AND MOUNTED FCS OF A380 PLANE

Keywords: Structural health monitoring, Guided waves, Dispersion equations and curves, Anisotropic composite laminates, Partial wave superposition approach, Transfer and global matrix methods, Dichotomy property, Viscoelastic damping models, Component of A380 nacelle

a small region. Mode shape analysis and MAC value tomography are used to analyze veering effect.

Results reveal that the closing modal curves exchange their mode shapes rapidly in these regions.

It is furthermore demonstrated that dispersion equations built with PWSA for a single layer plate and with TMM and GMM for a multi-layered plate system possess a dichotomy property in the case of pure elastic materials. This property states that these a priori complex-valued equations collapse to pure real or imaginary valued equations, depending on the behavior of some intermediate parameters. This is extremely helpful for overcoming numerical instabilities encountered during the equation solving process. With the dichotomy property, the large f d problem of TMM results from the vanishing of effective part between the real and imaginary parts of dispersion equations due to the poor conditioning of the monodromy transfer matrix.

Composite materials possess inherent viscoelasticity which produces guided wave amplitudes attenuating with propagation distance. To predict attenuation coefficients of anisotropic composite laminates, three damping models (Hysteretic, Kelvin-Voigt and Biot models) are integrated into GMM to form the damped GMM framework (dGMM). Attenuation coefficient is computed from the solutions of complex dispersion equations built with dGMM. Two aeronautic components of A380 plane both which are either mounted or unmounted on the plane are employed to validate the developed dGMM. The first one is a fan cowl structure having quasi-isotropic property and the second one an inner fixed structure being a sandwich plate. Theoretical and experimental analysis reveal that the attenuation coefficients of both structures hold anisotropic property in terms of spatial distribution despite the quasi-isotropic stacking layups of them. The existence of stress in the mounted case can greatly change the property of attenuation as compared to the unmounted case. In both structures, A0 mode undergoes much attenuation than S0 mode, which gives the guidance that selecting S0 mode for SHM of similar aeronautic structures is more desired in the selected frequency range.

The works conducted in the thesis provides inspirations for optimally deploying sensor networks by considering the influence of wave attenuation and for improving damage localization methods that takes advantage of frequency-wavenumber analysis.

I would like to thank my principal supervisor Prof. Nazih Mechbal for your kind guidance during the past four years. Your strict supervision expedited me so much for doing research with critical thinking and acute insight. For every manuscript I wrote, you can always give me the detailed comments and constructive revisions, which persistently improved my scientific writing skills. You also processed a lot of administrative affairs during the course of my Ph.D. study, like annually organizing the CSID audition, frequently communicating with the journal editor who was responsible for my manuscript, and managing the evaluation for my funding. Your patience to process these affairs liberated me that I can whole-heartedly concentrate on my research and complete my thesis smoothly. I would also like to appreciate my co-supervisor Dr. Marc Rébillat. We had a wonderful experience during my Ph.D. journey. For each time when I entered in a dilemma in my research, I always turned to find help from you because you can always give me meaningful suggestions. Each discussion with you is nutritive and I enjoyed it. You also taught me a lot of scientific skills, for example, teaching me how to read scientific paper and make academic presentation, training my experimental skills, improving my programming ability. You still financially supported me to attend international conference, which greatly broadened my horizon. Under your supervision for these four years, I have grown into an independent researcher that I can independently discover, study and analyze various problems in different angles. Thank you very much.

In the past four years, I have received many supports, help and guidance from the PIMM laboratory to conduct my research works. At this point, I would like to sincerely express my thanks to everyone in the laboratory, especially for Nassim Benbara, Xixi Li, Shaobo Yang, William Briand, Emmanuel Lizé, Erika Bayona, Quentin Jacquemin, Eric Monteiro, Mikhail Guskov, Christine Schrive, who helped me a lot during the course of completing this thesis.

Résumé

Les plaques composites anisotropes constituées de matériaux composites viscoélastiques sont de plus en plus utilisées dans l'industrie aéronautique et les stratégies de surveillance de la santé des structures (SHM) basées sur des ondes guidées apparaissent comme un outil très prometteur pour surveiller leur état de santé. Dans ce contexte, cette thèse est dédiée à l'étude de la propagation et de l'atténuation des ondes guidées dans de telles plaques composites.

Les équations fondamentales de la théorie de l'élasticité tridimensionnelle (3D) sont d'abord passées en revue. Divers matériaux composites sont introduits en mettant l'accent sur les caractéristiques de leurs matrices de rigidité. L'approche par superposition d'ondes partielles (PWSA) basée sur l'élasticité 3D est utilisée pour dériver les équations de dispersion des ondes guidées se propageant dans une plaque composite monocouche.

Dans une plaque composite multi-couches, la PWSA est étendue à la méthode de la matrice de transfert (TMM) qui relie les variables de champ des couches voisines (déplacement et contrainte) à travers les matrices des couches locales. Pour un stratifié composite arbitrairement orienté, un problème d'incompatibilité de matrice est rencontré en raison de l'anisotropie des matériaux composites, et une stratégie originale de matrice hybride (HMS) est proposée pour résoudre ce problème en refondant les matrices locales des couches incompatibles en une formulation hybride garantissant le bon déroulement de la TMM. Cette méthodologie est validée dans une plaque quasi-isotrope et une plaque métalliquecomposite.

Pour améliorer l'efficacité de calcul de la méthode de la matrice globale (GMM), une approche GMM optimisée est développée qui introduit une condition de symétrie dans la matrice globale pour réduire son ordre et tire parti des techniques de calcul matriciel et parallèle pour promouvoir les performances de calcul. La GMM optimisée est appliquée à une plaque composite aérospatiale comportant RESUME 400 couches, ce qui est le plus grand nombre de couches signalé à ce jour, pour calculer ses courbes de dispersion. En raison de la complexité de cette structure, le diagramme de dispersion subit un fort effet de "veering" qui correspond à des courbes de dispersions de même symétrie s'approchant l'une de l'autre mais ne se croisant jamais. L'analyse de la forme des modes et la tomographie des valeurs MAC sont utilisées pour analyser cet effet. Les résultats révèlent que les courbes modales s'approchant l'une de l'autre échangent leurs formes modales rapidement dans ces régions.

Il est en outre démontré que les équations de dispersion construites avec PWSA pour une plaque monocouche et avec les approches TMM et GMM pour un système de plaques multicouches possèdent une propriété de dichotomie dans le cas de matériaux élastiques purs. Cette propriété indique que ces équations à valeurs complexes a priori deviennent des équations à valeurs réelles ou imaginaires pures, en fonction du comportement de certains paramètres intermédiaires. Ceci est extrêmement utile pour surmonter les instabilités numériques rencontrées lors du processus de résolution d'équations. Avec la propriété de dichotomie, le problème des grand fd de l'approche TMM qui résulte de la disparition de la partie utile entre les parties réelle et imaginaire des équations de dispersion en raison du mauvais conditionnement de la matrice de transfert de monodromie est résolu.

Les matériaux composites possèdent une viscoélasticité inhérente qui produit des amplitudes d'ondes guidées qui s'atténuent avec la distance de propagation. Pour prédire les coefficients d'atténuation des stratifiés composites anisotropes, trois modèles d'amortissement (modèles hystérétique, Kelvin-Voigt et Biot) sont intégrés dans l'approche GMM pour former l'approche GMM amortie (dGMM). Le coefficient d'atténuation est calculé à partir des solutions d'équations de dispersion complexes construites avec la dGMM. Deux composants aéronautiques de l'avion A380, tous deux montés ou non montés sur l'avion, sont utilisés pour valider la dGMM développée. La première est une structure de capot de soufflante ayant des propriétés quasi-isotropes et la seconde une structure interne fixe étant une plaque sandwich. Les analyses théoriques et expérimentales révèlent que les coefficients d'atténuation des deux structures ont des propriétés anisotropes en termes de distribution spatiale malgré leur empilement quasi-isotrope. L'existence de contraintes dans les structures montées peut grandement modifier les propriétés d'atténuation par rapport aux structures non-montées. Dans les deux structures, le mode A0 subit beaucoup plus d'atténuation que le mode S0, ce qui indique que la sélection du mode S0 pour le SHM de structures aéronautiques similaires est plus souhaitable dans la gamme de fréquences sélectionnée.

RESUME

Les travaux menés dans la thèse fournissent ainsi des inspirations pour déployer de manière optimale des réseaux de capteurs en considérant l'influence de l'atténuation des ondes et pour améliorer les méthodes de localisation des dommages qui tirent parti de l'analyse fréquence-nombre d'onde.

Mots

RESUME

Abstract

Anisotropic composite plates made up of viscoelastic composite materials are increasingly used in aeronautic industry and structural health monitoring (SHM) strategies based on guided waves appear as a very promising tool to monitor their health state. In that context, this thesis is dedicated to the study of guided waves propagation and attenuation in such composite plates.

The fundamental equations of three-dimensional (3D) elasticity theory are first reviewed. Various composite materials are introduced with an emphasis on their characteristics of stiffness matrix rotation. The partial wave superposition approach (PWSA) based on 3D elasticity is used to derive the dispersion equations of guided waves propagating in a single layer composite lamina.

In a multi-layered composite plate system, the PWSA is extended to the transfer matrix method (TMM) which connects the neighboring layers' field variables (displacement and stress) through the local lamina matrices. For an arbitrarily oriented composite laminate, the matrix incompatibility issue of TMM is encountered due to the anisotropy of composite materials, and an original hybrid matrix strategy (HMS) is proposed to address this issue that recasts the incompatible layers' local lamina matrices into a hybrid form to guarantee the successful proceeding of the transferring process of TMM.

The HMS is validated in a quasi-isotropic plate and a metallic-composite plate.

To improve the computational efficiency of the global matrix method (GMM), an optimized GMM is developed that introduces the symmetry condition into the global matrix to reduce matrix order and takes advantage of sparse matrix and parallel computing techniques to promote computing performance. The optimized GMM is applied to an aerospace composite plate having 400 layers, which is the largest number of layers reported so far, for computing its phase velocity dispersion curves. Due to the complexity of this structure, the dispersion diagram experiences heavy veering effect which is the phenomenon of dispersion curves belong to the same symmetry mutually closing but never crossing in One of the most important issues in engineering concerns the monitoring and the early detection and localization of structural damages in order to prevent catastrophic failures. This process is referred to as structural health monitoring (SHM) and is expected to provide considerable improvements with respect to safety and maintenance costs. A SHM procedure is generally made up of five steps: damage detection, localization, classification, quantification and prognosis [START_REF] Yuan | Structural health monitoring (SHM) in aerospace structures[END_REF][START_REF] Farrar | An introduction to structural health monitoring[END_REF]. Over half a century, modern SHM techniques have reached maturity in engineering practice, playing a significant role in evaluating the integrity and durability of engineered structures and assets [START_REF] Su | Identification of damage using Lamb waves: from fundamentals to applications[END_REF]. These techniques are typified by radioscopy, ultrasonic scanning, shearography, dye penetrant testing, magnetic resonance imagery, laser interferometry, acoustic holography, infrared thermography and eddy-current [START_REF] Balageas | Structural health monitoring r&d at the "european research establishments in aeronautics"(erea)[END_REF][START_REF] Gray | X-ray imaging methods over the last 25 years-new advances and capabilities[END_REF], leading to vast achievements in scientific and industrial areas. With efficient, continuous and automated SHM techniques it is possible to identify structural damage at an early stage so as to prevent further failure occurrence, producing huge economic and human benefit [START_REF] Sohn | A review of structural health monitoring literature: 1996-2001[END_REF].

List of acronyms

In aeronautic and aerospace industries, composite materials are increasingly used from the past decades to the future due to their high performances to other engineering materials, such as the high strength-to-weight ratio [2]. However, the complexity of damage mechanism of composite materials increases the difficulty to develop effective SHM techniques for aeronautic composite structures. Composite materials generally experience various damage types during the manufacturing and in-service process, for example delamination between layers, porosity in the matrix, impact induced matrix crack, fiber breakage, fiber-matrix debonding and fatigue caused defects [START_REF] Yuan | Structural health monitoring (SHM) in aerospace structures[END_REF]. All these occurred damages can The Fan Cowl structure is made of multi-layered composite materials (carbon fiber reinforced polymers) and the Inlet Cowl structure is made of sandwich materials (composite skins and aluminum alloy honeycomb core). These materials present heavy anisotropy which poses a challenge for monitoring the healthy state of nacelles. In the thesis, the anisotropic property of the Fan Cowl structure and the Inlet Cowl structure will be examined with an emphasis on the attenuation coefficient of composite materials. In the family of elastic waves, the theoretical basis of a number of wave types such as bulk waves, surface waves, interface waves, guided waves, etc., have been well-established and they have received a variety of applications in engineering problems [START_REF] Auld | Acoustic fields and waves in solids[END_REF]. Among all kinds of existing SHM techniques, ultrasonic guided wave strategy is particularly effective in accomplishing the task of monitoring the health and integrity state of aeronautic composite structures [START_REF] Mitra | Guided wave based structural health monitoring: A review[END_REF]. Guided waves possess the characteristic that they are able to propagate over broad distances and thus can cover a large area with few sensors and few testing durations. This benefit results in reduced labor and equipment to perform a test, and makes long-range inspection possible [START_REF] Su | Guided lamb waves for identification of damage in composite structures: A review[END_REF]. Moreover, some guided wave modes are rather sensitive to damage occurring within the inspected structure. Thus, by appropriately configuring the excitation frequency and transducer polarization, geometry, and shape, effective modes can be excited to match the application at hand, making ultrasonic wave strategies very versatile [START_REF] Lammering | Lamb-wave based structural health monitoring in polymer composites[END_REF][START_REF] Miao | Shear horizontal wave transducers for structural health monitoring and nondestructive testing: A review[END_REF].

Figure 1.4 -Guided waves based SHM system [START_REF] Lammering | Lamb-wave based structural health monitoring in polymer composites[END_REF].

When adopting guided waves as an inspection tool for aeronautic composite structures, dispersion properties of these waves are an essential aspect that should be considered carefully. More precisely, dispersion behavior describes the evolution of wave velocity with modes and frequency [START_REF] Wilcox | The effect of dispersion on long-range inspection using ultrasonic guided waves[END_REF]. Due to the dispersion property, multiple wave modes can propagate simultaneously in the same frequency range

THE HISTORY OF ELASTIC WAVES IN HOMOGENEOUS ISOTROPIC MEDIA

and thus superpose to each other, a combination of several propagating modes is probably unavoidable [START_REF] Luangvilai | Attenuation of ultrasonic Lamb waves with applications to material characterization and condition monitoring[END_REF]. In addition, aeronautic composite materials possess inherent anisotropy that causes directional dependence of guided waves properties, e.g. varying wave speed along different directions [START_REF] Lammering | Lamb-wave based structural health monitoring in polymer composites[END_REF].

Both properties of guided waves complexified the analysis of the received wave signal, because wave packets change their shape with frequency and propagation direction. As a consequence, the aim of this manuscript is to increase the knowledge of guided waves propagation in composite aeronautic structures for SHM purposes.

The history of elastic waves in homogeneous isotropic media

The earlier works of investigating elastic waves propagation in layered isotropic media were performed for geophysical applications in that era. There are mainly the following five wave types having the name of the corresponding discoverer.

As early as in 1885, Lord Rayleigh studied the wave propagation on the free surface of a semiinfinite solid, which is a surface wave and now named Rayleigh wave. His investigation determined the velocity of the propagating surface wave [START_REF] Rayleigh | On waves propagated along the plane surface of an elastic solid[END_REF][START_REF] Lord | The theory of sound[END_REF]. In 1911, Love solved the simplest problem of wave propagation in a finite thickness layer, which was attached to a semi-space solid having a different material to the thick layer, and now this wave has been defined as the horizontal polarized SH wave due to the nature of particles' horizontally shear motion in the plane of the layer in addition to the alternative name Love wave [START_REF] Love | Some Problems of Geodynamics: Being an Essay to which the Adams Prize in the University of Cambridge was Adjudged in 1911[END_REF]. In 1917, Horace Lamb deeply studied the classical free plate during its propagation [START_REF] Pilant | Elastic waves in the earth[END_REF].

The above introduced works mainly dedicated to the class of isotropic media. In these works, the Helmholtz decomposition method is widely used to study the corresponding wave propagation problem that decouples the equations of motion into two separate wave equations governing two potential [START_REF] Zhu | Elastic wavefield separation based on the helmholtz decomposition[END_REF]. However, in anisotropic media especially for composite materials, the partial wave superposition approach based on the linear 3D elasticity theory becomes the prevalent way since it demonstrates closer physical relations between guided and bulk waves, which are further extended to a class of matrix-based methods. In the thesis, partial wave superposition approach will be particularly adopted to study guided waves propagation in a single-layered composite lamina.

Modeling methods of guided waves in composite structures

Accurately and efficiently computing dispersion curves of guided waves is mandatory to design a guided wave-based SHM system of aeronautic composite structures. It is the basis of selecting appropriate excitation frequency and designing the optimal piezoelectric transducer (PZT) network (position and size of the PZT elements) [START_REF] Lizé | Optimal dual-pzt sizing and network design for baseline-free shm of complex anisotropic composite structures[END_REF][START_REF] Raghavan | Finite-dimensional piezoelectric transducer modeling for guided wave based structural health monitoring[END_REF]. However, strong anisotropy and multiple layers stacking of composite laminates complicate the theory of guided wave propagation in these structures, thus the conventional single-layered isotropic model, i.e. Rayleigh-Lamb equation [26], is no longer applicable to composite laminates. In the past two decades, multifarious methods have been developed

to realize this purpose. For anisotropic material its dispersion equations are efficiently built, and the several small errors [START_REF] Haskell | The dispersion of surface waves on multilayered media[END_REF]. The Thomson-Haskell formulation conceptualized the simplicity of TMM and thereafter received a significant number of applications in both seismological and ultrasonic fields [START_REF] Nayfeh | Wave propagation in layered anisotropic media: With application to composites[END_REF].

MODELING METHODS OF GUIDED WAVES IN COMPOSITE STRUCTURES

But one fatal flaw of TMM is the notorious large f d problem which means that at the large product value of frequency f and plate's thickness d, the dispersion curves obtained from the solutions of the dispersion equation built with TMM become unstable. The reason of this problem is attributed to the poor conditioning of the transfer matrices thanks to the coexisted growing and decaying wave types.

In contrast to TMM, an alternative global matrix method (GMM) was proposed originally by Knopoff in 1964 [START_REF] Knopoff | A matrix method for elastic wave problems[END_REF]. The principal idea of GMM is to assemble all the sub-matrices representing the continuity conditions of field variables in each layer into a global matrix, along with all layers' wave amplitudes to be determined by the traction-free boundary conditions. The main merit of GMM rests on its numerical stability even at large f d range but at the cost of increasing computational burden.

Limited by the poor computational capabilities in the twentieth century, when the application was focused on the media having massive number of layers, the global matrix possesses a large size, thus the issue of computational inefficiency is unavoidable.

In order to fix the large f d problem of TMM, [START_REF] Wang | Stable reformulation of transfer matrix method for wave propagation in layered anisotropic media[END_REF] developed a new stiffness matrix method (SMM) by recasting the layer's transfer matrix to form the stiffness matrix that relates displacement to stress at the top and bottom side of a layer [START_REF] Wang | Stable reformulation of transfer matrix method for wave propagation in layered anisotropic media[END_REF][START_REF] Rokhlin | Stable recursive algorithm for elastic wave propagation in layered anisotropic media: Stiffness matrix method[END_REF]. Then, the continuity conditions used in TMM and GMM are now transformed to the recursive procedure from the first to the last layer in SMM to produce a global stiffness matrix of the whole plate. Finally, the traction-free boundary condition is applied to the generated global stiffness matrix to obtain the dispersion equation. It has been proved that SMM is unconditionally stable. One shortcoming of SMM is the degeneration of the conceptual simplicity in comparison to TMM and GMM. Due to this reason, we do not consider SMM in the thesis.

The modern matrix-based methods

The modern matrix-based methods are the successors of the classical TMM, GMM and SMM as introduced above. There are a significant number of references in pursuit of theoretical improvements and practical applications to these methods. We cannot refer each reference but only concern the representatives.

Wang and Yuan (2007) exerted symmetry condition in TMM to study the anisotropic property of composite laminates in terms of the wave characteristic curves in polar coordinate system, i.e.

phase velocity, slowness and group velocity [37]. The improved TMM was successfully validated by

MODELING METHODS OF GUIDED WAVES IN COMPOSITE STRUCTURES

the experimental data. However, their works cast a veil over the large f d problem inherent in TMM as stated previously. [START_REF] Maghsoodi | Calculation of wave dispersion curves in multilayered composite-metal plates[END_REF] applied TMM to predict the dispersion curves of metallic-composite plates [38]. They decomposed the coupled numerical problem into two cases, wave propagation along principal and off-principal axes of material respectively. Whereas, their method, like any other attempts, is invalid when dealing with the special case that the wave propagation direction corresponds to the off-principal direction of the composite layers but always to the principal direction of the metallic layers due to the isotropic property of metal. This causes the underlying different elastic matrix types issue between the two materials. [START_REF] Nandyala | Effective stiffness matrix method for predicting the dispersion curves in general anisotropic composites[END_REF] proposed an effective stiffness matrix method (ESM) to compute dispersion curves by regarding a multi-layered laminate as a single monoclinic layer, viz. from lamination to homogenization, such that it improves the computational efficiency of the traditional TMM [START_REF] Nandyala | Effective stiffness matrix method for predicting the dispersion curves in general anisotropic composites[END_REF]. ESM is an approximation method. It fails to predict dispersion curves at larger frequencies and higher orders [START_REF] Nandyala | Effective stiffness matrix method for predicting the dispersion curves in general anisotropic composites[END_REF]. Lowe et al. (2003) developed the software Disperse via GMM which became the pioneering computing program in the past two decades [START_REF] Pavlakovic | Disperse: User's manual version 2.0. 11, imperial college, university of london[END_REF]. However, some problems still exist in Disperse such as missing roots or producing outliers in a mode [START_REF] Barazanchy | A comparative convergence and accuracy study of composite guided-ultrasonic wave solution methods: Comparing the unified analytic method, safe method and disperse[END_REF], as well as the limited computational capability that it can only cope with laminates containing no more than 64 layers [11] (this incapability does not work for a 400-layered laminate considered in [11]). [START_REF] Pant | Derivation and experimental validation of lamb wave equations for an n-layered anisotropic composite laminate[END_REF] recast the conventional GMM based on 3D elasticity and partial wave superposition approach [8] to compute the dispersion curves of fiber-metal laminate. Although their method was well examined by experiments, only lower modes' data were compared (S0, A0, SH0 modes) and the comparison was made in the lower frequency range (0-600 kHz). Besides, one deficiency of their method is that they did not technically solve the numerically unstable issue, when waves propagate in the metallic-composite plates, that mismatched stiffness matrix types are used for the two materials also encountered in [38] (see the comments to this reference in the prior paragraph). In Pant's paper, a pseudo-correction measure was employed to circumvent the issue of mismatched stiffness matrix type by subtracting 1 • from the wave propagation angle when the direction of wave propagation is at the material's principal axis. But this measure is still invalid for metallic-composite plates because composite layers remain orthotropic and monoclinic type of elastic matrix in the principal and off-principal axes, respectively; conversely, the metal layers always keep the isotropic type in all directions. To solve the same problem of incompatible elastic matrix type, Barazanchy and Giurgiutiu (2016) proposed a unified formulation [9] by taking all elastic

MODELING METHODS OF GUIDED WAVES IN COMPOSITE STRUCTURES

matrix types into consideration. However, their method failed to pass the benchmark test for an unidirectional 50-layered plate due to another numerical instability issue. [START_REF] Ramasawmy | Elasticmatrix: A matlab toolbox for anisotropic elastic wave propagation in layered media[END_REF] developed a toolbox to compute dispersion curves of composite materials implemented in MATLAB called ElasticMatrix based on GMM [START_REF] Ramasawmy | Elasticmatrix: A matlab toolbox for anisotropic elastic wave propagation in layered media[END_REF]. This toolbox has limited functionality that it can only deal with isotropic and transversely isotropic materials, as well as restricting the wave propagation direction along the principal axes of material, which is evidently not suitable for aeronautic composite materials having strong anisotropic property.

In order to get stable dispersion solutions, Kamal and Giurgiutiu (2014) combined SMM and TMM to form the stiffness transfer matrix method (STMM) [42]. In this method, SMM can produce stable solutions at higher wavenumber region and TMM can give correct solutions at lower wavenumber region. Combining two independent methods is a bit superfluous in the author's opinion. Huber et al. (2018) adopted SMM to compute the dispersion curves of anisotropic composites with large to 400 layers being a component of the rocket Ariane 6 [11]. Computing such complex structure is time consuming but that paper did not report the computational time, which is the sole inadequacy.

That work contributed to a stand-alone freeware Dispersion Calculator, which has received substantial updates by Huber after its creation since 2018.

Finite element methods

Another effective route of modeling dispersion relation is to use finite element methods (FEM), among which the mainstream is the semi-analytical finite element method (SAFE). Just as its name implied, SAFE assumes the displacement field of a laminate in two separated forms: in the crosssection of waveguide 1D or 2D finite element discretization is utilized and in the propagation direction an analytical harmonic function of propagation distance and time is applied [START_REF] Bartoli | Modeling wave propagation in damped waveguides of arbitrary cross-section[END_REF]. Benefiting from the flexible discretization in the cross-section, SAFE has natural advantages for modeling wave propagation in waveguides of arbitrary cross-section. [START_REF] Bocchini | Graphical user interface for guided acoustic waves[END_REF] developed the freeware GUIGUW [START_REF] Bocchini | Graphical user interface for guided acoustic waves[END_REF] based on SAFE to compute dispersion curves of various waveguides containing multi-layered plate, cylinder, circular tube, square tube and railroad tracks, etc. Recently, a spectral method similar to SAFE has received increasing attention called spectral collocation method (SCM) based on Chebyshev polynomials [44] that allocates spectral points in the thickness direction of the plate to discretize each layer of a laminate, instead of elements in SAFE. This method is promising but may still encounter

MODELING METHODS OF GUIDED WAVES IN COMPOSITE STRUCTURES

numerical issues such as unstable solutions caused by the non-uniform Chebyshev grid points [45]. In order to take full advantage of the commercial FEM software such as ANSYS, a wave finite element method (WFEM) was developed by some researchers, e.g. [START_REF] Mace | Modelling wave propagation in two-dimensional structures using finite element analysis[END_REF] [46] and Sorohan et al. (2011) [47]. In the thickness direction, WFEM discretizes the waveguide with finite elements as the SAFE strategy, but in the propagation direction, only a nominal length, usually a wavelength, is discretized by an element. In WFEM, the boundary condition is replaced by the Floquet periodic condition. Given that the wavelength changes with frequency, the elements in the propagation direction should be re-meshed for different frequencies to retain the consistent accuracy [47], thus WFEM is not as flexible as SAFE and SCM.

Figure 1.9 -SAFE model of wave propagation, (a) discretization along thickness direction, (b) threenode element and degrees of freedom of each node [START_REF] Bartoli | Modeling wave propagation in damped waveguides of arbitrary cross-section[END_REF].

It should be stated here that the focus of the thesis is the matrix based methods (specifically, TMM and GMM) since these methods are based on the linear 3D elasticity theory and thus become the standard manner to derive the analytical dispersion equations of multi-layered plate system [START_REF] Barazanchy | A comparative convergence and accuracy study of composite guided-ultrasonic wave solution methods: Comparing the unified analytic method, safe method and disperse[END_REF]. SAFE method will be adopted for validation.

Other methods

The remaining available methods to build dispersion equation for a plate waveguide are partly based on the higher order plate theory (HOPT) [START_REF] Wang | Lamb wave propagation in composite laminates using a higher-order plate theory[END_REF]49], Ritz-Rayleigh method [START_REF] Stoklasová | Forward and inverse problems for surface acoustic waves in anisotropic media: A ritz-rayleigh method based approach[END_REF][START_REF] Grabec | Application of the ritz-rayleigh method for lamb waves in extremely anisotropic media[END_REF], Green's function [52],

and Green's matrix [START_REF] Karmazin | Investigation of lamb elastic waves in anisotropic multilayered composites applying the green's matrix[END_REF]. Although the HOPT can take the higher order shear deformation of plates into consideration, it is still an approximation method. The number of modes accessible from HOPT cannot be larger than six (three symmetric and three anti-symmetric modes) due to the limited characteristic matrix orders [49]. Besides, in the higher order modes, e.g. S2 and A2, or at larger frequencies, HOPT 1.4. ORGANIZATION OF THE THESIS may produce a poor accuracy of dispersion solutions. The Ritz-Rayleigh and Green's function based methods have superiority on characterizing material parameters through ultrasonic acoustic waves, and have led to some applications for metallic austenite [START_REF] Stoklasová | Forward and inverse problems for surface acoustic waves in anisotropic media: A ritz-rayleigh method based approach[END_REF]52], but not for composite materials, which has been done by the Green's matrix-based method [START_REF] Karmazin | Investigation of lamb elastic waves in anisotropic multilayered composites applying the green's matrix[END_REF].

Organization of the thesis

The objective of the thesis is to study guided waves propagation and attenuation in an arbitrarily oriented composite plate for its potential application to SHM. The thesis is organized as follows:

In Chapter 2, classical equations of three-dimensional (3D) elasticity are reviewed with an emphasis on the characteristics of stiffness matrix rotation of composite materials.

In Chapter 3, the partial wave superposition approach is used to derive the classical dispersion equations of guided waves propagating in a single layer composite lamina.

In Chapter 4, the transfer matrix method is introduced to model guided waves propagation in a multi-layered composite plate. An original hybrid matrix strategy is proposed to cope with the matrix incompatibility issue encountered in quasi-isotropic plates and metallic-composite plates.

In Chapter 5, the standard global matrix method is introduced. Based on it, an optimized global matrix method is developed that can greatly improve the computational efficiency of the standard global matrix method. The optimized version is applied to an aerospace composite structure having 400 layers which is the largest number of layers reported so far.

