daage and seisic rupture share the sae source echanis, yet, at different teporal and length scales. A oent tensor for describing the different types of point sources (Vvedenskaya, 956; Aki, 966; Kostrov, 974 [39,4,6]) and equivalent body-forces for displaceent discontinuities on an internal surface (Maruyaa, 963; Burridge and Knopoff, 964 [7,7]) provide powerful aids in the description of seisic 6

19 source theory. Studies of elastic waves generated by seisic sources have been suarized by Aki and Richards (99) [5]. Fro the seisic wavefors, identification of the type of source as well as its tie history needs to be extracted. The seisic source has been assued to be atheatically characterized as buried self-equilibrating force dipoles. Each dipole can be represented atheatically by two ipulse delta functions acting in opposite directions, with an infinitesial separation distance either along, or perpendicular to, the ipulse direction or, in the liit, by the spatial derivatives of the ipulse with respect to the separation-distance paraeter. These force dipoles can be cobined to for the oent tensor (Aki and Richards, 99) [5]. Many research efforts have been devoted to theoretically deterining the AE source event fro the AE wavefors (Michaels et al., 98; Scruby et al., 985; Shah and Labuz, 995; and Ohtsu et al., 998 [8,35,36,37,3]). Generalized ray theory has been proposed to analyze the transient AE waves in plates (Pao et al., 979 and Chang and Sachse, 985 [3,8]). The oent tensor (or derived equivalent body-force) approach has been adopted in the AE field to describe the daage processes (Rice, 98 [34]). When considering the wave propagation fro AE sources, analytic or nuerical Green s functions that correspond to buried force dipoles in an infinite doain have been forulated as coponents of the oent tensor. Although the elastic properties of the aterial do change as daage accuulates, this effect is not considered for characterization purposes. For the case of the growth of a sudden icro-crack, if the doinant wavelengths are uch 7

20 larger than the size of the icro-crack growth, radiation patterns should not be significantly affected by these inhoogeneities. Note that this assuption is not valid in the case of icro-growth of an existing crack since radiation ay be affected by the presence of the existing crack unless the existing crack size is also sall enough. In order to recover inforation about the point source fro the AE wavefors, suitable signal deconvolution and decoposition procedures are often used (Michaels et al., 98 and Chang and Sachse, 986 [8,8]). Once calculated, the oent tensor is decoposed into different coponents, each one representing a particular type of daage echanis. This approach, requiring extensive atheatical calculations, has been applied to different aterials for very siple structural geoetry. As any practical structures are of plate-like geoetries, a viable approach using the guided waves propagating along the plane of the plate has been exained. The waves are inherently dispersive and ulti-odal. Due to the dispersive nature of the guided waves, the wavefors change as they propagate away fro the source. Therefore it is necessary to understand the detailed nature of the guided waves in structures in order to extract inforation fro the sensed wavefors. Considering the specific odes of propagation, the deterination of the guided AE response signals can iprove the source location accuracy and enable the discriination of daage echaniss. This approach is now known as odal acoustic eission (MAE). Goran (99 [7]) first deonstrated how the extensional and flexural odes could be recognized in AE signals by perforing pencil lead-breaks on both aluinu and 8

21 unidirectional coposites. Further work has concentrated on source orientation (Goran and Ziola, 99 [6]), atrix crack detection (Goran and Ziola, 99 [5]), and source location and attenuation (Hastad and Downs, 995 and Prosser, 996 [,33]). A nuber of inherent advantages of the MAE approach were identified in the literature: () easier recognition and discriination of true AE daage signals; () better noise eliination; and (3) ore accurate source location. However, a large part of the published results have only dealt with siulated AE sources such as pencil lead break or ball ipact to validate the approach. These sources represent a onopole surface loading rather than the typical AE buried self-equilibrating dipole source. Since of all the studies have focused on the first two fundaental bending and extensional odes, A and S respectively, ost of these advantages have thus far reained largely on the conceptual level (Hastad, []). Guo et al. (996 []) presented plate loads based on first-order plate theory and odeled acoustic eission fro different daage echaniss in coposite lainates. Wave propagation fro atrix cracking and fiber fracture in lainated beas was odeled by Aberg and Gudundson ( []) eploying higher-order bea theory. Nuerical siulation of experients on cracked speciens was conducted by Aizawa et al. (987 [3]). Hastad et al. ( [3,43]) siulated AE signals fro dipole sources in an aluinu plate with a 3-D finite eleent code. 9

22 .3 Research Objectives In the current work at NCSU transient waves eitted fro AE sources, such as the foration of icro-daage in isotropic plates, are investigated. Using plate theory to predict extensional and flexural otions in thin plates, equivalent plate loads for different AE sources are first derived. Depending on the orientations of the icro-cracks (horizontal, vertical, and inclined) and the displaceent discontinuities across the fracture surfaces represented by one noral and two shear coponents, a total of nine types of transient wave disturbance sources and their responses are presented. Eploying the joint integral transfor, that is, a double Fourier transfor on the spatial variables and a Laplace transfor on tie, the transient wave response for the different sources in isotropic plates is forulated in copact for in ters of an inverse Fourier transfor (double integration). Further, it can be shown that the transient response to the various point sources can be reduced to inverse -D Hankel transfors (single integral) for isotropic plates. Asyptotic solutions obtained fro the -D transfor can be easily derived. Nine types of AE sources representing different icro-daage orientations and their corresponding plate loads are discussed. Nuerical results for four of the nine AE point sources with Heaviside loading tie histories are presented and copared with predictions fro the first order theories. The work builds upon previous work as follows. First Order extensional and bending AE theory (Bogert, Yuan, Yang 9 [44]) Considered true AE self-equilibrating dipole loading with first order displaceent field Sei-analytical solution for PC (not FEA) with reduced order integration via Bessel functions

23 Higher (second) Order Extensional Theory (Yuan, Yang, 5 [4]) for surface loading with axisyetric solution only In the current work: Extend the higher order extensional theory to include AE loading (oent tensor of self equilibrating dipoles) Extend sei-analytical reduced integration Bessel function transient solution concept to the third order theory Develop general sei-analytical Cartesian transient solution for the extensional theory Develop higher (third) order bending theory and copatible AE loading Extend sei-analytical reduced integration Bessel function transient solution concept to the third order theory Develop AE transient general sei-analytical Cartesian solution for bending.4 Organization In order to retain the flow of the presentation and to focus on parts of the developent that are new, very detailed steps are generally not shown in the body of the dissertation. Typical operations and anipulations that ust be perfored on equations, 5 extensional and 6 flexural, are shown for one or two equations in a series of Appendices so as not to bog down the flow of the derivations in Chapters through 7.

24 CHAPTER. EQUIVALENT BODY-FORCES FOR DISPLACEMENT DISCONTINUITIES. Moent Density Tensor Consider the icro-daage process which involves generation of a displaceent discontinuity on an internal surface of an elastic body. Fro the 3-D elastodynaic representation theore for an internal crack surface with displaceent discontinuities (crack opening and/or slip), the displaceent field, generated by the displaceent discontinuities coonly used in seisology, can be expressed as (Aki and Richards [5])) Gnp un (x, t) pq d, (n, p =,, 3) (.) q where (x,t) = (x, x, x 3, t) is the general position and tie at which the displaceent u n is to be evaluated, the sybol * denotes the tie convolution integral, G np (x,t;,) is the Green s function, which represents the nth coponent of displaceent at (x,t) due to a unit ipulse applied at x =, t = in the p-direction, is the general position on. G np and its derivatives are continuous across, and pq is the oent density tensor, naely, the syetric second-order tensor, depicted in Fig..3, defined by c u ( ξ, ) n (.) pq pqij i j where n is the unit outward noral vector to the crack surface, c pqij are elastic constants, and u i represents the discontinuity of the i th displaceent coponent (Note that here and throughout the paper a bolded quantity represents a vector or a atrix).

25 . Equivalent Body Force Another approach is to introduce equivalent body-forces (Aki and Richards, 99 [5]). Using the Dirac delta function (-) in describing the displaceent discontinuities on within the elastic body with volue V, where varies throughout V and is located on the discontinuity surface, using a property of the Green s functions, and introducing the equivalent body-force f for a displaceent discontinuity on, the contribution of the discontinuity to the displaceent at (x,t), in Eq. (.), can be written in the for t un ( x, t) d f p ( η, ) Gnp ( x, t ; η,) dv ( η) (.3) V where f p are the equivalent body-forces for the displaceent discontinuities given by f p (, ) ui ( ξ, ) n jcijpq ( η ξ) d pq ( η ξ) d (.4) q Eq. (.3) has a precise for of a coon body-force contribution. Clearly, this forulation requires Green s functions and derivatives of delta functions, rather than the spatial derivatives of Green s functions which are often not readily available. Since fracture within the volue V is an internal process, the total oentu and total angular oentu ust be conserved. It follows that the total force due to f, and the total oent of f about any fixed point, ust be zero. These requireents can be easily verified. q The above results are developed for a fracture surface of finite extent, but in practice data is often good only at periods for which the whole of is effectively a point source. 3

26 For these waves, the contributions fro different surface eleents d are all approxiately in phase, and the whole surface can be considered as a syste of dipoles (derivatives of the delta function) located at a point. Clearly, the point source eans that a/r, a/ <<, where r = x, the source-sensor distance, is the doinant wavelength of the AE signal and a is typical radius of the source region, assuing the origin of the coordinates is in the source region. For an effective point source, Eq. (.) can be approxiately reduced to where: u (x, t) M n pq G np q M pq pqd pq c pqijui ( ξ, ) n j (.5) is area of the point source region, M ij is the oent tensor of the source (syetric second-order tensor). In this case, the equivalent body-force f to a point source at with oent tensor M pq is given, by using the for of Eq. (.4), f p ( x, t) M pq ( t) ( x ξ) (.6a) x For p=,,3 on left side q=,,3 on right side in defining the forces f, f and f 3 which are the sae as those defined in the Mindlin theory [45]. For the extra loading generated fro the higher order theory, p=4,5,6 on the left side and the corresponding right side subscripts are p =,,3 respectively where the 6,3 pair is only needed for bending. q 4

27 q=,,3 for the higher order ters on the right side just as in the lower order theory. In other words the oent tensor stays 3x3. Expanding Eq. (.6a) yields: f M ( x ξ) M ( x ξ) M ( x ξ) 3 x x x3 f M ( x ξ) M ( x ξ) M ( x ξ) 3 x x x3 f M ( x ξ) M ( x ξ) M ( x ξ) x x x3 f M ( x ξ) M ( x ξ) M ( x ξ) 4 3 x x x3 f M ( x ξ) M ( x ξ) M 3 ( x ξ) x x 5 x 3 f M ( x ξ) M ( x ξ) M ( x ξ) x x z (.6b) Though the last three equations look just like the first 3 on the right side, as will be seen subsequently in Section 3., the f 4, f 5 and f 6 forces will be defined fro higher order integrals and this will change the integration of the right side ters, hence, giving rise to higher order forces that are different than f and f. Note that the 6 th equation is only required for bending. The first five equations apply for both bending and extension. With M c u n. pq pqij i j Eq. (.6b) indicates that each coponent of the oent tensor is associated with spatial derivatives of the delta function and that each coponent f p of the body force includes three dipoles. There are a total of nine dipoles that for the equivalent body-force 5

28 f for a generally oriented displaceent discontinuity. The oent tensor coponents M ij are shown in Fig.. and the corresponding dipoles are illustrated in Fig... M M M M M M M M M M Figure. Moent Tensor Coponents M ij x 3 x 3 x 3 x x x x x x M M M3 x 3 x 3 x 3 x x x x x x M M M 3 x 3 x 3 x 3 x x x x x x M 3 M 3 M 33 Figure. Moent tensor coponent corresponding dipoles 6

29 A point source displaceent discontinuity in an elastic plate, which can be characterized by its oent tensor M ij (the derived 3-D elasticity equivalent body-forces) as shown in Fig. 3., can be transfored into the corresponding plate loads based on plate theory. With the equivalent plate loads the equations of otion can be solved by eploying integral transfors. Using the equivalent plate loads and the plate theory, instead of the equivalent body-forces and 3-D elasticity, analysis of AE wave propagation in plates will be greatly siplified. u νu Figure.3 A point source with oent tensor M ij and inclined surface at (,, z ) in plate In Fig..3 above, u = u is the displaceent discontinuity, is a unit vector, n is the outward noral to the crack surface. A displaceent discontinuity u in the direction of any given unit vector ay be written as (Rice, 98[34]) 7

30 u u(, ) (.7) where u is the agnitude of the discontinuity. Hence, a general displaceent discontinuity can be resolved into an opening displaceent and two tangential slip displaceents. With Eq. (.7), the oent tensor in Eq. (.5) can be expressed as M pq c pqij i n ju (.8) In the isotropic case, Eq. (.8) becoes M [ n G( n n)] u (.9) ij k k ij i j j i where and G (ore traditionally ) are the Lae constants and ij is Kronecker s delta. For the point source, u can be written as u = Af(t), here A denotes the aplitude of the discontinuity and f(t) source tie function. Thus, the oent tensor ay be expressed as M ij M ij f ( t) (.) where the spatial coponent of the loading M ij c ijpq n A p q results in constants representing the aplitudes of M ij, for the flat fracture surfaces being considered here. 8

31 CHAPTER 3. EQUIVALENT PLATE LOADS FOR A POINT SOURCE IN A PLATE 3. Plate Equations of Motion Integral solutions fro plate theory for the point sources ay be derived if the equivalent body forces can be transfored into the corresponding plate loads. For a point source with displaceent discontinuities in a plate, there are nine possible dipoles, each dipole with ultiplier M ij (corresponding to the coponent of the oent tensor) representing a part of the equivalent body-force fro 3-D elastodynaics. The equivalent body forces can be replaced by plate loads via plate theory by integrating the forces though the thickness. Thus, the waves in the plate excited by M can be evaluated. The derivation is outlined below, starting with the displaceent fields and the equations of otion and finally the incorporation of the equivalent plate loads. For ore detail see Appendix A. The approxiate displaceent field described by the third-order plate theory can be expressed as u ( x, z, t) u( x, t) z ( x, t) z ( x, t) z ( x, t) 3 u ( x, z, t) v( x, t) z ( x, t) z ( x, t) z ( x, t) 3 u (, x z,) t w(,) x t z (,) x t z (,) x t (3.),,, where x = (x, x ) and ( u, v, 3,, ), (, w, 3) are the generalized extensional and flexural displaceents and are uncoupled in linear strain probles with 9

32 isotropic plates. Figure 3. Lower order plate forces The higher order plate forces are not shown on Figure 3. and are ore of a atheatical construct rather than physical like those fro classical theory. The equations of otion of a plate ay be written in ters of the plate resultant forces, by Hailton s principle or by integration of the elastodynaic equations of otion as shown in Appendix A, Eq. (A.4) - Eq.( A.), which results in, Extensional otion N N q I u I,, N N q I v I,, R, R, N I 33 3 n I I4 S, S, R u n I I4 S, S, R v (3.)

33 Flexural otion M M Q I I,, 4 M M Q I I,, 4 Q Q q I w I xx, yy, T T 3P r I I,. 4 6 T T 3P r I I,. 4 6 P, P, M3 nz Iw I 4 (3.3) where h/ I z dz ( j,,4,6) j h/ j (3.4) (N,, R,, S,, R) and (M,, Q,, T,,P,,M 3 ) are stress resultants which are related to the extensional displaceents ( u, v, 3,, ) and the flexural displaceents,,, (, w, 3) respectively. Additionally, ( q, q,, n, n ) and (,, q, r, r, n ) are the extensional and flexural plate loads. In the absence of a the z localized disturbance inside the plate, they are given by the surface tractions q [ 3( h/ ) 3( h/ )] n h 4 q [ 33( h/) 33 ( h/)] n h 4 h [ 3( h/) 3( h/)] 3 r h 8 [ ( h/) ( h/)] h (3.5) 33 33

34 3. AE Loading - Replaceent of Tractions with Body Forces If there is a source of disturbance inside the plate (see Figure.3), which is the case with an AE dipole loading, the source will contribute to the plate loads through its equivalent body forces. In other words, the plate forces due to surface tractions shown above will be replaced with body forces as shown below. The equations of otion with body forces fro 3-D elasticity, with the coa indicating differentiation, are ij, j f i ui (3.6) where f i are introduced as the equivalent body forces to the point source with oent tensor M ij (t) in the plate. Multiplying the f i ters in the above stress equations of otion by z ( =,,,3), consistent with the power of ultiplying the other ters to produce the equations of otion in ters of plate resultants, and then integrating with respect to z (the plate thickness), the ters h / h / f i z dz z (=,,,3) (3.7) represent the plate loads due to the internal point source in the equations of otion of the plate if the equivalent body force is used for f i. The plate loads generated by the point source can be written as: h/ h/ h/ q f dz, q f dz, f zdz, h/ h/ h/ 3 h/ z h/ z n f / dz, n h f / dz h (3.8) for extensional otion and,

35 h/ h/ h/ f zdz, f zdz, q f dz, h/ h/ h/ 3 h/ h/ h/ 3 3 / 4, / 5, h h z h/ 6 r f z dzr f z dzn f z dz (3.9) for flexural otion, where f i are given by Eq. (.6b). Perforing the integration in Eq. (3.8) and (3.9) in conjunction with the first five of Eqs. (.6b), and for a single point source in an infinite plate, conveniently letting the source be located at (,, z ) without loss of generality (see Fig..3), so that the z axis is a syetry axis through the point source, and after using the following equality, h / ( n) n ( n) f ( z) ( z z ) dz ( ) f ( z ) (3.) h / the plate loads becoe, for: Extensional otion (before noralization) q M ( x ) ( x ) M ( x ) ( x ) M 3 q M ( x ) ( x ) M ( x ) ( x ) M 3 M z ( x ) ( x ) M z ( x ) ( x ) M ( x ) ( x ) M M n ( x ) ( x ) z ( x ) ( x ) z M ( x ) ( x ) z M M n ( x ) ( x ) z ( x ) ( x ) z M ( x ) ( x ) z 3 3 (3.) Flexural otion (before noralization) After perforing the integration of all six of Eqs. (.6b), where the right side of the sixth equation is identical to that of the third equation, to accoodate the extra (sixth) 3

36 bending equivalent plate force n z, in a anner siilar to the extensional proble detailed above the, un-noralized bending force equations are: M z ( x ) ( x ) M z ( x ) ( x ) M ( x ) ( x ) 3 M z ( x ) ( x ) M z ( x ) ( x ) M ( x ) ( x ) 3 q M ( x ) ( x ) M ( x ) ( x ) M r M z ( x ) ( x ) M z ( x ) ( x ) 3 M z ( x ) ( x ) r 3 3 M z ( x ) ( x ) M z ( x ) ( x ) 3 M z ( x ) ( x ) 3 n M z ( x ) ( x ) M z ( x ) ( x ) M z ( x ) ( x ) z (3.) To provide a little ore clarity see Appendix B where the first extensional and the last bending equations are derived in ore detail. 3.3 Diensionless Plate Loads Once the equivalent plate loads for the point source are deterined, the plate response to the source can be deterined via the plate theory equations of otion via the joint integral transfor approach. To calculate the transient response of the plate to the point source, the required equivalent plate loads given by Eq. (3.) for extension, and Eq. (3.) for bending, can be conveniently ade diensionless, along with the other variables by noralizing the plate equations of otion. See Appendix A, Eq. (A.5) for a list of noralization constants. The equations for, n and n all require noralization. Using the noralization constants defined in Appendix A, and their derivatives as required, the noralized forces can be suarized as: 4

37 Extension q M ( x ) ( x ) M ( x ) ( x ) M 3 q M ( x ) ( x ) M ( x ) ( x ) M 3 3 M z ( x ) ( x ) 3 M z ( x ) ( x ) 3 M ( x ) ( x ) n 5 M ( x ) ( x ) z 5 M ( x ) ( x ) z M ( x ) ( x ) z n 3 5 M ( x ) ( x ) z 5 M ( x ) ( x ) z M ( x ) ( x ) z 3 (3.3) Bending After perforing noralization anipulations siilar to those in the extensional proble, the noralized bending equivalent plate forces can be suarized as; 3 M z ( x ) ( x ) 3 M z ( x ) ( x ) 3 M ( x ) ( x ) 3 3 M z ( x ) ( x ) 3 M z ( x ) ( x ) 3 M ( x ) ( x ) 3 q M ( x ) ( x ) M ( x ) ( x ) M r 7 Mz ( x) ( x) 7 Mz ( x) ( x) M3z ( x) ( x ) 3 3 r 7 M z ( x ) ( x ) 7 M z ( x ) ( x ) M z ( x ) ( x ) 3 n 5 M z ( x ) ( x ) 5 M z ( x ) ( x ) M z ( x ) ( x ) z (3.4) where all quantities are ade diensionless by using characteristic length l = h/, characteristic tie = l/c T ( c G/ ), and by noralizing the oent tensor as T M M Gl ij 3 ij /( ). Note that the pries of the diensionless oent tensor are dropped in Eq. (3.3) and (3.4) for convenience. An exaple of the noralization procedure is shown in Appendix C. 5

38 3.4 Nine Types of AE Sources and Their Moent Tensor Coponents Micro-daage can occur in the for of shear or slip (shear crack), or opening of the crack surfaces (tensile crack), or both (ixed crack). The crack ay have different orientations and locations. Nine types of AE point sources are considered, naely, horizontal shear and tensile crack (crack surfaces parallel to iddle plane of the plate), vertical shear and tensile crack (crack surfaces are perpendicular to the iddle plane), and inclined shear and tensile crack, with each case of the shear crack having two types of shearing directions. For each source, the corresponding plate responses can be evaluated nuerically. The shear crack and the tensile crack will be discussed separately. If a fracture surface includes both shear and opening siultaneously (ixed ode crack), the elastic wave eission (AE) fro the source can be obtained by using the principle of superposition, as long as linear theory applies. Without loss of generality in an infinite plate, a point source for an infinite plate located at (,, z ) can be excited by one of the nine types of icro-daage. For each case the equivalent plate loads, prior to non-diensionalization, can be obtained fro Eq. (3.) and Eq. (3.) respectively, where the oent tensor coponents are defined by Eq. (.9). The oent tensor coponents are given below and will be used for nuerically siulating the response of the plate to the point source excitations. 6

39 Horizontal Cracks Vertical Cracks Inclined Cracks 45 o 45 o 45 o Figure 3. AE source orientations (a) Vertical Tensile crack (crack surface perpendicular to the iddle plane) x 3 z n u νu h/ x x h/ Figure 3.3 Vertical tensile crack If the crack surface is perpendicular to the x axis, and the displaceent discontinuity is in the x direction, u = u [,, ] T, 7

40 M u To provide ore insight, the use of equation.9 for calculation the oent tensor is deonstrated for the vertical tensile crack in Appendix D. (b) Horizontal Tensile Crack x 3 f ( ) M x x x f ( ) M x x x f ( ) 3 M 33 x x x 3 z n x u νu h/ h/ x ( x x) ( x) ( x ) ( x3 z) Figure 3.4 Horizontal tensile crack (axi-syetric w.r.t. x 3 -axis) If the horizontal crack is tension induced with displaceent discontinuity is in the z direction, u = u [,, ] T, then M becoes; M u G 8

41 (c) Horizontal Shear Crack x 3 z n u νu h/ x x h/ Figure 3.5 Horizontal shear crack If the horizontal crack (n = [,, ] T ), referencing Fig 3, is shear induced with displaceent discontinuity in the x direction, that is, u = u [,, ] T, the crack is labeled as a horizontal shear crack I, and the oent tensor, prior to noralization, defined by Eq. (.9) is; G M u G If u u(,, ), the crack is labeled a horizontal shear crack II which gives M G u G (d) Vertical Shear Crack If the crack surface is perpendicular to the x axis (n = [,, ] T T ). If u u[,, ], this crack will be called a vertical shear crack I and 9

42 G M u G In another case, the discontinuity is in the x direction, with u = u [,, ] T, the crack is labeled as vertical shear crack II, and G M G u Now consider an inclined icro-crack with noral n to the crack surface in the plate (see Fig..3). Let be angle between the noral n and the x 3, or, z-axis, and n = [sin,,cos ] T. Then =, 9 o represent the horizontal and vertical cracks respectively. The crack can be tensile if u = n u, or a shear crack if n u =. (e) Inclined shear crack T If u u[cos,, sin ], this crack will be called an inclined shear crack I in this study, and Gsin Gcos M u Gcos Gsin 3

43 T If u u[,, ], the crack is labeled as inclined shear crack II, and G sin M Gsin Gcos u G cos (f) Inclined tensile crack Finally, for an inclined tensile crack Gsin Gsin M u Gsin Gcos 3

44 CHAPTER 4. VECTORIZED EQUATIONS OF MOTION/HOMOGENEOUS INTEGRAL TRANSFORM SOLUTIONS 4. Copact Equation of Motion-Extensional Wave The diensionless extensional equations of otion in ters of displaceents with plate extensional load f E (x, t) and zero initial conditions as expressed in the developent in Appendix A, in Eqs. (A.7), can be written in atrix notation as V V V V V V t x xx x x x I T T T T T TV f, t V V, at t t E (4.) Where V uv,,, [ Φ, Φ ] T 3 f [,,,, ] T E q q n n The initial conditions reflect that the plate is unloaded at the tie the disturbance occurs. There are no boundary conditions because the plate is assued to be infinite in the x -x plane. This assuption iplies that reflections fro the boundary are not considered in the forulation. This is a reasonable assuption if the sensors can detect the first wave passage distinctly fro later reflections in a large body such as an airplane fuselage. 3

45 The Fourier transforation of Eq. (4.) with respect to the spatial variables x and x yields V Tk ( ) V f, T t t V V, at t t (4.) where x = (x, x ) and the transfored variables k = (k, k ), and ~ V ( k, t ) V ( x, t)exp( ik x) dx (4.3) ~ f T ( k, t) ft ( x, t)exp( ik x) dx and T, T, T, T, T, T in Eq. (4.) are 5x5 constant atrices fored siply by isolating the coefficients of each displaceent degree of freedo and its derivatives as indicated in Eq. (4.) above. These can be further cobined into a single atrix T, after transforing Eq. (4.) into k space, to put the equations in their ost copact for. The definition of the T ij ters and the derivation of the T atrix are given in Appendix A. See Eq. (A.38). 4. Diensionless Dispersion Relation-Extension To obtain the dispersion relationship we set all echanical loads to zero and seek the plane wave eleentary solutions to Eq. (4.) of the for V aexp[( i kx t)] (4.4) 33

46 where V [ u, v, 3,, ] T, k = [k, k ] T is the wave vector, is the frequency, and a is a coplex-valued vector (or wave aplitude). Substituting Eq. (4.4) into Eq. (4.), we have the following generalized Eigenvalue proble T M a (4.5) ( ) Setting the deterinant of the coefficient atrix in Eq. (4.5) to zero for nontrivial solutions of a, k and have to be related by T(k) M (4.6) k k (4.7) 4 5 and s ( k), s ( k), s ( k), 3 3 s ( k), s ( k) (4.8) The explicit analytical expressions for 3 (k), 4 (k) and 5 (k) are uch too lengthy to list and were calculated sybolically with the MATHEMATICA code [44]. All 5 odes however can be readily visualized by plotting the as a function of the wave nuber k, after substituting typical values for aluinu for the aterial constants in T(k). It is usual to plot i rather than i so MATHEMATICA is used to calculate the square roots of the Eigenvalues. These are plotted in Figure 4.. ω is a horizontal shear ode propagating transverse to the longitudinal wave and is a straight line with respect to k 34

47 passing through the origin with a slope of. It is designated SH. represents the first extensional ode which is designated as the S ode which also starts at the origin of the vs. k space. The third ode is another horizontal shear ode designated SH. The fourth and fifth odes are the higher order extensional odes S and S. As will be seen shortly, the contribution of these odes will not be calculated directly as they fall above the cutoff frequency. However they still couple into the proble in the transient solution which requires inversion of the T atrix which is 5x5 and affects the calculation of the S response. In fact, it is the inclusion of the SH, S, and S odes that ake the solution different than the one calculated for the first order Mindlin-Kane theory that only includes three degrees of freedo and hence three odes. The cross-sectional ode shapes for extension, shown in Fig. 4., are syetric, eaning that the response above and below the neutral axis of the plate is such that no transverse deforation or rotation of the cross-section occurs. The ode shapes for flexure, also shown in Fig. 4., are antisyetric, eaning that they give rise to transverse deforation and rotation of the cross-section. Further, the extensional odes S and S are called even syetric odes as they involve degrees of freedo ultiplied by even powers of z in the displaceent field. S and S 3 are odd syetric odes as they involve degrees of freedo ultiplied by odd powers of z in the displaceent field. The antisyetric odes can also be odd or even according to the sae logic. The flexural odes A and A are even while A and A 3 are odd with deforation patterns as shown in Fig

48 Figure 4. Mode shapes for extension bending and transverse shear k S (k) 8 SH S 6 4 S SH k c =.84 k Figure 4. Higher order theory extensional dispersion relationships - frequency vs. wave-nuber 36

49 . c =.84 S S Cg.5 S. SH.5 SH Figure 4.3 Higher order theory extensional theory dispersion group velocity vs. frequency It is also useful to express the hoogeneous solution in ters of the group velocities C vs., where CCk, C. This is shown in Fig In Figs. 4. and 4.3 k the diensionless variables, k, ω and C g, are defined by k k l,, C C / c j j g g T where l h/, ct G/, l / ct and k,, Cg are the physical wave nuber, frequency and group velocity with corresponding diensions, respectively. In the Figures as in the derivation the asterisks are dropped for convenience, which ust be reebered when un-noralizing the solution for a real proble with physical aterial constants and diensions. 37

50 The dispersion relationships fro the first order theory (Mindlin Theory) are also shown in Figure 4.6 (Ref. 45). The dispersion relationships for the lower order odes fro the elasticity theory are shown in Figure 4.7 [4]. For extensional otion, with both the first and third order theories, due to the noralized cutoff frequency of.84 based on the Lab Wave elasticity theory S iniu, shown as the dashed line in Figure 4.6 (a), and consistent with piezoelectric gage capability around MHz, the extensional transient solution will only be calculated for the S ode. The iproveent realized in the third order theory coes fro the iproveent in the S ode dispersion curve. The S ode fro the third order theory as observed in Fig. 4.3 agrees uch ore closely with the elasticity theory result sown in Fig. 4.7 than does the first order ode seen in Fig. 4.6 (b). 4.3 Copact Equation of Motion-Flexural Wave The diensionless flexural equations of otion in ters of displaceents with plate bending loads f B (x, t) and zero initial conditions, as expressed in the developent in Appendix A, in Eq. (A.8), can be written in atrix notation as with the extensional proble. U U U U U U I A A A A A AU f, t t x xx x x x U U, at t t B (4.9) Where the flexural degrees of freedo and plate forces are:,,, T U [, w, 3], B f [,, q, r, r, n ] T, z 38

51 The Fourier transforation of Eq. (4.9) with respect to the spatial variables x and x yields U ) A(k U f B, t t U U, at t t (4.) where x = (x, x ) and the transfored variables k = (k, k ), and Uk (, t) Ux (, t)exp( ikx) dx f ( k, t) f ( x, t)exp( ikx) dx B B (4.) A is fored in a anner siilar to its extensional counterpart T and is given explicitly in Appendix A, Eq. (A.43). 4.4 Diensionless Dispersion Relation-Flexure For the linear equations, Eqs. (4.) with zero loads, the wave eleentary solutions,,, T for the flexural wave otion U [, w, 3] take the basic for: U a exp[( i k x t)] (4.) Fro Eq. (4.) and (4.), we have the atrix equation ( A- I) a (4.3) fro which, for non-trival solutions, A (4.4) ( k) I Letting s = - in the above equation, we have 39