In Chapter 6, an important property of dispersion equations, named dichotomy property, is originally explored, which states that the a priori complex-valued dispersion equations in the case of pure elastic materials collapse to pure real or imaginary valued equations, depending on the behavior of some intermediate parameters. This property is extremely helpful for overcoming numerical instabilities encountered during the equation solving process.

In Chapter 7, the methodology of predicting the frequency and spatially dependent attenuation is proposed by using the damped global matrix method. Two engineering case studies are conducted to validate the effectiveness of the methodology. The two involved structures are the components of AIRBUS A380. Using the actual and in-service data of the practical aeronautic structures makes a great contribution to the state-of-the-art and thus this chapter escorts the initiative of closing the gap between research and industrial deployment for SHM [START_REF] Cawley | Structural health monitoring: Closing the gap between research and industrial deployment[END_REF].

The general conclusions and major contributions of the thesis are presented in Chapter 8, along with several perspectives for future works. To finish the thesis, a short list of publications is provided in the end of this chapter.

Chapter 2

Fundamentals of wave propagation in solids

In this chapter, some fundamentals of elastic wave propagation in solids will be reviewed [2,[START_REF] Lammering | Lamb-wave based structural health monitoring in polymer composites[END_REF][START_REF] Nayfeh | Wave propagation in layered anisotropic media: With application to composites[END_REF],

including the basic equations of 3D elasticity, stiffness matrix types of various materials, stiffness matrix rotation, the expanded field equations of triclinic material and bulk wave propagation in solids. These contents are the basis of the subsequent chapters.

Basic equations of 3D elasticity

The mechanism of wave propagation in solid media is governed by the 3D elasticity theory. Thus, the basic equations of 3D elasticity will be introduced firstly, including stress-strain relation, straindisplacement relation and elastodynamic equations of motion.

Stress-strain relation

For a cubic element representing a tiny volume of a certain solid material, its 3D stress state can be illustrated in Figure 2.

] T (2.1) ε = [ ε 11 , ε 22 , ε 33 , ε 23 , ε 13 , ε 12 ] T (2.2) u = [ u 1 , u 2 , u 3 ] T (2.3)
where, T is the matrix transpose operator.

Figure 2.1 -The stress state of a cubic element of a solid material.

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ σ 11 σ 22 σ 33 σ 23 σ 13 σ 12 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ C 11 C 12 C 13 C 14 C 15 C 16 C 22 C 23 C 24 C 25 C 26 C 33 C 34 C 35 C 36 C 44 C 45 C 46 sym C 55 C 56 C 66 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ε 11 ε 22 ε 33 2ε 23 2ε 13 2ε 12 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (2.4)
where, the coefficients matrix in Eq. (2.4) is denoted by C ∈ C 6×6 which is the stiffness matrix of the material having symmetric property. Since it characterizes the elastic or viscoelastic behavior of stress and strain, it is also called elastic matrix in pure elastic materials or viscoelastic matrix in viscoelastic materials.

STIFFNESS MATRIX TYPES OF VARIOUS MATERIALS

Strain-displacement relation

The relation between strain and displacement is characterized by the geometric equations as formalized in Eq. (2.5).

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ε 11 = ∂u 1 ∂x 1 , ε 12 = 1 2 ( ∂u 1 ∂x 2 + ∂u 2 ∂x 1 ), ε 22 = ∂u 2 ∂x 2 , ε 23 = 1 2 ( ∂u 2 ∂x 3 + ∂u 3 ∂x 2 ), ε 33 = ∂u 3 ∂x 3 , ε 13 = 1 2 ( ∂u 1 ∂x 3 + ∂u 3 ∂x 1 ).
(2.5)

Elastodynamic equations of motion

Inherently, the dynamic behavior of particles in material is governed by its elastodynamic equations of motion, representing the equilibrium condition in the dynamic sense, as presented in Eq. (2.6), where ρ is the mass density.

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ∂σ 11 ∂x 1 + ∂σ 12 ∂x 2 + ∂σ 13 ∂x 3 = ρ ∂ 2 u 1 ∂t 2 ∂σ 21 ∂x 1 + ∂σ 22 ∂x 2 + ∂σ 23 ∂x 3 = ρ ∂ 2 u 2 ∂t 2 ∂σ 31 ∂x 1 + ∂σ 32 ∂x 2 + ∂σ 33 ∂x 3 = ρ ∂ 2 u 3 ∂t 2 (2.6)

Stiffness matrix types of various materials

In Eq. (2.4), there are 21 elements in the stiffness matrix C. However, for different materials, the number of elements in C and its types are different. In material science, materials are physically classified into five types according to the number of symmetry axes inherent in it, i.e. triclinic, monoclinic, orthotropic, transversely isotropic and isotropic materials [START_REF] Gibson | Principles of composite material mechanics[END_REF]. Mathematically, this classification is equivalent to count the different number of independent elements in the stiffness matrix C, in other words, depending on the stiffness matrix type. During deriving the wave propagation equations in the next chapter, the stiffness matrix types play a significant role for the wave propagation property.

Thus, the five materials along with their specific stiffness matrix types should be introduced firstly.

Triclinic material

For triclinic material, there are 21 independent constants in the stiffness matrix C T , which directly succeeds from the general type of the stiffness matrix C in Eq. (2.4). The set of all stiffness matrices

STIFFNESS MATRIX TYPES OF VARIOUS MATERIALS

of triclinic materials is denoted as C T .

C T = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ C 11 C 12 C 13 C 14 C 15 C 16 C 22 C 23 C 24 C 25 C 26 C 33 C 34 C 35 C 36 C 44 C 45 C 46 sym C 55 C 56 C 66 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (2.7)

Monoclinic material

For monoclinic material, the number of independent constants in its stiffness matrix C M reduces to 13, as presented in Eq. (2.8). All stiffness matrices of monoclinic materials constitute the set C M .

C M = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ C 11 C 12 C 13 0 0 C 16 C 22 C 23 0 0 C 26 C 33 0 0 C 36 C 44 C 45 0 sym C 55 0 C 66 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (2.8)

Orthotropic material

For orthotropic material, there are nine independent constants in its stiffness matrix C O , which belongs to the set of stiffness matrices of orthotropic materials denoted as C O .

C O = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ C 11 C 12 C 13 0 0 0 C 22 C 23 0 0 0 C 33 0 0 0 C 44 0 0 sym C 55 0 C 66 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (2.9) In engineering, C O is generally obtained from the elastic constants (E 1 , E 2 , E 3 , G 23 , G 31 , G 12 ,
ν 23 , ν 13 , ν 12 ) measured through the standard mechanical test for the normal shaped specimens [START_REF] Gibson | Principles of composite material mechanics[END_REF].

C O = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1/E 1 -ν 12 /E 1 -ν 13 /E 1 0 0 0 1/E 2 -ν 23 /E 2 0 0 0 1/E 3 0 0 0 1/G 23 0 0 sym 1/G 31 0 1/G 12 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ -1
(2.10)

Transversely isotropic material

For transversely isotropic material, there are five independent constants in its stiffness matrix C TI .

The widely used unidirectional composite materials in aeronautic and aerospace engineering just belong to this type. This material possesses the transversely isotropic property, i.e. in the plane transverse to the fiber direction, material properties are isotropic [2]. In the thesis, we classify C TI ∈ C O based on the fact that C TI has the same shape as C O in terms of number and positions of zero elements but with less number of independent coefficients.

C TI = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ C 11 C 12 C 12 0 0 0 C 22 C 23 0 0 0 C 22 0 0 0 C 22 -C 23 2 0 0 sym C 55 0 C 55 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (2.11)
In engineering, C TI can be obtained from the experimentally measured elastic constants

(E 1 , E 2 , G 12 , ν 12 , ν 23 ). C TI = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1/E 1 -ν 12 /E 1 -ν 12 /E 1 0 0 0 1/E 2 -ν 23 /E 2 0 0 0 1/E 2 0 0 0 1/G 23 0 0 sym 1/G 12 0 1/G 12 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ -1 (2.12)
where, G 23 = E 2 2(1+ν 23 ) .

Isotropic material

Isotropic materials only have two independent constants in its stiffness matrix C I . Most metallic materials belong to isotropic type, which is denoted by the set notation C I . In Section 2.3.3, we further

classify C I ∈ C I ⊆ C O . C I = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ C 11 C 12 C 12 0 0 0 C 11 C 12 0 0 0 C 11 0 0 0 C 11 -C 12 2 0 0 sym C 11 -C 12 2 0 C 11 -C 12 2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
(2.13)

STIFFNESS MATRIX ROTATION

The Lamé constants λ and µ are widely used to represent the elastic properties of isotropic materials.

C I = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ λ + 2µ λ λ 0 0 0 λ + 2µ λ 0 0 0 λ + 2µ 0 0 0 µ 0 0 sym µ 0 µ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (2.14)
where, the Lamé constants can be computed from Young's modulus E and Poisson's ratio ν according

to λ = Eν (1+ν)(1-2ν) , µ = E 2(1+ν)
. E and ν are generally measured from experiments.

Stiffness matrix rotation

Eqs. (2.9)(2.11)(2.13) show the stiffness matrix types of orthotropic, transversely isotropic and isotropic materials, which are measured along their respective material's symmetry axes, i.e. the principal axis x 1 illustrated in Figure 2.2. However, in plate, guided waves can propagate at a general off-principal direction. Thus, it is necessary to study the stiffness matrix types of these materials along the off-principal axis direction (x ′ 1 as illustrated in Figure 2.2). In elasticity theory, this task can be completed based on the tensor rotation rule, which is introduced in this section.

Voigt notation and tensor rotation rule

The first step is to transform the general stiffness matrix C pq into the fourth order tensor c ijkl via the Voigt notation, which is examplified in Figure 2.3 and detailed in Eq. (2.15).

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⇔ C = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (2.15)
The second step is to compute the rotation matrix shown in Eq. (2.16) that is based on the rotation of coordinate system around the axis x 3 as illustrated in Figure 2.2.

R(θ)

= ⎡ ⎢ ⎣ cos(θ) sin(θ) 0 -sin(θ) cos(θ) 0 0 0 1 ⎤ ⎥ ⎦ (2.16)
where, θ is the rotation angle being positive for counterclockwise direction and negative for clockwise direction.

The third step is to compute the rotated tensor c ′ mnop based on the tensor multiplication rule formalized in Eq. (2.17a), where the Einstein summation convention is implied for the repeated indices.

To ease computation, the tensor multiplication form in Eq. (2.17a) is expanded to the linear summation form in Eq. (2.17b).

c ′ mnop = R mi R nj R ok R pl c ijkl (2.17a) c ′ mnop = 3 ∑ i=1 3 ∑ j=1 3 ∑ k=1 3 ∑ l=1 R mi R nj R ok R pl c ijkl (2.17b)
Finally, by inversely applying the Voigt notation shown in Eq. (2.15), the rotated tensor c ′ mnop can be transformed back to the matrix form C ′ pq , and the rotated stiffness matrix C ′ is just obtained. Note that although the original stiffness tensor c ijkl in Eq. (2.17a) represents the type of triclinic material, it can be written for any other types.

Example of stiffness matrix rotation

The orthotropic, transversely isotropic and pure isotropic materials are widely used in aeronautic engineering. Their stiffness matrices are employed here to demonstrate the process of stiffness matrix rotation. To this end, the four rotation steps are firstly applied to the stiffness matrix of the orthotropic material C O defined in Eq. (2.9) with an arbitrary angle θ. The rotated matrix C ′ O is obtained in Eq. (2.18), with the detailed elements listed in Eq. (2.19). Among all possible rotation angles, there is a special one θ = 90 • which makes the elements .20). This angle corresponds to the transverse direction in relation to the fibers in the plate plane, in other words, the axis x ′ 1 is rotated to coincide with the axis x 2 in Figure 2.2.

C ′ 16 , C ′ 26 , C ′ 36 , C ′ 45 = 0 in the rotated C ′ O , see Eq. ( 2
C O = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ C 11 C 12 C 13 0 0 0 C 22 C 23 0 0 0 C 33 0 0 0 C 44 0 0 sym C 55 0 C 66 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ Rot{C O ,θ} -------→ C ′ O = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ C ′ 11 C ′ 12 C ′ 13 0 0 C ′ 16 C ′ 22 C ′ 23 0 0 C ′ 26 C ′ 33 0 0 C ′ 36 C ′ 44 C ′ 45 0 sym C ′ 55 0 C ′ 66 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (2.18)
where, Rot{C O , θ} is the rotation operator that rotates the stiffness matrix C O with angle θ.

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ C ′ 11 =
C O = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ C 11 C 12 C 13 0 0 0 C 22 C 23 0 0 0 C 33 0 0 0 C 44 0 0 sym C 55 0 C 66 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ Rot{C O ,90 • } --------→ C ′ O = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ C 22 C 12 C 23 0 0 C 11 C 13 0 0 C 33 0 0 C 55 0 sym C 44 0 C 66 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (2.20)
For the rotated stiffness matrix of the transversely isotropic material C ′ TI , it can be directly obtained based on Eq. (2.19)

C I = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ C 11 C 12 C 12 0 0 0 C 11 C 12 0 0 0 C 11 0 0 0 C 11 -C 12 2 0 0 sym C 11 -C 12 2 0 C 11 -C 12 2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ Rot{C I ,θ} ------→ C ′ I = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ C 11 C 12 C 12 0 0 0 C 11 C 12 0 0 0 C 11 0 0 0 C 11 -C 12 2 0 0 sym C 11 -C 12 2 0 C 11 -C 12 2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
(2.21)

Characteristics of stiffness matrix rotation

From the presented results of examples, the following characteristics can be observed. These characteristics is the basis of Chapter 3 and 4.

1. After rotation from the principal axis x 1 to the off-principal axis x ′ 1 when θ ̸ = 0 • and 90 This characteristic manifests that for monoclinic and triclinic materials, the rotation process does not change their respective stiffness matrix types.

The above characteristics of stiffness matrix rotation are summarized in Table 2.1.

Table 2.1 -The characteristics of stiffness matrix rotation

Matrix form in the (minor-)principal axis direction

Number of independent coefficients

Matrix form in the off-principal axis direction

C T ∈ C T 21 C ′ T ∈ C T C M ∈ C M 13 C ′ M ∈ C M C O ∈ C O 9 C ′ O ∈ C M C TI ∈ C O 5 C ′ TI ∈ C M C I ∈ C I ⊆ C O 2 C ′ I = C I ∈ C I ⊆ C O 2.

Expanded elastodynamic equations of motion of triclinic material

The elastodynamic equations of motion, Eq. ( Lamb waves as well as their coupling) involved in the next chapters, all the counterparts can be

EXPANDED ELASTODYNAMIC EQUATIONS OF MOTION OF TRICLINIC MATERIAL

simplified from the equations corresponding to triclinic material [START_REF] Nayfeh | Wave propagation in layered anisotropic media: With application to composites[END_REF]. )

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ σ 11 = C
(2.22)

C 11 ∂ 2 u 1 ∂x 2 1 + C 66 ∂ 2 u 1 ∂x 2 2 + C 55 ∂ 2 u 1 ∂x 2 3 + 2C 16 ∂ 2 u 1 ∂x 1 ∂x 2 + 2C 15 ∂ 2 u 1 ∂x 1 ∂x 3 + 2C 56 ∂ 2 u 1 ∂x 2 ∂x 3 + C 16 ∂ 2 u 2 ∂x 2 1 + C 26 ∂ 2 u 2 ∂x 2 2 + C 45 ∂ 2 u 2 ∂x 2 3 + (C 12 + C 66 ) ∂ 2 u 2 ∂x 1 ∂x 2 + (C 14 + C 56 ) ∂ 2 u 2 ∂x 1 ∂x + (C 46 + C 25 ) ∂ 2 u 2 ∂x 2 ∂x 3 + C 15 ∂ 2 u 3 ∂x 2 1 + C 46 ∂ 2 u 3 ∂x 2 2 + C 35 ∂ 2 u 3 ∂x 2 3 + (C 14 + C 56 ) ∂ 2 u 3 ∂x 1 ∂x + (C 13 + C 55 ) ∂ 2 u 3 ∂x 1 ∂x 3 + (C 36 + C 45 ) ∂ 2 u 3 ∂x 2 ∂x 3 = ρ ∂ 2 u 1 ∂t 2
(2.23)

C 16 ∂ 2 u 1 ∂x 2 1 + C 26 ∂ 2 u 1 ∂x 2 2 + C 45 ∂ 2 u 1 ∂x 2 3 + (C 12 + C 66 ) ∂ 2 u 1 ∂x 1 ∂x 2 + (C 14 + C 56 ) ∂ 2 u ∂x 1 ∂x 3 + (C 46 + C 25 ) ∂ 2 u 1 ∂x 2 ∂x 3 + C 66 ∂ 2 u 2 ∂x 2 1 + C 22 ∂ 2 u 2 ∂x 2 2 + C 44 ∂ 2 u 2 ∂x 2 3 + 2C 26 ∂ 2 u 2 ∂x 1 ∂x 2 + 2C 46 ∂ 2 u 2 ∂x 1 ∂x 3 + 2C 24 ∂ 2 u 2 ∂x 2 ∂x 3 + C 56 ∂ 2 u 3 ∂x 2 1 + C 24 ∂ 2 u 3 ∂x 2 2 + C 34 ∂ 2 u 3 ∂x 2 3 + (C 46 + C 25 ) ∂ 2 u 3 ∂x 1 ∂x 2 + (C 36 + C 45 ) ∂ 2 u 3 ∂x 1 ∂x 3 + (C 23 + C 44 ) ∂ 2 u 3 ∂x 2 ∂x 3 = ρ ∂ 2 u 2 ∂t (2.24) C 15 ∂ 2 u 1 ∂x 2 1 + C 46 ∂ 2 u 1 ∂x 2 2 + C 35 ∂ 2 u 1 ∂x 2 3 + (C 14 + C 56 ) ∂ 2 u 1 ∂x 1 ∂x 2 + (C 13 + C 55 ) ∂ 2 u 1 ∂x 1 ∂x 3 + (C 36 + C 45 ) ∂ 2 u 1 ∂x 2 ∂x 3 + C 56 ∂ 2 u 2 ∂x 2 1 + C 24 ∂ 2 u 2 ∂x 2 2 + C 34 ∂ 2 u 2 ∂x 2 3 + (C 46 + C 25 ) ∂ 2 u 2 ∂x 1 ∂x + (C 36 + C 45 ) ∂ 2 u 2 ∂x 1 ∂x 3 + (C 23 + C 44 ) ∂ 2 u 2 ∂x 2 ∂x 3 + C 55 ∂ 2 u 3 ∂x 2 1 + C 44 ∂ 2 u 3 ∂x 2 2 + C 33 ∂ 2 u ∂x 2 3 + 2C 45 ∂ 2 u 3 ∂x 1 ∂x 2 + 2C 35 ∂ 2 u 3 ∂x 1 ∂x 3 + 2C 34 ∂ 2 u 3 ∂x 2 ∂x 3 = ρ ∂ 2 u 3 ∂t 2
(2.25) 2.5. BULK WAVE

Bulk wave

Bulk wave is the basis of the partial wave superposition approach (PWSA) used to derive the dispersion equations of guided waves in a single-layered plate studied in the next chapter. Thus, the concept of bulk wave in solids will be introduced briefly in this section for monoclinic material. For other kind of materials, all the counterparts can be simplified from the corresponding equations.

Derivation of bulk wave

In the stiffness matrix of monoclinic material, the following elements are zero: C 14 = 0, C 15 = 0,

C 24 = 0, C 25 = 0, C 34 = 0, C 35 = 0, C 46 =
C 11 ∂ 2 u 1 ∂x 2 1 + C 66 ∂ 2 u 1 ∂x 2 2 + C 55 ∂ 2 u 1 ∂x 2 3 + 2C 16 ∂ 2 u 1 ∂x 1 ∂x 2 + C 16 ∂ 2 u 2 ∂x 2 1 + C 26 ∂ 2 u 2 ∂x 2 2 + C 45 ∂ 2 u 2 ∂x 2 3 + (C 12 + C 66 ) ∂ 2 u 2 ∂x 1 ∂x 2 + (C 13 + C 55 ) ∂ 2 u 3 ∂x 1 ∂x 3 + (C 36 + C 45 ) ∂ 2 u 3 ∂x 2 ∂x 3 = ρ ∂ 2 u 1 ∂t 2
(2.26)

C 16 ∂ 2 u 1 ∂x 2 1 + C 26 ∂ 2 u 1 ∂x 2 2 + C 45 ∂ 2 u 1 ∂x 2 3 + (C 12 + C 66 ) ∂ 2 u 1 ∂x 1 ∂x 2 + C 66 ∂ 2 u 2 ∂x 2 1 + C 22 ∂ 2 u 2 ∂x 2 2 + C 44 ∂ 2 u 2 ∂x 2 3 + 2C 26 ∂ 2 u 2 ∂x 1 ∂x 2 + (C 36 + C 45 ) ∂ 2 u 3 ∂x 1 ∂x 3 + (C 23 + C 44 ) ∂ 2 u 3 ∂x 2 ∂x 3 = ρ ∂ 2 u 2 ∂t 2
(2.27)

(C 13 + C 55 ) ∂ 2 u 1 ∂x 1 ∂x 3 + (C 36 + C 45 ) ∂ 2 u 1 ∂x 2 ∂x 3 + (C 36 + C 45 ) ∂ 2 u 2 ∂x 1 ∂x 3 + (C 23 + C 44 ) ∂ 2 u 2 ∂x 2 ∂x 3 + C 55 ∂ 2 u 3 ∂x 2 1 + C 44 ∂ 2 u 3 ∂x 2 2 + C 33 ∂ 2 u 3 ∂x 2 3 + 2C 45 ∂ 2 u 3 ∂x 1 ∂x 2 = ρ ∂ 2 u 3 ∂t 2
(2.28)

For bulk wave propagation in a solid in the direction provided by the wave vector ξ ⃗ as illustrated in Figure 2.4,2 the displacement fields can be expressed as the sinusoidal form shown in Eq. (2.29). where, i is the imaginary number unit, ξ and v are wavenumber and velocity, respectively, t is the time variable, U j is the amplitude of u j along axis x j , and n 1 , n 2 , n 3 are the directional cosines of the wave vector ξ ⃗ .

u j = U j e iξ(n 1 x 1 +n 2 x 2 +n 3 x 3 -vt) (j = 1, 2, 3) (2
With Eq. (2.29), the first order partial derivative of u j can be easily computed as follows [2,[START_REF] Nayfeh | Wave propagation in layered anisotropic media: With application to composites[END_REF].

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂u j ∂x 1 = iξn 1 U j e iξ(n 1 x 1 +n 2 x 2 +n 3 x 3 -vt) = iξn 1 u j ∂u j
∂x 2 = iξn 2 U j e iξ(n 1 x 1 +n 2 x 2 +n 3 x 3 -vt) = iξn 2 u j ∂u j ∂x 3 = iξn 3 U j e iξ(n 1 x 1 +n 2 x 2 +n 3 x 3 -vt) = iξn 3 u j ∂u j ∂t = -iξvU j e iξ(n 1 x 1 +n 2 x 2 +n 3 x 3 -vt) = -iξvu j (j = 1, 2, 3) (2.30) Furthermore, Eq. (2.31) shows the second order partial derivative of u j .

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂ 2 u j ∂x 2 1 = -ξ 2 n 2 1 u j , ∂ 2 u j ∂x 1 ∂x 2 = -ξ 2 n 1 n 2 u j , ∂ 2 u j ∂x 2 2 = -ξ 2 n 2 2 u j , ∂ 2 u j ∂x 2 ∂x 3 = -ξ 2 n 2 n 3 u j , ∂ 2 u j ∂x 2 3 = -ξ 2 n 2 3 u j , ∂ 2 u j ∂x 1 ∂x 3 = -ξ 2 n 1 n 3 u j , ∂ 2 u j ∂t 2 = -ξ 2 v 2 u j , (j = 1, 2 , 3).
(

⎡ ⎢ ⎣ A D E D B F E F C ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ U 1 U 2 U 3 ⎤ ⎥ ⎦ = ρv 2 ⎡ ⎢ ⎣ U 1 U 2 U 3 ⎤ ⎥ ⎦ (2.32)
where, the elements of the matrix K are listed below, λ = ρv 2 and U = [U 1 , U 2 , U 3 ] T are the eigenvalue 2.5. BULK WAVE and eigenvector, respectively.

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ A = C 11 n 2 1 + C 66 n 2 2 + C 55 n 2 3 + 2C 16 n 1 n 2 , B = C 66 n 2 1 + C 22 n 2 2 + C 44 n 2 3 + 2C 26 n 1 n 2 , C = C 55 n 2 1 + C 44 n 2 2 + C 33 n 2 3 + 2C 45 n 1 n 2 , D = C 16 n 2 1 + C 26 n 2 2 + C 45 n 2 3 + (C 12 + C 66 )n 1 n 2 , E = (C 13 + C 55 )n 1 n 3 + (C 36 + C 45 )n 2 n 3 , F = (C 36 + C 45 )n 1 n 3 + (C 23 + C 44 )n 2 n 3 .
(2.33)

For pure elastic media,3 K is a real symmetric matrix since all the parameters in Eq. (2.33) are real numbers, which immediately leads to the two properties based on the fundamentals of linear algebra that all the three eigenvalues (λ 1 , λ 2 , λ 3 ) are real numbers, and the three eigenvectors (U 1 , U 2 , U 3 ) are mutually orthogonal [START_REF] Jeffrey | Matrix operations for engineers and scientists: an essential guide in linear algebra[END_REF].

Each pair of eigenvalue and eigenvector of the Christoffel equation corresponds to a certain wave type amongst the shear horizontal wave (SH), shear vertical wave (SV) and longitudinal wave (L) that are characterized by the dominated component of polarization U 2 , U 3 and U 1 , respectively. In some situations, if there is not a dominated component for the solved eigenvector [U 1 , U 2 , U 3 ] T (that can always happen in anisotropic materials), a prefix quasi or q could be attached to the abbreviations SH, SV and L to represent the coupled waves, namely, qSH, qSV and qL waves. This usage is accepted by some references [37,[START_REF] Nadella | Characterization of guided-wave propagation in composite plates[END_REF]8], but we do not emphasize it in the thesis, thus the prefix quasi or q is omitted unless stated otherwise.

Numerical example

A transversely isotropic material is adopted to illustrate the three elementary bulk waves. Its material properties are cited from [START_REF] Nayfeh | Free wave propagation in plates of general anisotropic media[END_REF] and listed in Eq. (2.34) for convenience. The (minor-)principal axes of this material are consistent with the coordinate system of Figure 2.4. We study the bulk waves propagating in the x 1 -x 2 plane, thus n 3 = 0, E = 0 and F = 0 in Eq. (2.33), which leads to the decoupled polarization U 3 from U 1 and U 2 . n 1 and n 2 can be easily computed from the specific directional angle of the wave vector ξ ⃗ in Figure 2.4. Thus, for each direction angle, the three bulk wave velocities can be obtained by solving the eigenvalue problem in Eq. (2.32). By rotating the wave vector ξ ⃗ from 0 • to 360 • , the polar plot of the three bulk wave velocities, characterizing the anisotropic 2.6. CONCLUSION effect of the three waves, is illustrated in Figure 2.5(a). In literature, the slowness curves being the inverse of wave velocity are also widely used [37], and they are presented in Figure 2.5(b). One can see that SH and L waves hold an anisotropic effect, and SV wave is quasi-isotropic. This is due to the reason that SV wave polarizes along U 3 direction which is decoupled from U 1 and U 2 directions.

C = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
155. 6

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ GPa, ρ = 1600 kg/m 3 .
(2.34)

Conclusion

This chapter reviews the fundamental equations of 3D elasticity including stress-strain relation, strain-displacement relation and elastodynamic equations of motion. Five widely used composite materials are introduced in terms of their respective stiffness matrix, specifically, triclinic, monoclinic, orthotropic, transversely isotropic and isotropic materials. Stiffness matrix rotation is achieved by using Voigt notation and tensor rotation rule, along with the characteristics of the rotated matrix 2.6. CONCLUSION shown in Table 2.1. The expanded field equations of triclinic materials lead to bulk wave equations in solids. This chapter defines many concepts and presents some fundamental equations and characteristics, and thus establishes the basis of subsequent chapters for studying guided waves propagation in plate structure.

Chapter 3

Guided waves propagation in a single-layered plate

This chapter mainly introduces the partial wave superposition approach (PWSA) which is based on 3D elasticity of Chapter 2 to model guided waves propagation in a single-layered composite plate.

Traction-free boundary condition and symmetry condition will be employed to derive dispersion equations. For different composite materials, Lamb and SH waves will be coupled or separated according to the specific material characteristics. This chapter is a prerequisite for the following chapters that focus on guided waves propagation in a multi-layered composite laminate.

Wave propagation model of a single layer plate

The guided waves propagating in a single-layered plate schematically shown in Figure 3.1(a) is considered in this chapter. The plate extends infinitely in both directions of x 1 and x 2 axes, and the wave is guided for propagation along the direction of x 1 axis. In this condition, the plane-strain state in x 2 direction is satisfied which leads to the displacement field being invariant to x 2 . Thus, the time harmonic displacement expression in Eq. (2.29), which characterizes bulk waves, is now changed to the following one.

u = ⎡ ⎢ ⎣ u 1 u 2 u 3 ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ U 1 U 2 U 3 ⎤ ⎥ ⎦ e iξ(x 1 +αx 3 -vt) = ⎡ ⎢ ⎣ U V W ⎤ ⎥ ⎦ e iξ(x 1 +αx 3 -vt) (3.1)
where, α is the ratio of wavenumbers between x 3 and x 1 direction as illustrated in Considering the traction-free boundary condition (BC) at the two surfaces of plate as illustrated in Furthermore, the displacement and stress fields have symmetry condition (SC) at the midplane of the plate [37] that is presented in Eqs. (3.3) and (3.4).

[σ 33 , σ 23 , σ 13 ] T x 3 =±h = [0, 0, 0] T for boundary condition (3.2)

[u 3 , σ 13 , σ 23 ] T x 3 =0 = [0, 0, 0] T for symmetric condition (3.3) [u 1 , u 2 , σ 33 ] T x 3 =0 = [0, 0, 0] T for anti-symmetric condition (3.4)
where, h is the plate's half thickness and x 3 = 0 indicates the midplane of the plate.