52 ( k) s I A (4.4) where s ( k), s ( k), s ( k), s ( k) s ( k),and s ( k) are roots of the equation This is the dispersion relation of the plate with k being the wave nuber, and the frequency. The roots associated with the bending dispersion relationship are odes, A, A, A, A 3, SH and SH 3 respectively ay be expressed in the for ( k), ( k), ( k), ( k), ( k), ( k) (4.6) i then s ( k), s ( k), s ( k), 3 3 s ( k), s ( k), s ( k) (4.7) As in the case of extension the actual roots in sybolic for are too lengthy to include explicitly but are plotted below with the aterial constant based on ν.33. The dispersion relationships fro the first order theory (Mindlin Theory) are also shown in Figure 4.6. The dispersion relationships fro the 3D elasticity theory, siplified with Lab Wave theory, are shown in Figure 4.7 [4]. For bending otion, with both first and second order theories, due to a noralized cutoff frequency based on the elasticity theory S iniu value of.84, the bending solution will only be calculated for the A and A odes, though the higher order odes couple into the calculation of the lower ode response. By exaination of Figs. 4.5, 4.6(b) and 4.7 it can be seen that the A ode result is siilar for all three theories. However the second bending ode, A is uch ore accurately represented by the third 4

53 order theory than with the first order theory when coparing with the elasticity theory. The iproveent realized in the third order theory coes fro the iproveent in the A ode dispersion relationship profile. 4

54 k A 3 A (k) 5 SH 3 SH 5 A A c = k k Figure 4.4 Higher order theory bending dispersion relationships frequency vs. wave nuber. c =.84.5 A A 3 A Cg. A.5 SH SH Figure 4.5 Higher order theory bending dispersion relationships group velocity vs. frequency 4

55 5 S (Elasticity) 4 S S 3 =.839 A SH SH A S (a) k First Order Plate Theory First Order Plate Theory A.5 C g SH A A.5 SH S Figure 4.6 First order theory dispersion relationships (a) - frequency vs. wave nuber (b) group velocity vs. frequency (b) 43

56 3 D Elasticity S S C g SH A A SH S Figure 4.7 Dispersion relation of lower order odes for an isotropic plate with =.33 fro 3-D elasticity 44

57 CHAPTER 5. NON HOMOGENEOUS INTEGRAL TRANSFORM SOLUTIONS 5. Extensional Wave Transfored Solution Eploying the joint integral transfors, first a double Fourier transfor on the spatial variables (x, x ) and then a Laplace transfor on the tie, t, and perforing the inverse Laplace transfor, explicit expressions for the transient response of the isotropic plate to the point source can be obtained in the integral for of -D Fourier transfors. The inverse transfor evaluation then yields the solution. For isotropic plates, after soe anipulation, the transient solution can be further reduced to a su of ters which are in the for of inverse Hankel transfors, naely, integrals of products of functions of k and Bessel functions of kr. This -D integral solution can be evaluated efficiently and accurately by nuerical integration or by asyptotic expansion techniques such as the stationary phase ethod. solution; Starting with Eqs. (4.) and (4.3), repeated below for copleteness of the transient V Tk ( ) V f, E t t V V, at t t where x = (x, x ) and the transfored variables k = (k, k ), and 45

58 ~ V ( k, t ) V ( x, t)exp( ik x) dx Where the single aterial constant f ( k, t) f ( x, t)exp( ikx) dx E E ( G) / G ( ) / ( ) where is Poisson s ratio, and and G are the Lae constants. T is fored by superiposing the coefficient sub atrices T ij and T i as shown in Appendix A. For extensional otion then, the application of the Laplace transforation of Eq. (4.) with respect to t leads to I T V f (5.) (s ) E where V( k,s) V ( k,t)exp( st)dt, f ( k,s) f ( k,t)exp( st)dt (5.) The solution of Eq. (5.) is E E (5.3) V Ts I fe ( ) adj T s I adj T s I f T ( )( )( )( )( ) ( ) ( ) f E s I a s s s s s s3 s s4 s s5 E where s ( k), s ( k), s ( k), s ( k)and s ( k) are roots of the equation, developed above,

59 T( k) s I (5.4) and a is the leading coefficient of the fifth order polynoial that represents the extensional Eigenvalues. Though it is divided out in the hoogeneous proble and the Eigenvalue equations are further partitioned into a fourth and a second order polynoial to facilitate the solution, it is still needed for the transient solution. It is carried with D. This ter is typically for the lower order theories. Letting Ds ( ) T si, Q [ Q ij ] adj( T s I), Eq. (5.3) can be rewritten as ij [ Q ( s )] f E Ds ( ) V (5.5) where Q ij (s ) and D(s ) are polynoials in s and the degree of D is higher than Q ij. To carry out the inverse Laplace transfor of Eq. (5.5), the ethod of partial decoposition is used, which yields Q Q Ds ( ) ( s s ) (5.6) ij 5 ij ( s ) D Where s ( k) ij ij ( ) Q Q s dd D a D ( s ) ( s s ) ( ) 5 5 n n ds ( ) s s n, n n, n (5.7) With denoting a ultiplication of the ters over n rather than a suation. With Eq. (5.6), Eq. (5.5) becoes 47

60 V Q f E (5.8) 5 ij D ( s s ) 5x5 Now it is easy to apply the inverse Laplace transfor (Debnath, 995 [9]) to the above equation via the convolution theore, and Q t t ij 5 5x5sin( ) * fe( k, ) V (5.9) D where * denotes convolution with respect to tie, t sin( t) * f ( k. t) f ( k, )sin ( t ) d E E Finally, applying the inverse Fourier transfor to Eq. (5.9), the solution is given by T V ( x, t) [ u, v, 3,, ] V ( t, k)exp( ikx) dk 4 (5.) Vk (, t) 5 ij [ Q] 5x5sin( t)* f E( k, t) D sin( t) [ Q q Q q Q Q n Q n]* D (5.) where Q i (k) is a 5 atrix Q Q i i i 3i Q Q i,,3, 4,5 (5.) Q Q 4i 5i 48

61 where the ~ denotes the -D Fourier transfor of the quantities in space (x, x ), f [,,,, ] T E q q n n is the diensionless plate loading for extensional otion, the sybol * denotes the convolution between the transposed plate loading f E and sinwt with respect to tie, W (k) is the dispersion relation of the th extensional wave ij i ode, and all the other sybols, D ( k), W ( k), Q ( k), Q have been defined above. 5. Extensional Transfored Plate Loads For the point source at (,, z ) with oent tensor M ij (t), the Fourier transfor of the diensionless plate loads, Eq. (3.3), ust be coputed. The result is, f [,,,, ] T E q q n n, where q ik M ik M q ik M ik M i3k M z ik 3 M z 3M n 5ik M z 5ik M z M z 3 n 5ik M z 5ik M z M z 3 (5.3) In the derivation of Eq. (5.3) the property of the Fourier transfor ( n) n exp( ikx ) ( x) dx ( ik) (5.6) has been used. Note that F i is used later and is siply f E with a i factored out. To provide ore clarity into the transforing of the noralized plate loads the third of Eqs. (3.3) is transfored in Appendix E to produce the third of Eqs. (5.3). 49

62 5.3 Flexural Wave Transfored Solution Proceeding in siilar fashion to the extensional proble for bending, starting with Eqs. (4.) and (4.), repeated below for copleteness of the transient solution; U ), A(k U f B t t U U, at t t where x = (x, x ) and the transfored variables k = (k, k ), and Uk (, t) Ux (, t)exp( ikx) dx f ( k, t) f ( x, t)exp( ikx) dx B B then application of the Laplace transforation of Eq. (4.) with respect to t leads to AU f (5.5) ( s I ) B where U( k, s) U ( k, t)exp( st) dt, f ( k, s) f ( k, t)exp( st) dt (5.6) The solution of Eq. (5.5) is B B (5.7) adj( s ) A s I f B f B U ( ) A A s I adj( T s I) f b ( s s )( s s )( s s )( s s )( s s )( s s ) I B where s ( k), s( k), s3( k), s4( k), s5( k) ands6( k) are roots of the equation, developed above, 5

63 and b is the leading coefficient of the sixth order polynoial that represents the flexural Eigenvalues. Though it is divided out in the hoogeneous proble and the Eigenvalue equations are further partitioned into a fourth and a second order polynoial to facilitate the solution, it is still needed for the transient solution. As in the extensional case it is carried with D. This ter is typically for the lower order theories. A ( k) s I Letting D( s ) T s I, Q [ Q ij ] adj( T s I), Eq. (5.7) can be rewritten as ij [ Q ( s )] f B Ds ( ) U (5.9) where Q ij (s ) and D(s ) are polynoials in s and the degree of D is higher than Q ij. To carry out the inverse Laplace transfor to Eq. (5.9), the ethod of partial decoposition is used, which yields Q Q Ds ( ) ( s s ) (5.) ij 6 ij ( s ) D where s ( k) ij Q Q s ij ( ) dd D b D ( s ) ( s s ) ( ) 6 6 n n ds ( ) s s n, n n, n (5.) With denoting a ultiplication of the ters over n rather than a suation. With Eq. (5.), Eq. (5.9) becoes 5

64 Q U f B (5.) 6 ij D ( s s ) 6x6 Now it is easy to apply the inverse Laplace transfor (Debnath, 995 [9]) to the above equation via the convolution theore, and Q t t ij 6 6x6sin( ) * fb( k, ) U (5.3) D where * denotes convolution with respect to tie, t sin( t) * f ( k. t) f ( k, )sin ( t ) d B B Finally, applying the inverse Fourier transfor to Eq. (A.8), the solution is given by T Ux (, t) [,, w,,, 3] (, t)exp( i ) d 4 Uk k x k (5.4) with Uk (, t) 6 ij [ Q] 6x6sin( t)* f B( k, t) D sin( t) [ Q Q Q q Q r Q r Q nz]* D (5.5) where Q i (k) is the 6 atrix, j Q j Q 3 j Q i Q,, 3, 4, 5, 6 4 j i (5.6) Q 5 j Q 6 j Q 5

65 5.4 Flexural Transfored Plate Loads f B [,, q, r, r, n ] T is the diensionless plate load for flexural otion. In the z case of a point source at (,, z ) with oent tensor M ij (t), the Fourier transfor of the diensionless plate load, f [,, q, r, r, n ] T (5.7) B z as expressed in Eqs. (3.4) is: i3k M z i3k M z 3M 3 i3k M z i3k M z 3M 3 q ik M ik M 3 3 r 7ik M z 7ik M z M z r 7ik M z 7ik M z M z n 5ik M z 5ik M z M z z (5.8) The transforation process is siilar to that eployed for the extensional plate forces and will not be shown here for brevity. Note that F i is used later and is siply f B with a i factored out. 5.5 Transient Solution by Application of Inverse Fourier Transfor Thus, the total transient response of the plate to the point source is expressed in a copact integral for as the su of Eqs. (5.) and (5.4). Note that the derivations for the extensional and flexural otions follow the sae procedures and that for convenience, Eq. (5.) and (5.4) are written in the sae for except for the loadings. Note however 53

66 that the sybols D (k), ( k ), Q ij ( k), Q, are associated with different otions, and are i defined separately for extension and bending. Recall that the oent tensor M ij (t) for a point source can be expressed as M ij M ij f ( t) (Eq. (.)) where M ij, the spatial coponents, are constants for a given source. With Eq. (5.) and (5.4), the inverse Fourier transfor integral solution can also be rewritten in the for 5 i H V [ F F F3 F4 F5 ] exp( i ) d 4 Q Q Q Q Q k x k (5.9) D 6 i H U [ F F F3 F4 F5 F6 ] exp( i ) d 4 Q Q Q Q Q Q k x k (5.3) D With F f / ifor extension and F f / i for flexure, which essentially factors the i E i iaginary nuber i out of the transfored forces and places it as a ultiplier on the solution vectors as seen above. B The function H (k, t) is defined by sin t H t f t f t d i (5.3) t ( k, ) ( ) ( )sin ( ),,,3 Fro the above solutions, the total surface strains, ( ) zh /, which are easily easured fro the experients using piezoelectric sensors, can be derived fro the strain displaceent relationships. 54

67 5.6 Strain Forulation Extensional otion 5 5 i 3i i 3i H [( k( Q z Q ) k( Q z Q )]{ Fi} exp( i ) d 4 kx k (5.3) i D Flexural otion H [( k ( z ) k ( z z )]{ } exp( i ) d 4 k x k (5.33) 6 6 i 3 4i i 3 5i Q Q Q Q F i i D Again, the Q ters are different for extension and flexure even if they carry the sae subscripts and superscripts. A derivation of the strain expressions is detailed in Appendix G 5.7 Cartesian to Polar Transforation Reduction to D Hankel Transfors The transient solutions, Eq. (5.9), (5.3), (5.3), and (5.33), are given in ters of the two-diensional inverse Fourier transfor, where the integration is a double integral with respect to k and k when the wave nuber vector k is expanded. For isotropic plates the solutions ay be expressed as a series involving the integral of products of Bessel functions of kr and other functions of a single variable k which take the for of inverse Hankel transfors. This reduction fro a -D to -D integral is developed as follows. In ters of polar coordinates k k cos, k k sin (5.34) 55

68 The exponential ter becoes x r cos, x sin (5.35) r exp( ik x) exp[ ikr cos( )], dk dkdk kdkd Eq. (5.) and (5.4), along with (5.9) and (5.3), indicate that the Fourier transfor of any displaceent coponent, for exaple, the extensional displaceent u ~, can be written as H u ( k, t) i[ FQ FQ F3Q F4Q F5Q] (5.36) D ij Note that Q, j,, 3, 4,5 for u, v, 3,, respectively, picking the proper row of the Q ij atrix, according to the atrix notation, Eq. (5.6). Substituting Eq. (5.34) into Eq. (5.36), u ~ can be expressed as a suation H u ( k, t) [ C ( k)cos n S ( k)sin n ] 5 5 n n n D ( k, t) ( k) H ( k, t) [( )exp( ) ( )exp( )] ( k ) 5 5 Cn isn in Cn isn in n D (5.37) Where the coefficients C ( k) and S ( k) are coplex and deterined by carrying out the n n ultiplications indicated in Eq. (5.36) and then reducing all powers sin n and cos n to sin(n) and cos(n). These considerable algebraic and trigonoetric anipulations lead to Eq. (5.4). Note that in Eq. (5.37) C n, S, H, D are independent of. n These anipulations cannot be done reasonably by hand but were accoplished with the MATHEMATICA code for the third order theory. This process can be carried out 56

69 anually, with difficulty, for sipler probles. As an exaple, for a D plane stress proble with only displaceent degrees of freedo, u and v, which is the subject of Appendix J, the coefficients are calculated in Appendix J.A in order to provide insight into how they are coputed here. Appendix J is included to provide insight into the higher order theories using a sipler proble where all quantities can be anually coputed. Substituting Eq. (5.37) into the integrand of Eq. (5.9) for extension, and integrating with respect to first, the integration with respect to k leads to a series of inverse Hankel transfors. For exaple, the displaceent u in extensional otion can be expressed as 5 5 C n isn H ur (,, t) exp[ cos( )] kdk in ikr d 4 n D C n isn H kdk exp[ in ikr cos( )] d } D (5.38) Making a change of variable /, and using the integral representation of the Bessel function of order n, J n ( kr) exp[ i( n kr sin)] d n and using the definition J n ( kr) ( ) J n ( kr) (where the upper liit ( / ) ), Eq. (5.38) is reduced to H ( k, t) u U n ( k, ) J n ( kr) kdk (5.39) 5 5 n D ( k) where 57

70 U n C in( ) n n S ( k, ) [ e ( ) cc] i n i ( C cos n S sin n ) n n n [ e in( ) n ( ) cc] (5.4) and cc stands for the coplex conjugate of the preceding ters. Therefore, for isotropic plates the transient wave solutions are reduced to a series of the inverse Hankel transfors. Perforing siilar anipulation on v, and, and 3 the displaceents for extensional otion are, u U v V ( ) n ( k, ) ( k, ) ( k, ) J ( kr) kdk n 5 n ax n H ( k, t) n n (5.4) n D k n ( k, ) ( k, ) where, Un Unc ( k) Uns ( k) V n Vnc ( k) Vns ( k) n nc ( k) cos n ins ( k) sinn n nc ( k) ns ( k) n nc ( k) ns ( k) (5.4) The response of the surface strains to the point source due to extensional otion can be obtained fro Eq. (5.3) by perforing siilar anipulations or fro Eq. (5.4) by applying spatial differentiation to the displaceent field. Fro Eq. (5.3), it follows that 58