To facilitate the following derivation of dispersion equations, the first and second order partial

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂u j ∂x 1 = iξU j e iξ(x 1 +αx 3 -vt) = iξu j ∂u j ∂x 2 = 0 ∂u j ∂x 3 = iξαU j e iξ(x 1 +αx 3 -vt) = iξαu j ∂u j ∂t = -iξvU j e iξ(x 1 +αx 3 -vt) = -iξvu j (j = 1, 2, 3) (3.5) ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂ 2 u j ∂x 2 1 = -ξ 2 u j , ∂ 2 u j ∂x 1 ∂x 2 = 0, ∂ 2 u j ∂x 2 2 = 0, ∂ 2 u j ∂x 2 ∂x 3 = 0, ∂ 2 u j ∂x 2 3 = -ξ 2 α 2 u j , ∂ 2 u j ∂x 1 ∂x 3 = -ξ 2 αu j , ∂ 2 u j ∂t 2 = -ξ 2 v 2 u j , (j = 1, 2 , 3).
(3.6)

Dispersion equations of coupled Lamb and SH waves in triclinic material

With the specific form of wave field in Eq. (3.1), the boundary condition in Eq. (3.2) and the symmetry condition in Eqs. (3.3)(3.4), dispersion equations of guided waves propagating in a singlelayered plate with triclinic material will be derived in this chapter. The same derivation procedure will be applied to other materials.

Christoffel equation

Substituting Eq. (3.6) into Eqs. (2.23)(2.24)(2.25), the following Christoffel equation K(α)U = 0 can be derived.

⎡ ⎢ ⎣ K 11 (α) K 12 (α) K 13 (α) K 12 (α) K 22 (α) K 23 (α) K 13 (α) K 23 (α) K 33 (α) ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ U V W ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ 0 0 0 ⎤ ⎥ ⎦ (3.7)
where,

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ K 11 (α) = C 11 -ρv 2 + 2C 15 α + C 55 α 2 , K 12 (α) = C 16 + (C 14 + C 56 )α + C 45 α 2 , K 13 (α) = C 15 + (C 13 + C 55 )α + C 35 α 2 , K 22 (α) = C 66 -ρv 2 + 2C 46 α + C 44 α 2 , K 23 (α) = C 56 + (C 36 + C 45 )α + C 34 α 2 , K 33 (α) = C 55 -ρv 2 + 2C 35 α + C 33 α 2 . (3.8)
To make sure that Eq. (3.7) has nontrivial amplitudes of displacement U, the matrix K(α) should be singular, namely det{K(α)} = 0, where det{•} means the operator computing the determinant for a square matrix. Expanding this determinant results in a sixth order polynomial equation in terms of α as expressed in Eq. (3.9).

A 6 α 6 + A 5 α 5 + A 4 α 4 + A 3 α 3 + A 2 α 2 + A 1 α + A 0 = 0 (3.9)

DISPERSION EQUATIONS OF COUPLED LAMB AND SH WAVES IN TRICLINIC MATERIAL

where, the seven polynomial coefficients are presented in Figure 3.2 and obtained via the software Mathematica due to its powerful capability of symbolic computing. Once the elastic coefficients C pq and mass density ρ are known, meanwhile phase velocity v is provided as a specific value v = v 0 , the seven polynomial coefficients can be computed based on Figure 3.2. Then, the six α's roots can be solved via polynomial root-finding algorithms [START_REF] Muller | A method for solving algebraic equations using an automatic computer[END_REF][START_REF] Niu | A method for finding the zeros of polynomials using a companion matrix[END_REF], or by directly calling the MATLAB built-in function roots.

Displacement and stress fields

For each α r (r = 1, • • • , 6), there will be a corresponding solution vector of displacement amplitude

U r = [U r , V r , W r ]
T according to Eq. (3.7), which is one degree on indeterminacy according to the theory of linear algebra [START_REF] Jeffrey | Matrix operations for engineers and scientists: an essential guide in linear algebra[END_REF]. The three displacement amplitudes can be determined by assigning a specific value for a certain term, for example U r = 1, and V r and W r are found from the resulted equation set via Cramer's Rule as presented in Eq. (3.10).

V r = K 11 (α r )K 23 (α r ) -K 12 (α r )K 13 (α r ) K 13 (α r )K 22 (α r ) -K 12 (α r )K 23 (α r ) , W r = K 11 (α r )K 23 (α r ) -K 12 (α r )K 13 (α r ) K 12 (α r )K 33 (α r ) -K 13 (α r )K 23 (α r ) (3.10)
With the predefined form of displacements in Eq. (3.1) and the obtained amplitudes in Eq. (3.10), the displacement field u = [u 1 , u 2 , u 3 ] T now can be written as the superposition of the six solutions as shown in Eq. (3.11), in which each term within the summation represents a partial wave. Therefore,

DISPERSION EQUATIONS OF COUPLED LAMB AND SH WAVES IN TRICLINIC MATERIAL

the denomination partial wave superposition approach (PWSA) is chosen.

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ u 1 = ( ∑ 6 r=1 η r e iξαrx 3
) e iξ(x 1 -vt)

u 2 = ( ∑ 6 r=1 η r V r e iξαrx 3
) e iξ(x 1 -vt)

u 3 = ( ∑ 6 r=1 η r W r e iξαrx 3
) e iξ(x 1 -vt)

(3.11)
where, η r is the participation factor of partial wave to be determined, which can be organized into a

vector η = [η 1 , η 2 , η 3 , η 4 , η 5 , η 6 ] T .
Substituting Eq. (3.11) into Eq. (2.22), the PWSA is also applied to the stress tensor as presented in Eq. (3.12).

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ σ 11 = { ∑ 6 r=1 [(C 11 + C 15 α r ) + (C 16 + C 14 α r )V r + (C 15 + C 13 α r )W r ] e iξαrx 3 η r } iξe iξ(x 1 -vt) σ 22 = { ∑ 6 r=1 [(C 12 + C 25 α r ) + (C 26 + C 24 α r )V r + (C 25 + C 23 α r )W r ] e iξαrx 3 η r } iξe iξ(x 1 -vt) σ 33 = { ∑ 6 r=1 [(C 13 + C 35 α r ) + (C 36 + C 34 α r )V r + (C 35 + C 33 α r )W r ] e iξαrx 3 η r } iξe iξ(x 1 -vt) σ 23 = { ∑ 6 r=1 [(C 14 + C 45 α r ) + (C 46 + C 44 α r )V r + (C 45 + C 34 α r )W r ] e iξαrx 3 η r } iξe iξ(x 1 -vt) σ 13 = { ∑ 6 r=1 [(C 15 + C 55 α r ) + (C 56 + C 45 α r )V r + (C 55 + C 35 α r )W r ] e iξαrx 3 η r } iξe iξ(x 1 -vt) σ 12 = { ∑ 6 r=1 [(C 16 + C 56 α r ) + (C 66 + C 46 α r )V r + (C 56 + C 36 α r )W r ] e iξαrx 3 η r } iξe iξ(x 1 -vt) (3.12)
Considering that in the stress tensor, only the three terms (σ 33 , σ 23 , σ 13 ) are useful because they correspond to the boundary conditions (see Eq. (3.2)), the three terms are extracted into Eq. (3.13).

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ σ 33 = ( ∑ 6 r=1 β 1r e iξαrx 3 η r ) iξe iξ(x 1 -vt) σ 23 = ( ∑ 6 r=1 β 2r e iξαrx 3 η r ) iξe iξ(x 1 -vt) σ 13 = ( ∑ 6 r=1 β 3r e iξαrx 3 η r ) iξe iξ(x 1 -vt) (3.13)
where, β ir is the amplitude of partial wave in terms of stress and is defined in Eq. (3.14).

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ β 1r = (C 13 + C 35 α r ) + (C 36 + C 34 α r )V r + (C 35 + C 33 α r )W r β 2r = (C 14 + C 45 α r ) + (C 46 + C 44 α r )V r + (C 45 + C 34 α r )W r β 3r = (C 15 + C 55 α r ) + (C 56 + C 45 α r )V r + (C 55 + C 35 α r )W r (r = 1, • • • , 6) (3
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ η 1 η 2 η 3 η 4 η 5 η 6 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 0 0 0 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (3.15) D(v, ξ) = D R (v, ξ) + iD I (v, ξ) ≜ det{D(v, ξ
)} = 0 for sym. and anti. modes (3.16) Note that D(v, ξ) is a complex-valued matrix due to the existence of complex exponential terms in it. Thus, its determinant D(v, ξ) is a complex number in general and can be separated as real part D R (v, ξ) and imaginary part D I (v, ξ) as indicated in Eq. (3.16).

If the SC is added along with the BC to derive dispersion equation, the option

x 3 = -h in Eq. ( 3
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ η 1 η 2 η 3 η 4 η 5 η 6 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 0 0 0 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (3.17) D s (v, ξ) = D R s (v, ξ) + iD I s (v, ξ) ≜ det{D s (v, ξ)} = 0 for symmetric modes (3.18)
where, the subscript 's' denotes the symmetric modes.

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 1 1 1 1 1 V 1 V 2 V 3 V 4 V 5 V
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ η 1 η 2 η 3 η 4 η 5 η 6 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 0 0 0 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (3.19)

DISPERSION EQUATIONS OF COUPLED LAMB AND SH WAVES IN TRICLINIC MATERIAL

D a (v, ξ) = D R a (v, ξ) + iD I a (v, ξ) ≜ det{D a (v, ξ)} = 0 for anti-symmetric modes (3.20)
where, the subscript 'a' denotes the anti-symmetric modes.

Once the dispersion equation is built regardless of which SC being considered, for example, D(v, ξ) = 0 in Eq. (3.16), it can be solved through some root-finding methods such as Newton-Raphson method or bisection method. From numerical realization perspective, Newton-Raphson method is not suitable for this problem because it requires the derivative information of the characteristic function D(v, ξ) which is not available since D(v, ξ) is transcendental. Thus, bisection method is the desired one. With such method, the solutions can be found by fixing v = v 0 to sweep the roots of ξ (D(v 0 , ξ) = 0) or by fixing ξ = ξ 0 to sweep the roots of v (D(v, ξ 0 ) = 0). The two sweeping schemes will be comprehensively investigated in Chapter 6.

The continuous loci of solutions (v, ξ) form the dispersion curves in v -ξ domain in which each individual branch represents a single modal curve. The dispersion curves in ω -ξ and ω -v domains can be obtained from the relation ω = ξv, where ω is the angular frequency. For (quasi-)isotropic materials, the group velocity c g is computed from each individual branch of dispersion curves based on Eq. (3.21a) in ω-ξ domain or Eq. (3.21b) in ω-v domain. For anisotropic materials, the steering effect should be taken into account when computing group velocity in these materials. The related work was reported in [37]. In the subsequent chapters of the thesis, we directly present the computed dispersion curves of group velocity in anisotropic materials by taking advantage of the approach reported in [37] without a statement.

c g = ∂ω ∂ξ (3.21a) c g = v 2 v -ω ∂v ∂ω (3.21b)

Mode shape

When a solution point (v, ξ) of the symmetry-omitted dispersion equation D(v, ξ) = 0 in Eq. (3.16) is obtained, it can be substituted back in Eq. (3.15) to compute the vector η which corresponds to the zero-eigenvalue correlated eigenvector of the matrix D(v, ξ) in Eq. (3.15). Once all the required quantities are obtained, the displacement and stress fields described in Eqs. (3.11) and (3.13) can be

DISPERSION EQUATIONS OF COUPLED LAMB AND SH WAVES IN MONOCLINIC MATERIAL

reconstructed. At a fixed section of the waveguide, i.e. omitting the phase term e iξ(x 1 -vt) in Eqs. (3.11) and (3.13), the displacement and stress mode shapes of the plate can be portrayed along the plate thickness direction x 3 through Eqs. (3.11) and (3.13), respectively, which are further classified into symmetric and anti-symmetric modes by checking the symmetry condition at the midplane of the plate stated in Eqs. (3.3) and (3.4). Undoubtedly, if the solution point (v, ξ) is substituted into Eq. (3.17) or (3.19), the computed displacement and stress mode shapes will meet the symmetry condition automatically.

Dispersion equations of coupled Lamb and SH waves in monoclinic material

The dispersion equations of monoclinic material is a simpler case of triclinic material, which will be obtained by making a series of simplifications for the equations of triclinic material.

Christoffel equation

For monoclinic material, the following stiffness coefficients are zero in addition to the triclinic

⎡ ⎢ ⎣ K 11 (α) K 12 (α) K 13 (α) K 12 (α) K 22 (α) K 23 (α) K 13 (α) K 23 (α) K 33 (α) ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ U V W ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ 0 0 0 ⎤ ⎥ ⎦ (3.22) { K 11 (α) = C 11 -ρv 2 + C 55 α 2 , K 12 (α) = C 16 + C 45 α 2 , K 13 (α) = (C 13 + C 55 )α, K 22 (α) = C 66 -ρv 2 + C 44 α 2 , K 23 (α) = (C 36 + C 45 )α, K 33 (α) = C 55 -ρv 2 + C 33 α 2 . (3.23)
The singular condition of matrix K(α) leads to det{K(α)} = 0. Expanding this determinant produces a third order polynomial equation in terms of α 2 as expressed in Eq. (3.24).

A 6 α 6 + A 4 α 4 + A 2 α 2 + A 0 = 0 (3.24)
where, the coefficients A 6 , A 4 , A 2 , A 0 are presented in Eq. (3.25), and can be computed from the

DISPERSION EQUATIONS OF COUPLED LAMB AND SH WAVES IN MONOCLINIC MATERIAL

constant elastic coefficients C pq and mass density ρ at a specified phase velocity v = v 0 .

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ A 6 = C 33 C 44 C 55 -C 33 C 2 45 A 4 =(C 44 C 55 -C 2 45 )(C 55 -ρv 2 ) + C 33 C 55 (C 66 -ρv 2 ) + C 33 C 44 (C 11 -ρv 2 ) -2C 16 C 45 C 33 + 2(C 36 + C 45 )(C 13 + C 55 )C 45 -(C 13 + C 55 ) 2 C 44 -(C 45 + C 36 ) 2 C 55 A 2 =C 33 (C 11 -ρv 2 )(C 66 -ρv 2 ) + C 44 (C 11 -ρv 2 )(C 55 -ρv 2 ) + C 55 (C 66 -ρv 2 )(C 55 -ρv 2 ) -(C 11 -ρv 2 )(C 45 + C 36 ) 2 -(C 66 -ρv 2 )(C 13 + C 55 ) 2 -2(C 55 -ρv 2 )C 16 C 45 + 2C 16 (C 45 + C 36 )(C 13 + C 55 ) -C 2 16 C 33 A 0 = [ (C 11 -ρv 2 )(C 66 -ρv 2 ) -C 2 16 ] (C 55 -ρv 2 ) (3.25)
The three α 2 roots can be solved from Eq. (3.24) via Cardano's Formula or polynomial root-finding algorithms. The six α's roots are further obtained by satisfying the conditions in Eq. (3.26).

α 2 = -α 1 , α 4 = -α 3 , α 6 = -α 5 (3.26)

Displacement and stress fields

For each α r (r = 1, • • • , 6), the three displacement amplitudes (U r , V r , W r ) can be solved from Eq. (3.22) by assigning U r = 1, then V r and W r are found from the resulted equation set via Cramer's Rule as presented in Eq. (3.27).

V r = K 11 (α r )K 23 (α r ) -K 12 (α r )K 13 (α r ) K 13 (α r )K 22 (α r ) -K 12 (α r )K 23 (α r ) , W r = K 11 (α r )K 23 (α r ) -K 12 (α r )K 13 (α r ) K 12 (α r )K 33 (α r ) -K 13 (α r )K 23 (α r ) (3.27)
With the negative reciprocity in Eq. (3.26), it is easy to get the relation of Eq. (3.28) if one substitutes Eq. (3.26) and the detailed expression of K ij (α) of Eq. (3.23) into Eq. (3.27).

V 2 = V 1 , V 4 = V 3 , V 6 = V 5 ; W 2 = -W 1 , W 4 = -W 3 , W 6 = -W 5 (3.28)
By applying PWSA, the displacement field of monoclinic material is now presented in Eq. (3.29), which remains the same form as Eq. (3.11) but has different forms of V r and W r thanks to Eq. (3.23).

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ u 1 = ( ∑ 6 r=1 η r e iξαrx 3
)

e iξ(x 1 -vt)

u 2 = ( ∑ 6 r=1 η r V r e iξαrx 3
) e iξ(x 1 -vt)

u 3 = ( ∑ 6 r=1 η r W r e iξαrx 3
) e iξ(x 1 -vt)

(3.29)

DISPERSION EQUATIONS OF COUPLED LAMB AND SH WAVES IN MONOCLINIC MATERIAL

The stress field (σ 33 , σ 23 , σ 13 ) of monoclinic material is expressed in Eq. (3.30) based on PWSA, which keeps the same form as Eq. (3.13) but has different forms of (β 1r , β 2r , β 3r ) as shown in Eq. (3.31).

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ σ 33 = ( ∑ 6 r=1 β 1r e iξαrx 3 η r ) iξe iξ(x 1 -vt) σ 23 = ( ∑ 6 r=1 β 2r e iξαrx 3 η r ) iξe iξ(x 1 -vt) σ 13 = ( ∑ 6 r=1 β 3r e iξαrx 3 η r ) iξe iξ(x 1 -vt) (3.30) ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ β 1r = C 13 + C 36 V r + C 33 α r W r β 2r = C 45 α r + C 44 α r V r + C 45 W r β 3r = C 55 α r + C 45 α r V r + C 55 W r (r = 1, • • • , 6) (3.31)
Like in Eq. (3.28), β ir exhibits some relationships described in Eq. (3.32).

β 12 = β 11 , β 14 = β 13 , β 16 = β 15 β 22 = -β 21 , β 24 = -β 23 , β 26 = -β 25 β 32 = -β 31 , β 34 = -β 33 , β 36 = -β 35 (3.32)

Dispersion equations

Under the condition of Eqs. (3.26)(3.28)(3.32), the linear homogeneous equation set of the symmetric modes of triclinic material in Eq. (3.17) is now changed to Eq. (3.33). Thus, the dispersion equation of symmetric modes of monoclinic material is generated in Eq. (3.34) in implicit form.

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ W 1 -W 1 W 3 -W 3 W 5 -W 5 β 21 -β 21 β 23 -β 23 β 25 -β 25 β 31 -β 31 β 33 -β 33 β 35 -β 35 β 11 e iξα 1 h β 11 e -iξα 1 h β 13 e iξα 3 h β 13 e -iξα 3 h β 15 e iξα 5 h β 15 e -iξα 5 h β 21 e iξα 1 h -β 21 e -iξα 1 h β 23 e iξα 3 h -β 23 e -iξα 3 h β 25 e iξα 5 h -β 25 e -iξα 5 h β 31 e iξα 1 h -β 31 e -iξα 1 h β 33 e iξα 3 h -β 33 e -iξα 3 h β 35 e iξα 5 h -β 35 e -iξα 5 h ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ η 1 η 2 η 3 η 4 η 5 η 6 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 0 0 0 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (3.33) D s (v, ξ) = D R s (v, ξ) + iD I s (v, ξ) ≜ det{D s (v, ξ)} = 0 for symmetric modes (3.34)
where, the subscript 's' denotes the symmetric modes.

For the dispersion equation of anti-symmetric modes of monoclinic material, it can be derived from Eq. (3.19) which belongs to the triclinic material by applying Eqs. (3.26)(3.28)(3.32). Thus, Eq. (3.35)

DISPERSION EQUATIONS OF COUPLED LAMB AND SH WAVES IN MONOCLINIC MATERIAL

is generated and the desired dispersion equation is expressed as implicit form in Eq. (3.36).

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 1 1 1 1 1 V 1 V 1 V 3 V 3 V 5 V 5 β 11 β 11 β 13 β 13 β 15 β 15 β 11 e iξα 1 h β 11 e -iξα 1 h β 13 e iξα 3 h β 13 e -iξα 3 h β 15 e iξα 5 h β 15 e -iξα 5 h β 21 e iξα 1 h -β 21 e -iξα 1 h β 23 e iξα 3 h -β 23 e -iξα 3 h β 25 e iξα 5 h -β 25 e -iξα 5 h β 31 e iξα 1 h -β 31 e -iξα 1 h β 33 e iξα 3 h -β 33 e -iξα 3 h β 35 e iξα 5 h -β 35 e -iξα 5 h ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ η 1 η 2 η 3 η 4 η 5 η 6 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 0 0 0 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (3.35) D a (v, ξ) = D R a (v, ξ) + iD I a (v, ξ) ≜ det{D a (v, ξ)} = 0 for anti-symmetric modes (3.36)
where, the subscript 'a' denotes the anti-symmetric modes.

To facilitate computation, the explicit3 formula of the determinant of D s (v, ξ) in Eq. due to its powerful capability of symbolic computing. Note that during manipulation using Mathematica, the Euler's equation should be used, for example, e iξα 1 h = cos(ξα 1 h) + i sin(ξα 1 h).

D s (v, ξ) =[B 1 cos(ξα 1 h) sin(ξα 3 h) sin(ξα 5 h) + B 2 sin(ξα 1 h) cos(ξα 3 h) sin(ξα 5 h) + B 3 sin(ξα 1 h) sin(ξα 3 h) cos(ξα 5 h)]B 4 (3.37a) D a (v, ξ) =[B 1 sin(ξα 1 h) cos(ξα 3 h) cos(ξα 5 h) + B 2 cos(ξα 1 h) sin(ξα 3 h) cos(ξα 5 h) + B 3 cos(ξα 1 h) cos(ξα 3 h) sin(ξα 5 h)]B 5 (3.37b)
where,

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ B 1 = β 11 (β 25 β 33 -β 23 β 35 ), B 2 = β 13 (β 21 β 35 -β 25 β 31 ), B 3 = β 15 (β 23 β 31 -β 21 β 33 ) B 4 = 8 [W 1 (β 25 β 33 -β 23 β 35 ) + W 3 (β 21 β 35 -β 25 β 31 ) + W 5 (β 23 β 31 -β 21 β 33 )] B 5 = 8i [V 1 (β 15 -β 13 ) + V 3 (β 11 -β 15 ) + V 5 (β 13 -β 11 )] (3.38)
With Eqs. (3.37a) and (3.37b), the implicit dispersion equations, Eqs. (3.34) and (3.36), of the monoclinic material have been exposed in Eqs. (3.39a) and (3.39b), which are classically presented in many textbooks [START_REF] Nayfeh | Wave propagation in layered anisotropic media: With application to composites[END_REF][START_REF] Lammering | Lamb-wave based structural health monitoring in polymer composites[END_REF]. Considering that the tangent functions are discontinuous, it is preferred to adopt Eqs. (3.37a) and (3.37b) for computation.

B 1 cot(ξα 1 h) + B 2 cot(ξα 3 h) + B 3 cot(ξα 5 h) = 0 for symmetric modes (3.39a) B 1 tan(ξα 1 h) + B 2 tan(ξα 3 h) + B 3 tan(ξα 5 h) = 0 for anti-symmetric modes (3.39b)

DISPERSION EQUATIONS OF DECOUPLED LAMB AND SH WAVES IN ORTHOTROPIC MATERIAL

Once a solution point (v, ξ) of a certain dispersion equation, specifically D s (v, ξ) = 0 in Eq. (3.34) or D a (v, ξ) = 0 in Eq. (3.36), is found, the displacement and stress mode shapes of monoclinic material can be obtained easily by following the same strategy as triclinic material stated in section 3.2.4.

Note that due to the complexity of triclinic materials, there is not a simple explicit form of the dispersion equations in Section 3.2.3 like for monoclinic materials presented in this section.

Dispersion equations of decoupled Lamb and SH waves in orthotropic material

In orthotropic plate, guided waves will be decoupled into Lamb and SH waves, which will be demonstrated in this section.

Christoffel equation

For orthotropic material, the following stiffness coefficients are zero in addition to the mono-

⎡ ⎢ ⎣ C 11 -ρv 2 + C 55 α 2 0 (C 13 + C 55 )α 0 C 66 -ρv 2 + C 44 α 2 0 (C 13 + C 55 )α 0 C 55 -ρv 2 + C 33 α 2 ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ U V W ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ 0 0 0 ⎤ ⎥ ⎦ (
(C 66 -ρv 2 + C 44 α 2 )V = 0 (3.41) [ C 11 -ρv 2 + C 55 α 2 (C 13 + C 55 )α (C 13 + C 55 )α C 55 -ρv 2 + C 33 α 2 ] [ U W ] = [ 0 0 ] (3.42)
In the next subsections, the dispersion equations of Lamb and SH waves will be derived separately.

DISPERSION EQUATIONS OF DECOUPLED LAMB AND SH WAVES IN ORTHOTROPIC MATERIAL

Displacement and stress fields of Lamb wave

The nontrivial solution of [U 1 , U 3 ] T in Eq. (3.42) requires the singular condition of the coefficient matrix as presented in Eq. (3.43). It is the possibility to expand the determinant results in a quadratic equation in terms of α 2 in Eq. (3.44) and the polynomial coefficients are listed in Eq. (3.45).

⃓ ⃓ ⃓ ⃓ ⃓ C 11 -ρv 2 + C 55 α 2 (C 13 + C 55 )α (C 13 + C 55 )α C 55 -ρv 2 + C 33 α 2 ⃓ ⃓ ⃓ ⃓ ⃓ = 0 (3.43) A 4 α 4 + A 2 α 2 + A 0 = 0 (3.44) ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ A 4 = C 33 C 55 A 2 = (C 11 -ρv 2 )C 33 + (C 55 -ρv 2 )C 55 -(C 13 + C 55 ) 2 A 0 = (C 11 -ρv 2 )(C 55 -ρv 2 ) (3.45)
From Eq. (3.44), the four α's roots can be easily obtained that have the negative reciprocity

α 2 = -α 1 , α 4 = -α 3 .
For each α r (r = 1, 2, 3, 4), the displacement amplitude vector [U r , W r ] T can be solved from Eq. (3.42) by letting U r = 1, and W r is calculated using the first row of Eq. (3.42).

W r = ρv 2 -C 11 -C 55 α 2 r (C 13 + C 55 )α r (r = 1, 2, 3, 4) (3.46)
It is easy to get the relation

W 2 = -W 1 and W 4 = -W 3 under the fact α 2 = -α 1 , α 4 = -α 3 .
For the displacement field equations of Lamb wave, the four α's solutions manifest that there are four partial waves to compose Lamb wave with two longitudinal waves and two shear vertical waves.

⎧ ⎨ ⎩ u 1 = ( ∑ 4 r=1 η r e iξαrx 3
) e iξ(x 1 -vt)

u 3 = ( ∑ 4 r=1 η r W r e iξαrx 3
)

e iξ(x 1 -vt) (3.47)
where, η r (r = 1, 2, 3, 4) is the participation factors of partial waves and they can be organized into a

vector η = [η 1 , η 2 , η 3 , η 4 ] T .
Unlike Eqs. (3.13)(3.30), the stress field of Lamb wave only involves in σ 33 and σ 13 terms which are stated in Eq. (3.48) based on PWSA.

⎧ ⎨ ⎩ σ 33 = ( ∑ 4 r=1 β 1r η r e iξαrx 3 ) iξe iξ(x 1 -vt) σ 31 = ( ∑ 4 r=1 β 3r η r e iξαrx 3 ) iξe iξ(x 1 -vt) (3.48)

DISPERSION EQUATIONS OF DECOUPLED LAMB AND SH WAVES IN ORTHOTROPIC MATERIAL

where, the stress amplitude of partial wave β ir is derived into Eq. (3.49), from which the relationship among β ir can be easily obtained:

β 12 = β 11 , β 14 = β 13 , β 32 = -β 31 , β 34 = -β 33 .
{

β 1r = C 13 + C 33 α r W r β 3r = C 55 α r + C 55 W r (r = 1, 2, 3, 4) (3.49)

Dispersion equations of Lamb wave

For the displacement and stress field equations shown in Eqs.

⎡ ⎢ ⎢ ⎢ ⎣ W 1 -W 1 W 3 -W 3 β 31 -β 31 β 33 -β 33 β 11 e iξα 1 h β 11 e -iξα 1 h β 13 e iξα 3 h β 13 e -iξα 3 h β 31 e iξα 1 h -β 31 e -iξα 1 h β 33 e iξα 3 h -β 33 e -iξα 3 h ⎤ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎣ η 1 η 2 η 3 η 4 ⎤ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎣ 0 0 0 0 ⎤ ⎥ ⎥ ⎥ ⎦ (3.50) D s (v, ξ) = D R s (v, ξ) + iD I s (v, ξ) ≜ det{D s (v, ξ)} = 0 for symmetric modes (3.51)
By using the software Mathematica, the determinant of the coefficient matrix D s (v, ξ) in Eq. (3.50) is simplified into Eq. (3.52).

det{D s (v, ξ)} = -4i(W 3 β 31 -W 1 β 33 ) [cos(ξα 3 h) sin(ξα 1 h)β 13 β 31 -cos(ξα 1 h) sin(ξα 3 h)β 11 β 33 ] (3.52)
Upon simplification, the Lamb wave dispersion equation of symmetric modes for a single-layered orthotropic plate is explicitly stated in Eq. (3.53), which is also presented in [START_REF] Nayfeh | Wave propagation in layered anisotropic media: With application to composites[END_REF].

tan(ξα 1 h) tan(ξα 3 h) = β
⎡ ⎢ ⎢ ⎢ ⎣ 1 1 1 1 β 11 β 11 β 13 β 13 β 11 e iξα 1 h β 11 e -iξα 1 h β 13 e iξα 3 h β 13 e -iξα 3 h β 31 e iξα 1 h -β 31 e -iξα 1 h β 33 e iξα 3 h -β 33 e -iξα 3 h ⎤ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎣ η 1 η 2 η 3 η 4 ⎤ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎣ 0 0 0 0 ⎤ ⎥ ⎥ ⎥ ⎦ (3.54)

DISPERSION EQUATIONS OF DECOUPLED LAMB AND SH WAVES IN ORTHOTROPIC MATERIAL

D a (v, ξ) = D R a (v, ξ) + iD I a (v, ξ) ≜ det{D a (v, ξ)} = 0 for anti-symmetric modes (3.55)
With the software Mathematica, the determinant of the coefficient matrix D a (v, ξ) in Eq. (3.54) is simplified into Eq. (3.56).

det{D a (v, ξ)} = 4i(β 11 -β 13 ) [-cos(ξα 1 h) sin(ξα 3 h)β 13 β 31 + cos(ξα 3 h) sin(ξα 1 h)β 11 β 33 ] (3.56)
Upon simplification, the Lamb wave dispersion equation of anti-symmetric modes for a singlelayered orthotropic plate is explicitly stated in Eq. (3.57), still presented in [START_REF] Nayfeh | Wave propagation in layered anisotropic media: With application to composites[END_REF].

tan(ξα 1 h) tan(ξα 3 h) = β
α 2 = ρv 2 -C 66 C 44 ⇒ α 5 = √ ρv 2 -C 66 C 44 , α 6 = - √ ρv 2 -C 66 C 44 (3.58) u 2 = ( 6 ∑ r=5 η r e iξαrx 3
) e iξ(x 1 -vt) (3.59)

σ 23 = ( 6 ∑ r=5 β 2r η r e iξαrx 3 ) iξe iξ(x 1 -vt) (3.60)
where,

α 6 = -α 5 , β 25 = C 44 α 5 , β 26 = C 44 α 6 .