71 E H ( k, t) En ( k, ) Jn ( kr) kdk (5.43) 5 5 n D ( k) Note that the 5 extensional odes are used and as any non-zero coefficients are used as result fro the trigonoetric reduction, up to 5. The coefficients contain coponents, E n E ( k)cos n E ( k) sin n. These expressions in sybolic for nc ns are extreely lengthy for the third order theory and will not be listed here. However, the expressions for E nc and E ns are given explicitly in Appendix I for the first order theory. After perforing siilar laborious calculations eploying MATHEMATICA for flexural otion the resulting displaceents are, n ( k, ) 6 ax n ( k, ) w n H ( k, t) n n D( k) n ( k, ) n ( k, ) 3 n ( k, ) ( k, ) J n ( kr) kdk (5.44) ( k) ( k) n nc ns n nc ( k) ns ( k) n nc ( k) cos n ns ( k) sinn ( k) ( k) n nc ns n nc ( k) ns ( k) n nc ( k) ns ( k) (5.45) 59

72 The surface-strain response can be derived fro Eqs. (5.33) or fro Eq. (5.44). Using Eq. (5.33) for extension it follows that B H ( k, t) En ( k, ) Jn ( kr) kdk (5.46) 6 5 n D( k) Though this expression is very siilar to its extensional counterpart note that 6 odes are included and the appropriate odes for bending are used here. Where E n E nc ( k)cos n E ( k) sin n. The expressions for E nc and E ns are too lengthy ns to list for the third order theory but are written explicitly for the first order Mindlin theory in Appendix I, which suarizes the key equations for the first order theory. A derivation of Eqs. (5.43) and (5.46) is the subject of Appendix G. 5.8 Outgoing AE waves and Asyptotic Solutions The integral representation of the displaceent field includes non-propagating (evanescent), outgoing propagating and incoing propagating waves. In order to distinguish outgoing and incoing waves it is useful to replace J (kr) in the integrands by J () () ( kr) [ H ( kr) H ( kr)]/ (5.47) () () where H ( x) and ( x) are the coplex Hankel functions of the first and second kind of order defined as H v H H () () ( x) J ( x) J ( x) iy ( x) iy ( x) ( x) where Y (x) is the Bessel function of the second kind of order. (5.48) For large values of kr, the asyptotic approxiations to these coplex Hankel functions are 6

73 ( ) i( kr / 4 / ) () i( kr / 4 / ) H ( kr) ~ e, H ( kr) ~ e (5.49) k r k r In conjunction with the factor exp(-it) contained in the loading ter H (k,t), these functions represent the outgoing and incoing waves respectively. In general, the response function H (k,t) in the integrand of the integral solutions, Eqs. (5.4) or (5.43) for extension, and Eqs. (5.44) or (5.46) for bending, contain the ters, sin t or cos t, which, as is shown in Appendix H, are related to t, for exaple H ( cos t) ( k, t) (5.5) for source tie function described by Heaviside step function. Then the integrand can be siplified by use of the following two expressions [ H [ H () () ( kr) H ( kr) H () () ( kr)]cos t Re{[ H n ( kr)]sin t Re{[ H n () () ( kr) H ( kr) H () () ( kr)] e ( kr)] ie int int } } (5.5) () () () Fro the asyptotic approxiations for H ( kr) and H ( kr), the ter with ( kr) () represents the outgoing waves while the ter with ( kr) represents the incoing waves. If only outgoing waves are desired, the surface strain responses, Eq. (5.43) for extension and (5.46) for bending becoe, after utilizing Eqs. (5.47), (5.5), and (5.5), H H 5 5 n D ( k) cos t E ( k, ) J ( kr ) kdk E ( k, ) 4 ( ) 5 5 n () it Re[ H ( ) ] n kr e kdk n D k n n (5.55) 6

74 where the appropriate coefficients are used for extension and bending respectively and the suation over includes 6 odes in bending. It is usually not feasible to evaluate the integral solutions exactly. The alternatives are nuerical evaluation, or an approxiate analytical evaluation. 6

75 CHAPTER 6. NUMERICAL RESULTS 6. Results Overview An isotropic plate with Poisson s ratio =.33 was chosen for the nuerical studies. Now consider the transient response. Assue that the sources are located at (,, z ) with the oent tensor M ij. The tie dependence of the oent tensor M ij (t) is assued to be the Heaviside step function H(t) in the calculation. Note that a rap function with a rise tie (Guo et al., 996 [9, ]) can also be used to odel the tie dependence of M ij (t). For the source with a unit Heaviside step function, the convolution function H (k,t) defined in Eq. (5.3) is H ( cos t) ( k, t) (6.) A derivation of the forcing function in Eq. 6. is shown in Appendix H. The transient response to the local sources in isotropic plates has been evaluated by nuerical integration. Since the solutions are given in integral for with infinite upper and lower liits, a high-frequency cutoff (or high-wave nuber cutoff) should be eployed in the nuerical calculation. In general, the frequencies of acoustic eissions that occur in structures are often band-liited. Suppose a wavelet of AE signals has its energy confined ostly within < ax. Therefore, a proper axiu frequency ay be chosen for a given proble in the 63

76 nuerical integration. With a given axiu frequency, a finite nuber of wave odes (wave nubers) can be deterined. Then fro the dispersion relation, = ω (k), the corresponding axiu wavenuber (or cut-off wavenuber) for each ode can be deterined. Thus the integration can be carried out using an adaptive step size in the wavenuber doain for each wave ode. A axiu frequency was often used in stationary phase approxiations in Fourier inversions. The frequency of the second propagating extensional ode of a coposite bea was considered as the axiu frequency (Aberg and Gudundson, []), and f.5 MHz and. MHz, were applied to the first extensional and flexural odes in ax coposite plates, respectively (Guo et al., 996 [9]). For isotropic plates, the cut-off frequency for the S ode is.5( ) /( ) It is also the iniu c propagating frequency for the S ode according to the plate theory. However, 3-D elasticity gives the iniu frequency, ( s ) in.839, for the corresponding ode (see Fig. 4.7). Thus, (k) < ( s ) in =.839 is used for all wave odes in the inverse Hankel transforation calculation. Obviously, in the frequency region < < ( s ) in five odes: S, SH, A, A and SH, coexist and the nuerical solution is valid up to this axiu frequency. For exaple, if an aluinu plate with thickness h =.6 is used, then the diensionless frequencies =.839 and 3.8 correspond to f =.75 and.93 MHz respectively. Hence, the solution is valid up to this cutoff frequency in AE wavefor analysis. In other nuerical studies, siilar bounds are used: Guo et al., 996 [9] (< MHz); Hastad et al., [3] (< MHz). 64

77 with For a point source the oent tensor can be can be expressed as M M M M (6.) 3 /( Gl ), M f() t f ( G) A, for shear crack f (, G) A, for tensile crack As shown in section 3.4 where A is the aplitude of the displaceent discontinuity, is the source (crack) area, and f(t) is the source tie function without diension. In general, the values of, A and M are unknown. For convenience it is assued M /( Gl 3 ) in the following nuerical calculations. Thus, the siulated strain wavefor due to a point source provides relative aplitudes of different wave odes. If M Gl, the /( 3 ) wave shape is the sae, and the aplitude can be obtained siply by ultiplying the 3 nuerical result by the factor M /( ) Gl. The relative values of the oent tensor coponents M were detailed in section Results General Figures 6. through 6.4 and 6.6 through 6. give the transient responses of the surface strains ε kk due to four of the nine source types, vertical tension crack I, horizontal tension crack, horizontal shear crack I and inclined shear crack I at z = h/4 (diensionless z = ½), that is, one-fourth plate thickness fro the top plate surface. Figure 6.5 shows the u displaceent at θ= o and 45. The response location distance is at diensionless distance r= fro the source for all of the figures. The wave fors for strain fro the 65

78 third order theory are copared with those fro the first order Mindlin theory which is described in [45] and outlined in Appendix I. For the vertical tensile crack I, results are calculated at = o and 45 o to show the angular dependence. The responses are due to a transient step function H(t), the Heavyside ipact loading approxiation with a constant force requiring no rise tie. The nuerical results for the nine types of fundaental AE sources, with the tie function being a rap function, are also readily obtainable fro the third order plate theory, but are not presented here for brevity. For each source, the calculated surface strain response, kk, that can be easily easured fro a piezoelectric sensor in an experient, is presented. Note that for the horizontal tensile crack source the response is axisyetric, ε kk = ε kk (r,t). The wavefor due to the vertical shear crack I can be obtained siply by changing the sign of the wavefor due to the horizontal shear crack I (see the oent tensors for the two sources in Section 3.4). Thus, the wavefor due to the vertical shear crack I is not shown but akes up a fifth source type. The wavefor of the surface strains changes with crack orientation. The effects of the orientation on the wavefor are also shown in Figs. 6.8 and 6. which are for a horizontal shear crack I and a shear crack inclined at 3 o. When is varied, the ratio between the aplitude of extensional and flexural waves will change. Fro the figures discussed above, the following phenoena can be observed. The group velocity for the A ode according to the elasticity and the third order theories is C g. as can be seen in Figs. 4.5 and 4.7. Note that the arrival tie is roughly the distance 66

79 divided by the average group velocity, in this case, /. or diensionless t=. Actually the average group velocity is slightly less due to the rise tie. This supports the observed arrival tie of about 95 observed for the A ode. The group velocity for the A ode according to the elasticity and the third order theories is C g.8 as can be seen in Figs. 4.5 and 4.7. Again, noting that the arrival tie is roughly the distance divided by the axiu group velocity, or /.8 here, for a diensionless t=7. This supports the observed arrival of about 5 to 7 observed for the A ode. Hence, the arrival tie for the A ode is not close to the arrival tie for S (axiu C g.73 for the S ode). In the Mindlin theory the group velocities and hence arrival ties for the S and A odes are close. This is a shortcoing of the first-order plate theory. Although the plate theory uses the exact cut-off frequency for the A ode, the wave behavior of this ode is poorly described beyond this frequency when coparing with the corresponding ode fro the elasticity theory. The ost striking difference between the wave fors generated by the third order theory and the first order theory, in addition to the variation in arrival ties, is the relative doinance of the S extensional contribution to the total response. The S ode group velocity is quite different than the one generated fro the first order theory being larger in agnitude and longer in duration. Such a change is not unexpected in light of the differences in the dispersion curves. Soe particular observations fro the individual disturbance source types are as follows. 67

80 6.3 Vertical Tensile Crack Looking at the = o and =45 o surface strain results, both for the third order theory, Figs. 6. and 6.3, and for the first order theory, Figs. 6. and 6.4, it is observed that the strain aplitude is greater at = 45 o rather than at = o at diensionless r= for the source. This is for a dipole loading that siulates a discontinuity along the x axes corresponding to = o. It is expected that ost of the responses would be the greatest along the line of action of the force, as this loading is not axisyetric. This is the case for the ore fundaental responses such as the u displaceent which coincides with the x direction. This can be seen clearly in Fig. 6.5, which is a plot of the u displaceents at = o and =45 o for the bending ode A. The u displaceent here fro the flexural forulation involves the degrees of freedo associated with bending fro Eq. (A.), which are the ters ultiplying odd powers of z in the total displaceent field. u ( x, z, t) z ( x, t) z ( x, t) 3 The response along the line of action is significantly greater. The reason that this is not the case with the strain ε kk is inherent in the definition of the strain. ε kk represents an arithetic su of the coponent strains ε and ε, which is the total strain that would be sensed by a non-directional piezoelectric strain gage typically used in AE applications. The ε coponent which is siply du /dx is larger for the sensor located along the x axis corresponding to = o. However, the ε coponent which is du /dx, is virtually zero along the x axis because the transverse coponent u is nearly zero there. This is not the 68

81 case at =45 o where the transverse displaceent u is nearly 6% of the u displaceent, giving rise to the second ter in the strain definition. Hence this total strain ε kk is larger at = 45 o unlike the ore fundaental response quantities that coincide with the intuitive result of being larger along the line of action of the force application. Though the solution accounts for transverse shear waves, the x axis can be thought of as a line of syetry for the transverse horizontal waves which only build up response away fro the line of action of the force. Along the line of action of the force behavior is doinated by the longitudinal waves. 6.4 Horizontal Tensile Crack The horizontal tensile crack, which accounts for a discontinuity in the x 3 direction can be thought of as having a vertical force line of action. This is soewhat like the pebble in the water scenario except for the dipole nature of the loading, but siilar in the sense that the resulting response is axi-syetric. The ε kk responses can be seen in Fig. 6.6 for the third order theory and Fig. 6.7 for the first order theory. The third order response is characterized by a large, late peaking S ode which clearly couples with the dispersing bending response in later tie. This varies substantially fro the first order response where the extensional ode is less pronounced, shorter in duration and earlier arriving which largely uncouples it for the bending ode response in its contribution to the total response, as seen in Fig

82 6.5 Shear crack I The wavefors of the individual strains ε and ε both have SH and SH 3 ode coponents as expected, since these are not axi-syetric waves. However, the wavefor of the total surface strain ε kk = ε + ε has no SH ode coponent at all. In fact, according to the plate theory, the SH odes ake no contribution to the surface strains ε kk in isotropic plates, and this can be readily proven fro the expressions for the surface strain response to a general point source. Unlike the vertical and horizontal tensile crack responses, the horizontal shear crack response is not as heavily influenced by the extensional S ode. The bending response has a greater agnitude and therefore is ore clearly discerned in the total response. 6.6 Inclined Shear Crack I The inclined shear crack is in the x - x 3 plane inclined at an angle of β =3 with the x axis as defined in Fig..3 and depicted in Fig. 3. with the oent tensor as defined in Section 3.4. This discontinuity exhibits a strong extensional coponent S both for the first and third order theories, particularly so in the third order theory where the extensional response doinates the overall response. 7

83 6.7 Coparison with Elasticity Theory The third order theory was copared with an elasticity forulation developed in Ref. 4. The elasticity solution was derived for non-dipole loadings but adaptable to dipole loading, but only for an axi-syetric loading. Therefore it was copared with the horizontal tensile loading case fro the third order theory. The coparison is seen in Fig. 6.. The agreeent is not perfect but the third order theory wavefors are uch closer to the elasticity solution than those predicted by the first order theory. 7

84 Figure 6. Surface strain transient response to source: Vertical Tensile Crack =. Third Order Theory The arrows indicate the corresponding odes near arrival. r=, z =h/4 7

85 Figure 6. Surface strain transient response to source: Vertical Tensile Crack =. First Order Theory. The arrows indicate the corresponding odes near arrival. r=, z =h/4 73

86 Figure 6.3 Surface strain transient response to source: Vertical Tensile Crack = 45 Third Order Theory. The arrows indicate the corresponding ode near arrival r=, z =h/4 74

87 Figure 6.4 Surface strain transient response to source: Vertical Tensile Crack = 45. First Order Theory. The arrows indicate the corresponding odes near arrival. r=, z =h/4 75

88 Figure 6.5 Surface displaceent transient response to source: Vertical Tensile Crack = and = 45 for A flexural ode. Third Order Theory. The arrows indicate the corresponding odes near arrival. r=, z =h/4 76

89 Figure 6.6 Surface strain transient response to source: Horizontal Tensile Crack (axi-syetric). Third Order Theory. The arrows indicate the corresponding odes near arrival. r=, z =h/4 77

90 Figure 6.7 Surface strain transient response to source: Horizontal Tensile Crack (axi-syetric). First Order Theory. The arrows indicate the corresponding odes near arrival. r=, z =h/4 78

91 Figure 6.8 Surface strain transient response to source: Horizontal Shear Crack I =. Third Order Theory. The arrows indicate the corresponding odes near arrival. r=, z =h/4 79

92 Figure 6.9 Surface strain transient response to source: Horizontal Shear Crack I =. First Order Theory The arrows indicate the corresponding odes near arrival. r=, z =h/4 8

93 Figure 6. Surface strain transient response to source: Inclined Shear Crack I =3, =. Third Order Theory. The arrows indicate the corresponding odes near arrival. r=, z =h/4 8

94 Figure 6. Surface strain transient response to source: Inclined Shear Crack I =3, =. First Order Theory. The arrows indicate the corresponding odes near arrival. r=, z =h/4 8

95 8 x 4 Horizontal Tensile Crack kk 8 z = /4 r = 3 D Elasticity 5 x 4 S kk 5 Extensional Motion 8 x 4 A kk A Flexural Motion t Figure 6. Surface strain wavefor odeled by Theory of Elasticity and Third Order Theory fro AE source: Horizontal Tensile Crack at z = h/4, wavefor observed at r = 83

96 CHAPTER 7. DISCUSSION AND CONCLUSIONS The siulation of transient wave response for the point sources in an isotropic plate based on third-order plate theory provides physically reasonable solutions up to the given axiu frequency iposed in the nuerical evaluation. This is evidenced by the fact that the higher order plate theory provides good approxiation to the first arrival tie fro the AE source for the two fundaental odes (A, S ) and also with the A ode, unlike the first order theory. These wave radiation patterns (or wavefors) due to point sources ay serve as a set of references to locate and recognize the active daage in an isotropic plate. In general, the flexural otion always plays an iportant role in the wavefor eitted by AE source in plates. Thus, in ost cases, the plate theory ay give reasonable AE wavefors, or at least flexural wavefors, up to the first cut-off frequency c for the higher-order ode A. The axiu frequency ipleented in the nuerical calculation ay vary for generation of different icro-daage echaniss. If the higher frequencies of the AE signals generated by the sources can be detected, such as signals fro the higher-order S ode if a sensor (receiver) with high-frequency bandpass is used for easureent, then the value of axiu frequency should be increased in the plate theory for valid coparison with experients. 84

97 Away fro the low frequency region the third-order plate theory gives good results for the S ode. The wave behavior of the higher-order odes S and A provided by the first-order plate theory are not in good agreeent with the corresponding odes fro the elasticity theory. Substantial iproveent in the dispersion relations are observed for both the lowest odes and the higher-order odes with the third order theory. Therefore, ore accurate wavefors eitted fro AE sources can are realized fro a higher-order plate theory. Applying the joint integral transfors, the transient response of the coposite plate to a point source can be expressed as double inverse Fourier transfor as well, although it cannot be reduced to the for of Hankel transfor because of differing stiffnesses in the fiber and atrix directions. Perforing nuerical inversion of the integral solutions, the detailed wave patterns radiated fro the sources in a coposite plate can also be calculated with the higher order plate theory. In suary, the higher order extensional theory has been extended to include AE loading, the reduced integration Bessel function transient solution concept was applied to the higher order extensional and bending solutions, and a higher order bending theory with general Cartesian AE loading solutions was developed and ipleented. Future developent could include a general 3D elasticity sei-analytical solution with dipole loading, an anisotropic solution for coposite aterial systes, verification by testing of the current theory starting with the plane stress solution developed in Appendix J, and ost iportantly, the solution of the inverse proble of reconstructing dipole daage 85

98 sources fro easured strain signals. This, coupled with odern wireless sensor technology and current real-tie data acquisition systes, could lead to a robust structural health onitoring syste for future generation aircraft. Such a syste would pinpoint and onitor daage forulation and accuulation continuously rather than requiring tie-consuing scanning during infrequent tear down inspections. Early identification of potentially dangerous daage could enhance aircraft safety and iniize repair costs. 86

99 REFERENCES. M. Aberg and P. Gudundson, Microechanical Modeling of Transient Waves fro Matrix Cracking and Fiber Fracture in Lainated Beas, International Journal of Solids and Structures, Vol. 37, pp ,.. J. D. Achenbach, Wave Propagation in Elastic Solids, North-Holland Publishing Copany, Asterda, T. Aizawa, T. Kishi, and F. Mudry, Acoustic Eission Wave Characterization: A Nuerical Siulation of the Experients on Cracked and Uncracked Speciens, Journal of Acoustic Eission, Vol. 6, pp. 85-9, K. Aki, Estiation of Earthquake Moveent, Released Energy, and Stress-strain Drop fro G Wave Spectru, Bulletin of the Earthquake Research Institute, Tokyo University, Vol. 44, pp. 3-88, K. Aki and P. G. Richards, Quantitative Seisology, University Science Books, Sausalito, California, D. E. Bray and D. McBride, Nondestructive Testing Techniques, John Wiley & Sons, Inc., New York, R. Burridge and L. Knopoff, Body Force Equivalents for Seisic Dislocations, Bulletin of Seisological Society of Aerica, Vol. 54, pp , C. Chang and W. Sachse, Analysis of Elastic Wave Signals fro an Extended Source in a Plate, Journal of the Acoustical Society of Aerica, Vol. 77, pp , L. Debnath, Integral Transfors and Their Applications, CRC Press Inc., Boca Raton, FL, K. S. Downs and M. A. Hastad, Acoustic Eission fro Depressurization to Detect/Evaluate Significance of Ipact Daage to Graphite/Epoxy Pressure Vessels, Journal of Coposite Materials, Vol. 3, No. 3, pp ,

100 . Y. A. Dzenis and J. Qian, Analysis of Microdaage Evolution Histories in Coposites, International Journal of Solids and Structures, Vol. 38, pp ,.. J. P. Favre and J. C. Laizet, Aplitude and Counts per Event Analysis of the Acoustic Eission Generated by the Transverse Cracking of Cross-ply CFRP, Coposites Science and Technology, Vol. 36, pp. 7-43, T. J. Fowler, J. A. Blessing, P. J. Conlisk, and T. L. Swanson, The MONPAC Syste, Journal of Acoustic Eission, Vol. 8, No. 3, pp. -8, S. Ghaffari and J. Awerbuch, On the Correlation between Acoustic Eission and Matrix Splitting in a Unidirectional Graphite/Epoxy Coposite, Acoustic Eission: Current Practice and Future Directions, ASTM STP 77, W, Sachse et al., Eds., ASTM, Philadelphia, pp , M. R. Goran, Plate Wave Acoustic Eission, Journal of the Acoustic Society of Aerica, Vol. 9, No., pp , M. R. Goran and S. M. Ziola, AE Source Orientation by Plated Wave Analysis, Journal of Acoustic Eission, Vol. 9, No. 4, pp , M. R. Goran and S. M. Ziola, Plate Waves produced by Transverse Matrix Cracking, Ultrasonics, Vol. 9, pp. 45-5, P. J. de Groot, P. A. M. Wijnen, and R. B. F. Janssen, Real-tie Frequency Deterination of Acoustic Eission for Different Fracture Mechaniss in Carbon/Epoxy Coposites, Coposites Science and Technology, Vol. 55, pp. 45-4, D. Guo, A. Mal, and K. Ono, Wave Theory of Acoustic Eission in Coposite Lainates, Journal of Acoustic Eission, Vol. 4, pp. 9-46, D. Guo, Lab Waves fro Microfractures in Coposite Plate, Doctoral thesis, University of California, Los Angeles, M. A. Hastad and K. S. Downs, On Characterization and Location of Acoustic Eission Sources in Real Size Coposite Structures A Wavefor Study, 88

101 Journal of Acoustic Eission, Vol. 3, No. /, pp. 3-4, M. A. Hastad, Thirty Years of Advances and Soe Reaining Challenges in the Application of Acoustic Eission to Coposite Materials, Acoustic Eission Beyond the Millenniu, T. Kishi et al., Eds., Elsevier Science Ltd., UK, pp. 77-9,. 3. M. A. Hastad, A. O Gallagher, and J. Gary, Effects of Lateral Plate Diensions on Acoustic Eission Signals fro Dipole Sources, Journal of Acoustic Eission, Vol. 9, pp ,. 4. N. A. Haskell, Radiation Pattern of Surface Waves fro Point Sources in Multi-layered ediu, Bulletin of the Seisological Society of Aerica, Vol. 54, pp , T. R. Kane and R. D. Mindlin, High-frequency Extensional Vibrations of Plates, ASME, Journal of Applied Mechanics, Vol. 3, pp , B. V. Kostrov, Seisic Moent and Energy of Earthquake, and Seisic Flow of Rock, Izvestiya, Physics of the Solid Earth, Vol., pp. 3-, T. Maruyaa, On the Force Equivalents of Dynaic Elastic Dislocations with Reference to the Earthquake Mechanis, Bulletin of the Earthquake Research Institute, Tokyo, University, 4, , J. E. Michaels, T. E. Michaels, and W. Sachse, Application of Deconvolution to Acoustic Eission Signal Analysis, Materials Evaluation, Vol. 3, pp. 3-36, R. D. Mindlin, Influence of Rotatory Inertia and Shear in Flexural Motion of Isotropic Elastic Plate, ASME, Journal of Applied Mechanics, Vol. 8, pp. 3-38, R. D. Mindlin and M. A. Medick, Extensional Vibration of Elastic Plate, ASME, Journal of Applied Mechanics, Vol. 6, pp , M. Ohtsu, T. Okaoto, and S. Yuyaa, Moent Tensor Analysis of Acoustic Eissions for Cracking Mechaniss in Concrete, ACI Structures Journal, Vol. 95, 89

102 No., pp , Y. H. Pao, R. R. Gajewski, and A. N. Ceranoglu, Acoustic Eission and Transient Waves in an Elastic Plate, Journal of the Acoustical Society of Aerica, Vol. 65, pp. 96-5, W. H. Prosser, Applications of Advanced, Wavefor based AE Techniques for Testing Coposite Materials, Proceedings of the SPIE Conference on Nondestructive Evaluation Techniques for Aging Infrastructure and Manufacturing: Materials and Coposites, SPIE, Scottsdale, AZ, pp , J. R. Rice, Elastic Wave Eission fro Daage Processes, Journal of Nondestructive Evaluation, Vol., No. 4, pp. 5-4, C. B. Scruby, Quantitative Acoustic Eission Techniques, Research Techniques in Nondestructive Testing, Acadeic Press, Inc., London, Vol. 8, Chapter 8, pp. 4-8, C. B. Scruby, G. R. Baldwin, and K. A. Stacey, Characterization of Fatigue Crack Extension by Quantitative Acoustic Eission, International Journal of Fracture, Vol. 8, pp. -, K. R. Shah and J. F. Labuz, Daage Mechaniss in Stressed Rock fro Acoustic Eission, Journal of Geophysical Research, Vol., No. R8, pp , J. J. Stanes, Waves in Focal Regions, IOP Publishing Liited, England, A. V. Vvedenskaya, The Deterination of the Displaceent Fields by eans of Dislocation Theory, Izvestiya Akas, Nauk, S.S.S.R., Ser. Geofiz., pp. 7-84, S. Yang and F. G. Yuan, Transient Wave Propagation of Isotropic Plates using a Higher-order Plate Theory, International Journal of Solids and Structures, Vol. 4, pp , R. L. Weaver and Y. H. Pao, Axisyetric Elastic Waves Excited by a Point Source in a Plate, ASME, Journal of Applied Mechanics, Vol. 49, pp , 9

103 G. B. Whitha, Linear and Nonlinear Waves, John Wiley & Sons, Inc, M. A. Hastad, A. O Gallagher, and J. Gary, Modeling of Buried Monopole and Dipole Sources of Acoustic eission with a finite eleent Technique, Journal of Acoustic Eission, Vol. 7, pp. 97-, MATHEMATICA VERSION 7, Wolfra Research, Inc., P. Bogert, F.G. Yuan, S. Yang, Transient Lab waves fro AE Sources in Isotropic Plates, Internal Journal of Solids and Structures (pending review), 9 9

104 APPENDIX 9

105 APPENDIX A FORMULATION OF THE HIGHER-ORDER PLATE THEORY Displaceent Field In odeling the transient wave propagation in thin plate-like structures using two-diensional plate theory, rather than three-diensional elasticity theory, it is possible, in principle, to expand the displaceent field of a plate in ters of the thickness coordinate up to any desired degree. To account for the effects of transverse shear deforation and rotary inertia and to iprove the accuracy of the extensional wave otion by taking the third order noral and transverse shear strains into consideration, a consistent displaceent field ay be developed as follows. The extensional and bending degrees of freedo are: [,, uv,, 3 ] [ w,,, ],, 3 Respectively. The displaceent field is: u ( x, zt, ) u( x, t) z ( x, t) z ( x, t) z ( x, t) 3 u ( x, zt, ) v( x, t) z ( x, t) z ( x, t) z ( x, t) 3 u ( x, zt, ) w( x, t) z ( x, t) z ( x, t) (A.) where x = (x, x ). The x -x plane is chosen to lie along the id-plane of the plate. The extensional and bending degrees of ters are recognizable by the degrees of freedo they contain. The displaceent coponents (u, v, w,,, 3 ) have the sae physical eaning as in the first-order shear deforation theory (Mindlin, 95[9]; Yuan and Yang[4]), and five additional linear, quadratic and cubic ters of z associated with, and 93

106 , 3 for extension and, for bending are added to the expansion of the displaceent field. Note that the displaceent field in Eq. (.) suggests that the transverse noral to the id-plane will induce elongation or contraction (Kane and Mindlin, 956[5]; Mindlin and Medick, 959[3]). The linear strains associated with the displaceent field Eq. (.) are z z z () () () 3 (3) () () () 3 (3) z z z () () () () () 3 (3) z z z () () () 3 3 z 3 z 3 () () () 3 3 z 3 z 3 where u,,,,,,, () () () (3) () () () (3) v,,,,,,., () () 33 3, 33 w, 3 z3 () () () (3) u, v,,,,,,,,, () () () 3 w,, 3 3,, 3 3 3, () () () 3 w,, 3 3,, 3 3 3, (A.) (A.3) The ters with extensional degrees of freedo are part of the extensional proble while the ters with bending degrees of freedo are part of the bending proble. Further, the discrepancies between the actual displaceent field and that of the approxiate plate theory need to be corrected by aking the following substitutions related to the thickness strains For extension for, () () () i () i () () 4 3 for 3 4 3, 4 3 And for bending i i for,,, () () () () () () for () () (A.4a) (A.4b) 94

107 where i are the corrected coefficients siilar to those introduced by Mindlin and Medick (959). Matching the cut-off frequencies of the following odes: () plate extensional odes and the first two syetric odes of 3-D elasticity theory; () plate flexural ode with the second antisyetric ode of 3-D elasticity theory, the corrected coefficients are chosen as 3 /, 4 6 /5, / ( 9 65 ) fro atching cutoff frequencies. Calculation of the shear correction factors requires characteristic equation for the syste and the dispersion relationship fro elasticity. A derivation is shown In Appendix F. Constitutive Equations for the Third Order Theory The constitutive equations for a linear elastic plate ay be derived fro the strain energy density in the 3-D elasticity theory with the corrections ade by Eqs. (A.4). V G[ ( 3,633)] G[ ( 4, 3 ) ( 4, 3 )] (A.5) Where the and indicate the appropriate correction factor for the extensional and bending forulations, respectively. In Eq. (A.5) for the extensional forulation, using the correction factor subscript 3; ( ) ( ) ( ) ( ) (A.6) and are Lae s constants for the aterial. Note that the second Lae constant, is the shear odulus G. The two can be used interchangeably. Looking at the first plate extensional resultant force N for exaple, starting with the strain energy, 95

108 V ( ) G 3 (A.7) Now integrating through the thickness, N h/ dx (A.8) h/ Using the extensional strain contributions fro Eq. (A.3),,,, and the integral can be expanded to: () () () () () 33 h/ h/,,,, 33 h/ h/ (A.9) N ( G) ( u z ) dz ( v z ) dz which after evaluation leads to the first plate constitutive equation in Eq. (A.a), that is, 3 3 h h N ( G)[ hu,, ] [ hv,, h 33 ] (A.) The other extensional and flexural plate forces are developed in siilar fashion and can be suarized for an isotropic plate as Extensional Wave N G u, 3, h N G h v,, h3 3 N G u, v,,, (A.a) 3 h N3 h[ 3 ( u, v, ) 3 ( G) 3] 3 (,, ) (A.b) 96

109 R R 4 3 h 3, G 3, S G u, 3 5, 3 h h h S G v,, S G u, v,,, (A.c) (A.d) Flexural Wave M G, G, 3 5 h h M G, G, 8 M G,, G,, 3 h Q w, h 3( 3, ) hg G Q w, 3(, ) 3 5 P h w, h 3( 3, ) G G P w, 8 3( 3, ) (A.a) (A.b) (A.c) M h h h ( G) ( ( ) (A.d) ,,) 6,, T G 5, G 7, h h T G, G, T G,, G,, 5 h (Ae) where h is the thickness of the plate, and G are the Lae s constants of elasticity, and the stress resultants are defined by 97