Dispersion relation of SH wave

For deriving dispersion equation of SH wave, only the traction-free boundary condition of Eq. (3.2)

(only σ 23 is evaluated) can be applied. ] [

σ 23 | x 3 =±h = 0 ⇒ { α 5 e -iξα 5 h η 5 + α 6 e -iξα
η 5 η 6 ] = [ 0 0 ] ⇒ ⃓ ⃓ ⃓ ⃓ ⃓ α 5 e -iξα 5 h α 6 e -iξα 6 h α 5 e iξα 5 h α 6 e iξα 6 h ⃓ ⃓ ⃓ ⃓ ⃓ = 0 (3.62)
Expand the determinant in Eq. (3.62) to get Eq. (3.63),

e iξ(α 5 -α 6 )h = e iξ(α 6 -α 5 )h α 6 =-α 5 , d=2h ---------→ cos(ξα 5 d) + i sin(ξα 5 d) = cos(ξα 5 d) -i sin(ξα 5 d) (3.63) sin(ξα 5 d) = 0 ⇒ ξα 5 d = nπ ⇒ ξd √ ρv 2 -C 66 C 44 = nπ (n = 0, 1, 2, . . .) (3.64)
The dispersion relation of SH wave linking phase velocity v and wavenumber ξ can be obtained in Eq. (3.65).

v = √ C 66 ρ + C 44 ρ ( nπ ξd ) 2 (n = 0, 1, 2, . . .) (3.65)
Or substitute ξ = ω v into Eq. (3.65) to get another form of Eq. (3.66) linking v and ω. to get the corresponding dispersion equations of the two materials, respectively.

v = √ d 2 ω 2 C 66 ρd 2 ω 2 -C 44 n 2 π 2 (n = 0, 1, 2, . . .) (3.

DERIVATION OF THE CLASSICAL RAYLEIGH-LAMB EQUATION VIA PWSA

Derivation of the classical Rayleigh-Lamb equation via PWSA

The classical Rayleigh-Lamb equation is originally derived through the Helmholtz decomposition method [26]. In this section, we dedicate to retrieve it by following Eqs.

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ A 4 = (λ + 2µ)µ A 2 = [ (λ + 2µ) -ρv 2 ] (λ + 2µ) + (µ -ρv 2 )µ -(λ + µ) 2 A 0 = [ (λ + 2µ) -ρv 2 ] (µ -ρv 2 ) (3.68)
The discriminant of the quadratic equation on α 2 in Eq. (3.44) can be derived easily via the software Mathematica.

∆ = A 2 2 -4A 4 A 0 = (λ + µ) 2 ρ 2 v 4 (3.69)
The two solutions about α 2 can be further obtained for which the smaller one is assigned as α 2

L and the larger one

α 2 T . ⎧ ⎨ ⎩ α 2 L = α 2 1 = -A 2 - √ ∆ 2A 4 = ρv 2 -(λ+2µ) λ+2µ = v 2 c 2 L -1 α 2 T = α 2 3 = -A 2 + √ ∆ 2A 4 = ρv 2 -µ µ = v 2 c 2 T -1 (3.
W L = W 1 = ρv 2 -C 11 -C 55 α 2 L (C 13 + C 55 )α L = ρv 2 -(λ + 2µ) -µ ρv 2 -(λ+2µ) λ+2µ (λ + µ)α L = ρv 2 -(λ + 2µ) (λ + 2µ)α L = v 2 -c 2 L c 2 L α L = ( v 2 c 2 L -1 ) 1 α L = α 2 L α L = α L (3.71) W T = W 3 = ρv 2 -C 11 -C 55 α 2 T (C 13 + C 55 )α T = ρv 2 -(λ + 2µ) -µ ρv 2 -µ µ (λ + µ)α T = -(λ + µ) (λ + µ)α T = - 1 α T ( 3
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ β 11 = λ + (λ + 2µ)α L W L = λ + (λ + 2µ)α 2 L = λ + (λ + 2µ) ρv 2 -(λ+2µ) λ+2µ = ρv 2 -2µ β 31 = µα L + µW L = µα L + µα L = 2µα L β 13 = λ + (λ + 2µ)α T W T = λ + (λ + 2µ)α T -1 α T = -2µ β 33 = µα T + µW T = µα T + µ -1 α T = µ α 2 T -1 α T (3.73)
The right hand side of Eq. (3.53) can be computed by substituting Eq. ( 3.73) into it. For symmetric modes,

β 11 β 33 β 13 β 31 = (ρv 2 -2µ)µ α 2 T -1 α T (-2µ)(2µα L ) = (ρv 2 -2µ)(α 2 T -1) -4µα L α T = (v 2 -2 µ ρ )(α 2 T -1) -4 µ ρ α L α T = (v 2 -2c 2 T )(α 2 T -1) -4c 2 T α L α T = ( v 2 c 2 T -2)(α 2 T -1) -4α L α T = (α 2 T -1)(α 2 T -1) -4α L α T = (α 2 T -1) 2 -4α L α T (3.74) Let p = ξα L , q = ξα T .
tan(ph) tan(qh) = (α 2 T -1) 2 -4α L α T = (α 2 T -1) 2 ξ 4 -4α L α T ξ 4 = [ (ξα T ) 2 -ξ 2 ] 2 -4ξ 2 (ξα L )(ξα T ) = (q 2 -ξ 2 ) 2 -4ξ 2 pq (3.75)
For anti-symmetric modes,

tan(ph) tan(qh) = -4α L α T (α 2 T -1) 2 = -4ξ 2 pq (q 2 -ξ 2 ) 2 (3.76)
where,

p 2 = ω 2 c 2 L -ξ 2 , q 2 = ω 2 c 2 T -ξ 2 .
Eventually, the classical Rayleigh-Lamb equation is retrieved.

Conclusion

PWSA is a unified framework to derive dispersion equations of guided waves propagating in a single-layered composite lamina which integrates simultaneously the traction-free boundary condition and symmetry condition. In orthotropic, transversely isotropic and isotropic materials, guided waves will be decoupled into Lamb and SH waves, whereas in triclinic and monoclinic materials, both wave types are coupled to each other. The classical Rayleigh-Lamb equation is retrieved by using PWSA, which serves as an alternative to the Helmholtz decomposition method.

Chapter 4

Guided wave propagation in a multi-layered plate: transfer matrix method

This chapter comprehensively studies the various properties of guided waves propagation in a multi-layered composite plate through the classical transfer matrix method (TMM) [2,[START_REF] Nandyala | Effective stiffness matrix method for predicting the dispersion curves in general anisotropic composites[END_REF][START_REF] Nayfeh | Wave propagation in layered anisotropic media: With application to composites[END_REF]38]. The core contribution of this chapter to the state-of-the-art is that a hybrid matrix strategy is originally proposed to address the matrix incompatibility issue that is usually encountered for arbitrarily oriented composite laminates. The hybrid matrix strategy is theoretically derived by following the PWSA of Chapter 3 and is numerically assessed for three commonly used composite laminates. In the laminate, each lamina is reinforced by the carbon fibers within the composing matrix, thus producing orthotropic material property from macroscopical view. In the lamina plane, the fiber direction and transverse fiber direction are defined as the principal and minor-principal direction of the lamina, respectively. Naturally, all other directions are defined as its off-principal direction. The perfect interface condition between adjacent layers is assumed and thus leads to the continuity of field variables (displacement and stress) across all the interfaces, otherwise delamination problem [START_REF] Zhu | Waves in a generally anisotropic viscoelastic composite laminated bilayer: Impact of the imperfect interface from perfect to complete delamination[END_REF] occurs which is not the concern of the thesis.

The multi-layered composite plate model

Guided waves propagation in a plate waveguide emanating from a point source is at arbitrary angles. The wave front behaves like a circular shape, see In Sections 3.3 and 3.4, we have studied that C O and C ′ O (or C M ) will lead to different coupling properties of Lamb and SH waves in a single lamina. Thus, in a laminate system, the mismatch issue between wave propagation direction and the fiber (principal) direction of a lamina will lead to different wave types among laminae. As a consequence, to model guided waves propagation in a unified multi-layered plate system, how to unite the various wave types in different laminae becomes an essential issue, which can be tacked through TMM (in this chapter) and GMM (in the next chapter) frameworks. In the following section, the displacement and stress equations of a specific wave type will be reorganized into a series of local lamina matrices which lay the foundation of TMM and GMM.

Then, these local matrices are transferred into a monodromy one based on the TMM framework.

Finally, the dispersion equation of the laminate system is generated and solving this equation just offers various wave solutions.

LOCAL LAMINA MATRICES ALONG THE WAVE PROPAGATION DIRECTION

Local lamina matrices along the wave propagation direction

We consider a general n-layered laminate with various stacking angles of layups

[ϕ 1 /ϕ 2 / • • • /ϕ n ]
and the interested wave propagation direction at angle θ, as illustrated in

C ′ k = Rot{C k , θ -ϕ k },
where C k is the stiffness matrix of layer l k in its principal direction, and the operator Rot{•, •} is defined in Eq. (2.18). According to the study in Sections 3.3 and 3.4, the different matrix type of C ′ k will lead to different wave types in layer l k . If C ′ k is monoclinic type, the resulted waves are the coupled Lamb and SH waves in layer l k (see Section 3.3); if C ′ k is orthotropic type, the guided waves are decoupled into pure Lamb and SH waves in layer l k (see Section 3.4). Thus, in a multi-layered plate system, the wave type present in each layer should be determined separately.

Local lamina matrices of coupled Lamb and SH waves

When the interested wave propagation direction is along the off-principle direction of a general layer l k , the rotated stiffness matrix of this layer C ′ k ∈ C M produces the coupled Lamb and SH waves in this layer. In this case, the three displacement quantities (u 1 , u 2 , u 3 ) should be simultaneously solved, whose expressions are presented in Eq. (3.29). Rewrite it as the matrix form in Eq. (4.1) and the compact form in Eq. (4.2) is obtained.

LOCAL LAMINA MATRICES ALONG THE WAVE PROPAGATION DIRECTION

⎡ ⎢ ⎣ u 1 u 2 u 3 ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ 1 1 1 1 1 1 V 1 V 2 V 3 V 4 V 5 V 6 W 1 W 2 W 3 W 4 W 5 W 6 ⎤ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ e iξα 1 x 3 e iξα 2 x 3 e iξα 3 x 3 e iξα 4 x 3 e iξα 5 x 3 e iξα 6 x 3 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ η 1 η 2 η 3 η 4 η 5 η 6 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ e iξ(x 1 -vt) (4.1) u 3×1 = U 3×6 Λ 6×6 (x 3 )η 6×1 e iξ(x 1 -vt) (4.2)
where, u 3×1 , U 3×6 , Λ 6×6 (x 3 ) and η 6×1 are defined as follows. The subscript "m × n" designates the matrix dimension.

u 3×1 = [u 1 , u 2 , u 3 ] T (4.3)
U 3×6 = ⎡ ⎢ ⎣ 1 1 1 1 1 1 V 1 V 2 V 3 V 4 V 5 V 6 W 1 W 2 W 3 W 4 W 5 W 6 ⎤ ⎥ ⎦ (4.4) Λ 6×6 (x 3 ) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ e iξα 1 x 3 e iξα 2 x 3 e iξα 3 x 3 e iξα 4 x 3 e iξα 5 x 3 e iξα 6 x 3 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (4.5) η 6×1 = [η 1 , η 2 , η 3 , η 4 , η 5 , η 6 ] T (4.6)
Correspondingly, the three stress quantities (σ 33 , σ 23 , σ 13 ) presented in Eq. (3.30) are rewritten as the matrix form in Eq. (4.7) and the compact form in Eq. (4.8) by suppressing the common factor iξ since this factor has no effect on the traction-free boundary condition to be evaluated in Section 4.3

for the TMM framework.

LOCAL LAMINA MATRICES ALONG THE WAVE PROPAGATION DIRECTION

⎡ ⎢ ⎣ σ 33 σ 23 σ 13 ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ β 11 β
⎤ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ e iξα 1 x 3 e iξα 2 x 3 e iξα 3 x 3 e iξα 4 x 3 e iξα 5 x 3 e iξα 6 x 3 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ η 1 η 2 η 3 η 4 η 5 η 6 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ e iξ(x 1 -vt) (4.7) σ 3×1 = β 3×6 Λ 6×6 (x 3 )η 6×1 e iξ(x 1 -vt) (4.8)
where, σ 3×1 and β 3×6 are defined as follows.

σ 3×1 = [σ 33 , σ 23 , σ 13 ] T (4.9) β 3×6 = ⎡ ⎢ ⎣ β 11 β
Eqs. (4.1) and (4.7) can be concatenated into a single equation. Then, we have the matrix form in Eq. (4.11) and the compact form in Eq. (4.12).

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ u 1 u 2 u 3 σ 33 σ 23 σ 13 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 1 1 1 1 1 V 1 V 2 V 3 V 4 V 5 V 6 W 1 W 2 W 3 W 4 W 5 W 6 β 11 β 12 β 13 β 14 β 15 β 16 β 21 β 22 β 23 β 24 β 25 β 26 β 31 β 32 β 33 β 34 β 35 β 36 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ e iξα 1 x 3 e iξα 2 x 3 e iξα 3 x 3 e iξα 4 x 3 e iξα 5 x 3 e iξα 6 x 3 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ η 1 η 2 η 3 η 4 η 5 η 6 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ e iξ(x 1 -vt) (4.11) S 6×1 = Γ 6×6 Λ 6×6 (x 3 )η 6×1 e iξ(x 1 -vt) = Z 6×6 (x 3 )η 6×1 e iξ(x 1 -vt) (4.12)
where, S 6×1 = [u 1 , u 2 , u 3 , σ 33 , σ 23 , σ 13 ] T represents the field variables of the coupled Lamb and SH waves in the layer l k ; Z 6×6 (x 3 ) = Γ 6×6 Λ 6×6 (x 3 ) and Γ 6×6 is stated in Eq. (4.13).

It should be noted that those matrices, Γ 6×6 , Λ 6×6 (x 3 ), Z 6×6 (x 3 ), only depend on the lamina properties of layer l k . Thus, they are uniformly defined as the local lamina matrices. This terminology is applicable to the subsequent subsections.

LOCAL LAMINA MATRICES ALONG THE WAVE PROPAGATION DIRECTION

Γ 6×6 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 1 1 1 1 1 V 1 V 2 V 3 V 4 V 5 V 6 W 1 W 2 W 3 W 4 W 5 W 6 β 11 β 12 β 13 β 14 β 15 β 16 β 21 β 22 β 23 β 24 β 25 β 26 β 31 β 32 β 33 β 34 β 35 β 36 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (4.13)

Local lamina matrices of pure Lamb waves

When the interested wave propagation direction is just along the (minor-)principal direction of layer l k , the rotated stiffness matrix of this layer C ′ k ∈ C O according to the Characteristic 2 of Section 2.3.3 leads to the decoupled Lamb and SH waves in this layer. The current subsection aims to derive the local lamina matrices of pure Lamb waves and the next subsection pure SH waves. For pure Lamb waves, only the two displacement quantities (u 1 , u 3 ) are involved, whose expressions are presented in Eq. (3.47). Rewrite it as the matrix form in Eq. (4.14) and the compact form in Eq. (4.15) is obtained.

[ u 1 u 3 ] = [ 1 1 1 1 W 1 W 2 W 3 W 4 ] ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ e iξα 1 x 3 e iξα 2 x 3 e iξα 3 x 3 e iξα 4 x 3 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎣ η 1 η 2 η 3 η 4 ⎤ ⎥ ⎥ ⎥ ⎦ e iξ(x 1 -vt) (4.14) u 2×1 = U 2×4 Λ 4×4 (x 3 )η 4×1 e iξ(x 1 -vt) (4.15)
The stress terms regarding to pure Lamb waves are (σ 33 , σ 13 ) which are presented in Eq. (3.48) and are rewritten as the matrix form in Eq. (4.16) and the compact form in Eq. (4.17) by suppressing the common factor iξ.

[ and the compact form in Eq. (4.19).

σ 33 σ 13 ] = [ β 11 β 12 β 13 β 14 β 31 β 32 β 33 β 34 ] ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ e iξα 1 x 3 e iξα 2 x 3 e iξα 3 x 3 e iξα 4 x 3 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎣ η 1 η 2 η 3 η 4 ⎤ ⎥ ⎥ ⎥ ⎦ e iξ(x 1 -vt) (4.16) σ 2×1 = β 2×4 Λ 4×4 (x 3 )η 4×1 e iξ(x 1 -vt) (4.

LOCAL LAMINA MATRICES ALONG THE WAVE PROPAGATION DIRECTION

⎡ ⎢ ⎢ ⎢ ⎣ u 1 u 3 σ 33 σ 13 ⎤ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎣ 1 1 1 1 W 1 W 2 W 3 W 4 β 11 β 12 β 13 β 14 β 31 β 32 β 33 β 34 ⎤ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ e iξα 1 x 3 e iξα 2 x 3 e iξα 3 x 3 e iξα 4 x 3 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎣ η 1 η 2 η 3 η 4 ⎤ ⎥ ⎥ ⎥ ⎦ e iξ(x 1 -vt) (4.18) S 4×1 = Γ 4×4 Λ 4×4 (x 3 )η 4×1 e iξ(x 1 -vt) = Z 4×4 (x 3 )η 4×1 e iξ(x 1 -vt) (4.19)
The local lamina matrices, Γ 4×4 , Λ 4×4 (x 3 ), Z 4×4 (x 3 ), of the pure Lamb waves in layer l k have been obtained according to Eq. (4.19).

Local lamina matrices of pure SH waves

The current subsection continues to derive the local lamina matrices of pure SH waves in layer l k .

The involved displacement is only u 2 , whose expression is presented in Eq. (3.59). It can be rewritten as the matrix form in Eq. (4.20) and the compact form in Eq. (4.21).

u 2 = [ 1 1 ] [ e iξα 5 x 3 e iξα 6 x 3 ] [ η 5 η 6
] e iξ(x 1 -vt) (4.20)

u 1×1 = U 1×2 Λ 2×2 (x 3 )η 2×1 e iξ(x 1 -vt) (4.21)
The stress term regarding pure SH waves is σ 23 which is presented in Eq. (3.60) and is rewritten as the matrix form in Eq. (4.22) and the compact form in Eq. (4.23) by suppressing the common factor iξ.

σ 23 = [ β 25 β 26 ] [ e iξα 5 x 3 e iξα 6 x 3 ] [ η 5 η 6 ] e iξ(x 1 -vt) (4.22) σ 1×1 = β 1×2 Λ 2×2 (x 3 )η 2×1 e iξ(x 1 -vt) (4.23)
Eqs. (4.20) and (4.22) can be concatenated into a single equation to form the field variables.

[

u 2 σ 23 ] = [ 1 1 β 25 β 26 ] [ e iξα 5 x 3 e iξα 6 x 3 ] [ η 5 η 6
] e iξ(x 1 -vt) (4.24)

S 2×1 = Γ 2×2 Λ 2×2 (x 3 )η 2×1 e iξ(x 1 -vt) = Z 2×2 (x 3 )η 2×1 e iξ(x 1 -vt) (4.25)
The local lamina matrices, Γ 2×2 , Λ 2×2 (x 3 ), Z 2×2 (x 3 ), of the pure SH waves in layer l k have been obtained based on Eq. (4.25).

TRANSFER MATRIX METHOD

Transfer matrix method

In this section, the local lamina matrices corresponding to each wave type will be transferred into a monodromy one under the TMM framework by applying the continuity condition.

Local transfer matrix in a lamina

The

S l k top = S l k | x 3 =0 = Γ l k Λ l k (0)η l k e iξ(x 1 -vt) = Γ l k η l k e iξ(x 1 -vt) (4.26) S l k bot = S l k | x 3 =d k = Γ l k Λ l k (d k )η l k e iξ(x 1 -vt) (4.27)
where, the superscript 'l k ' denotes that the associated terms belong to a general layer l k ; the subscript 'bot' and 'top' represents bottom and top side, respectively; d k is the thickness of layer l k . Note that in Eq. (4.26), Λ l k (0) = I, where I is the identity matrix.

The common term η l k e iξ(x 1 -vt) in Eqs. (4.26) and (4.27) can be eliminated and thus the field variables of layer l k at its top and bottom sides can be linked as shown in Eq. (4.28).

S l k bot = T l k S l k top (4.28)
where, T l k is stated in Eq. (4.29) and is naturally defined as the local transfer matrix of layer l k .

T l k = Γ l k Λ l k (d k ) ( Γ l k ) -1 ∈ C p×p (4.29)
where, C p×p represents the set of complex matrix with dimension p × p and p is the number of partial waves, specifically, for the coupled Lamb and SH waves in Section (4. The transferring rule:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ S l 1 bot = T l 1 S l 1 top S l 2 bot = T l 2 S l 2 top . . . S ln bot = T ln S ln top (4.30)
On the other hand, the continuity condition (CC) at each interface as illustrated in Figure 4.2 should be satisfied. Thus, we have Eq. (4.31).

The continuity condition:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ S l 2 top = S l 1 bot S l 3 top = S l 2 bot . . . S ln top = S l n-1 bot (4.31)
S ln bot in the last equation of Eq. (4.30) can be consecutively transferred to S l 1 top by recurrently substituting Eq. (4.31) into Eq. (4.30). This process is illustrated in Eq. (4.32).

S ln bot = T ln S ln top = T ln T l n-1 S l n-1 top . . . = T ln T l n-1 T l n-2 • • • T l 2 T l 1 S l 1 top (4.32)
Thus, the field variables at the bottom-and top-most surfaces of the laminate are linked with a monodromy matrix T mono as stated in Eq. (4.33) which characterizes the transfer matrix of the laminate system.

S ln bot = T mono S l 1 top (4.33)

T mono = T ln T l n-1 T l n-2 . . . T l 2 T l 1 (4.34)
It is clear that T mono ∈ C p×p has the same dimension as T l k (k = 1, . . . , n).

Dispersion equation

The monodromy transfer matrix of the laminate is then used to derive the dispersion equation.

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ u 1 u 2 u 3 σ 33 σ 23 σ 13 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ln bot = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ u 1 u 2 u 3 σ 33 σ 23 σ 13 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ l 1 top (4.35)
The traction-free boundary condition at the bottom-and top-most surfaces of the multi-layered plate system is presented in Eq. (4.36), see Figure 4.2(b) for the illustration.

The boundary condition:

⎡ ⎢ ⎣ σ 33 σ 23 σ 13 ⎤ ⎥ ⎦ ln bot = ⎡ ⎢ ⎣ σ 33 σ 23 σ 13 ⎤ ⎥ ⎦ l 1 top = ⎡ ⎢ ⎣ 0 0 0 ⎤ ⎥ ⎦ (4.36)
Substituting Eq. (4.36) into Eq. (4.35) leads to Eq. (4.37).

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ u 1 u 2 u 3 0 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ln bot = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ u 1 u 2 u 3 0 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ l 1 top (4.37)
For the purpose of deriving dispersion equation through TMM, only the bottom left corner of T mono is useful because this submatrix relates to the boundary condition. Extracting out this submatrix from Eq. (4.37) leads to Eq. (4.38).

⎡ ⎢ ⎣ 0 0 0 ⎤ ⎥ ⎦ ln bot = ⎡ ⎢ ⎣ T 41 T 42 T 43 T 51 T 52 T 53 T 61 T 62 T 63 ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ u 1 u 2 u 3 ⎤ ⎥ ⎦ l 1 top (4.38)
Note that the displacement in the top-most surface, u l 1 top , should be non-trivial, thus the bottom left corner of T mono in Eq. (4.38) should have a vanishing determinant. This finally produces the dispersion equation of the coupled Lamb and SH waves linking v and ξ as presented in Eq. (4.39).

TRANSFER MATRIX METHOD

D T (v, ξ) = D R T (v, ξ) + iD I T (v, ξ) ≜ ⃓ ⃓ ⃓ ⃓ ⃓ ⃓ ⃓ T 41 T 42 T 43 T 51 T 52 T 53 T 61 T 62 T 63 ⃓ ⃓ ⃓ ⃓ ⃓ ⃓ ⃓ = 0 (4.39)
where, the subscript 'T' denotes that the dispersion equation is built with TMM.

2. The pure Lamb waves for

T mono ∈ C 4×4 When T mono ∈ C 4×4 ,
⎤ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎣ u 1 u 3 σ 33 σ 13 ⎤ ⎥ ⎥ ⎥ ⎦ l 1 top (4.40)
By applying the traction-free boundary condition at the bottom-and top-most surfaces of the plate,

[ σ 33 σ 13 ] ln bot = [ σ 33 σ 13 ] l 1 top = [ 0 0 ]
, Eq. (4.40) becomes Eq. (4.41).

⎡ ⎢ ⎢ ⎢ ⎣ u 1 u 3 0 0 ⎤ ⎥ ⎥ ⎥ ⎦ ln bot = ⎡ ⎢ ⎢ ⎢ ⎣ T 11 T 12
⎤ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎣ u 1 u 3 0 0 ⎤ ⎥ ⎥ ⎥ ⎦ l 1 top (4.41)
The requirement of the non-trivial displacement in the top-most surface,

[ u 1 u 3 ] l 1 top ̸ = [ 0 0 ] , finally
produces the dispersion equation of the pure Lamb waves linking v and ξ as presented in Eq. (4.42).

D T (v, ξ) = D R T (v, ξ) + iD I T (v, ξ) ≜ ⃓ ⃓ ⃓ ⃓ ⃓ T 31 T 32 T 41 T 42 ⃓ ⃓ ⃓ ⃓ ⃓ = 0 (4.42)
3. The pure SH waves for [

T mono ∈ C 2×2 When T mono ∈ C 2×2 , it
u 2 σ 23 ] ln bot = [ T 11 T 12 T 21 T 22 ] [ u 2 σ 23 ] l 1 top (4.43)
By applying the traction-free boundary condition at the bottom-and top-most surfaces of the plate,

[ σ 23 ] ln bot = [ σ 23 ] l 1 top = [ 0 ]
, the dispersion equation of the pure SH waves linking v and ξ just come into being, as presented in Eq. (4.44).

D T (v, ξ) = D R T (v, ξ) + iD I T (v, ξ) ≜ T 21 = 0 (4.44)

TRANSFER MATRIX METHOD

To this stage, the derivation process of TMM for various guided wave types has been completed.

The pseudo-code for processing the three wave types is presented in Algorithm 2, 3 and 4, respectively.

But before running the three algorithms, Algorithm 1 should be run to determine the correct wave type existing in the laminate system. Note that though this chapter concentrates on the multi-layered plate, the TMM framework is compatible with the single layer case given that for a single layer plate

T mono = T l 1 in Eq. (4.34).
Algorithm 1 Judge wave type according to the rotated laminae's stiffness matrix type.

Input: Each layer's material properties (C k , ρ k , d k , ϕ k ) for (k = 1, . . . , n);
The wave propagation angle of interest θ.

1: Compute the rotated stiffness matrix C ′ k =Rot{C k , θ -ϕ k } for (k = 1, . . . , n). 2: if All C ′ k ∈ C M then % Branch 1 3:
Call Algorithm 2 to process the coupled Lamb and SH waves.

4: else if All C ′ k ∈ C O then % Branch 2 5:
Call Algorithm 3 and 4 to process the pure Lamb and SH waves.

6: else if Some C ′ k ∈ C M and other C ′ k ∈ C O then % Branch 3 7:
Call Algorithm 5 to process the hybrid Lamb and SH waves.

. , n);

The wave propagation angle of interest θ;

The minimum and maximum wavenumber of interest (ξ min , ξ max );

The minimum, maximum and incremental velocity of interest (v min , v max , ∆v).

1: Compute C ′ k =Rot{C k , θ -ϕ k } for (k = 1, . . . , n). 2: if Any C ′ k / ∈ C M then 3:
Call Algorithm 1 to determine the correct wave type. 4: end if 5: for v 0 = v min : ∆v : v max do % Solve the dispersion equation at each fixed v 0 .

6:

for k = 1 : 1 : n do % Loop through each layer.

7:

Compute (α r , V r , W r , β 1r , β 2r , β 3r ) for (r = 1, . . . , 6) via Eqs. (3.24)(3.27)(3.31) using the data (C ′ k , ρ k , v 0 ). 8:
Compute the local lamina matrices

( Γ l k 6×6 , Λ l k 6×6 (d k ) )
via Eq. (4.11).

9:

Compute the local transfer matrix T l k 6×6 via Eq. (4.29). Compute the monodromy transfer matrix T mono 6×6 via Eq. (4.34).