110 N Q M h/ z h/ N3 h/ d z, R z / / 3 dz, h / 33dz S z / h M h 3 z P z 3 T z (A.3) Equations of Motion With the linear strain-displaceent relation, the equations of otion of the higher-order plate theory can be derived using the principle of virtual displaceents or Hailton s principle (e.g., Washizu, 98) t ( U V K) dt (A.4) t where U is the virtual strain energy, V virtual work done by applied forces, and K the virtual kinetic energy. Alternatively the Equations of otion can be derived using the elastodynaic equilibriu equations ultiplied by powers of z and then integrated through the thickness to convert stresses to plate allowable. Starting with the classic elastodynaic equations of elasticity, 3 x x z 3 f x x z f u v f3 w z x x (A.5) The five extensional plate equations of otion are developed by integrating the first two equations through the thickness with respect to z, then ultiplying the third equation by z and integrating through the thickness with respect to z, and finally by ultiplying the first and second equations by z and integrating through the thickness with respect to z. As an exaple look at the third equation of otion ultiplied by z, noting that fro Eq. (A.) that the only inertial ter associated with the total transverse displaceent u 3 in extension is z 3 so w = z 3 z dz z z wzdz z3 zdz (A.6) z x x h / / / / / 3 h 3 h 3 h h h/ h/ h/ h/ h/ 98

111 After integration by parts the first ter yield the plate force N 3 and the surface tractions (which will be subsequently replaced by the equivalent body forces) which by definition will be called the loading ter., i.e. h/ h/ 33 h/ h h h h z dz Nz / (A.7) The second ter after integration by parts yields the plate force derivative definition of R, as defined in Eq. (a.3), which is: h/ h/ 3 z / / 3zdz R h h R, x x x (A.8) The third ter yields R, in siilar fashion. Putting the contributions of the four ters together and using the definition of inertial oents, h/ I j ( z ) dz ( j,,4,6), the result yields the third extensional equation of j h/ otion in Eqs. (A.) which is. R R N I (A.9),, z 3 In siilar fashion the other four extensional equations of otion are developed and the six bending equations of otion. For an isotropic plate with id-plane parallel to the plane z =, the coplete equations of otion for the third order theory are: Extension N N q I u I,, N N q I v I,, R, R, N I 33 3 n I I4 S, S, R u n I I4 S, S, R v (A.) 99

112 Flexure M M Q I I,, 4 M M Q I I,, 4 Q Q q I w I xx, yy, T T 3P r I I,. 4 6 T T 3P r I I,. 4 6 P P M n I w I,, 3 z 4 (A.) with a nuber of boundary conditions which ust be specified on the plate edges; q, q,,, and are the loads related to the surface tractions and defined by q h n [ 3( h/) - 3( h/)] h /4, [ 3( h/) 3( h/)] 3 r h /8 h q 33( h/ ) 33( h/ ), [ 33( h/ ) 33 ( h/ )] z [ 33( /) 33 ( /)] 4 n h h h (A.) where the subscript =, and is the ass density. Equations of Motion in Ters of Displaceent The equations of otion in Eq. (A.) and (A.) can be expressed in ters of the displaceents by substituting for the stress resultants fro the constitutive equations, Eq. (A.). The equations of otion and the plate constitutive equations indicate that the extensional or syetric waves with five degrees of freedo (u, v, 3,, ) are decoupled fro the flexural or anti-syetric waves with three degrees of freedo (,, w,,, 3 ). The syetric and anti-syetric waves refer to the wave displaceent profile syetric and anti-syetric with respect to the id-plane of the plate (z = ) respectively as shown in fig. 4.. In the following these two wave otions are exained separately. For isotropic plates, the equations of otion take the for:

113 Extensional wave I I [( Gu ), ( Gv ), Gu, ] [( G), ( G), G, ] I 3 3, q Iu I (A.3a) I I [ Gv, ( G) u, ( G) v, ] [ G, ( G), ( G), ] I 3 3, q Iv I (A.3b) I I [ 4G( 3, 3, ) ( 4G3)(,, )] 3[ ( u, v, ) ( G) ] I (A.3c) I [( Gu ), ( Gv ), Gu, ( 3 4 G) 3, 4 4 G ] I4 I I4 [( G), ( G), G, ] n u (A.3d) I [ Gv, ( G) u, ( G) v, ( 3 4 G) 3, 4 4 G ] I4 I I4 [ G, ( G), ( G), ] n v (A.3e) Flexural Wave I I [( G) ( G) G (3 )] [( G) 4,,, 3,, I I, G(,, )] 6 3, G( w, ) I I 4 (A.4a)

114 I I [( G) ( G) G (3 )] [( G) 4,,, 3,, I I, G(,, )] 6 3, G( w, ) I I 4 I I Gw (, w,,, ) G[3(,, ) ( 3, 3, )] q I w I 3 I4 I [( G) ( G) G 3 (3 )] [( G) 6,,, 3,, I 4 I, G(,, )] 6 3, 3 G( w, ) r I 4 I 6 I4 I [( G) ( G) G 3 (3 )] [( G) 6,,, 3,, I 4 I, G(,, )] 6 3, 3 G( w, ) r I 4 I 6 I { Gw (, w,,,) 4 6 ( G) 3 6(,,)]} I { [3( ) ( )] ( )} z 4 G,, 3, 3, 6,, n Iw I 43 (A.4b) (A.4c) (A.4d) (A.4e) (A.4f) In the following analysis it is convenient to ake the variables diensionless. To accoplish this step, we ay introduce a representative length scale h/, a typical tie scale,.5h / ct, where c T G / is the velocity of bulk transverse or shear waves), and define the following non-diensional variables and quantities

115 xi t w x i, t, w h/ h/ u v u, v, h/ h/ h/ 3 h h i ' i, ', pi ' pi 3 q n q, n, G Gh / Gh/3 q q, G Gh/3 h h kj kjh/,, CT G ' ( x ) ( x ), l ( x ) ( x ) l '() () (A.5) where k, j are the nondiensional wave nuber and frequency which will be used in the dispersion relation. According to the non-diensionalization, the phase velocity and group velocity are noralized by ct G/. The bulk longitudinal wave velocity ( G) /, plate velocity c /[( P E ) ] and c R (Rayleigh wave velocity) are also used and their diensionless representations noralized by c T are cr c,,, L cp cr ct (A.6) ct G where ( cl / ct) / G and is Poisson s ratio. Note that the G non-diensional quantities will be used in the following wave analysis and all the pries will be dropped for convenience unless stated otherwise. Then the diensionless equations of otion can be derived. To do so, the non-diensionalized values are substituted into Eqs. (A.3) and (A.4) including the non-diensionalized teporal and spatial derivitives, of the quantities in Eq. (A.5) as required. The equations are then divided by the shear odulus which gives rise to factors that are in the for, -, or -, which siplify the equations. The inertial ters are then expressed as defined earlier, j c L I 3 h, I h, etc. as h/ j (,,4,6). These substitutions along with h/ I z dz j 3

116 soe algebraic anipulation lead to the non-diensionalized equations of otion in ters of displaceents: Extensional wave u, ( ) v, u, [, ( ),, ]/3 ( ), q u /3 3 3 (A.7a) v, ( ) u, v, [, ( ),, ]/3 ( ), q v /3 3 3 (A.7b) (,, ) [ ( ) ](,, )] ( )( u, v, ) { u, ( ) v, u, [ 3( ) 4 ] 3, 4 4 } 3 5, ( ),, n u 3 v u v {, ( ),, [ 3 ( ) 4 ] 3, 4 4} 3 5, ( ),, n v (A.7c) (A.7d) (A.7e) Flexural wave 3 ( ) (3 ) ( ) 5 3 6( ) 3, 3 ( w, ) 5,,, 3,,,, (A.8a) 3, ( ),, (3 3, ) (, ( ),,) 5 3 6( ) 3, 3 ( w, ) 5 (A.8b) 4

117 ( w, w,,, ) [,, ( 3, 3, )] q w (A.8c) ( ) 3 (3 ) ( ) , 6( ) 3, 5 ( w, ) r 7 7 7,,, 3,,, (A.8d) 5 5 ( ) 3 (3 ) ( ) , 6( ) 3, 5 ( w, ) r 7 7 7,,, 3,,, (A.8e) 9 ( w, w,,,) ( )(,,) (,,) ( 3, 3,) 6( )(,,) nz w (A.8f) The diensionless equations depend only on a single paraeter except loading paraeters, and thus the dispersion relation discussed below depends solely on the single paraeter or Poisson s ratio. This is one advantage of this non-diensionalization. Equations of Motion in Vector and Matrix For Next, the vectorized equations of otion in ters of displaceent will be established and the coefficients of the above equations will be used to for the constitutive atrices for extension and bending leading to a very copact for which will expedite the solution. The plate otion, extensional or flexural, is governed by a syste of second-order linear partial differential equations (PDE s). Transient wave solutions ay be obtained by integral transfors. Extensional otion The diensionless equations of otion of the plate with load f E (x, t) and zero initial conditions, as expressed in Eqs. (A.7) above can be written in the atrix notation as 5

118 V V V V V V I T T T T T TV f, t t x xx x x x V V, at t t E (A.9) Where T V [ u, v, 3,, ] and f [,,,, ] T E q q n n, and T, T, T, T, T, T are 5x5 constant atrices fored siply by isolating the coefficients of each displaceent degree of freedo and its derivatives as indicated in Eqs. (A.7) to for Eq. (A.9) above. Note that the bolded quantities represent a vector or a atrix. To illustrate the foration of the coefficient atrices consider the eleent T. This atrix picks up the coefficients of all ters in the noralized extensional displaceent equations of otions, Eqs. (A.7a-A.7e), that ultiply second derivatives with respect to x of any of the five displaceent degrees of freedo that coprise V. Matheatically then, the second ter of Eq. (A.9) can be expressed as: 3 u 3 v V T 4 3 x (A.3) 5 3, 5 3 With T being the coefficient atrix. Also, consider the inertial ters that for the I atrix that ultiply the teporal derivates of the displaceent which is the first ter of Eq. (A.9), 6

119 I ext u 3 3 v u v 3 3 (A3) The reaining coefficients, T, T, T, T, and T are assebled in siilar fashion to those associated with the T ter. The bending forulation uses the six bending noralized equations of otion, Eqs. (A.8a-A.8f) and will be suarized in the next section. Equation of Motion in Copact For The equations of otion can be assebled into a very copact for that, albeit, a syste of equations, has the appearance of a single degree of freedo syste and is easier to anipulate in seeking a solution. In this for the coefficient atrices Tij can be fored into a single constitutive atrix T via the application of the Fourier transfor. This cannot be done prior to taking the transfor in x,x space by siply superiposing the ters fro all of the individual Tij atrices in their assigned row and colun siilar in anner to the foration of a stiffness atrix in finite eleent analysis. The reason is, that the individual coefficient atrices ultiply displaceent derivatives that don t have a coon denoinator. This incopatibility will vanish with the application of the Fourier transfor, which enables the establishent of the copact for. Transfored Equations of Motion in Copact For In seeking both the hoogeneous and inhoogeneous solutions of the syste of equations of otion, a double transfor approach will be eployed. First the Fourier transfor will be applied. The Fourier transforation of Eq. (A.9) with respect to the spatial variables x and x leads to: 7

120 V Tk ( ) V f, T t t (A.3) V V, at t t (A.33) where x = (x, x ) and the transfored variables k = (k, k ), and ~ V ( k, t ) V ( x, t)exp( ik x) dx (A.34) f ( k, t) f ( x, t)exp( ikx) dx E E Addressing the aforeentioned proble with assebling a copact coefficient atrix, advantage will be taken here of the forula for the n th derivative of the transfor, ( n ) n V ( x) ( ik i ) V(k) (A.35) For the extensional proble this results in the following transforations of the coefficients in k space when the ( ik ) n i ters ultiplying V(k) are grouped with the T ij ters. T ( k) = -k T T ( k) = -k k T T ( k) = -k T T( k) = -ik T T ( k) = -ik T T ( k) = T (A.36) When the above transforations and groupings are applied to each ter of Eq. (A.9), the derivatives vanish and ters with like rows and coluns ters can be superiposed fro the five coefficient atrices into a single constitutive atrix. 8

121 Following the previous exaple then T ( ) k, after applying the first of Eqs. (A.36) becoes; k k 3 k k 3 -k T 4 k (A.37) 5 k k 3 5 k k 3 The other individual coefficient atrices are transfored in siilar anner and then added together, resulting in the transfored aster constitutive atrix. k k ( ) k k ( ) kk i3( ) k kk 3 3 kk k k ( ) kk k k i3( ) k ( ) 3 3 Tk ( ) 3 i3( ) k 3 i3( ) k 4 ( k k ) 3 3 i[ 3( ) 4 ] k i[ 3( ) 4 ] k ( k k ) ( ) kk i[ 3( ) 4 ] k k k 4 ( ) kk ) 5 ( ) kk ( k k i[ 3( ) 4 ] k ( ) kk k k (A.38) The ass, or inertial, atrix is: 3 3 M=I (A.39) Where ( G) / G ( ) / ( ), where is Poisson s ratio, and and G are the Lae constants where the shear odulus G is identical to, the ore traditional Lae constant. 9

122 Flexural otion Eploying the Fourier transfors and following siilar procedures, the copact for of the equations of otion for flexure in the isotropic plate can be obtained fro Eqs. (A.8). The equation of otion and the transfored solution for the flexural otion of the plate with bending load vector f B (x, t) and zero initial conditions are U U U U U U t x xx x x x I A A A A A AU f, t U U, at t t Where the flexural degrees of freedo and plate forces are: T U [,, w,,, 3], f [,, q, r, r, n ] T, B z B (A.4) The Fourier transforation of Eq. (A.4) with respect to the spatial variables x and x yields U ), A(k U f B t t (A.4) U U, at t t where x = (x, x ) and the transfored variables k = (k, k ), and Uk (, t) Ux (, t)exp( ikx) dx (A.4) f ( k, t) f ( x, t)exp( ikx) dx B B Where A ij are 6x6 coponent atrices fored siply by isolating the coefficients of each displaceent degree of freedo and its derivatives as indicated in Eqs. (A.8) and (A.4) above. These can be further cobined into a single atrix A(k) to put the equations in their ost copact for with A(k) representing the cobined coefficient atrices with the k i factors resulting fro the transfored derivatives of U analogous to T for the extensional proble. This procedure is as outlined for the extensional wave using the extensional displaceent equation of otion and will not be shown here for brevity. The results are:

123 3 k 3k 3 k k 3 kk ( ) 3ik 3 kk ( ) ik(( ) 6 ) k 3k kk ( ) k k 3 3 ik kk ( ) 3 ik(( ) 6) k k ik ik k k ik ik 3 3 A 5 k 5k 5 5 k k 5 kk ( ) 5ik 9 kk ( ( ) ) ik(( ) 6 3 ) k 5k kk ( ) k k 5 5 ik kk ( )( ) 9 ik(( ) 63 ) ik( ( ) 6) ik( ( ) 6) k 9 6( ) k 3 ik( 6) ik( ( ) 6) k k (A.43) and I M (A.44)

124 APPENDIX B SAMPLE PLATE LOAD DERIVATION Extensional Plate Load Exaple Consider q ore closely. Before noralizing (actually for q it can be shown that q =q, but all of the other loads ust be noralized). h/ h/ () () () h/ h/ 3 q fdz [ M ( x) ( x ) ( z z ) M ( x) ( x ) ( z z ) - M ( z z ) ( x) ( x )] dz h/ h/ () () h/ h/ 3 () M ( x) ( x ) () ( zz ) dz M ( x) ( x ) () ( zz ) dz M ( x) ( x ) () ( zz ) dz h/ h/ where the superscripted pries indicating differentiation have been replaced by the order of differentiation in parentheses. Using the equality for the integral of a function ties a derivative of a delta function, we note again that with f(z )being the delta function, then f(z)= f(z )= f () (z ) = f () (z )= h/ h/ 3 h/ h/ q f dz [ M ( x ) ( x )[ ()] M ( x ) ( x )[ ()] M ( x ) ( x )[ ()] M ( x ) ( x ) M ( x ) ( x ) As shown in Eq. (3.) above. Bending plate load exaple consider n z ore closely. Before noralizing, n z is defined as f6 M3 ( x) M3 ( x) M33 ( x) x x z h/ h/ nz fzdz / 6 [ M / 3 ( ) M3 ( ) M33 ( )] zdz h x x x h x x z Where the vector quantity ( x ) ( x3) ( x) ( z). Then using the equality for the integral of a function ties a derivative of a delta function we note that Then n z can be expressed as: f(z)=z f(z )= f () (z ) = z f () (z )=z