12:

Solve D T (v 0 , ξ) = 0 in Eq. (4.39) to get multiple roots (ξ 1 , ξ 2 , . . . ) ∈ [ξ min , ξ max ]. 13: end for Output: All solution points (v i , ξ i ) in the range [v min , v max ] × [ξ min , ξ max ].
Plot dispersion curves of the coupled Lamb and SH waves in (v, ξ), (ω, ξ), (ω, v) and (ω, c g ) domains where ω = ξv and c g = ∂ω ∂ξ is the group velocity.

TRANSFER MATRIX METHOD

Algorithm 3 Process the pure Lamb waves via TMM

Input: Each layer's material properties (C k , ρ k , d k , ϕ k ) for (k = 1, . . . , n);
The wave propagation angle of interest θ;

The minimum and maximum wavenumber of interest (ξ min , ξ max );

The minimum, maximum and incremental velocity of interest (v min , v max , ∆v).

1: Compute C ′ k =Rot{C k , θ -ϕ k } for (k = 1, . . . , n). 2: if Any C ′ k / ∈ C O then 3:
Call Algorithm 1 to determine the correct wave type. 4: end if 5: for v 0 = v min : ∆v : v max do % Solve the dispersion equation at each fixed v 0 .

6:

for k = 1 : 1 : n do % Loop through each layer.

7:

Compute (α r , W r , β 1r , β 3r ) for (r = 1, . . . , 4) via Eqs. (3.44)(3.46)(3.49) using the data (C ′ k , ρ k , v 0 ).

8:

Compute the local lamina matrices

( Γ l k 4×4 , Λ l k 4×4 (d k ) )
via Eq. (4.18).

9:

Compute the local transfer matrix T l k 4×4 via Eq. (4.29).

10:

end for 11:

Compute the monodromy transfer matrix T mono 4×4 via Eq. (4.34).

12:

Solve D T (v 0 , ξ) = 0 in Eq. (4.42) to get multiple roots (ξ 1 , ξ 2 , . . . ) ∈ [ξ min , ξ max ]. 13: end for Output: All solution points (v i , ξ i ) in the range [v min , v max ] × [ξ min , ξ max ].
Plot dispersion curves of the pure Lamb waves in (v, ξ), (ω, ξ), (ω, v) and (ω, c g ) domains.

Algorithm 4 Process the pure SH waves via TMM

Input: Each layer's material properties (C k , ρ k , d k , ϕ k ) for (k = 1, . . . , n);
The wave propagation angle of interest θ;

The minimum and maximum wavenumber of interest (ξ min , ξ max );

The minimum, maximum and incremental velocity of interest (v min , v max , ∆v).

1: Compute C ′ k =Rot{C k , θ -ϕ k } for (k = 1, . . . , n). 2: if Any C ′ k / ∈ C O then 3:
Call Algorithm 1 to determine the correct wave type. 4: end if 5: for v 0 = v min : ∆v : v max do % Solve the dispersion equation at each fixed v 0 .

6:

for k = 1 : 1 : n do % Loop through each layer.

7:

Compute (α r , β 2r ) for (r = 5, 6) via Eqs. (3.58)(3.60) using the data (C ′ k , ρ k , v 0 ).

8:

Compute the local lamina matrices

( Γ l k 2×2 , Λ l k 2×2 (d k ) )
via Eq. (4.24).

9:

Compute the local transfer matrix T l k 2×2 via Eq. (4.29). Compute the monodromy transfer matrix T mono 2×2 via Eq. (4.34).

12:

Solve D T (v 0 , ξ) = 0 in Eq. (4.44) to get multiple roots (ξ 1 , ξ 2 , . . . ) ∈ [ξ min , ξ max ]. 13: end for Output: All solution points (v i , ξ i ) in the range [v min , v max ] × [ξ min , ξ max ].
Plot dispersion curves of the pure SH waves in (v, ξ), (ω, ξ), (ω, v) and (ω, c g ) domains.

Wave mode shapes

Wave mode shapes are the displacement and stress fields across the thickness of the laminate at a fixed point in the wave propagation path. The key step to compute wave mode shapes is to compute the participation factors η l k of each layer, which can be regarded as, to some extent, the inverse process of the transfer matrix derivation in Section 4.3.1 and 4.3.2. In this section, we take the example of computing mode shapes of the coupled Lamb and SH waves. For other wave types, it can be achieved by a similar way.

For a known dispersion solution point (v i , ξ i ) solved through Algorithm 2, each layer's various kinds of matrices are known as well,

( U l k 3×6 , β l k 3×6 , Λ l k 6×6 (d k ), Γ l k 6×6 , T l k 6×6 , T mono 6×6 )
. Then, u l 1 top in Eq. (4.38) can be solved from the three-order linear homogeneous equation set, which is the zero-eigenvalue correlated eigenvector of the submatrix T mono 6×6 (4 : 6, 1 : 3). Thanks to the traction-free boundary condition at the top side of layer l 1 (see Eq. (4.36)), σ l 1 top = 0, the field variables of layer l 1 at its top side are then obtained

S l 1 top = [ u l 1 top 0 ]
. For other layer's field variables at its top side, it can be computed recurrently from S l 1 top as follows based on Eq. (4.28).

S l k top = S l k-1 bot = T l k-1 S l k-1 top (k = 2, . . . , n) (4.45)
Each layer's participation factors η l k are then computed as follows based on Eq. (4.26) by omitting the phase term e iξ(x 1 -vt) .

η l k = ( T l k ) -1 S l k top (k = 1, . . . , n) (4.46)
Finally, substitute η l k into Eqs. (4.2) and (4.8) to compute the layer l k 's mode shapes of displacement and stress, respectively. The whole laminate's mode shapes are then concatenated from all layers' ones. waves in this layer). This issue is very common and it is usually present in multi-layered anisotropic composite plates owing to the various stacking angles of layups. However, this issue has not been given sufficient attention in the literature and only a few works attempted to tackle this issue.

The first work was presented by S. Pant in [8]. In this reference, a pseudo correction measure was adopted to avoid this issue. Specifically, along the interested wave propagation direction, when the rotated stiffness matrix in the layer l i behaves like the orthotropic type but in another layer l j like the monoclinic type, namely,

C ′ i ∈ C O and C ′ j ∈ C M , C ′ i is further rotated with an additional minor angle ∆θ = -1 • such that it transitions to C ′′ i ∈ C M .
The screenshot of Pant's pseudo correction measure is presented in Figure 4.3. As the additional rotation angle is relatively small, the pseudo correction has very little effect on the overall stiffness of the laminate. Thus, it is doable but not orthodox from a mathematical perspective. Besides, there is a vital problem that, in metallic-composite structures, the stiffness matrix of the metallic layer always keeps the isotropic type regardless of the rotation angle, and the composite layers can be orthotropic or monoclinic type along an arbitrary propagation direction. Thus, in metallic-composite structures Pant's pseudo correction measure is invalid. , the corresponding eigenvectors are linearly dependent. Thus, the vector orthogonality analysis was used to sort these eigenvectors to generate the six linearly independent eigenvectors being the admissible displacement amplitudes of the six partial waves. The screenshot of Barazanchy's 4.4. HYBRID MATRIX STRATEGY vector orthogonality analysis is presented in Figure 4.4. This method is only applicable to isotropic materials and it encountered numerical instability issue when applied to a 50-layered plate [9].

⎡ ⎢ ⎣ C 11 -ρv 2 + C 11 -C 12 2 α 2 0 C 11 +C 12 2 α 0 C 11 -C 12 2 -ρv 2 + C 11 -C 12 2 α 2 0 C 11 +C 12 2 α 0 C 11 -C 12 2 -ρv 2 + C 11 α 2 ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ U V W ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ 0 0 0 ⎤ ⎥ ⎦ (4.47) α 2 1 = ρv 2 C 11 -1, α 2 3 = 2ρv 2 C 11 -C 12 -1, α 2 5 = 2ρv 2 C 11 -C 12 -1. (4.48)
Figure 4.4 -Barazanchy's vector orthogonality analysis to deal with the incompatibility issue [9].

In the current thesis, we propose a new hybrid matrix strategy (HMS) to solve the incompatibility issue, which is effective for various material types including isotropic and orthotropic materials. The most important aspect of HMS is that it is directly derived from the layer-wise PWSA similar to the coupled Lamb and SH waves in Section 4.2.1. Thus, HMS is mathematically rigorous and numerically stable.

Hybrid local lamina matrices of pure Lamb and SH waves

The incompatibility issue mathematically corresponds to the case that in some layers their local transfer matrices are six-order T l i 6×6 (producing the coupled Lamb and SH waves) and in other layers they have two decoupled four-order and two-order of local transfer matrices T l j 4×4 (producing the pure Lamb waves) and T l j 2×2 (producing the pure SH waves). The process of deriving the HMS is to 'recouple' T l j 4×4 and T l j 2×2 to yield a new six-order local transfer matrix T ˜lj 6×6 , which is compatible with

T l i
6×6 in terms of the matrix order.

HYBRID MATRIX STRATEGY

The single-layer system studied in Chapter 3 is the basis to derive the equations regarding to the multi-layer system. Specifically, the displacement expressions of (u 1 , u 3 ) in Eq. (3.47) and u 2 in Eq. (3.59) can be merged in Eq. (4.49).

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ u 1 = ( ∑ 4 r=1 η r e iξαrx 3
) e iξ(x 1 -vt)

u 2 = ( ∑ 6 r=5 η r e iξαrx 3
)

e iξ(x 1 -vt)

u 3 = ( ∑ 4 r=1 η r W r e iξαrx 3
)

e iξ(x 1 -vt) (4.49)
It is equivalent to Eq. (4.50) under the concept of layer-wise PWSA.

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ u 1 = ( ∑ 4 r=1 1 × η r e iξαrx 3 + ∑ 6 r=5 0 × η r e iξαrx 3
) e iξ(x 1 -vt)

u 2 = ( ∑ 4 r=1 0 × η r e iξαrx 3 + ∑ 6 r=5 1 × η r e iξαrx 3
) e iξ(x 1 -vt)

u 3 = ( ∑ 4 r=1 W r × η r e iξαrx 3 + ∑ 6 r=5 0 × η r e iξαrx 3
)

e iξ(x 1 -vt) (4.50)
Rewrite Eq. (4.50) as the matrix form in Eq. (4.51) and the compact form in Eq. (4.52).

⎡ ⎢ ⎣ u 1 u 2 u 3 ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ 1 1 1 1 0 0 0 0 0 0 1 1 W 1 W 2 W 3 W 4 0 0 ⎤ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ e iξα 1 x 3 e iξα 2 x 3 e iξα 3 x 3 e iξα 4 x 3 e iξα 5 x 3 e iξα 6 x 3 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ η 1 η 2 η 3 η 4 η 5 η 6 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ e iξ(x 1 -vt) (4.51) u 3×1 = U ˜3×6 Λ 6×6 (x 3 )η 6×1 e iξ(x 1 -vt) (4.52)
where, U ˜3×6 is used to distinguish U 3×6 defined in Eq. (4.4), and it is regarded as the hybrid matrix of U 2×4 in Eq. (4.14) and U 1×2 in Eq. (4.20). The tilde '˜' represents the hybrid sense. Physically, the zero elements in U ˜3×6 denote that the amplitudes of the corresponding partial waves are zero.

The stress equations of (σ 33 , σ 13 ) in Eq. (3.48) and σ 23 in Eq. (3.60) can be merged in Eq. (4.53)

and are further equivalently written as the superposition form in Eq. (4.54).

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ σ 33 = ( ∑ 4 r=1 η r β 1r e iξαrx 3
) e iξ(x 1 -vt)

σ 23 = ( ∑ 6 r=5 η r β 2r e iξαrx 3
) e iξ(x 1 -vt)

σ 13 = ( ∑ 4 r=1 η r β 3r e iξαrx 3
)

e iξ(x 1 -vt) (4.53) ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ σ 33 = ( ∑ 4 r=1 β 1r × η r e iξαrx 3 + ∑ 6 r=5 0 × η r e iξαrx 3
) e iξ(x 1 -vt)

σ 23 = ( ∑ 4 r=1 0 × η r e iξαrx 3 + ∑ 6 r=5 β 2r × η r e iξαrx 3
) e iξ(x 1 -vt)

σ 13 = ( ∑ 4 r=1 β 3r × η r e iξαrx 3 + ∑ 6 r=5 0 × η r e iξαrx 3
) e iξ(x and the compact form in Eq. (4.58).

⎤ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ e iξα 1 x 3 e iξα 2 x 3 e iξα 3 x 3 e iξα 4 x 3 e iξα 5 x 3 e iξα 6 x 3 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ η 1 η 2 η 3 η 4 η 5 η 6 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ e iξ(x 1 -vt) (4.55) σ 3×1 = β ˜3×6 Λ 6×6 (x 3 )η 6×1 e iξ(x
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ u 1 u 2 u 3 σ 33 σ 23 σ 13 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 1 1 1 0 0 0 0 0 0 1 1 W 1 W 2 W 3 W 4 0 0 β 11 β 12 β 13 β 14 0 0 0 0 0 0 β 25 β 26 β 31 β 32 β 33 β 34 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ e iξα 1 x 3 e iξα 2 x 3 e iξα 3 x 3 e iξα 4 x 3 e iξα 5 x 3 e iξα 6 x 3 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ η 1 η 2 η 3 η 4 η 5 η 6 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
e iξ(x 1 -vt) (4.57)

S 6×1 = Γ ˜6×6 Λ 6×6 (x 3 )η 6×1 e iξ(x 1 -vt) = Z ˜6×6 (x 3 )η 6×1 e iξ(x
T ˜6×6 = Γ ˜6×6 Λ 6×6 (d k ) ( Γ ˜6×6 ) -1 ∈ C 6×6 (4.59)
With the compatible transfer matrices between T ˜lj 6×6 and T l i 6×6 , the hybrid monodromy transfer matrix T ˜mono 6×6 is computed based on Eq. (4.34). The above derivations for the HMS is programmatically interpreted in Algorithm 5, which is logically accompanied by Algorithm 1 to make sure that the 4.4. HYBRID MATRIX STRATEGY current guided wave type correctly corresponds to the hybrid case. Specifically, the Branch 3 of Algorithm 1 should be targeted. In response to the HMS, the wave solutions that are obtained via Algorithm 5 are classified into the hybrid Lamb and SH waves to distinguish the terminology of the coupled waves in Algorithm 2 and the pure waves in Algorithm 3 and 4.

Algorithm 5 HMS to process the matrix incompatibility issue of the hybrid Lamb and SH waves

Input: Each layer's material properties (C k , ρ k , d k , ϕ k ) for (k = 1, . . . , n);
The wave propagation angle of interest θ;

The minimum and maximum wavenumber of interest (ξ min , ξ max );

The minimum, maximum and incremental velocity of interest (v min , v max , ∆v).

1: Compute C ′ k =Rot{C k , θ -ϕ k } for (k = 1, . . . , n). 2: if All C ′ k ∈ C M or all C ′ k ∈ C O then 3:
Call Algorithm 1 to determine the correct wave type. 4: end if 5: for v 0 = v min : ∆v : v max do 6:

for k = 1 : 1 : n do 7: if C ′ k ∈ C M then %
Compute (α r , V r , W r , β 1r , β 2r , β 3r ) for (r = 1, . . . , 6) via Eqs. (3.24)(3.27)(3.31) using the data (C ′ k , ρ k , v 0 ).

9:

Compute the local lamina matrices

( Γ l k 6×6 , Λ l k 6×6 (d k ) )
via Eq. (4.11).

10:

Compute the local transfer matrix T l k 6×6 via Eq. (4.29).

11:

else if C ′ k ∈ C O then % Process the hybrid Lamb and SH waves.

12:

Compute (α r , W r , β 1r , β 3r ) for (r = 1, . . . , 4) via Eqs. (3.44)(3.46)(3.49) using the data (C ′ k , ρ k , v 0 ).

13:

Compute the local lamina matrices

( Γ l k 4×4 , Λ l k 4×4 (d k ) )
via Eq. (4.18).

14:

Compute (α r , β 2r ) for (r = 5, 6) via Eqs. (3.58)(3.60) using the data (C ′ k , ρ k , v 0 ).

15:

Compute the local lamina matrices

( Γ l k 2×2 , Λ l k 2×2 (d k ) )
via Eq. (4.24).

16:

Compute the hybrid local lamina matrices Γ ˜lk 6×6 using the obtained

( Γ l k 4×4 , Γ l k 2×2
) based on Eq. (4.57).

17:

Compute the hybrid local transfer matrix T ˜lk 6×6 using the obtained

( Γ ˜lk 6×6 , Λ l k 6×6
) based on Eq. (4.59).

(v i , ξ i ) in the range [v min , v max ] × [ξ min , ξ max ].
Plot dispersion curves of the hybrid Lamb and SH waves in (v, ξ), (ω, ξ), (ω, v) and (ω, c g ) domains.

FOUR CLASSES OF COMPOSITE LAMINATES

Four classes of composite laminates

In aeronautic and aerospace industry, an arbitrarily oriented composite laminate can be classified into four classes according to the stacking angles and material compositions as listed in Table 4.1, in which the unidirectional, cross-ply and quasi-isotropic laminates are composed solely by orthotropic materials (the transversely isotropic materials are included in the type of orthotropic materials according to the Section 2.2.4.) and the metallic-composite laminate consists of orthotropic (for composite layers) and isotropic (for metallic layers) materials. The second column of Table 4.1 lists the representative for each laminate class. According to the remark below Table 4.1, the unidirectional laminate can be mathematically modeled as the single layer plate which is the focus of Chapter 3. Thus, in the next sections for numerical examples, only the latter three laminates are adopted to validate the developed TMM adapted to the coupled, pure and hybrid wave types as indicated in Table 4.1.

In an arbitrarily oriented composite laminate, if a certain wave propagation angle θ will lead to the incompatibility issue (causing the hybrid Lamb and SH waves), this angle is defined as the singular angle of this laminate. For example, θ = 45 • in the quasi-isotropic laminate of Table 4.1.

[0] 8 θ ∈ {0 • , 90 • } √ θ / ∈ {0 • , 90 • } √ Cross-ply [(0/90) s ] 2 θ ∈ {0 • , 90 • } √ θ / ∈ {0 • , 90 • } √ Quasi-isotropic [0/90/ + 45/ -45] s θ ∈ {0 • , 45 • , 90 • , -45 • } √ θ / ∈ {0 • , 45 • , 90 • , -45 • } √ Metallic-composite [Al/0/90/Al/90/0/Al] 2 θ ∈ {0 • , 90 • } √ θ / ∈ {0 • , 90 • } √
1 Remark on the unidirectional laminate class: since the field variables (displacement and stress) are considered to be continuous across the interface of adjacent layers (the perfect interface condition is assumed), the unidirectional laminate [0]8 is mathematically equivalent to the single layer plate [0]1 that has the same thickness as the [0]8 plate. 2 Al: the aluminum layer that is an isotropic material.

NUMERICAL EXAMPLE ON A CROSS-PLY LAMINATE

Numerical example on a cross-ply laminate

The first example is the cross-ply laminate consisting of eight transversely isotropic laminae with stacking angles of layups [(0/90) s ] 2 . The material properties of each ply are listed in Table 4.2 cited from [12]. The total thickness of the plate is 1.6 mm since the thickness of each ply is 0.2 mm. The stiffness matrix of each ply along its fiber direction is presented in Eq. (4.60) which is computed based on Eq. (2.12). To clearly illustrate the process of stiffness matrix rotation and the existing wave types in the cross-ply laminate that is adapted to the rotated stiffness matrix of each layer, the profile section of this laminate is illustrated in Figure 4.5 for the current wave propagation angle θ = 0 • . From this conceptual diagram, it can be seen that, after rotation of Eq. (4.60), the rotated stiffness matrix of each layer still keeps the orthotropic type, i.e. 8). According to the Algorithm 1, Branch 2 is targeted that corresponds to the case of the pure Lamb and SH waves. After running Algorithm 3 and 4, the resulted dispersion curves of the pure Lamb and SH waves are depicted in In Figure 4.9(a)(b), the u 2 component is zero, but in Figure 4.9(c), the u 1 and u 3 component is zero.

C ′ i ∈ C O (i = 1, . . . ,
𝐂 𝑖 ′ =
This is consistent with their respective wave types implied in For the coupled modes, the mode order is named by following the terminology of the pure modes with a prefix quasi to distinguish the pure counterparts in some references [37,[START_REF] Nadella | Characterization of guided-wave propagation in composite plates[END_REF]8],

namely, quasi-S0, quasi-A0 and quasi-SH0 modes. In the thesis, we do not emphasize this point. The prefix quasi is omitted in the next sections unless stated otherwise.

From

Numerical example on a quasi-isotropic laminate

In this section, a carbon-fiber epoxy laminate is used to demonstrate the developed HMS.

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ GPa (4.61)
The quasi-isotropic stacking sequence of layups results in that there is not a common principal axis among the laminae and thus no pure Lamb or SH waves present in the plate for any propagation angle.

For the specific stacking angles of layups [0/90/ + 45/ -45] s , when the interested wave propagation angle just coincides with one layer's private principal direction, i.e. θ ∈ {0 Immediately, we are encountering the incompatibility issue between T l i 6×6 and

( T l j 4×4 , T l j 2×2
)

. The HMS developed in Section 4.4 just dedicates to solve this issue, which is achieved by reconstructing

( T l j 4×4 , T l j 2×2
)

to the hybrid local transfer matrix T ˜lj 6×6 . By doing so, all layers' local transfer matrices are now compatible and they are assembled together to yield the hybrid monodromy transfer matrix

T ˜mono

6×6 based on the TMM framework. After running Algorithm 1 and 5, the dispersion solutions of Note that the experimental data points in Figure 4.12 are extracted from [8], and they are used in Figure 4.12 for the comparison purpose from the experimental perspective. The good agreement between the theoretically computed curves and the experimental data points shown in Figure 4.12, as a consequence, validates the feasibility and effectiveness of the combined TMM and HMS methodology.

To intuitively observe all the singular angles of the quasi-isotropic laminate, the polar plot of wave velocities are created in

∈ 𝒞 ℳ 𝐂 1 ′ ∈ 𝒞 ℳ 𝐂 3 ′ ∈ 𝒞 𝒪 𝐂 4 ′ ∈ 𝒞 𝒪 𝐂 5 ′ ∈ 𝒞 𝒪 𝐂 6 ′ ∈ 𝒞 𝒪 𝐂 7 ′ ∈ 𝒞 ℳ 𝐂 8 ′ ∈ 𝒞 ℳ 𝐓 6×6 𝑙 2 𝐓 6×6 𝑙 1 ෩ 𝐓 6×6 𝑙 3 ෩ 𝐓 6×6 𝑙 4 ෩ 𝐓 6×6 𝑙 5 ෩ 𝐓 6×6 𝑙 6 𝐓 6×6 𝑙 7 𝐓 6×6 𝑙 8
Accessible wave type: This measure is conceptually illustrated in Figure 4.14 for the angle θ = 0 • which has been discussed in As the additional angle is relatively small, this pseudo correction has very little effect on the overall stiffness of the laminate. Thus, it is doable for the quasi-isotropic laminate but not orthodox from the mathematical view. In the metallic-composite laminate, Pant's pseudo correction is invalid because the stiffness matrix of the metallic layer always remains the orthotropic type irrespective of the rotation angle. We will demonstrate this case in Section 4.8.3.

Correct 𝐂 𝑖 ′ ∈ 𝒞 𝒪 → 𝐂 𝑖 ′′ ∈ 𝒞 ℳ via 𝐂 𝑖 ′′ = Rot{𝐂 𝑖 ′ , Δ𝜃} with

Numerical example on a metallic-composite laminate

In this section, the HMS is applied to a metallic-composite laminate composed by three aluminum properties of the two materials are cited from [8] and listed in Table 4.4 and 4.5 for convenience. The stiffness matrices of the two materials along their respective principal-axis direction can be computed based on Eqs. (2.14) and (2.12). Thanks to the isotropic property of aluminum material, there exists a common principle-axis direction between the aluminum and the glass fiber layers, 0 • and 90 • . Thus, for the metallic-composite plate, the incompatibility issue occurs at any angle other than θ = 0 • or 90 (a) 6×6 (j = 2, 3, 5, 6) if without using the hybrid ones T ˜li 6×6 (i = 1, 4, 7) employed in Figure 4. 15(b). This illustration proves that Pant's pseudo correction measure fails to deal with the incompatibility issue of metallic-composite laminate but the HMS is the unique solution to address this issue if the matrix-based methodologies (TMM and GMM1 ) are adopted to analyze wave propagation characteristics.

[ kg/m 3 ] ) E 1 [GPa] E 2 [GPa] G 12 [GPa]

NUMERICAL EXAMPLE ON A METALLIC-COMPOSITE LAMINATE

𝐂 2 ∈ 𝒞 𝒪 𝐂 1 ∈ 𝒞 ℐ 𝐂 3 ∈ 𝒞 𝒪 𝐂 4 ∈ 𝒞 ℐ 𝐂 5 ∈ 𝒞 𝒪 𝐂 6 ∈ 𝒞 𝒪 𝐂 7 ∈ 𝒞 ℐ 𝜙 2 = 0 °𝜙1 = 0 °(Al) 𝜙 3 = 90 °𝜙4 = 0 °(Al) 𝜙 5 = 90 °𝜙6 = 0 °𝜙7 = 0 °(Al) 𝜃 2 = 0 °𝜃1 = 0 °𝜃3 = 90 °𝜃4 = 0 °𝜃5 = 90 °𝜃6 = 0 °𝜃7 = 0 °𝐂2 ′ ∈ 𝒞 𝒪 𝐂 1 ′ ∈ 𝒞 ℐ 𝐂 3 ′ ∈ 𝒞 𝒪 𝐂 4 ′ ∈ 𝒞 ℐ 𝐂 5 ′ ∈
𝐂 2 ∈ 𝒞 𝒪 𝐂 1 ∈ 𝒞 ℐ 𝐂 3 ∈ 𝒞 𝒪
∈ 𝒞 ℳ 𝐂 1 ′ ∈ 𝒞 ℐ 𝐂 3 ′ ∈ 𝒞 ℳ 𝐂 4 ′ ∈ 𝒞 ℐ 𝐂 5 ′ ∈ 𝒞 ℳ 𝐂 6 ′ ∈ 𝒞 ℳ 𝐂 7 ′ ∈ 𝒞 ℐ 𝐓 6×6 𝑙 2 ෩ 𝐓 6×6 𝑙 1 𝐓 6×6 𝑙 3 ෩ 𝐓 6×6
∈ 𝒞 ℳ 𝐂 1 ′ ∈ 𝒞 ℐ 𝐂 3 ′ ∈ 𝒞 ℳ 𝐂 4 ′ ∈ 𝒞 ℐ 𝐂 5 ′ ∈ 𝒞 ℳ 𝐂 6 ′ ∈ 𝒞 ℳ 𝐂 7 ′ ∈ 𝒞 ℐ 𝐓 6×6 𝑙 2 𝐓 4×4 𝑙 1 ,

Conclusion

This chapter reviewed in detail the classical TMM framework for modelling guided waves propagation in anisotropic composite laminates. For an arbitrarily oriented plate, the Lamb modes and SH modes will couple to each other, or completely separate depending on the stacking angles of layups and the observed wave propagation direction. For the quasi-isotropic laminate, if the observation angle of interest is just along one layer's principle axis, the matrix incompatibility issue will appear. This issue is more severe in metallic-composite laminate because it occurs at any direction in the plate plane except along the common principal axes of composite laminae. To address this issue, the HMS is theoretically derived by following the PWSA. It is thus possible to deal with wave propagation problem in an arbitrarily oriented composite laminate. Guided wave propagation in a multi-layered plate: global matrix method Although the effectiveness of TMM has been sufficiently validated in Chapter 4, one fatal flaw of TMM is the notorious large f d problem which states that at the large product value of frequency f and layer thickness d, the dispersion curves obtained from the solutions of the dispersion equation built with TMM become unstable. The usual way to address this issue is to adopt the global matrix method (GMM). But one side effect of GMM is its low efficiency when using it to cope with a multi-layered plate that has many layers.

In this chapter, three advanced techniques are jointly applied to the traditional GMM to improve the computational inefficiency issue. They are sparse matrix technique, parallel computing technique and the matrix order reduction technique inspired by the symmetry condition. The thoroughly optimized GMM is applied to compute the dispersion curves of a 400-layered composite structure that is the prototype of the rocket booster pressure vessels of the future launcher Ariane 6. The success of this application proves the possibility that by introducing advanced computing techniques, GMM can be applied to the aerospace composites that usually have a large number of layers.

The standard global matrix method

Before starting to derive the equations of GMM, we make the assumption that all layers have the compatible field variables in terms of matrix order, i.e.

( u p 2 ×1 , σ p 2 ×1 , S p×1 ) l k (k = 1, . . . , n),
where n is the total number of layers and p is the number of partial waves in each layer (p = 6 for the 5.1. THE STANDARD GLOBAL MATRIX METHOD coupled or hybrid Lamb and SH waves, p = 4 for the pure Lamb waves and p = 2 for the pure SH waves). This assumption leads to that all layers have the compatible local lamina matrices as well,

(Γ p×p , Λ p×p (x 3 ), Z p×p (x 3 )) l k (k = 1, . . . , n).
If not so, the HMS is employed to pre-process the incompatible layers such that the hybrid matrix

( Γ ˜6×6 , Λ 6×6 (x 3 ), Z ˜6×6 (x 3 )
) l i and the coupled matrix (Γ 6×6 , Λ 6×6 (x 3 ), Z 6×6 (x 3 )) l j are coexisting and compatible in terms of matrix order. In the sequel, the subscript indicating the matrix order p × p is suppressed for the sake of conciseness.

For a general layer l k in Figure 4.2(b), its field variables (displacement and stress) can be uniformly rewritten as the following matrix multiplication form.

u l k = U l k Λ l k (x 3 )η l k e iξ(x 1 -vt) (5.1a) σ l k = β l k Λ l k (x 3 )η l k e iξ(x 1 -vt) (5.1b) S l k = Γ l k Λ l k (x 3 )η l k e iξ(x 1 -vt) = Z l k (x 3 )η l k e iξ(x 1 -vt) (5.1c)
where,

Γ l k = [ U l k β l k ] ∈ C p×p and Z l k (x 3 ) = Γ l k Λ l k (x 3 ) ∈ C p×p .
In a laminate, we have the continuity condition (CC) at all interfaces of two adjacent layers and traction-free BC at the top and bottom surfaces of the laminate. Specifically, at a typical interface i k as illustrated in Figure 4.2(b), the field variables of layer l k at its bottom side should equal to the field variables of layer l k+1 at its top side as a result of rigid connection.

S l k bot = S l k+1 top (k = 1, • • • , n -1) (5.2)
where, the subscript 'bot' and 'top' represents bottom and top side, respectively. Substituting Eq. (5.1c) into Eq. (5.2) can lead to the following equation.

Z l k bot η l k -Z l k+1 top η l k+1 = 0 (k = 1, • • • , n -1) (5.3)
where,

Z l k bot = Z l k (d k ) ∈ C p×p , Z l k+1 top = Z l k+1 (0) ∈ C p×p .
For the traction-free BC at the top-and bottom-most surfaces as shown in Z σ,l 1 top η l 1 = 0 and Z σ,ln bot η ln = 0 (5.5)

where,

Z σ,l 1 top = β l 1 Λ l 1 (0) ∈ C p 2 ×p , Z σ,ln bot = β ln Λ ln (d n ) ∈ C p 2 ×p .
Sequentially applying Eq. (5.3) from interface i 1 to i n-1 as illustrated in BC at the top surface → CC at the interfaces

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ BC at the bot. surface → ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ -Z σ,l 1 top Z l 1 bot -Z l 2 top . . . . . . Z l k bot -Z l k+1 top . . . . . . Z l n-1 bot -Z ln top Z σ,ln bot ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ η l 1 . . . η l k . . . η ln ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = 0 (5.6) D G (v, ξ) = D R G (v, ξ) + iD I G (v, ξ) ≜ det{D G (v, ξ)} = 0 (5.7)
where, the subscript 'G' denotes that the dispersion equation is built with GMM.

If the studied structure is a single layer plate, the CC in Eq. (5.6) is diminished and only the two BC remain. In this case, Eq. (5.6) collapses to the following form, which has been systematically studied in Chapter 3.

[ Z σ,l 1 top Z σ,l 1 bot ] η l 1 = 0 (5.8)
Until this stage, the derivation of the standard GMM has been completed. The whole process of solving the dispersion equation built with GMM is similar to the TMM and summarized in Algorithm 6

for the coupled Lamb and SH waves. For other wave type, the corresponding algorithm can be easily designed.

It should be noted that the computational burden of GMM is more salient than TMM for a laminate having many layers because the order of global matrix D G (v, ξ) ∈ C np×np increases with the increasing number of layers n. However, the order of the monodromy transfer matrix T mono ∈ C p×p

BENCHMARK TEST OF GMM THROUGH THE DATASET OPEN GUIDED WAVES

in Eq. (4.34) remains unchanged regardless of the number of layers. But GMM is more robust than TMM at higher frequency range from the numerical analysis view. We will validate this point in the subsequent benchmark testing.

Algorithm 6 Process the coupled Lamb and SH waves via GMM

Input: Each layer's material properties (C k , ρ k , d k , ϕ k ) for (k = 1, . . . , n);
The wave propagation angle of interest θ;

The minimum and maximum wavenumber of interest (ξ min , ξ max );

The minimum, maximum and incremental velocity of interest (v min , v max , ∆v).

1: Compute C ′ k =Rot{C k , θ -ϕ k } for (k = 1, . . . , n). 2: if Any C ′ k / ∈ C M then 3:
Call Algorithm 1 to determine the correct wave type. 4: end if 5: for v 0 = v min : ∆v : v max do % Solve the dispersion equation at each fixed v 0 .

6:

for k = 1 : 1 : n do % Loop through each layer.

7:

Compute (α r , V r , W r , β 1r , β 2r , β 3r ) for (r = 1, . . . , 6) via Eqs. (3.24)(3.27)(3.31) using the data (C ′ k , ρ k , v 0 ).

8:

Compute the local lamina matrices

( Γ l k 6×6 , Λ l k 6×6 (d k ) )
via Eq. (4.11).

9:

Compute

( Z l k bot , Z l k+1 top )
via Eq. ( 5.3).

10:

end for 11:

Compute

( Z σ,l 1 top , Z σ,ln bot )
via Eq. (5.5).

12:

Assemble the global matrix D G based on Eq. (5.6).

13:

Solve D G (v 0 , ξ) = 0 in Eq. (5.7) to get multiple roots (ξ 1 , ξ 2 , . . . ) ∈ [ξ min , ξ max ]. 14: end for Output: All solution points

(v i , ξ i ) in the range [v min , v max ] × [ξ min , ξ max ].
Plot dispersion curves of the coupled Lamb and SH waves in (v, ξ), (ω, ξ), (ω, v) and (ω, c g ) domains where ω = ξv and c g = ∂ω ∂ξ is the group velocity.

Benchmark test of GMM through the dataset Open Guided Waves

In this section, the dataset of Open Guided Waves (OGW) [62] is used to validate the standard GMM. We choose this composite structure because it is a benchmark platform for studying guided waves-based subjects including wave propagation simulation models, damage detection methods and structural health monitoring techniques. OGW is a 16-layered carbon fiber reinforced polymer (CFRP) plate with the dimensions of 500 mm × 500 mm and a total thickness of 2 mm. The stacking angles of layups is [45/0/ -45/90/ -45/0/45/90] s , indicating the quasi-isotropic property. The material properties of OGW laminae are listed in Table 5.1. The orthotropic type stiffness matrix of this material is presented in Eq. (5.9).

BENCHMARK TEST OF GMM THROUGH THE DATASET OPEN GUIDED WAVES

GPa

(5.9)

Dispersion curves comparison between GMM and TMM

Both TMM and GMM are adopted to compute the dispersion curves of OGW at the propagation angle θ = 0 • . The coarse dispersion curves computed via the two methods are generated in Figure 5.1 for comparison. It shows that in a whole both methods produce consistent results but TMM has unstable solutions in the higher frequency range. The unstable issue of TMM corresponds to the large f d problem [6]. We will investigate this issue in detail in Section 6.6.

Dispersion curves comparison between GMM and experimental data

The OGW project has reported the experimental wave field signals. We analyzed these signals to identify the wave velocities that can be used as a reference to compare with the numerically computed dispersion curves. The experimentally obtained wave velocity data at several excitation frequencies is presented in Figure 5.

Dispersion curves comparison between GMM and ESM

As stated previously, when dealing with a laminate having many layers, TMM has superiority of computational efficiency in comparison with GMM, but it usually suffers from the large f d problem.

An alternative of TMM named the effective stiffness matrix method (ESM) is briefly introduced here to compute the dispersion curves of a multi-layered plate system.

ESM was originally studied in [START_REF] Nandyala | Effective stiffness matrix method for predicting the dispersion curves in general anisotropic composites[END_REF]. The core idea of ESM is the equivalence. It regards a multilayered plate system as a single-layered homogeneous plate with the same thickness. But the material The elements of stiffness matrix in the n-th layer:

BENCHMARK TEST OF GMM THROUGH THE DATASET OPEN GUIDED WAVES

The simplified effective stiffness matrix (monoclinic type)

THE OPTIMIZED GLOBAL MATRIX METHOD

property of the equivalent plate is changed to the monoclinic class. The mass density of the equivalent plate is the weighted average of each layer's density on their respective layer thickness. The equivalent (effective) stiffness matrix is computed from each layer's stiffness matrix in the weighted average sense.

The computation equations are concisely provided in Figure 5.4. These equations were obtained from the mechanical analysis for composite materials [START_REF] Sun | Three-dimensional effective elastic constants for thick laminates[END_REF].

Figure 5.4 shows that the averaged effective stiffness matrix possesses the monoclinic property.

Wave propagation equations on the single-layered monoclinic plate has been studied in detail in Section 3.3. Thus, we directly present the computed dispersion curves of OGW plate in Figure 5.2(a), accompanied by the GMM computation for comparison. One can see that ESM can produce consistent dispersion curves comparable with GMM for the three fundamental modes (S0, A0 and SH0 modes) especially at lower frequency range. The difference at higher frequency range may be caused by the inaccurate stiffness matrix that is computed by using the equations in Figure 5.4. Based on this demonstration, one can recognize that the validity of ESM is limited to the fundamental modes and lower frequency range.

The optimized global matrix method

It has long been recognized that GMM has lower computational efficiency than TMM when dealing with a laminate having large number of layers [6]. In this section, we demonstrate that by making three optimizations for the standard GMM, the optimized GMM can receive a great promotion on the computational efficiency even for a 400-layered laminate which has the largest number of layers reported so far in reference [11]. The three optimizations are outlined: (1) addition of symmetry condition to the global matrix, (2) applying the sparse matrix technique, (3) endowing the parallel computing attribute.

The symmetry condition in the global matrix

For the dispersion equation expressed in Eqs. (5.6) and (5.7), the symmetric and anti-symmetric modes are mutually coupled. On the other hand, the dimension of the global matrix

D G (v, ξ) ∈ C 6n×6n
is huge for a large number of layers n, which causes heavy computational burden. For most aerospace composite structures, symmetrically stacked plies are commonly applied in composite laminates, this where, the superscript 'l m ' represents the middle layer that embraces the midplane and the subscript 'mp' implies the location of the midplane, both which can be identified according to the odevity of a laminate [START_REF] Quintanilla | The symmetry and coupling properties of solutions in general anisotropic multilayer waveguides[END_REF] as illustrated in Figure 5.5. bot and ς X,lm mid , the superscript X = s and a represent the symmetric and anti-symmetric condition, respectively.

In view of the odevity of a n-layered symmetric laminate, the midplane places at different locations for n being an even or odd number. Thus, the SC should be evaluated separately for the two types of laminate.

• The laminate has an even number of layers.

For this laminate type, its midplane locates at the bottom side of the middle layer l m as illustrated in (5.11), meanwhile setting x 3 = d m and rearranging the resulted expressions to matrix form, the following compact form of SC can be obtained.

Z s,lm

bot η lm = 0 and Z a,lm bot η lm = 0 (5.12)

THE OPTIMIZED GLOBAL MATRIX METHOD

where, Z s,lm bot = χ lm s Λ lm (d m ) ∈ C 3×6 , Z a,lm bot = χ lm a Λ lm (d m ) ∈ C 3×6 . χ lm s and χ lm a are defined in Eq. (5.14). Λ lm (•) is defined in Eq. (4.5). The superscripts 's' and 'a' denote the symmetric and anti-symmetric condition, respectively. The subscript 'bot' implies the location of the midplane at the bottom side of layer l m .

• The laminate has an odd number of layers.

For this laminate type, the midplane locates at the mid-height of the layer l m as illustrated in

= χ lm s Λ lm (d m /2) ∈ C 3×6 , Z a,lm mid = χ lm a Λ lm (d m /2) ∈ C 3×6 .
The subscript 'mid' implies the location of the midplane at the mid-height of layer l m .

χ lm s = ⎡ ⎢ ⎣ W 1 W 2 W 3 W 4 W 5 W
⎤ ⎥ ⎦ lm , χ lm a = ⎡ ⎢ ⎣ 1 1 1 1 1 1 V 1 V 2 V 3 V 4 V 5 V 6 β 11 β 12 β 13 β 14 β 15 β 16 ⎤ ⎥ ⎦ lm (5.14)
Regardless of symmetry and odevity, Eqs. (5.12) and (5.13) can be abstractly written as follows by designating the symmetry script 'X' and the midplane location script 'mp'.

Z X,lm

mp η lm = 0 (5.15)

Following the indication in Figure 5.5, the CC (see Eq. (5.3)) can be sequentially applied from interface i 1 to and only to i m-1 . Combining the BC (see Eq. (5.5)) and the SC (see Eq. (5.15)), the SCinduced global matrix D X G (v, ξ) ∈ C 6m×6m can be generated into Eq. (5.16), which yields the implicit dispersion equation Eq. (5.17) in accordance with the superscript X = s and a for the symmetric and anti-symmetric modes, respectively, and the subscript mp=bot and mid for the laminate having an even and odd number of layers, respectively.

BC at the top surface →

CC at the interfaces

{ SC at the midplane → ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ -Z σ,l 1 top Z l 1 bot -Z l 2 top . . . . . . Z l m-1 bot -Z lm top Z X,lm mp ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ η l 1 η l 2 . . . η l m-1 η lm ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = 0 (5.16) D X G (v, ξ) = D X,R G (v, ξ) + iD X,I G (v, ξ) ≜ det{D X G (v, ξ)} = 0 (5.17)
There are two merits of the SC-induced GMM. The first one is the halved size of the SC-induced global matrix comparing to the original one, turning out to reduce the computational burden. The second one is that the symmetric and anti-symmetric modes' solutions are separated beforehand. As a consequence, the post separation can be exempted and the mode jumping problem studied in [START_REF] Quintanilla | The symmetry and coupling properties of solutions in general anisotropic multilayer waveguides[END_REF] can be avoided.

Sparsity of global matrix

The global matrix has sparsity property. Observing Eqs. (5.6) and (5.16), the non-zero entries of the global matrices are regularly distributed along the diagonal band. This fact indicates that the sparse matrix technique can be applied to the GMM. Using sparse matrix can play an important role when dealing with a laminate having a large number of layers which causes a large sized global matrix. By transforming this matrix into its sparse counterpart, the storage space can be saved and the computational speed can be greatly accelerated during processing the large sized matrix. We use MATLAB built-in function sparse to achieve the sparse matrix technique.

Parallelization of GMM

Even if not so apparent, GMM can be perfectly parallelized. Mathematically, the dispersion equation D(v, ξ) = 0 defined in Eq. (5.7) can be solved by fixing v = v 0 to sweep the roots of ξ (D(v 0 , ξ) = 0) or by fixing ξ = ξ 0 to sweep the roots of v (D(v, ξ 0 ) = 0). The two sweeping schemes are mutually independent.

Moreover, in both sweeping schemes, the sweeping process between the different fixed variables are mutually independent as well. For the example of sweeping ξ, the two equations D(v

0 , ξ) = 0 and D(v (2) 0 , ξ) = 0 can be solved concurrently as there is no any communication between the two equations. It is also true when sweeping v, the two equations D(v, ξ (2) 0 ) = 0 can be solved concurrently. This fact indicates that the two sweeping processes belongs to the classical Perfectly Parallel Problem [START_REF] Herlihy | The art of multiprocessor programming[END_REF]. Thus, execution of GMM can be parallelized during programming the algorithm to intentionally cope with the computational efficiency issue. Applying the parallel

APPLICATION OF THE OPTIMIZED GMM TO AN AEROSPACE COMPOSITE LAMINATE WITH 400 LAYERS

computing techniques for GMM can be easily achieved since there are many numerical computing toolboxes or packages available from widely used programming languages such as Matlab and Python.

By introducing the three advanced computing techniques to the standard GMM, the optimized GMM is now high efficiency and it is able to deal with aerospace composite laminate that usually has a large number of layers.

Application of the optimized GMM to an aerospace composite laminate

with 400 layers

Dispersion curves

In this section, we apply the optimized GMM to a 400-layered aerospace composite laminate with layups [0/90/45/ -45] 50s which is the prototype of the booster pressure vessels of the rocket Ariane 6 under development [11]. The composing material of the laminate is CFRP. The stiffness matrix of this material along the fiber direction is provided in Eq. (5.18). The mass density is ρ = 1550 kg/m 3 .

The plate total thickness is d = 50 mm. Eq. (5.17) is adopted to compute the dispersion curves of this laminate in the frequency-phase velocity domain through the relation ω = ξv. It can be thought that if the original global matrix was used, the size of the matrix will be large to 2400 orders because there is 400 layers and each layer has six partial waves (p = 6). By using the SC-induced global matrix, the matrix size has been halved to 1200 orders. Given that the studied laminate has even number of layers, Figure 5.5(a) should be focused. Thus, Eq. (5.12) should be adopted to apply SC for the full sized global matrix.

C = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 154
The computed dispersion curves through the optimized GMM are depicted in Figure 5.6. It shows that in the ellipses, the curves belonging to the same symmetry kind can mutually close but never cross. This phenomenon is termed the veering effect [START_REF] Manconi | On the effect of damping on dispersion curves in plates[END_REF], which will cause mode jumping problem in some computing software of dispersion curves [START_REF] Quintanilla | The symmetry and coupling properties of solutions in general anisotropic multilayer waveguides[END_REF]. In our computing program, we solve this problem by refining the solution points of the loci of one branch in the veering effect regions.

APPLICATION OF THE OPTIMIZED GMM TO AN AEROSPACE COMPOSITE LAMINATE WITH 400 LAYERS

Study the veering effect to avoid mode jumping problem

The veering effect is also termed as the repulsion behavior of dispersion curves in some references [START_REF] Überall | Repulsion of phase-velocity dispersion curves and the nature of plate vibrations[END_REF][START_REF] Zharnikov | Repulsion of dispersion curves of quasidipole modes of anisotropic waveguides studied by finite element method[END_REF][START_REF] Quintanilla | The symmetry and coupling properties of solutions in general anisotropic multilayer waveguides[END_REF] or the osculation of spectral lines in other reference [START_REF] Kausel | Osculations of spectral lines in a layered medium[END_REF]. It refers to the phenomenon that under certain coupling conditions two or more dispersion curves of different modes but belonging to the same symmetry kind close firstly and then veer away to finally diverge instead of crossing in the repulsion region. The extent of closing depends on the degree of coupling and anisotropy. This phenomenon happens constantly in Figure 5.6. The qualitative analysis has been made in some references [START_REF] Überall | Repulsion of phase-velocity dispersion curves and the nature of plate vibrations[END_REF][START_REF] Zharnikov | Repulsion of dispersion curves of quasidipole modes of anisotropic waveguides studied by finite element method[END_REF]. In the repulsion regions, the dispersion curves are usually accompanied by rapid exchanging of their mode shapes [START_REF] Manconi | On the effect of damping on dispersion curves in plates[END_REF].

The frequent occurrence of veering effect complicates the generation of continuous dispersion curves especially for the extrapolation-based modal curve tracing technique [6,[START_REF] Lowe | Disperse User's Manual Version 2.0.20a[END_REF] and the correlation-based mode shape analysis [START_REF] Nandyala | Effective stiffness matrix method for predicting the dispersion curves in general anisotropic composites[END_REF] for which the mode jumping problem is a big trouble. For example, the crossed A 1 and SH 1 curves in Fig. 3(c) of [37], the crossed S 0 and S ′ 0 curves in Fig. 5 of [11], the crossed S 0 and SH 0 curves in Fig. 3 the mode type of each master point. As for the color code in Figure 5.7, it will be discussed latter.

In Figure 5.8A, the shear horizontal displacement u 2 is predominates over u 1 and u 3 , indicating that point A should belong to the SH S0 mode, though not so pure. In Figure 5.8B, no displacement component is dominating among u 1 , u 2 , u 3 , thus S0 and SH S0 modes coexist in the turning point. In though not so pure. The mode shape analysis based on the three subfigures reveals that although points A, B, C locate at the same curve they have different behaviors. For points D, E, F, a similar behaviors can be observed and mode shapes at those points belong to S0, S0+SH S0 and SH S0 mode, respectively. In summary, after analyzing the veering effect in the small region R1, the two modal curves belong to the same symmetric modes and are very close but never cross. This is accompanied by a dramatically exchange of their displacement mode shapes.

APPLICATION OF THE OPTIMIZED GMM TO AN AEROSPACE COMPOSITE LAMINATE WITH 400 LAYERS

Special attention should be paid to Figure 5.8CD, though the two points belong to the same S0 mode, their minor components u 2 have opposite phases, namely, positive and negative amplitudes of u 2 in Figure 5.8C and D, respectively. The same phenomenon also occurs in Figure 5.8AF and BE, for which the latter is more prominent. This phenomenon reveals that, during the exchange of the major component of mode shapes, the minor components have a rollover in its phase.

The mode shape analysis was also used to produce continuous dispersion curves in some references [9]. However, if the turning points are not identified in the repulsion regions, point A will mistakenly jump to F, so does for point C to D, thus causing the illusion that the two modal curves cross over in the repulsion region. This illusion occurred in the Fig. 3(b) of [START_REF] Nandyala | Effective stiffness matrix method for predicting the dispersion curves in general anisotropic composites[END_REF], causing the mode jumping problem in this reference.

MAC value tomography of displacement mode shapes

A thorough solution to this problem is to compute the modal assurance criteria (MAC) values [START_REF] Ewins | Modal testing: theory, practice and application[END_REF] between the mode shapes of the consecutive dispersion solutions and the master points in the two modal curves. This has been achieved in Figure 5.7 in which the color code represents the MAC values of the loci points in one curve to the master point of the ellipse.

From these diagrams, it can be seen that the line segments near to the respective master points have larger MAC values. This is because the loci of the dispersion solutions is continuous, thus the closer to the master point, the more similar the mode shape will be, and causing a more larger MAC value. This phenomenon manifests that the turning point is a good indicator to distinguish mode branches in the repulsion regions to avoid mode jumping problem.

However, in

APPLICATION OF THE OPTIMIZED GMM TO AN AEROSPACE COMPOSITE LAMINATE WITH 400 LAYERS

Stress mode shape analysis

For the six master points in the repulsion region R1 of symmetric modes, we also plot their stress mode shapes in Figure 5.10 and perform the MAC value tomography in Figure 5.9. It can be seen that the stress mode shapes of the six master points are mutually similar except points A and F. Moreover, Figure 5.9 demonstrates that the MAC value of stress mode shapes is not an ideal indicator to distinguish mode branches because the minimum MAC value is 0.96, indicating the nonorthogonality of stress mode shapes. In fact, among the three components of stresses (σ 33 , σ 23 , σ 13 ), the σ 33 component is always the major one and the six points have the same σ 33 mode shapes. In short, it is not recommended to use stress mode shapes for distinguishing mode branches of dispersion solutions, and the displacement mode shapes are the best choice.

Computational efficiency

The computational efficiency of the optimized GMM is reported in this subsection for the 400layered laminate. The adopted solving methods to perform the test is bisection method and phase change method that will be introduced in Chapter 6. Computations were performed in MATLAB COMPOSITE LAMINATE WITH 400 LAYERS R2016a hosted on a Dell Precision T3500 workstation with the processor Intel Xeon W3530@2.8GHz, 4 CPU cores, 18G RAM. Bi-directional sweeping scheme is used, namely, fixing v = v 0 to sweep the roots of f and fixing f = f 0 to sweep the roots of v. In each direction, the sweeping line is equally offsetted 100 times from the minimum to the maximum, and the precision of solutions is set to be the fifth decimal place. The sweeping step length of phase velocity and frequency is 10 m/s and 1 kHz, respectively. Each solving method runs 10 times and the averaged computational time of the symmetric and anti-symmetric modes is listed in Table 5.2.

Table 5.2 shows that sweeping v costs more time than sweeping f . The explanations to this phenomenon are given in Chapter 6. By using the optimized GMM, the computations for symmetric and anti-symmetric modes can be completed within half an hour, which is an encouraging result given that the studied laminate has the largest number of layers reported so far (400 layers). We also adopted the standard GMM to perform the computation with the same running parameters listed above. Within a reasonable duration (24 hours), computations did not complete even for once running of program. We had to cut the program down and thus no any computational time can be supplied in Table 5.2 for the standard GMM. This fact proves that the thoroughly optimized GMM can greatly improve the computational efficiency when applying it to cope with a laminate having a large number of layers. s The computational time corresponds to the symmetric modes.

a The computational time corresponds to the anti-symmetric. modes.

Conclusion

This chapter reviews the standard GMM and derives the optimized GMM, and applying it to compute the dispersion curves of the Open Guided Waves plate and Ariane 6. The comparison between GMM computation and the experimental data validates the effectiveness of GMM framework.

In the higher frequency range, GMM is more stable than TMM because the latter will suffer from the large f d problem. On the contrary, ESM is only valid at lower frequency range. To improve the computational efficiency of GMM, three optimizations are introduced including (1) addition of symmetry condition to the global matrix, (2) applying the sparse matrix technique, (3) endowing the parallel computing attribute. The numerical study on a 400-layered aerospace composite laminate validates the effectiveness of the optimized GMM.

Chapter 6

Dichotomy property of dispersion equation of guided waves propagating in anisotropic composite plates

The previous chapters mainly focus on the way to build complex-valued dispersion equations for guided waves propagating in a single-layered plate via PWSA [72] of Chapter 3 or a multi-layered plate via TMM [START_REF] Nayfeh | Wave propagation in layered anisotropic media: With application to composites[END_REF]38] of Chapter 4 and GMM [2,[START_REF] Lammering | Lamb-wave based structural health monitoring in polymer composites[END_REF] of Chapter 5. Little attention is paid to develop efficient and stable numerical solving methods associated with the derived complex-valued dispersion equations. In this chapter, the conditions under which complex-valued dispersion equations are either real-or purely imaginary-valued equations (termed as dichotomy property) are derived for both singleand multi-layered composite plates. With such a property, the complex-valued dispersion equations can be efficiently numerically solved within the real number field via the standard bisection method or the corrected phase change method. It is thus now possible to overcome numerical issues frequently reported in literature. The proposed methodology of this chapter provides a new standard framework to solve the dispersion equations which is stable, multipurpose, and numerically efficient.

The works conducted in this chapter have been published in the journal Mechanical Systems and Signal Processing, see [START_REF] Guo | Dichotomy property of dispersion equation of guided waves propagating in anisotropic composite plates[END_REF].

Problem statement

In Chapter 3, 4 and 5, dispersion equations are built with PWSA for the single layer plate and TMM and GMM for the multi-layered system, respectively. It can be recognized that even if significant 6.1. PROBLEM STATEMENT efforts are carried out to find ways of building the dispersion equation via different methods, little work was conducted to study the property of the derived dispersion equation and to find efficient and stable associated numerical solving methods. In most references, once the derivation of dispersion equation is completed, one just says that the solution to this equation usually requires the use of numerical methods (see [37,8,38,[START_REF] Nandyala | Effective stiffness matrix method for predicting the dispersion curves in general anisotropic composites[END_REF]12]), but little details are provided.

Mathematically, the derived dispersion equation belongs to the family of complex-valued equations, which is more difficult to solve than a real-valued equation, due to the existence of complex exponential terms e iξαrx 3 presented in the related matrices. In general, the determinant in a certain dispersion equation, D(v, ξ), has real, D R (v, ξ), and imaginary, D I (v, ξ), parts. Thus, the intuitive way of solving the dispersion equation is to find the zero points of both parts and then take their intersections as the true solutions as presented in Eq. (6.1a), which is easy to implement because both D R (v, ξ) and D I (v, ξ) are real-valued functions such that many real variables-based root-finding algorithms can be used like the bisection method.

D(v, ξ) ≜ D R (v, ξ) + iD I (v, ξ) = 0 ⇔ { D R (v, ξ) = 0 D I (v, ξ) = 0 (6.1a) D(v, ξ) ≜ D R (v, ξ) + iD I (v, ξ) = 0 ⇔ abs{D(v, ξ)} = 0 (6.1b)
However, the majority of methods mathematically transform the solving process to search the global minimal moduli of the dispersion function that should be zero in theory, see Eq. (6.1b). A representative example is the pioneering software Disperse which employed a path-dependent minimal moduli search strategy combined with the convergence criteria of checking the included phase angle of the complex-valued characteristic function that should be greater than 90 • [START_REF] Lowe | Plate waves for the ndt of diffusion bonded titanium[END_REF][START_REF] Lowe | Disperse User's Manual Version 2.0.20a[END_REF]. The toolbox Elas-ticMatrix makes use of MATLAB built-in function fminbnd to achieve the minimal moduli searching purpose [START_REF] Ramasawmy | Elasticmatrix: A matlab toolbox for anisotropic elastic wave propagation in layered media[END_REF]. Another complex strategy adopted by the software Dispersion Calculator is to check, in a small interval, both the occurrence of minimum moduli and sign change of the dispersion function [11], see Figure 6.1. Recently, Zhu and Qian et al. developed an iterative method to solve the dispersion equation by also coping with the modulus of the dispersion function, which provides a strict convergence condition to distinguish the local and global minima [START_REF] Zhu | A numerical algorithm to solve multivariate transcendental equation sets in complex domain and its application in wave dispersion curve characterization[END_REF].

In comparison with the multi-layered anisotropic system, the standard bisection method is widely used for the single-layered isotropic Rayleigh-Lamb equation. The solving process of the Rayleigh-Lamb equation is achieved by dividing the whole solution domain into three regions, i.e., region 1 and c L are the velocities of transverse and longitudinal modes, respectively. Then in each region, the complex-valued Rayleigh-Lamb equation is derived to be a real one such that the standard bisection method can be adopted to solve it in an easy way [START_REF] Hakoda | Using the partial wave method for wave structure calculation and the conceptual interpretation of elastodynamic guided waves[END_REF]12]. Figure 6.2 just explains this process.

In addition to the two strategies introduced above, another interesting method is by consecutively monitoring the phase change of the dispersion function, which is actually evolved from the convergence criteria of Disperse. However, this method suffers from numerical instability issues when applied to multiple layered plates having 50 layers, see Fig. 7(c) of [9].

Inspired by the strategy of solving the Rayleigh-Lamb equation introduced above, the same idea of converting the complex-valued equation to a real one, which can be easily solved via standard root-finding algorithms, has been applied here to the single-layered and multi-layered anisotropic dispersion equation built with PWSA, GMM and TMM. Finally an important property of dispersion equations, termed as dichotomy property in this chapter, is demonstrated: they are either real-or purely imaginary-valued equations. On the basis of this property, it is shown that some numerical issues can be overcome and as a consequence the obtained dispersion curves become very stable.

PROPERTIES OF CHRISTOFFEL EQUATION OF MONOCLINIC MATERIALS

Properties of Christoffel equation of monoclinic materials

The investigation starts with the Christoffel equation of monoclinic material K(α)U = 0 that is presented in Eq. (3.22) with the detailed coefficients K ij (α) in Eq. (3.23). For other material classes, specifically, orthotropic, transversely isotropic and isotropic materials, the investigation made in this section is naturally compatible with these materials. But for triclinic materials, we directly take advantage of numerical validation due to the complexity of the derived equations from the triclinic materials.