125 z h/ () () () [ 3( ) ( / ) ( ) 3( ) ( ) ( ) - 33 ( ) ( ) ( )] h n M x x zz M x x zz M zz x x z dz M ( x) ( x )[( ) ( z ) ] M ( x) ( x )[( ) ( z ) ] M ( x) ( x )[( ) ( z ) M ( x) ( x ) z M ( x) ( x ) z M ( x) ( x ) z Above, the superscripted pries and the superscripted nubers in parentheses indicate differentiation, with the nubers in parentheses being the order of differentiation. The result is identical to the last equation in Eqs. (3.). 3

126 APPENDIX C PLATE LOAD NORMALIZATION EXAMPLE Looking at for exaple fro Eq. (3.) note that, fro Appendix A, Eq. (A.5), that ' z x ' '() (), z', x', ( x) ( x) l, ( x) ( x) l l l G l 3 Here the pries indicate the noralized variable, not a derivative. The superscripted brackets with delta, i.e., () indicate the first derivative of the delta function. A ore coplete list of noralization paraeters is contained in Appendix A. Solving the above for the unpried variables and subbing the into the third of (Eqs. 3.) yields ' l ' (() ' ' ' '() G M3z l ' ( x ) ( x ) M3 zl ( x ) ( x ) M33 ( x ) ( x ) 3 l l l l l l After collecting ters the above becoes ' M3 ' (() ' M3 ' '() ' ' 3 z ' ( x 3 ) ( x) 3 z 3 '( x) ( x) 3 M33 ( x) ( x) Gl Gl M ' ij Now we will define the noralized oent tensor coponent M ij 3 Gl Leaving the noralized equation, after dropping the pries for convenience, 3 M3z ( x ) ( x ) 3 M3z( x ) ( x ) 3 M33( x ) ( x ) which becoes The third of Eqs.(3.3). 4

127 APPENDIX D EXAMPLE CALCULATION OF MOMENT TENSOR FOR VERTICAL TENSILE CRACK Coputation of the oent tensor will be deonstrated for the vertical tensile crack case. Recalling, Mij [ nk k ij G( ni j nji )] u and defining the vectors in Fig..3 T T and above for this load case, n [,, ], u [,, ] [,, ] lead to k M ij M[ nn n33] G[ nn] u ( [ ] G[ ]) u ( G) u In siilar fashion it can be shown that M [ () G] u u M [ ( ) G] u M M For the reaining off diagonal ters M3 M3 ( [ n n n33 ] 3 G[ n3 n3 ]) u ( [ ] G[ ]) u Finally for the last diagonal ter, M33 [ () G] u u With a unit displaceent discontinuity agnitude, the oent tensor for the vertical tensile crack with x discontinuity is k G Mij 5

128 APPENDIX E TRANSFORMING OF NORMALIZED PLATE FORCES EXAMPLE Consider the third equation in Eq. (3.3). The plate force coponent was used in Appendix C to illustrate the plate force noralization to it will be used as well to illustrate this final step. Applying the Fourier transfor to the third of Eqs. (3.3) [ [ 3 M z ( x ) ( x ) 3 M z ( x ) ( x ) 3 M ( x ) ( x )] e dx ] e dx ik x ikx Starting with the inside integral and using the identity Eq. (5.6) [ 3 M z ( x ) ik 3 M z ( x )() 3 M ( x )()] e dx ikx And now integrating ters that are a function of x the transfored force is 3 M z ( x ) ik () 3 M z ik () 3 M () i3k M z i3k M z 3M This is the third of Eqs. (5.3). The other four extensional noralized plate forces are transfored in siilar anner. 6

129 APPENDIX F - SHAPE CORRECTION FACTORS Extensional wave Defining the characteristic deterinant of the syste using the isotropic for of the extensional constitutive atrix, since the odes are not a function of orientation in an isotropic aterial, k ik( 3 ) ( k ) 3 k ( k ) 3 3 ik( ) k 4 ik(( ) 3 4 ) 5 5 ( k ) ik(( ) 3 4 ) k ( k ) k Now, letting k=o which is the value corresponding to the cutoff value of oega, ω c Now, setting the non trivial equations equal to the cutoff frequencies fro the elasticity forulation for the corresponding odes as shown in Ref. (4) and Fig. 4.7; Cutoff frequency for extensional odes SH and S { c ( /), c }

130 Gives the equations, 3 5 c 3 c 4 They yield the correction factors Which are identical to those used y Mindlin and Medick [3]. Flexural wave Defining the characteristic deterinant of the syste using the isotropic for of the bending constitutive atrix, since the odes are not a function of orientation in an isotropic aterial; 3 k 3 3 ik ( k 5 ) ik(( ) 6 ) 5 3 k 3 ( k 5 ) 5 ik k ik ( k ) 3 5 k 5 5ik 9 ( k ) ik(( ) 6 3 ) 7 5 k 5 9 ( k ) 7 ik( ( ) 6) k ik(3 ( ) 6) k Now setting the non trivial equations equal to the cutoff frequencies fro the elasticity forulation for the corresponding odes, Gives the equations 8

131 {{ },{ ( 65 45) },{ ( 65 45) }, { (45 65) },{ (45 65) },{ 5 6 } Using the cutoff frequencies for bending odes A and A { c, c /} in c (45 65) and c 5 6 yields the shape correction factors

132 APPENDIX G STRAINS FROM DISPLACEMENTS FORMULATION Extensional Surface Strain For an isotropic aterial strain gages are likely to be located on the surface. Piezoelectric gages easure the total strain ε kk = ε + ε where ε = du/dx and ε = dv/dx. It is iportant to note that u and v are functions of the transverse variable z and involve ultiple degrees of freedon as per the displaceent field equations, Eq. (A.). Selecting only the ters required to define ε and ε associated with extensional otion leaves: u ( x, zt, ) u( x, t) z ( x, t) u ( x, zt, ) v( x, t) z ( x, t) (G.) Looking at the individual degrees of freedo above contributing to the displaceents fro which ε kk is defined, prior to the conversion to polar coordinates, starting with Eq. (5.9): (G.) i H u x zt Q zq F ik x dk 5 5 i 3i (,, ) ( [ ]{ }) exp( ) i 4 i D (G.3) i H u x zt Q zq F ik x dk 5 5 i 4i (,, ) ( [ ]{ }) exp( ) i 4 i D Recognizing that the Q ij coponents fro (Eq. 5.9) not present above are considered zero above, only the F i ters that ultiply the coponents of Q ij that are present produce a nonzero contribution. Cobining the u and v displaceent coponents and differentiating according to the strain definition, and expanding the dot products in the exponential superscript, and the integration variable k: du 5 i 3i [ Q 5 z Q]{ Fi} exp( i[ kx kx]) i kk = dx i D 4 dk k 5 du i 4i H [ z ]{ Fi} exp( i[ kx kx]) Q Q dx i D H (G.4)

133 Observing that only the exponential ters are function of the differentiation variables x and x, the expression becoes ii () H = k [ Q z Q ] k [ Q z Q ]{ F} exp( i[ kx]) dk (G.6) 5 5 i 3i i 4i kk i 4 i D Expanding the ters of the inner suation the extensional surface strain becoes k [ QF QF QF3 QF4 QF5] k z [ Q F Q F Q F Q F Q F ] H = exp( i[ kx]) dk k z [ F F F3 F4 F5] Q Q Q Q Q kk k [ QF QF QF3 QF4 QF5] D (G.7) With length scale l=h/ or h=l then the z coordinate, when noralized, becoes z =z/l=z/(h/) so at the surface with z =h/ the noralized value becoes z (h/)=(h/)/(h/)=. Therefore it is not necessary to carry the z ters, which are unity, in the extensional displaceent field equations above for surface strains. Of course for strains at other locations, such as z=h/4, the noralized z coordinate would not be and z ust reain in the general strain forula. Eq. (5.3) is the above expression written ore copactly by not expanding the inner suation over i. The strain expression can be expressed even ore copactly by following the procedure outlined in Section 5.7 where a polar conversion reduces the order of integration to the single variable k with the trigonoetric reduction of the products in the strain expression above, when coupled with the introduction of a change of variable siilar to that of the displaceent forulation. The resulting Bessel function for and coefficients that are a function of the polar output location variable θ are: E H ( k, t) En ( k, ) Jn ( kr) kdk (G.8) 5 5 n D ( k) As in Eq. 5.43, where the E subscript indicates extension and the strain coefficients E E ( k)cos n E ( k)sinn (G.9) n nc ns are very lengthy for the third order theory and are calculated sybolically by MATHEMATICA.

134 Flexural Surface Strains Following the sae procedure using the bending ters fro the displaceents field equations: u (, x zt,) z (,) x t z (,) x t 3 u ( x, zt, ) z ( x, t) z ( x, t) (G.) 3 The expression for bending strain after differentiation is H = k [ zq z Q ] k [ zq z Q ]{ F} exp( i[ kx]) dk (G.) 6 5 i 3 4i i 3 5i kk i 4 i D Where the degrees of freedo in the first of each superscript pair above are those corresponding to the flexural degrees of freedo that contribute to u and v which are those degrees of freedo ultiplying odd powers of z shown in Eq. (G.) above, excerpted fro the total displaceent field equations in Eq. (A.), naely, degrees of freedo,,,. The expanded expression analogous to the extensional forulation becoes: k z[ QF QF QF3 QF4 QF5 QF6] k z [ Q F Q F Q F Q F Q F Q F ] H exp( ikx [ ]) dk D k z [ QF Q F QF3 QF4 QF5 Q 56 F6 ] kk = k z [ QF QF QF3 QF4 QF5 QF 6] (G.) Eq. (5.33) is the above expression written ore copactly by not expanding the inner suation over i. The strain expression can be expressed even ore copactly as in the case of extension with the reduction of integration order of Section 5.7. The result is B H ( k, t) En ( k, ) Jn ( kr) kdk (G.3) 6 5 n D( k) which is Eq

135 APPENDIX H-HEAVYSIDE TEMPORAL LOADING The Heavyside loading function represents an instantly applied force with no rise tie that is applied constantly throughout the loading history. It as a atheatical construct as alost all forces have soe rap up tie even if very short. It is a useful atheatical representation that tends to excite virtually all of the frequencis of a structure. Recall the total loading f(x,t)=m ij f(t) where the spatial part M ij was defined in Eqs. (.9) and (.) and further developed in Section 3.4. Mij is the oent tensor and contributes to the equivalent body forces Eqs. (.6b) and (.6b) and the to the diensionless equivalent plate loads eqs. (3.3) and (3.4) for extension and flexure respectively. This contribution is already in the forulation through the plate loads and it is the teporal portion that is of concern here. f(t). t For the unit Heavyside loading shown above f(t)= and its Laplace transfor Looking at Eq. (5.8) for extension, f s V Q 5 ij D ( s s ) 5x5 f E noting that the Laplace transfor of f E (k,t) which applies to the teporal portion t, for the Heavyside loading, is siply /s, then the inverse Laplace transfor is given by the convolution theore. Fro Eq. (5.8) recogniozing the for sin ( t ) (H.) t L * ( ) f d s s 3

136 Exaining the portion of Eq. (5.9) for extension, (or Eq. (5.3) for flexure), that coes fro the inverse Laplace transfor of Eq. (5.8), it reains to evaluate the resulting convolution integral for the Heavyside loading. Which by definition is: sin( t) * f ( k, t) B B E (H.) t sin( t) * f ( k. t) f ( k, )sin ( t ) d (H.3) But, as can be seen fro Eq. (5.) for extension (or Eq. (5.5) for bending), with Eq. (5.) repeated here for convenience, 5 ij [ Q] 5x5sin( t)* f ( k, ) Vk E t (, t) D (H.4) sin( t) [ Q q Q q Q Q n Q n]* D the transfored spatial portion of f E or f B ultiplies the Q atrix, leaving only the teporal portion of f in the convolution so what reains is, sin( ) t t f ( t) * f( ) *sin ( ) t d (H.5) This can be evaluated by introducing a change in variable, T= t- and consequently d =-dt. Then, calling the result H (k,t), since the wave nuber k is still present in the roots, the teporal portion of the force is given by, H t - TdT= t t k, sin cos ( cos t ) cos t Where the first ter represents the stead state load and the second ter represents the ore doinant transient portion of the teporal loading. (H.6) 4

137 APPENDIX I SUMMARY OF MINDLIN (FIRST ORDER) KEY EQUATIONS Displaceent Field u ( x, z, t) u( x, t) z ( x, t) u ( x, z, t) v( x, t) z ( x, t) u ( x, z, t) w( x, t) z ( x, t) 3 Extensional Degrees of freedo Flexural Degrees of Freedo Plate Equations of Motion - Extension N N q I u,, N N q I v,, R, R, N I 33 Plate Resultants Definitions Extension and Flexure N h/ Q h/ h/ M z dz, / / 3 dz, N h h 3 h/ 33dz R z S z / Plate Forces - Extensional q M ( x ) ( x ) M ( x ) ( x ) M 3 q M ( x ) ( x ) M ( x ) ( x ) M 3 M z ( x ) ( x ) M z ( x ) ( x ) M ( x ) ( x )

138 Plate Forces Extensional Transfored Diensionless q~ ikm ik M q~ ikm ik M ~ i3k M z ik 3M z 3M Plate Equation of Motion - Flexure M M Q I xx, x xy, y x M M Q I xy, x yy, y y Q Q q I w xx, yy, Plate Forces Flexural M z ( x ) ( x ) M z ( x ) ( x ) M ( x ) ( x ) 3 M z ( x ) ( x ) M z ( x ) ( x ) M ( x ) ( x ) 3 q M ( x ) ( x ) M ( x ) ( x ) M Plate Forces Flexural Transfored Diensional ~ i3km z i3k M z 3M 3 ~ i3km z i3k M z 3M 3 q~ ik M ik M 3 3 Constitutive Equations Extension T k k ( ) kk ip( ) k ( ) kk k k ip( ) k i3 p( ) k i3 p( ) k k k 3p Constitutive Equations Flexural 6

139 k k 3 A ( ) kk ik k ( ) k k k ik 3 ( k i3k i3ik k ) Transient Displaceent Solution Extension 3 i 3 H V [ F F 3( zf3 im33) ] exp( i ) d 4 Q Q Q k x k D Transient Displaceent Solution Extension U i 3 3 [3z ( F im 3) Q 3 ( 3) 3 ] exp( ) z F im Q F Q ik x 4 D Transient Strain Solution Extension H dk Transient Strain Solution Extension 3 [3( kq kq 4 3( k Q 3 kk [( kq kq ) F ( kq kq ) F H 3( kq kq )( zf3 im33] exp( ik x) dk D k Q )( z F im 3 )( z F im ) ) ( k Q 3 3 k Q 3 H ) F3 ] D exp( ik x) dk 7

140 Transient Strain Solution Extension Bessel Representation u U ( k, ) v V ( k, ) J ( kr) kdk 3 5 n H ( k, t) n n D ( k) n ( k, ) n Un Unc ( k) Uns ( k) Vn Vnc ( k) cos n Vns ( k) sinn n nc ( k) ns ( k) Transient Strain Solution Extension Bessel- Representation ( k, t) E ( k, ) J ( kr) kdk 3 kk n n n D ( k) H E E ( k)cos n E ( k)sinn n nc ns Transient Displaceent Solution Flexure - Bessel Representation 3 5 n ( k, ) H ( k, t) (, ) ( ) n k J n kr kdk n D ( k) w n ( k, ) n nc ( k) ns ( k) n nc ( k) cos n ns ( k) sinn n nc ( k) ns ( k) Transient Strain Solution Flexure - Bessel Representation ( k, t) E ( k, ) J ( kr) kdk 3 kk n n n D( k) H E E ( k)cos n E ( k)sinn n nc ns 8

141 Explicit Expressions for Strain Coefficients E n The total response of the surface strains to the point source is the suation of two parts: one due to extensional otion, and the other due to flexural otion. The two responses can be conveniently expressed as in the for 3 4 H ( k, t) En ( k, ) J n ( kr) kdk n D ( k) where En Enc ( k)cos n Ens ( k) sin n However, the quantities, H( k, t), D ( k), ( k), Enc( k), Ens( k), are different for each otion. Hence, the expressions for E nc and E ns are given separately: Extensional otion Fro Eq. (4.), E c / k [ M ( q q ) M q ] 3M 33q3 / k E / k i 3z M 3 q3, E / k i3 z M 3 q c s 3 E / k [ M ( q q ) M q ] c E / k M ( q q q ) s with 4 q ( k A33) [ A33 3( a ) p ] k q [( )( A33) 3( ) p ] k 4 q ( k A33 ) A33k q3 ( )( k ) pk A33 k 3p where ( k ) is the dispersion relation for the th extensional ode. Flexural otion Fro Eq. (4.4), with E E E E E 3 / k z[ M ( q q ) M ] c q c s / k / k i M i M 3 3 [ q 3 ( q 3 3 k 3 k ( q q q 3 / k z[ M ( q q ) M ] 3 k z M ( q q ) c q s / q 4 4 q ( k 3 p p k ) p k 4 q [( )( p k ) 3 p ] k q ( k 3 p p k ) ( k 3 p ) p k q3 3 p ( k 3 p ) k ) 4 )] 9

142 Note that in the above forulation the equivalent plate loads for the point source are noralized according to the plate theory, as are the coponents of the oent tensor. 3

143 APPENDIX J - PLANE STRESS FORMULATION Using the concept of oent tensor (or derived equivalent body-forces) in seisology for a point source with displaceent discontinuities on the fracture surface in three-diensional elastodynaics, this paper first presents equivalent loads for the point source and then derives transient wave response to a general acoustic eission (AE) point source with an arbitrary oent tensor in isotropic plates under plane stress condition. The transient response which represents waves propagating fro the general AE point source in the plate is expressed in an explicit integral for. It is shown that the transient response which is given by the double inverse Fourier transfors can be siplified into a finite series involving inverse Hankel transfors which is one-diensional inversion in isotropic plates. Exact solutions of the AE sources have been obtained. Three types of AE sources representing different icro-daages and their corresponding plate loads are discussed. Nuerical results for nine of the AE point sources with Heaviside tie history are presented..the equivalent body-forces in the thin plate In the elastodynaic proble of a thin plate under plane stress condition, the displaceent field (u, v) is a function of x = (x, x ) and t. The equations of otion are written as,, f u (J..),, f v where (,, ) are stress coponents, and (f, f ) are the equivalent body-forces acting at expressed as f M,( x ξ) M,( x ξ) (J..) f M,( x ξ) M,( x ξ) where ξ (, ). Without loss of generality it is assued that the dipoles are applied at the origin (, ), the body forces can be conveniently expressed as f M( x) ( x) M( x) ( x) (J..3) f M( x) ( x) M( x) ( x). AE waves in thin isotropic plates The -D equations of otion under plane stress can be expressed in ters of displaceents as ( u, u, ) ( )( u, v, ) f u (J..) ( v, v, ) ( )( u, v, ) f v or in a atrix for u u u f u x v xx v (J..) x v f v 3

144 To ake the variables diensionless, a representative length scale l h(h is the thickness of the plate) and a typical tie scale l/ c T ( ct / ) is the velocity of bulk transverse or shear vertical waves) are introduced and the following nondiensional variables and quantities are defines by x i xi / l, tt/, uu/ l, vv/ l, k i kl i,, f i fl i / (J..3) The nondiensional quantities will be used and the pries will be dropped hereafter. The equations of otion Eq. (J..) can be written in nondiensional for together with zero initial conditions: V V V V I T T T f, t T t x xx x (J..4) V V, at t t where [, ] T T V uv, f [ f, f],and T, T and T are constant atrices. The Fourier transforation of Eq. (J..4) with respect to x and x yields V T( k) V f, t (J..5) V V, at t where k ( k, k), Vk (, t) Vx (, t)exp( ikx) dx, fk (, t) fx (, t)exp( ikx ) dx (J..6) ( ) kk kk k k k k T ( ) where ( ) / ( ) /( ), is Poisson s ratio, and and are the Lae constants. The application of Laplace transforation of Eq. (J..5) with respect to t leads to ( T s I) V f (J..7) where Vk (, s) Vk (, t)exp( st) dt, f ( k, s) (, s)exp( st) dt f k (J..8) The solution of Eq. (J..7) can be siply obtained by atrix inversion: adj( Ts I) adj( Ts I) V ( Ts I) f f f (J..9) T s I ( s s)( s s) where s ( k ) and s ( ) k are roots of the equation T s I (J..) Let s, we have Tk ( ) I (J..) which is the dispersion relation of the plate with k being the wave nuber vector, and ω the frequency. The roots associated with two wave odes ay be expressed in the for 3

145 W ( k) k (J..) W ( k) k The phase velocities of the two non-dispersive wave odes are and respectively. Let Ds ( ) TsI ( s k)( s k ) ij s ( k k) ( ) kk (J..3) Q[ Q ] adj( Ts I) ( ) kk s ( kk) and Eq.(.9) can be written as ij Q ( s ) V f (J..4) Ds ( ) ij where Q ( s ) and Ds ( ) are polynoials in s and the degree of D is higher than ij that of Q. To carry out the inverse Laplace transfor to Eq. (J..4), the ethod of partial decoposition is eployed, which yields ij ij ij ij Q ( s ) Q Q Q (J..5) Ds ( ) D ( s s) D ( s k) D ( s k ) where ij ij s W ( k ), Q Q ( s ) dd n n ds ( ) s s n, n n, n D D( s ) ( s s ) ( W W ) Substituting Eq. (J..3) into Eq. (J..5), the following equations can be obtained: D ( ) k, D ( ) k ij ( ) k ( ) kk ij ( ) k ( ) kk (J..6) Q, Q ( ) kk ( ) k ( ) kk ( ) k With Eq. (J..5), Eq. (J..4) becoes ij Q V f (J..7) D ( s s) Now the inverse Laplace transfor is applied to the above equation via the convolution theore, and ij [ Q] sin( Wt) fk (, t) sin( Wt) Vk (, t) [ f f] D Q Q (J..8) W D W where denotes convolution with respect to tie, t sin( Wt) fk (, t) fk (, )sin W( t) d (J..9) i and Q ( k ) is colun vector, Q i [ i, i ] T Q Q, i =,; that is 33

146 Q Q ( ) k ( ) kk, Q ( ) kk ( ) k ( ) k ( ) kk, Q ( ) kk ( ) k (J..) 3. Transient stress wave solution For the oent tensor at the origin, the Fourier transfor of the diensionless equivalent body-force, f [ f, f ], can be expressed as f ikm ikm f ikm ikm (J.3.) using the following identity: ( ) exp( ) n ikx ( x) dx ( ik) (J.3.) With Eq. (J.3.), the integral solution, Eq. (J..8) can be written in a copact for as H V i[ FQ FQ] D (J.3.3) where Fi kmi kmi. (J.3.4) The function H( k, t) is defined as sin( Wt) t H( k, t) I( t) W I( )sin W ( ), t d W, (J.3.5) where I(t) is the source tie function for the AE source. Finally, applying the inverse Fourier transfor to Eq. (J.3.3), the solution is given by T i H V [ uv, ] [ F F ] exp( i ) d 4 Q Q kx k D (J.3.6) 4. Solution in the polar coordinate syste The transient solutions, Eq. (J.3.6), are given by the two-diensional inverse Fourier transfor. For isotropic plates they ay be expressed as a series involving inverse Hankel transfors. In ters of polar coordinates: k kcos, k ksin and x rcos, x rsin, one has exp( ikx) exp[ ikrcos( )], dk dkdk kdkd Eq. (J.3.3) indicates that the Fourier transfor of the displaceent field, for exaple, u can be written as u ( k, t) i ( FQ FQ ) H / D (J.4.) u can be further expressed in ters of trigonoetric functions. 3 3 i( ) k u ( k, t) [ Cn cos( n) Sn sin( n)] H / D 4 n 3 3 i( ) k [( Cn isn )exp( in) ( Cn isn )exp( in)] H / D 8 n (J.4.) 34

147 where the coefficients Cn and S n are obtained by substituting Eq. (J..6) and Eq. (J.3.4) into Eq. (J.4.). This process is tedious even for the two degree of freedo plane stress proble. For the u displaceent for ode this process is carried out in Appendix J.A. Once the coefficients have been deterined, substituting Eq. (J.4.) into the integrand of Eq. (J.3.6) and integrating with respect to first, the integration with respect to k leads to a series of inverse Hankel transfors. For exaple, the displaceent u can be expressed as 3 i( ) C n isn H 4 u( r,, t) exp[ cos( )] 6 k dk in ikr d n D (J.4.3) C n isn H 4 + kdk exp[ in ikr cos( )] d D Making a change of variable /, and using the integral representation of the Bessel function of order n, Jn( kr) exp[ i( n kr sin )] d n and Jn( kr) ( ) Jn( kr) ( ( /)), Eq. (J.4.3) is reduced to 3 i( ) Cn isn H 4 n ur (,,) t exp( ) ( ) 6 kdk i in J n kr n D Cn isn H 4 n + kdk i exp( in ) J ( ) n kr D n 3 i ( ) H Cn isn (cos n isin n 8 ) n D In suary, where i ( ) Cn isn + cos sin ( ) H ( cos sin ) ( ) 4 n i n Jn kr k dk n 3 4 C 8 n n Sn n Jn kr k dk n D 3 ( ) H 4 u ( ) ( ) 8 U n Jn kr k dk n D n n( ) ( n cos n sin ) and each ter of U n is U i C n S n U( ) U( ) U( ) ( M M)cos M sin U3( ) ( M M)cos3 M sin3 U( ) (3 M M)cos M sin U ( ) U ( ) 3 3 (J.4.4) (J.4.5) More detail of this derivation can be found in Appendix J.B. In a siilar anner, the displaceent v can be obtained by 35

148 v ( k, t) i [ FQ FQ ] H / D (J.4.6) The ore copact for can be expressed by 3 3 i( ) k v ( k, t) [ cn cos( n) sn sin( n)] H / D 4 n (J.4.7) 3 3 i( ) k [( cn isn)exp( in) ( cn isn )exp( in)] H / D 8 n where coefficients cn and s n are obtained fro Eq. (J..6) and Eq. (J.3.4) in a anner siilar to that sown for u: The inverse Fourier transfor of v gives 3 ( ) H 4 v V ( ) ( ) 8 n Jn kr k dk D where n n n ( ) ( n cos n sin ) V i c n s n and each ter of V n is V ( ) V ( ) V ( ) M cos ( M M )sin V ( ) M cos3 ( M M )sin3 3 V ( ) M cos ( M 3 M )sin (J.4.8) (J.4.9) V3( ) V3( ) In suary, for isotropic plates the transient wave solutions are reduced to a series of the inverse Hankel transfors. Perforing siilar anipulation, the displaceents are 3 u H ( ) U n 4 J ( ) 8 n kr k dk v (J.4.) n D Vn ( ) 5. Closed For Solution for Heavyside Loading For the source tie function described by Heaviside step function, Eq. (J.3.5) becoes H ( k, t) ( cos W t)/ W. Then ( cos kt) and H/ D H 4 ( ) k then Eq. (J.4.4) can be reduced to H ( k, t ) ( cos kt )/( k ) (J.5.a) H k t kt k (J.5.b) (, ) ( cos )/( ) ( cos kt) / D ( ) k and 4 (J.5.) 36

149 u (cos kt)[ U J( kr) U3J3( kr)] dk 8 ( cos kt)[ U J( kr) U 3J3( kr)] dk 8 U (cos kt) J ( kr) dk U3 (cos kt) J 3( kr) dk 8 U (cos ktj ) ( krdk ) U3 (cos ktj ) 3( kr) 8 dk The first ter above represents the ode solution and the second ter the ode contribution To u. Using the following forulas: Jn( kr) dk r One has Or t cosn arcsin r, t r r t Jn( kr)cosktdk or, t r n n r sin, t r n t r t t r, (J.5.3) (J.5.4) (cos kt) J( kr) dk, t r (J.5.5) t r t r t (cos kt) J( kr) dk r t r,, t r t r (cos kt) J( kr) dk, t r (J.5.6) t r t r t r, t r 37

150 , t r (cos kt) J3( kr) dk, t r (J.5.7) 3 ( t t r ), 3 r t r t r Putting it together for the u contribution for t r fro ode for exaple, is given by; 3 t t t r u (r,,t) U U3 8 3 r t r r t r t ( M M )cos ( M )sin (J.5.8) r t r 3 8 t t r M M )cos3 ( M )sin3 3 r t r Once the displaceents are derived, applying spatial differentiation, strains are readily obtained. The piezoelectric sensor to easure the surface strains, kk, generated by the source is often adopted in structural health onitoring. Therefore the surface strain is given as follows: H kk [( kq kq ) F ( kq kq ) F] exp( i ) d 4 kx k (J.5.9) D The response of the surface strains to the point source can be obtained fro Eq. (J.5.9) by perforing siilar anipulation and it follows that H 5 kk E (, ) ( ) n k J n kr k dk (J.5.) D n where E E ( k)cos n E ( k)sin n (J.5.) n nc ns E ( k) E ( k) E ( k) E ( k), E ( k) ( M M ) E ( k) [( M M )cos ( M M )sin ] 6. Dispersion relation First, the relation between frequency and nuber can be obtained fro Eq.(J..9), Tk ( ) I, and it follows that W ( k) k W ( k) k 38

151 In the polar coordinates, the above relation becoes k, k (J.6.) where and are the analytical fors of the frequencies associated with the two stress wave odes. According to Eq.(6.), the phase velocity associated with the SH and S odes can be obtained as / k, (J.6.) cp As the frequency changes, the phase velocity doesn t change. Therefore, it could be seen that the phase velocity is not dispersive. 7. Results The vertical tensile crack will be considered., In the case of plane stress the discontinuity extents through the thickness of the thin body, as opposed to the higher order theories, where the discontinuity can have a z location soewhere through the thickness. The closed for solution for ode has been Considering a vertical tensile crack loading in the x direction the only non zero ters in the oent tensor are the diagonal ters for which M =β and M =. For this case the ode one solution for u given in Eq. (5.8) can be siplified and expressed explicitly as t t t r (r,, t) ( ) cos ( )cos3 3 u (J.7.) 8 3 r t r r t r This result will be plotted directly and copared with the result using Eq. (J.4.), shown again below, with nuerical integration in MATHEMATICA. u H U ( ) v 8 3 n 4 J ( ) n kr k dk n D Vn ( ) (J.4.) The result shown in Fig. J. is a plot showing the closed for solution (Eq. (J.7.) in dashed lines and the nuerical solution Eq. (J.) fro MATHEMATICA in the solid line. The closed for solution is singular at the arrival tie and the atheatical solution has a axiu response of.65 diensionless displaceent. Mode is the shear horizontal ode SH. Mode, the S longitudinal extensional ode, is shown in Fig. J. and has a axiu response of.575 diensionless displaceent and an earlier diensionless arrival tie. 39

152 Figure J. Coparison of closed for and nuerical solutions for the ode u displaceent for a vertical tensile crack. r=5, θ=45 θ Figure J. Coparison of closed for and nuerical solutions for the ode u displaceent for a vertical tensile crack. r=5, θ=45 Figure J.3 provides four snapshots in increasing tie and shows the u displaceent as a function off r and θ for the shear horizontal ode near the origin. It is observed that 4

153 the response increases off axis as the θ= line is a syetry axis for the SH ode with sall u response. X 3 X X Figure J.3 MATHEMATICA aniation of u displaceent for SH ode as a function of r,, at four snapshots in tie 4

154 Figure J.4 provides four snapshots in increasing tie and shows the u displaceent as a function off r and θ for the longitudinal ode near the origin. It is observed that the response decreases off axis, as the θ= line is directly along the load direction for a vertical tensile crack and should experience the strongest response for the S ode. X 3 X X Figure J.4 MATHEMATICA aniation of u displaceent for S ode as a function of r,, at four snapshots in tie 4