Property of the polynomial discriminant ∆

The third order polynomial equation in terms of α 2 corresponding to the Christoffel equation of monoclinic material is presented in Eq. (3.24) and we duplicate this equation into Eq. (6.2) for convenience.

A(α 2 ) ≜ A 6 α 6 + A 4 α 4 + A 2 α 2 + A 0 = 0 (6.2)

PROPERTIES OF CHRISTOFFEL EQUATION OF MONOCLINIC MATERIALS

where, the coefficients A 6 , A 4 , A 2 , A 0 are presented in Eq. (3.25) which depends on the constant elastic coefficients C pq and mass density ρ as well as phase velocity v.

∆ = p 3 27 + q 2 4 , p = 3A 6 A 2 -A 2 4 3A 2 6 , q = 27A 2 6 A 0 -9A 6 A 4 A 2 + 2A 3 4 27A 3 6 (6.3)
Once the polynomial coefficients (A 6 , A 4 , A 2 , A 0 ) are known, the three α 2 roots can be solved from Eq. ( 6.2) via Cardano's Formula or polynomial root-finding algorithm, for which their behavior is determined by the discriminant ∆ defined in Eq. ( 6.3) according to the following three cases [9,77].

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
(1) ∆ < 0, all α 2 roots are real and unique;

(2) ∆ = 0, all α 2 roots are real and at least two are equal;

(3) ∆ > 0, one α 2 root is real and the other two are complex conjugates.

It is of great importance to state that the case ∆ > 0 will be automatically excluded for isotropic materials (see Ch. 5.6 of [START_REF] Nayfeh | Wave propagation in layered anisotropic media: With application to composites[END_REF] for more information). Actually, the absence of ∆ > 0 is not limited to isotropic materials, after a number of numerical investigations on various composite materials including transversely isotropic, orthotropic and monoclinic materials studied in the next subsection, only a few materials will lead to ∆ > 0 in a small phase velocity range. That is to say, for most composite materials, the case ∆ > 0 will be also automatically excluded. One convincing evidence catering this perspective comes from the Fig. 2 of [9]. In the following numerical examples, the existence of ∆ > 0 will be checked firstly for the sake of strictness. For ∆ ≤ 0, the three α 2 roots are all real numbers and can be sorted in Eq. (6.4a). The six α roots regardless of which case are obtained by satisfying the condition in Eq. (6.4b).

α 2 1 ≤ α 2 3 ≤ α 2 5 , if ∆ ≤ 0 (6.4a) α 2 = -α 1 , α 4 = -α 3 , α 6 = -α 5 (6.4b)

Illustration of the property of polynomial discriminant ∆

The typical ∆ -v relation, in two cases respectively representing the absence of ∆ > 0 and the presence of ∆ > 0, is plotted in Figure 6.3 based on Eqs. (3.25) and (6.3).

From Figure 6.3(b), it can be seen that the case ∆ > 0 only happens in a narrow phase velocity range for this particular composite material. Actually, most of composite materials reported in references will automatically exclude the case ∆ > 0 (similarly to Figure 6.3(a)). Only a few of ). After we performed a number of numerical investigations on various composite materials from references, a list denoting for various materials the absence or presence of ∆ > 0 is generated in Table 6.1 and 6.2, respectively. In the two tables, the listed materials have been rotated

PROPERTIES OF CHRISTOFFEL EQUATION OF MONOCLINIC MATERIALS

to the 45 • direction to keep the monoclinic type of stiffness matrix, otherwise it is indicated in the table.

Based on the results of the numerical investigations in Table 6.1 and 6.2 achieved on various composite materials, the occurrence of ∆ > 0 is application-dependent (and occurs only for some special materials). For most composite materials, the case ∆ > 0 will be automatically excluded. For sake of strictness, in each numerical example of this chapter, we firstly check whether or not the case ∆ > 0 existed.

Properties of intermediate parameters

In this subsection, we only focus on the case ∆ ≤ 0 to theoretically explore the property of some intermediate parameters that are used to generate dispersion equation. There are three reasons to exclude the case ∆ > 0 in this subsection: (1) the theoretical study on the case ∆ ≤ 0 is enough to show the essence of dichotomy property; (2) the theoretical derivation on the case ∆ > 0 is untractable, thus numerical validation is necessary and will be conducted in Section 6.5.2; (3) the case ∆ > 0 is an

DICHOTOMY PROPERTY OF DISPERSION EQUATION OF A SINGLE LAYER PLATE

extremely rare phenomenon for composite materials according to the explanation in Section 6.2.2.

For the case ∆ ≤ 0, all three α 2 roots are real numbers. Based on this, if α 2 r ≥ 0 then α r (= ± √ α 2 r ) is a real number regardless of being positive or negative, while if α 2 r < 0 then α r is a purely imaginary number. In order to investigate the dichotomy property of dispersion equations, three properties of the intermediate parameters V r , W r , β 1r , β 2r , β 3r , B 5 , which are the intermediate parameters to generate dispersion equation, depending on α r are listed below and the proof is provided in Appendix A.

Property 1: α r ∈ R =⇒ (V r , W r , β 1r , β 2r , β 3r ) ∈ R 5 . Property 2: α r ∈ iR =⇒ (V r , β 1r ) ∈ R 2 and (W r , β 2r , β 3r ) ∈ iR 3 . Property 3: ∀α r , B 5 ∈ iR.
When the six roots of Eq. ( 6.2) are obtained for a specific monoclinic material and v 0 , one can sort them to satisfy the condition in Eq. (6.4a

) (α 2 1 ≤ α 2 3 ≤ α 2 5
). This leads to the following four cases:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
(1) all α 1 , α 3 , α 5 are real numbers

( 0 ≤ α 2 1 ≤ α 2 3 ≤ α 2 5 ) ; (2) only α 1 is a purely imaginary number ( α 2 1 < 0 ≤ α 2 3 ≤ α 2 5 ) ; (3) only α 1 , α 3 are purely imaginary numbers ( α 2 1 ≤ α 2 3 < 0 ≤ α 2 5
) ; (4) all α 1 , α 3 , α 5 are purely imaginary numbers

( α 2 1 ≤ α 2 3 ≤ α 2 5 < 0 ) .
It is obvious that the boundary of the four cases is α 1 = 0, α 3 = 0, α 5 = 0, respectively, which can be reached by letting A 0 = 0 in Eq. (6.2). Keeping in mind the expression of A 0 in Eq. (3.25), this further leads to the definition of three critical velocities, which characterize the three bulk wave velocities (longitudinal, shear horizontal and shear vertical modes) propagating along the lamina plane.

Note that dispersion equation has a singularity at the bulk wave velocity, thus, some outliers in the dispersion solutions will occur. For example, the constant phase velocities appearing in Fig. 3 of [START_REF] Barazanchy | A comparative convergence and accuracy study of composite guided-ultrasonic wave solution methods: Comparing the unified analytic method, safe method and disperse[END_REF] and produced by the software Disperse can be interpreted as one of these outliers.

v I = √ A + √ B 2ρ , v II = √ A - √ B 2ρ , v III = √ C 55 ρ (6.5)
where, A = C 11 + C 66 and B = (C 11 -C 66 ) 2 + 4C 2 16 .

Dichotomy property of dispersion equation of a single layer plate

In this section, the single-layered paradigm will be concerned. The dispersion equation of antisymmetric modes derived in Section 3.3.3 will be adopted to study the dichotomy property. The

DICHOTOMY PROPERTY OF DISPERSION EQUATION OF A SINGLE LAYER PLATE

tools used for anti-symmetric modes are readily applicable to symmetric modes. For convenience, the related equations are duplicated here.

D a (v, ξ) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 1 1 1 1 1 V 1 V 1 V 3 V 3 V 5 V 5 β 11 β 11 β 13 β 13 β 15 β 15 β 11 e iξα 1 h β 11 e -iξα 1 h β 13 e iξα 3 h β 13 e -iξα 3 h β 15 e iξα 5 h β 15 e -iξα 5 h β 21 e iξα 1 h -β 21 e -iξα 1 h β 23 e iξα 3 h -β 23 e -iξα 3 h β 25 e iξα 5 h -β 25 e -iξα 5 h β 31 e iξα 1 h -β 31 e -iξα 1 h β 33 e iξα 3 h -β 33 e -iξα 3 h β 35 e iξα 5 h -β 35 e -iξα 5 h ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (6.6) D a (v, ξ) = D R a (v, ξ) + iD I a (v, ξ) ≜ det{D a (v, ξ)} = 0 implicit form (6.7)
where, the subscript 'a' denotes the anti-symmetric modes.

D a (v, ξ) =[B 1 sin(ξα 1 h) cos(ξα 3 h) cos(ξα 5 h) + B 2 cos(ξα 1 h) sin(ξα 3 h) cos(ξα 5 h) + B 3 cos(ξα 1 h) cos(ξα 3 h) sin(ξα 5 h)]B 5 = 0 explicit form (6.8)
where,

B 1 = β 11 (β 25 β 33 -β 23 β 35 ), B 2 = β 13 (β 21 β 35 -β 25 β 31 ), B 3 = β 15 (β 23 β 31 -β 21 β 33 ) B 5 = 8i [V 1 (β 15 -β 13 ) + V 3 (β 11 -β 15 ) + V 5 (β 13 -β 11 )]
(6.9)

Dichotomy property of dispersion equation when sweeping ξ

6.3.1.1 Case 1: all α 1 , α 3 , α 5 are real numbers.

For case 1, according to Property 1, all β 1r , β 2r , β 3r (r = 1, 3, 5) are real numbers. The three coefficients B 1 , B 2 , B 3 defined in Eq. (6.9) are real numbers as well. Besides, set B 5 = iP 5 where P 5 is a real number according to the Property 3. Thus, the dispersion function D a (v, ξ) defined in Eq. (6.8) can be written as follows.

D a (v, ξ) = i[B 1 sin(ξα 1 h) cos(ξα 3 h) cos(ξα 5 h) + B 2 cos(ξα 1 h) sin(ξα 3 h) cos(ξα 5 h) + B 3 cos(ξα 1 h) cos(ξα 3 h) sin(ξα 5 h)]P 5 = iD I a (v, ξ) ⇒ a purely imag. quantity (6.10)
It is evident from Eq. (6.10) that D a (v, ξ) should be a purely imaginary-valued function in case 1

given that all the terms in this function are real numbers other than the imaginary number unit. This is equivalent to say that its real part is identically vanishing, D R a (v, ξ) ≡ 0. Thus, the original complex-valued equation D a (v, ξ) = 0 now has been transformed to D I a (v, ξ) = 0, which is a real-valued With the new definitions, keeping in mind that sin(iξa

1 h) = i sinh(ξa 1 h), cos(iξa 1 h) = cosh(ξa 1 h),
where sinh(•) and cosh(•) is the hyperbolic sine and cosine function, respectively, D a (v, ξ) defined in Eq. (6.8) can be deduced as follows.

D a (v, ξ) = -[B 1 sinh(ξa 1 h) cos(ξα 3 h) cos(ξα 5 h) + P 2 cosh(ξa 1 h) sin(ξα 3 h) cos(ξα 5 h) + P 3 cosh(ξa 1 h) cos(ξα 3 h) sin(ξα 5 h)]P 5 ⇒ a real quantity (6.12)

This result reveals that in case 2, the imaginary part is identically vanishing, D I a (v, ξ) ≡ 0. Thus, the original complex-valued equation D a (v, ξ) = 0 now has been converted to D R a (v, ξ) = 0, which is a real-valued equation easy to be solved via bisection method. where, P 1 , P 2 are definitely real numbers.

DICHOTOMY PROPERTY OF DISPERSION EQUATION OF A SINGLE LAYER PLATE

Substitute Eq. (6.13) into Eq. (6.8) to lead to the following form. Obviously, as with case 1, the solving process should be focused on the imaginary part in case 3, D I a (v, ξ) = 0.

D a (v, ξ) =i[-P 1 sinh(ξa 1 h) cosh(ξa 3 h) cos(ξα 5 h) -P 2 cosh(ξa 1 h) sinh(ξa 3 h) cos(ξα 5 h) Under the new settings, Eq. ( 6.8) can be further deduced to Eq. (6.16). Hence, case 4 has the same dichotomy property as with case 2, i.e., the solving process should be focused on the real part,

+ B 3 cosh(ξa 1 h) cosh(
D R a (v, ξ) = 0. D a (v, ξ) = -[B 1 sinh(ξa 1 h) cosh(ξa 3 h) cosh(ξa 5 h) + B 2 cosh(ξa 1 h) sinh(ξa 3 h) cosh(ξa 5 h)
+ B 3 cosh(ξa 1 h) cosh(ξa 3 h) sinh(ξa 5 h)]P 5 ⇒ a real quantity (6.16)

Numerical validation of the dichotomy property when sweeping ξ

The chosen structure is a single layer plate with monoclinic type stiffness matrix shown in Eq. (6.17).

The mass density is ρ = 1500 kg/m 3 and the plate thickness is d = 1 mm. These material properties are cited from [9].

C = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
102.6 24.1 6.3 0 0 40 18.7 6.4 0 0 10 13.3 0 0 -0.1 3.8 0.9 0 sym 5.3 0 23.6

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ GPa (6.17)
The absence of ∆ > 0 has been checked in the Fig. 2(d) of [9] that shows the three real-valued α 2 roots. With the provided material properties, the three critical velocities can be computed based LAYER PLATE on Eq. (6.5), which, therefore, divide the whole domain of phase velocity v into four ranges that correspond to the four cases presented in Table 6.3.

Table 6.3 -The ranges of phase velocity v [m/s] that correspond to the four cases.

Case 1 75, +∞) [2142. 25, 8918.75) [1879.72, 2142.25) (0, 1879.72)

0 ≤ α 2 1 ≤ α 2 3 ≤ α 2 5 Case 2 α 2 1 < 0 ≤ α 2 3 ≤ α 2 5 Case 3 α 2 1 ≤ α 2 3 < 0 ≤ α 2 5 Case 4 α 2 1 ≤ α 2 3 ≤ α 2 5 < 0 [8918.

Analysis of Case 1:

Firstly, we assign a typical phase velocity v 0 = 9000 m/s according to Table 6.3, and substitute the specified material parameters into Eq. (3.25) to calculate the polynomial coefficients of Eq. ( 6.2).

Then, the three α 2 roots of the bi-cubic equation, A(α 2 ) = 0, can be solved via Cardano's Formula, from which the six α roots are finally obtained with the three predominant ones being α 1 = 0.5667, α 3 = 3.2007, α 5 = 5.8644. All the three terms are real numbers, which is in agreement with the precondition of case 1.

Once α 1 , α 3 , α 5 are obtained, all the intermediate parameters can be calculated. Then, the implicit dispersion equation for anti-symmetric modes defined in Eq. (6.7) can be generated, which is a complex-valued equation with respect to ξ at v 0 = 9000 m/s due to the existence of the complex exponential terms in matrix D a (v, ξ). Thus, the standard solving procedures should handle both real and imaginary parts, represented in Eq. (6.1a). We plot its function curves of real and imaginary parts in Figure 6.4 to intuitively study the distribution of solutions of the equation D a (v 0 = 9000, ξ) = 0 before solving it via bisection method.

Let us focus on the implicit curve firstly in both subfigures of Figure 6.4. For the real part, its curve is oscillating and chaotic, superficially indicating 'many spurious roots' in the presented x-axis range. For the imaginary part, its curve is continuous and smooth and the seven zero-points can be clearly recognized as marked by red circles. Comparing the order of magnitude for both curves, the one of real part (10 -8 ) is largely smaller than the one of imaginary part (10 7 ).

Then, we superimpose the explicit curves to make deep comparison with the implicit curves, where the explicit curves are plotted based on the explicit expression, Eq. (6.10). In the subfigure of imaginary part both curves are totally overlapped because the explicit expression just succeeds from the implicit one. In the subfigure of real part the zero-valued explicit curve reflects the inference of Eq. (6.10) in a numerical way. As for the chaotic implicit curve, which should have been totally zero-valued in theory like the explicit curve, the reason lies in the numerical errors, e.g. round-off error or machine epsilon, that is inevitable when performing numerical analysis.

Anyway, the solving process should be focused on the effective part (imaginary part in this case).

By sweeping ξ with a small enough step within an interested range, the accurate solutions of ξ can be solved via bisection method as shown in Figure 6.4(b). These solutions correspond to the points of dispersion curves, v versus ξ, of the anti-symmetric modes at v 0 = 9000 m/s. The conventional dispersion curves v versus ω can be transformed from v versus ξ through ω = ξv.

One positive effect of dichotomy property is the ability to overcome the numerical instability issue encountered in [9], which adopts a phase change method1 to solve the dispersion equation. In the thesis, we exemplify the solution of ξ 1 = 538.95 to illustrate our improvement to the phase change method adapted to dichotomy property.

In the implicit curve of Figure 6.4(b), there are two points 1 and 2 around the solution ξ 1 = 538.95.

We map the two points on the complex plane as schematically shown in Figure 6.5(a). It can be seen that each point may appear at any one of two positions in the complex plane due to the existence of tiny real part, see Figure 6.4(a), caused by numerical error which can be regarded as a random disturbance to the theoretic positions locating on the y-axis in Figure 6.5(a). In practice, no matter

DICHOTOMY PROPERTY OF DISPERSION EQUATION OF A SINGLE LAYER PLATE which 1

′ or 1 ′′ is identified, the phase angle of point 1 is approximately equal to -90 • , and the phase angle of point 2 is approximately equal to 90 • . In Figure 6.

DICHOTOMY PROPERTY OF DISPERSION EQUATION OF A SINGLE LAYER PLATE

Analysis of Case 2:

Adopting the previously introduced material properties and assigning a typical phase velocity v 0 = 5000 m/s according to Table 6.3, the three predominant α solutions can be easily obtained as α 1 = 3.6014i, α 3 = 1.5933, α 5 = 3.0868. Apparently, only α 1 is a purely imaginary number, which is in agreement with the precondition of case 2. With the solved α 1 , α 3 , α 5 , the implicit Eq. (6.7) and explicit Eq. ( 6.12) can be illustrated in Figure 6.6, with the real and imaginary part respectively, to intuitively study the distribution of its solutions. As predicted in Eq. ( 6.12), for case 2, the real part becomes effective whereas due to the existence of numerical error the imaginary part becomes chaotic.

Finally, the three solutions of ξ at v 0 = 5000 m/s can be solved using bisection method as marked in In case 2, particular attention should be paid to the phase change method due to the existence of tiny imaginary part which will give rise to disorder in the phase angle. We take the solution of ξ 2 = 3025.76 as the example to illustrate this issue. Map the two points 1 and 2 in Figure 6.6(a) on the complex plane as schematically shown in Figure 6.7. Like in case 1, each point may appear at any one of two points in the complex plane due to the disturbance of tiny imaginary part. Thus, point 1 ′ and 1 ′′ will be identified as two different phase angles respectively approaching to 180 • and -180 • , which will further give rise to oscillating in the phase angle curve as illustrated in Figure 6.8(a). If directly applying phase change method to such an disordered curve, instability issue like spurious or loss roots will be faced [9]. Thus, when sweeping ξ to monitor phase change, the spurious phase angle of -180 •

DICHOTOMY PROPERTY OF DISPERSION EQUATION OF A SINGLE LAYER PLATE

should be corrected as 180 • firstly, as displayed in Figure 6.8(b). Eventually, with the corrected phase angle curve, the ξ solutions can be obtained via phase change method as indicated in Figure 6.8(b), which are the same as by using bisection method in Figure 6.6(a).

The theoretical position of 1

The theoretical position of 2

Dichotomy property of dispersion equation when sweeping v

The efforts made above are related with the dichotomy property when sweeping wavenumber ξ.

Alternatively, this property is also owned when sweeping phase velocity v at a fixed ξ 0 but with more complexity, since at a fixed ξ 0 , the process of sweeping v will cross all the four cases of α 1 , α 3 , α 5 .

However, when sweeping ξ, a single v 0 only leads to an unitary case among the four possible ones.

Thanks to the theoretical derivations previously carried out, it is possible to automatically adapt when

DICHOTOMY PROPERTY OF DISPERSION EQUATION OF A SINGLE LAYER PLATE

sweeping phase velocity. We only take here a numerical example to illustrate the solving process of the dispersion equation D a (v, ξ 0 ) = 0 via both bisection and phase change methods which considers the dichotomy property when sweeping phase velocity.

It should be noted that the intermediate parameters, α r , V r , W r , β ir (r = 1, 3, 5; i = 1, 2, 3) only depend on phase velocity v for the given material parameters. Thus, when sweeping ξ at a fixed v 0 , all evaluations of D a (v 0 , ξ) at all stagnation steps of ξ share the same intermediate parameters.

However, when sweeping v at a fixed ξ 0 , these intermediate parameters should be computed repeatedly at different stagnation steps of v, thus decelerating the sweeping speed. This point will be assessed in Section 6.7.1 by comparing the computational time of the two sweeping schemes. It should be noted that the phase angle curve in case 2 and 4 has been corrected in Figure 6.9, and in case 3 there is no roots. In any case, the curves of real and imaginary parts should be fed to bisection method, or the corrected phase curve should be fed to phase change method, to get the solutions of v as indicated in Figure 6.9.

After comprehensively studying the dichotomy property of dispersion equation for the anti-symmetric modes, we computed its complete dispersion curves as shown in Figure 6.10(a), in which the horizontal and vertical dotted lines highlight the sweeping lines used in the previous paragraphs to exemplify the dichotomy property of different cases. The dichotomy property of symmetric modes, though not presented in this thesis, can be also investigated by following the methodology used for anti-symmetric modes, thus its dispersion curves have been depicted in Figure 6.10(b).

Comparing the two schemes of sweeping ξ and v from Figure 6.10, sweeping ξ will lose many data points in a mode's non-dispersive ranges, like A0 mode in higher wavenumber, S0 mode in lower wavenumber and SH0 mode in the whole range of wavenumber. However, sweeping v will lose data points near to a mode's cutoff wavenumber, like A1 mode near to 500 [1/m]. Therefore, in order to generate the complete dispersion curves, sweeping ξ and v simultaneously then combining both solution sets are necessary. In reference [9], an interpolation method is adopted to reconstruct the missing points in the discrete dispersion solutions. This is only valid for lower modes.

Dichotomy property of dispersion equation in f -ξ and f -v domains

The dispersion curves v versus ξ retrieved from the symmetry-ignored dispersion equation D(v, ξ) = 0 belong to actually the wavenumber-phase velocity domain. The conventional dispersion curves f versus ξ and v versus f can be retrieved from Eq. (6.18a) and Eq. (6.18b), respectively, through the change methods can be used to be the solver for the two equations, but the effective part should be identified firstly, and if the real part is the effective one, phase correction measure should be taken to avoid numerical instability issue.

Dichotomy property of evanescent waves

When it comes to evanescent waves, the wavenumber becomes purely imaginary ξ = iζ wherein ζ is a real number. In contrast to propagating waves, the energy of evanescent waves is quickly dissipated, as a result, its range of propagation is limited to the near field of source. The evanescent waves have received increasing attention in the recent years due to its high sensitivity to minor defects of composite structures [START_REF] Yan | Conversion of evanescent lamb waves into propagating waves via a narrow aperture edge[END_REF][START_REF] Giurgiutiu | Propagating, evanescent, and complex wavenumber guided waves in high-performance composites[END_REF].

In this subsection, we prove the dichotomy property of evanescent waves propagating in a single layer plate with monoclinic property, under the condition ∆ ≤ 0. The derivation directly starts from Eqs. (6.10)(6.12)(6.14)(6.16) which are the explicit expressions showing the dichotomy properties of the propagating waves for the four cases.

Substitution of ξ = iζ into Eqs. (6.10)(6.12)(6.14)(6.16) and upon simplification via the identical relations sinh(ix) = i sin(x) and cosh(ix) = cos(x), the dichotomy property of evanescent waves for the four cases are concluded as follows.

For case 1, all α 1 , α 3 , α 5 are real numbers.

D a (v, iζ) = -[B 1 sinh(ζα 1 h) cosh(ζα 3 h) cosh(ζα 5 h) + B 2 cosh(ζα 1 h) sinh(ζα 3 h) cosh(ζα 5 h) + B 3 cosh(ζα 1 h) cosh(ζα 3 h) sinh(ζα 5 h)]P 5 = D R a (v, ζ) ⇒ D I a (v, ζ) ≡ 0 (6.19)
For case 2, only α 1 is a purely imaginary number.

D a (v, iζ) = -i[B 1 sin(ζa 1 h) cosh(ζα 3 h) cosh(ζα 5 h) + P 2 cos(ζa 1 h) sinh(ζα 3 h) cosh(ζα 5 h) + P 3 cos(ζa 1 h) cosh(ζα 3 h) sinh(ζα 5 h)]P 5 = iD I a (v, ζ) ⇒ D R a (v, ζ) ≡ 0 (6.20)

DICHOTOMY PROPERTY OF DISPERSION EQUATION OF A MULTI-LAYERED PLATE

For case 3, only α 1 , α 3 are purely imaginary numbers.

D a (v, iζ) =[P 1 sin(ζa 1 h) cos(ζa 3 h) cosh(ζα 5 h) + P 2 cos(ζa 1 h) sin(ζa 3 h) cosh(ζα 5 h) -B 3 cos(ζa 1 h) cos(ζa 3 h) sinh(ζα 5 h)]P 5 = D R a (v, ζ) ⇒ D I a (v, ζ) ≡ 0 (6.21)
For case 4, all α 1 , α 3 , α 5 are purely imaginary numbers.

D a (v, iζ) = -i[B 1 sin(ζa 1 h) cos(ζa 3 h) cos(ζa 5 h) + B 2 cos(ζa 1 h) sin(ζa 3 h) cos(ζa 5 h) + B 3 cos(ζa 1 h) cos(ζa 3 h) sin(ζa 5 h)]P 5 = iD I a (v, ζ) ⇒ D R a (v, ζ) ≡ 0 (6.22)

Dichotomy property of dispersion equation of a multi-layered plate

Chapter 4 and 5 have presented the dispersion equation D X (v, ξ) = 0 in the context of a multilayered plate system, where the subscript 'X' represents T and G for TMM and GMM, respectively. In the context of a single monoclinic lamina, we have also observed the dichotomy property of D X (v, ξ), that is to say, in some ranges of v the real part is identically vanishing, D R X (v 0 , ξ) ≡ 0, while in other ranges of v, D I X (v 0 , ξ) ≡ 0. Strictly proving this property for an arbitrary n-layered laminate is not reachable since the explicit expression of D X (v, ξ) is no longer available. Furthermore, in extreme conditions, the total number of cases in a general n-layered laminate will be as large as 3n + 1 given that each layer has three critical velocities according to Eq. (6.5) such that the case by case deduction is no more realizable.

In order not to stuck in intractable mathematical pitfall, we propose a sampling strategy to investigate the dichotomy property of dispersion equation in a numerical way through making qualitative analysis. Inspired by the investigation on the single monoclinic lamina, the dichotomy property of dispersion equation in the context of laminate stems from the property of the intermediate parameters of each layer, which is actually controlled by the common phase velocity v once each layer's material properties are given. Thus, no matter what value the fixed v 0 is, the implicit function D X (v 0 , ξ) defined in Eq. (4.39) for TMM and Eq. (5.7) for GMM with ξ being the argument can be divided into real part D R X (v 0 , ξ) and imaginary part D I X (v 0 , ξ), no matter which part being the vanished one, it will not be totally zero in practice due to numerical error. However, we can uniformly sample m points for the two functions within the interested range [ξ min , ξ max ], and find the maximum absolute function value from these sampling points for each part.

M R = max 1≤i≤m |D R X (v 0 , ξ i )|, M I = max 1≤i≤m |D I X (v 0 , ξ i )| for ξ i ∈ [ξ min , ξ max ] (6.23)

DICHOTOMY PROPERTY OF DISPERSION EQUATION OF A MULTI-LAYERED PLATE

Finally, the dichotomy property is determined by comparing the true maximum values between the two parts, meanwhile, the effective part can be obtained by using the following rule.

-

If M R ≫ M I , the effective part is D R X (v 0 , ξ) and D I X (v 0 , ξ) ≡ 0; -If M R ≪ M I , the effective part is D I X (v 0 , ξ) and D R X (v 0 , ξ) ≡ 0.
The sampling strategy is achievable due to the fact that the vanished part, though polluted by numerical error, is lower than the effective part by several orders of magnitude. Furthermore, this sampling strategy is very easy to implement and it is also an universal strategy to determine the dichotomy property of dispersion equation not only applicable for TMM and GMM but also for SMM 2 , and the single lamina (for both cases ∆ ≤ 0 and ∆ > 0) as concluded in Figure 6.11(a). In this figure, using the enumeration strategy to determine the dichotomy property of a single-layered plate reflects the know-why of this property, and adopting the sampling strategy embodies the knowhow to characterize this property for a complex structure in practice.

Bisection method Phase change method

Comments on the numerical labels:

(1) The intermediate parameters keep constants for a fixed 𝑣 0 , which maximizes the computational efficiency of both methods.

(2) The intermediate parameters are computed repeatedly for different 𝑣, which slows down the computational efficiency of both methods. With the deterministic dichotomy property for any case, the dispersion equation can be solved by sweeping ξ at a fixed v 0 or sweeping v at a fixed ξ 0 via bisection or phase change methods, as summarized in Figure 6.11(b), in which two stars represent a higher computational efficiency than one star. This subfigure is also validated in Section 6.7.1.

OVERCOMING COMMON NUMERICAL ISSUES USING DICHOTOMY PROPERTY

Overcoming common numerical issues using dichotomy property

After studying the dichotomy property, some numerical issues frequently reported in literature can be overcome with the help of dichotomy property. Two typical issues are the instability of the phase change method and the instability of TMM.

Overcoming numerical instability of the phase change method

In [9] numerical instabilities were encountered when generating the dispersion curves of a 50-layered orthotropic laminate via GMM, see Figure 6.12(b). This laminate has a special layup [0] 50 , i.e. each layer orients at the same fiber direction. The authors of [9] alleged that the numerical instabilities were caused by the large number of layers under study, and the maximum ability of their method to compute a laminate must be no more than 13 layers in order to obtain satisfying results for GMM.

In this section, we recompute the dispersion curves of this laminate by using the same GMM but corrected phase change method (CPCM), as shown in Figure 6.12(a). The adopted material properties can be referred from [9,[START_REF] Guo | Dichotomy property of dispersion equation of guided waves propagating in anisotropic composite plates[END_REF]. Figure 6.12(a) shows that our results do not suffer from instability issue anymore. The reason is that when solving dispersion equation built with GMM via phase change method, at the region of missing roots in Figure 6.12(b), phase correction measure should be taken.

The comparison between the original phase change method and the corrected phase change method is conducted in Figure 6. [START_REF] Farrar | An introduction to structural health monitoring[END_REF]. It shows that the chaos uncorrected phase curve leads to the missing roots in Figure 6.12(b). In contrast, the corrected phase curve clearly shows the locations of solutions at each step of the corrected phase curve. Thus, by applying phase correction measure, the instability issue is just addressed.

Overcoming numerical instability of TMM

TMM usually suffers from instability issue known as the so-called large f d problem which refers to the unstable solutions at a large product value of frequency f and plate thickness d. One typical example citing from [42] is plotted on Figure 6.14(b) for illustration, in the frequency-wavenumber domain. In this graph, the red curve divides the graph into two parts, with the left part of solutions via TMM and right part via SMM, because either method suffered from instability in its opposite part as explained in [42]. and (c) initial results, for the proposed unified formulation to predict guided-ultrasonic wave (GUW) dispersion curves was given in this article.

OVERCOMING COMMON NUMERICAL ISSUES USING DICHOTOMY PROPERTY

The simple, and mathematically straightforward unified formulation was based on Christoffel's equation of a lamina. First, the a 2 values part of the formulated eigenvalue problem were solved using the bi-cubic equation. Second, a values were used to solve the eigenvalue problem that yielded three pair of eigenvalues and eigenvectors. The eigenvectors corresponding to a zero eigenvalue form the solution to the formulated eigenvalue problem.

The eigenvectors with their corresponding a values were used to produce the stress and displacement field matrices. The traction free boundary condition at the top and bottom surface of the medium were applied to the stress field matrix, and using the zero-determinant condition the wavenumber-wavespeed pairs which formed the dispersion curves were retrieved using a phase approach.

A phase a nants in the w a phase angl two consecut were within t curve pair wa and simple alt complex num change in the finding comm After the d shape analysi wavespeed pa shape analysi gonality of th used to obtai domain. Usin The presence of ∆ > 0 is checked for this material by making the curve of ∆ with respect to v as shown in Figure 6.15(a), as long as substituting the corresponding material parameters in [42,[START_REF] Guo | Dichotomy property of dispersion equation of guided waves propagating in anisotropic composite plates[END_REF] into Eq. (3.25) and Eq. (6.3). Figure 6.15(a) shows the presence of ∆ > 0 when v ∈ (0, v ∆ ) where v ∆ = 1648.05 such that ∆(v ∆ ) = 0. For this special case, there is a need to explore whether or not the dichotomy property still holds. Nonetheless, the complexity of this case hinders the theoretical manner on dichotomy property that is used in the enumeration strategy. Thus, making numerical validation is the sensible way. For this purpose, we arbitrarily assign v 0 = 1000 that consequently leads to ∆ = 24.05 and α 2 1 = -11.89, α 2 3 = -0.69 -0.20i, α 2 5 = -0.69 + 0.20i. Clearly, α 2 1 is a real number and α 2 3 and α 2 5 are mutually complex-conjugated. Then, the function curve of D a (v 0 = 1000, ξ) can be generated in Figure 6.15(b) as real and imaginary part, respectively. This subfigure clearly shows the preservation of dichotomy property in the special case ∆ > 0, with the real part effective that is identified automatically through the sampling strategy formalized in Eq. (6.23).

The frequency-wavenumber domain dispersion equation, Eq. (6.18a), is employed to recompute the dispersion curves of this plate as delineated in Figure 6.14(a) lying in the right part being the propagating waves. It can be seen that the instability issue does not occur in our computation, thus proving the stability and robustness of our method that accounts for the dichotomy property.

EXPLORING THE LARGE FD PROBLEM OF TMM USING DICHOTOMY PROPERTY

The curves of evanescent waves shown in Figure 6.14(a) lying in the left part are retrieved by considering the dichotomy property of evanescent waves which has been proved in Section 6.3.4. It can be seen that the evanescent waves connect with the propagating waves at zero wavenumber.

Furthermore, some evanescent modes connect two different propagating modes possessing the same symmetry. For example, the first anti-symmetric evanescent mode connects the propagating A0 and SH A0 modes, and the first symmetric evanescent mode connect the propagating SH S1 and SH S2 modes.

This phenomenon has also been observed in [START_REF] Giurgiutiu | Propagating, evanescent, and complex wavenumber guided waves in high-performance composites[END_REF] but via semi-analytical finite element (SAFE) method for most of non-isotropic materials. Our theoretical analysis on the unidirectional composite plate contribute a plus to the theory of guided waves propagation in composite materials.

Finally, the emphasis should be placed on the multi-layered composite laminate in terms of the dispersion equation of evanescent waves built by GMM, which possesses the dichotomy property as well. Thus, as the analysis for propagating waves in a laminate, the sampling strategy should be a simple and efficient tool to retrieve the dispersion curves of the evanescent waves.

Exploring the large fd problem of TMM using dichotomy property

In Figure 5.1(a), TMM suffers from the large f d problem. In this section, we adopt the methodology of studying dichotomy property to investigate this problem. The function curve of D X (v 0 , ξ) = 0 at v 0 = 1000 m/s are presented in Figure 6.16 for both TMM and GMM. From Figure 6.16(a)(b), both the real and imaginary parts of TMM become disordered and heavily oscillating in the high frequency range, which produces many spurious roots in the high frequency range of Figure 5.1(a). In other words, there is no effective part for TMM at v 0 = 1000 m/s. However, for GMM, the effective part (imaginary part in this case) is still observable. The unique root can be obtained from the effective part of GMM via bisection method.

The true reason of causing the large f d problem of TMM lies in the fact that the monodromy transfer matrix, T mono defined in Eq. (4.34), has a poor condition number. To illustrate this, the condition number of the monodromy transfer matrix and global matrix are presented in Figure 6.17.

It can be seen that for GMM, poor condition occurs only at the unique root location, but for TMM, poor condition occurs in high frequency range. Thus, TMM usually suffers from the large f d problem at higher frequency range.

DISCUSSION

After we recomputed the dispersion curves of composite structures in some classical references, which pertained to studying various computational models of dispersion curves of composite materials, a list that summarizes the presence or absence of large f d problem of TMM has been made in Table 6.4.

This table demonstrates that TMM does not always encounter the large f d problem. It is application dependent.

Table 6.4 -The presence or absence of large f d problem of TMM in classical references.

1 This item is used to generate Figures 6.16 and 6.17.

Note: the tested frequency range is up to 2000 kHz that is the highest frequency commonly used for guided waves-based structural health monitoring techniques.

Discussion

Comparison of computational efficiencies of two sweeping schemes

In this section, we compare the computational efficiencies of the two sweeping schemes for solving dispersion equation, i.e. fixing v = v 0 to sweep ξ roots and fixing ξ = ξ 0 to sweep v roots by using two solving methods (bisection and phase change methods). The tested structure is the 50-layered plate in Section 6.5.1. Both serial and parallel 3 computing techniques are applied to the standard GMM.

Computations were performed in MATLAB R2016a hosted on a Dell Precision T3500 workstation with the processor Intel Xeon W3530@2.8GHz, 4 CPU cores, 18G RAM. Computing range is shown 3. Parallel computing technique is introduced in Section 5.3.3.

DISCUSSION

in Figure 6.12. Bi-directional sweeping scheme is used. In each direction, the sweeping line is equally offsetted 100 times from the minimum to the maximum, and the precision of solutions is set to be the fifth decimal place. The sweeping step length of phase velocity and wavenumber is 10 m/s and 10 [1/m], respectively. Each solving method runs 10 times and the averaged computational time is listed in Table 6.5. Table 6.5 shows that sweeping v costs more time than sweeping ξ irrespective of the serial or parallel computing technique. The explanations to them are given in Section 6.3.2. Additionally, the speed-up ratio of the computational time between parallel and serial computing technique is as large as 2.8 for a computer with four CPU cores, which is desired. Finally, the computational time of CPCM is slightly greater than the counterpart of bisection method, which is due to the phase correction operation. The comparison result of Table 6.5 validates Figure 6.11(b).

Generalization of dichotomy property in other material classes

The methodology developed in this chapter can be also applied to other material classes like triclinic material, which is full anisotropy having 21 independent stiffness coefficients, and piezoelectric material, whose dispersion equations are more complex than composite materials. After investigation for a certain material, if dichotomy property does not present in it, both the real and imaginary parts of dispersion function are effective in this case such that the true roots of the dispersion equation are the intersected roots of real and imaginary parts, i.e. Eq. (6.1a) should be applied. An example of this case for a triclinic material 4 is illustrated in Figure 6.18(a). 5 However, for this case, phase change method is still applicable without need to take phase correction measure because phase angle changes smoothly with searching variable as illustrated in Figure 6.18(b).

4. The dispersion equation of a single layer plate with triclinic material is presented in Section 3.2. 5. The used material properties can be referred from [START_REF] Guo | Dichotomy property of dispersion equation of guided waves propagating in anisotropic composite plates[END_REF]45].

CONCLUSION

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

[1/m] -8 -6 -4 -2 0 2 4 6 8 10 12 -8 -6 -4 -

Conclusion

The dispersion equation of guided waves propagating in elastic composite plates are not a completely complex-valued equation, and it presents the dichotomy property of being either a real-or purely imaginary-valued equation. This property is strictly investigated firstly for a single-layered monoclinic lamina in the case ∆ ≤ 0 by using the enumeration strategy, showing that there exists four cases for α 1 , α 3 , α 5 depending on their properties of being real or purely imaginary number. The three parameters will lead to two mutually-exclusive effective parts, real and imaginary parts of the dispersion equation. Bisection and phase change methods are two efficient ways to solve the dispersion equations, but if the real part becomes to be effective one, phase correction measure should be taken toward to the phase change method in order to overcome some numerical instability issues.

The dichotomy property is further extended to a general multi-layered composite laminate through the proposed sampling strategy under the framework of GMM and TMM, but it is still applicable to SMM. The enumeration strategy and sampling strategy reflect the know-why and know-how aspects of the dichotomy property, respectively. Although this chapter comprehensively studies the dichotomy property of dispersion equation in wavenumber-phase velocity domain, the counterparts in frequencywavenumber and frequency-phase velocity domain still hold this property, as well as in the case of evanescent wave modes. Thus, this property has formed a closure characteristic on itself in mathematical sense as long as the damping effect is not involved.

BACKGROUND OF GUIDED WAVES ATTENUATION

attenuation prediction method thus paves the way to optimally deploy sensor network for SHM of anisotropic viscoelastic composite structures.

A manuscript related to this chapter has been submitted to the journal Structural Health Monitoring for peer review.

Background of guided waves attenuation

When adopting guided waves as an inspection tool for composite structures, the attenuation property of these waves is an essential aspect that should be considered carefully in addition to dispersion property [START_REF] Wilcox | The effect of dispersion on long-range inspection using ultrasonic guided waves[END_REF]. This is especially true for carbon fiber reinforced composites as such materials possess inherent viscoelasticity that causes attenuation and thus impacts significantly the sensor network deployment in terms of sensor geometry, installation, number, and position [START_REF] Schubert | On attenuation and measurement of lamb waves in viscoelastic composites[END_REF][START_REF] Asamene | Influence of attenuation on acoustic emission signals in carbon fiber reinforced polymer panels[END_REF]. In comparison with the network used for lightly damped structures, a larger sensor network (i.e. having a larger number of sensors) is required to cover highly damped structures with enough wave amplitudes [START_REF] Sreekumar | Attenuation of ao lamb mode in hybrid structural composites with nanofillers[END_REF][START_REF] Wilcox | Mode and transducer selection for long range lamb wave inspection[END_REF].

Furthermore, attenuation property plays an important role in amplitude-based damage identification methods [START_REF] Rébillat | Damage localization in geometrically complex aeronautic structures using canonical polyadic decomposition of lamb wave difference signal tensors[END_REF] and machine learning tools for modeling guided waves [START_REF] Haywood-Alexander | Structured machine learning tools for modelling characteristics of guided waves[END_REF].

Over the past decades, studies related to attenuation are mainly focused on numerical computations and experimental measurements. In composite structures, materials viscoelastic damping, usually characterized by Hysteretic (HR) or Kelvin-Voigt (KV) damping models, is the primary factor that causes attenuation [START_REF] Gresil | Prediction of attenuated guided waves propagation in carbon fiber composites using rayleigh damping model[END_REF][START_REF] Treviso | Damping in composite materials: Properties and models[END_REF]. The Rayleigh damping model is also adopted to investigate the damping effect on wave attenuation [START_REF] Ramadas | Modelling of attenuation of lamb waves using rayleigh damping: Numerical and experimental studies[END_REF][START_REF] Mardanshahi | Simulated lamb wave propagation method for nondestructive monitoring of matrix cracking in laminated composites[END_REF]. Theoretically, attenuation coefficients of guided waves can be computed from the corresponding dispersion equations (DEs). Thus, considerable efforts have been made to derive accurate or approximated DEs, being complex-valued in the viscoelastic media, and to develop efficient numerical algorithms to solve them. The simplest one is the classical Rayleigh-Lamb equation which represents waves propagation in a homogeneous isotropic viscoelastic plate [START_REF] Zhu | A numerical algorithm to solve multivariate transcendental equation sets in complex domain and its application in wave dispersion curve characterization[END_REF]93]. The partial wave superposition approach (PWSA) is generally employed to derive the DEs of the single layer anisotropic viscoelastic plate [72,[START_REF] Zhu | Accurate characterization of 3d dispersion curves and mode shapes of waves propagating in generally anisotropic viscoelastic/elastic plates[END_REF]. For the multi-layered system, PWSA is extended to a class of matrix-based methods, e.g. transfer matrix method (TMM) [START_REF] Castaings | Guided waves propagating in sandwich structures made of anisotropic, viscoelastic, composite materials[END_REF] and global matrix method (GMM) [96], which are based on the linear 3D elasticity theory and thus become the standard manner to derive the analytical DEs of this system. However, solving these equations is not

BACKGROUND OF GUIDED WAVES ATTENUATION

an easy task and it usually requires Lowe's dual-variable root-finding method [6] or Zhu-Qian's 2D

Module Ratio Convergence Method (2D MRCM) [START_REF] Zhu | A numerical algorithm to solve multivariate transcendental equation sets in complex domain and its application in wave dispersion curve characterization[END_REF]. Some alternative approaches turn to establish the approximated DEs but are solved by efficient root-searching algorithms such as the Newton-Raphson method and eigenvalue decomposition method. The representatives are the approaches based on Legendre polynomials [START_REF] Dahmen | Investigation of the coupled lamb waves propagation in viscoelastic and anisotropic multilayer composites by legendre polynomial method[END_REF][START_REF] Dahmen | Optimization of coupled lamb wave parameters for defect detection in anisotropic composite three-layer with kelvin-voigt viscoelasticity using legendre polynomial method[END_REF][START_REF] Zhang | Full dispersion and characteristics of complex guided waves in functionally graded piezoelectric plates[END_REF] and higher-order plate theory [START_REF] Calomfirescu | On the propagation of lamb waves in viscoelastic composites for shm applications[END_REF][START_REF] Torres-Arredondo | A viscoelastic plate theory for the fast modelling of lamb wave solutions in ndt/shm applications[END_REF][START_REF] Orta | Modelling lamb wave propagation in visco-elastic composite plates using a fifth-order plate theory[END_REF]]. Furthermore, discrete numerical methods are widely applied as well to solve the problem of complex wave in multilayered viscoelastic plates due to their easy implementation. For instance, one can cite the wave finite element method (WFEM) using the Floquet periodicity condition [START_REF] Manconi | Estimation of the loss factor of viscoelastic laminated panels from finite element analysis[END_REF][START_REF] Manconi | On the effect of damping on dispersion curves in plates[END_REF], the spectral collocation method (SCM) based on Chebyshev polynomials [44,80], and the semi-analytical finite element method (SAFE) suitable for arbitrary cross-section waveguides [START_REF] Bartoli | Modeling wave propagation in damped waveguides of arbitrary cross-section[END_REF][START_REF] Taupin | A detailed study of guided wave propagation in a viscoelastic multilayered anisotropic plate[END_REF]79]. The common characteristic of these methods lies in that discretization only takes place in the thickness direction of the plate, thus increasing computational efficiency compared to the traditional 3D finite element modeling.

In addition to the numerical approach, attenuation coefficients can be also measured experimentally, but the related works are scarce compared to the abundant theoretical advancements. The simplest approach consists in computing the attenuation rate of wave amplitudes between two sensors in an A-scan experiment [START_REF] Sreekumar | Attenuation of ao lamb mode in hybrid structural composites with nanofillers[END_REF][START_REF] Ramadas | Modelling of attenuation of lamb waves using rayleigh damping: Numerical and experimental studies[END_REF]. A comprehensive method is to use a curve fitting technique which estimates attenuation by fitting the experimental data of wave amplitudes versus propagation distances, having the merit of considering both geometric spreading and structural damping [START_REF] Gresil | Prediction of attenuated guided waves propagation in carbon fiber composites using rayleigh damping model[END_REF][START_REF] Ono | Attenuation of lamb waves in cfrp plates[END_REF]. Besides, two advanced identification methods are also used, including the matrix pencil method [93] and the estimation of signal parameters via rotational invariance techniques (ESPRIT) [78]. The common characteristic of both methods lies in that they can identify not only the attenuation coefficient but also the propagating wavenumber.

In the aforementioned works, much effort was put to obtain the 3D dispersion curves depending on frequency at a fixed propagation direction [START_REF] Zhu | Accurate characterization of 3d dispersion curves and mode shapes of waves propagating in generally anisotropic viscoelastic/elastic plates[END_REF][START_REF] Zhang | Full dispersion and characteristics of complex guided waves in functionally graded piezoelectric plates[END_REF]80]. Scarce work was devoted to study the directional property of attenuation especially for practical engineering structures, which has a great significance for sensors network operational deployment in anisotropic composite plates. To this end, this chapter proposes a new method to predict the frequency and spatially dependent attenuation of guided waves propagating in anisotropic viscoelastic composite laminates. Here frequency and spatially means that wave attenuation depends not only on frequency but also on the propagation direction. The proposed method extends the conventional damping free GMM, see Chapter 5, to the

WAVE PROPAGATION THEORY IN THE VISCOELASTIC COMPOSITE LAMINATE

damping case by incorporating viscoelastic damping models. This method is termed "dGMM" for damped global matrix method. It is supported by a two-step numerical root-solving algorithm, having stable and efficient merits. The proposed dGMM is first verified theoretically by numerical comparison with the SAFE method, then validated through experimentation on two aeronautic composite parts (at scale one) of an Airbus A380 nacelle as shown in Figure 7.1. For each structure, two cases were considered, unmounted and mounted parts on the plane. The two structures are rather different as one is made up of a multi-layered carbon epoxy composite plate with many stiffeners, fan cowl structure (FCS) as shown in Figure 7.1(a), which is 2.20 m high and 5.80 m in half-circumference; and the other one is a sandwich type structure, inner fixed structure (IFS) shown in Figure 7.1(b), with an aluminum alloy honeycomb core and two multi-ply carbon epoxy outer skins, which has a dimension of 2.75 m in height and 3.80 m in half-circumference [3]. It should be emphasized here, that unlike the experiments on small-scale plates [START_REF] Schubert | On attenuation and measurement of lamb waves in viscoelastic composites[END_REF][START_REF] Asamene | Influence of attenuation on acoustic emission signals in carbon fiber reinforced polymer panels[END_REF][START_REF] Sreekumar | Attenuation of ao lamb mode in hybrid structural composites with nanofillers[END_REF][START_REF] Ramadas | Modelling of attenuation of lamb waves using rayleigh damping: Numerical and experimental studies[END_REF]79], the results reported in this chapter are the unique work so far concerning attenuation prediction for practical aeronautic structures, and the most important contribution to the state-of-the-art is that in-service data of the mounted cases of both structures are used to validate the proposed method.

Viscoelastic damping models of composite materials

The HR and KV models are two common damping models able to describe the viscoelasticity of composite materials [12] as presented in Eqs. (7.1) and (7.2), respectively. For both models, the elastic modulus is considered as a complex number E * , in which the real part E is the storage modulus and the imaginary part is associated with two factors γ HR and γ KV that are defined as the respective loss factors of the two models. Besides, a less common damping model named Biot (BT) model is presented in Eq. (7.3), which is mainly applied to highly damped aerospace structures [START_REF] Mastroddi | Time-and frequency-domain linear viscoelastic modeling of highly damped aerospace structures[END_REF]. Amongst the three models, HR model generates a complex modulus E * independent of frequency ω, whereas KV and BT models produce a frequency dependent complex modulus. Thus, the dependency property on ω is attached to E * for the two models, i.e. E * (ω).

E * = E(1 -iγ HR ) HR model (7.1) E * (ω) = E ( 1 -iγ KV ω ω c ) KV model (7.
2)

E * (ω) = E ⎡ ⎣ 1 + 2 π γ BT ln √ 1 + ( ω ϵ ) 2 -i 2 π γ BT arctan ( ω ϵ ) ⎤ ⎦ BT model (7.3)

WAVE PROPAGATION THEORY IN THE VISCOELASTIC COMPOSITE LAMINATE

where, i is the unit of the imaginary number, and the superscript ' * ' denotes that the associated term belongs to the complex number family; ω c is the characteristic frequency of the KV model characterizing that γ KV is measured at ω c ; γ BT and ϵ are the loss factor and scaling factor of the BT model, respectively.

For a general layer composing the laminate, e.g. layer l 1 in Figure 7.2, the stiffness matrix of this layer has been introduced in Eq. (2.10) for the pure elastic case, which is characterized by nine independent elastic constants, i.e. three longitudinal moduli E

E * 1 (ω), E * 2 (ω), E * 3 (ω), G * 23 (ω), G * 31 (ω), G * 12 (ω)
. 1 In view of this, the usual stiffness matrix C becomes correspondingly a frequency dependent complex matrix C * (ω) as presented in Eq. (7.4), which shows the orthotropic anisotropy of composite materials.

C * (ω) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1/E * 1 (ω) -ν 12 /E * 1 (ω) -ν 13 /E * 1 (ω) 0 0 0 1/E * 2 (ω) -ν 23 /E * 2 (ω) 0 0 0 1/E * 3 (ω) 0 0 0 1/G * 23 (ω) 0 0 sym 1/G * 31 (ω) 0 1/G * 12 (ω) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ -1 (7.4)
where, the six Poisson's ratios are real numbers and ν 21 , ν 31 , ν 32 are not independent due to the symmetry of the stiffness matrix.

The stiffness matrix C * (ω) corresponds to each layer's fiber direction, for instance, the x 1 axis direction in the layer l 1 of Figure 7.2. It should be rotated to the currently focused wave propagation direction that is along the x θ 1 axis as shown in Figure 7.2, which has been introduced in Section 2.3. The rotated stiffness matrix is presented in Eq. (7.5) showing the monoclinic anisotropy of composite

Damped global matrix method

With the viscoelastic stiffness matrix C * (ω, θ) in a general layer l i , the damped global matrix method (dGMM) will be briefly derived by following the steps of the standard GMM to keep the completeness of dGMM. The displacement and stress vectors in a general layer l i are expressed in compact form in Eqs. (7.6) and (7.7) that represent the superposition of six partial waves derived from 3D elasticity theory in Chapter 3.

u l i = [u 1 , u 2 , u 3 ] T = U l i Λ l i (x 3 )η l i e ik(x 1 -vt) (7.6
)

σ l i = [σ 33 , σ 23 , σ 13 ] T = β l i Λ l i (x 3 )η l i e ik(x 1 -vt) (7.7)
where, the superscript "l i " implies that the associated term belongs to the layer l i . k is the (complex) wavenumber instead of ξ used in the previous chapters. U l i and β l i are the amplitude matrix of displacement and stress, respectively. Λ l i (x 3 ) is a diagonal matrix depending on the thickness direction coordinate x 3 . η l i is the partial wave participation vector. They are uniformly defined in Eqs. (7.8)(7.9)(7.10) (7.11).

U l i = ⎡ ⎢ ⎣ 1 1 1 1 1 1 V 1 V 2 V 3 V 4 V 5 V 6 W 1 W 2 W 3 W 4 W 5 W 6 ⎤ ⎥ ⎦ (7.8)
β l i = ⎡ ⎢ ⎣ β 11 β
Λ l i (x 3 ) = ⎡ ⎢ ⎣ e ikα 1 x 3 . . . e ikα 6 x 3 ⎤ ⎥ ⎦
(7.10)

η l i = [η 1 , η
⎤ ⎥ ⎦ ⎡ ⎢ ⎣ U V W ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ 0 0 0 ⎤ ⎥ ⎦
(7.12)

A 6 α 6 + A 4 α 4 + A 2 α 2 + A 0 = 0 (7.13)
where, the four polynomial coefficients are presented in Eq. (3.25).

For each solution of α j in Eq. (7.13), corresponds a pair of solution of V j , W j , β 1j , β 2j , β 3j as presented in Eqs. (7.14) and (7.15):

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ V j = ( C * 11 -ρv 2 + C * 55 α 2 j ) (C * 36 + C * 45 ) - ( C * 16 + C * 45 α 2 j ) (C * 13 + C * 55 ) (C * 13 + C * 55 ) ( C * 66 -ρv 2 + C * 44 α 2 j ) - ( C * 16 + C * 45 α 2 j ) (C * 36 + C * 45 ) W j = ( C *
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ β 1j = C * 13 + C * 36 V j + C * 33 α j W j β 2j = C * 45 α j + C * 44 α j V j + C * 45 W j β 3j = C * 55 α j + C * 45 α j V j + C * 55 W j (j = 1, . . . , 6) (7.15)
To this step, the displacement and stress vectors in Eqs. (7.6) and (7.7) can be combined to ease the application of the continuity condition at the interface of interlamination. We then have,

S l i = [ u l i σ l i ] = [ U l i β l i ] Λ l i (x 3 )η l i e ik(x 1 -vt) = Z l i (x 3 )η l i e ik(x 1 -vt) (7.16)
where

Z l i (x 3 ) = [ U l i β l i ]
Λ l i (x 3 ). At the interface between layers l i and l i+1 , the continuity condition requires that the displacement and stress fields should be continuous, as stated in Eq. (7.17). Besides, the traction-free boundary condition at the two outer surfaces should be satisfied due to the free surface of the plate, as expressed in Eq. (7.18). For the n-layered laminate system shown in Figure 7.2, the continuity condition of each interface can be consistently assembled to form a global matrix system, as presented in Eq. (7.19), in which the subscript "4:6" in Z l 1 (0) and Z ln (d n ) means that only the fourth to sixth rows of the two matrices are evaluated because these rows correspond to stress terms σ 33 , σ 23 , σ 13 and thus, represent the traction-free boundary conditions in Eq. (7.18).

S l i | x 3 =d i = S l i+1 | x 3 =0 (i = 1, . . . , n -1) (7.17
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ -Z l 1 4:6 (0) Z l 1 (d 1 ) -Z l 2 (0)
. . . . . .

Z l i (d i ) -Z l i+1 (0) . . . . . . Z l n-1 (d n-1 ) -Z ln (0) Z ln 4:6 (d n ) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ η l 1 . . . η l i η l i+1 . . . η ln ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 . . . 0 0 . . . 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (7.19)
The huge global matrix in Eq. (7.19) can be concisely rewritten as G(ω, k * , θ) ∈ C 6n×6n given that this matrix depends on frequency ω, wavenumber k and propagation angle θ on the one hand, and on the other hand k should be a complex number, i.e. k * ,2 owing to the effect of viscoelastic damping introduced from the stiffness matrix in Eq. (7.4). The non-trivial solution of η l i in Eq. (7.19) requires that the determinant of the global matrix G(ω, k * , θ) should vanish, which finally generates the DE of the n-layered laminate system, as presented in Eq. (7.20).

D(ω, k * , θ) = det{G(ω, k * , θ)} = 0 (7.20)
where, k * = k r + ik i . Both k r and k i are real positive numbers, and k r is the wave's propagating wavenumber and k i the attenuation coefficient given in Nepers per meter, Np/m (1 Np/m=8.69 dB/m) [12].

Eq. (7.20) is a nonlinear transcendental equation with complex, multivariate and multi-roots characteristics. Solving this equation is mathematically intractable. We will present and implement in detail an original solving algorithm in the subsequent section. Thus, once the solutions of Eq. (7.20) are obtained, the corresponding solutions of η l i in Eq. (7.19) can be extracted from the zero-eigenvalue through Eq. (7.22). But for heavily damped structures, the energy velocity c e defined in Eq. (7.23) should be adopted because in this case, the imaginary part k i is large enough such that the group velocity computed through Eq. (7.22) is physically meaningless [72,[START_REF] Bernard | Guided waves energy velocity in absorbing and nonabsorbing plates[END_REF].

c p = ω k r (7.21) c g = ∂ω ∂k r (7.22
)

c e = ∫ H 0 ⟨P 1 ⟩ dx 3 ∫ H 0 (⟨E k ⟩ + ⟨E p ⟩) dx 3 (7.23)
where, H is the plate's total thickness, ⟨P 1 ⟩ is the time-averaged Poynting vector along the wave propagation direction, i.e. x θ 1 in Figure 7.2, ⟨E k ⟩ and ⟨E p ⟩ are the time-averaged kinetic and potential energy density stored in the wave guide. The three terms are defined in Eq. (7.24), in which, the Einstein summation convention is implied for repeated index.

⟨P i ⟩ = - 1 2 Re{σ ij u ̇j} (i = 1, 2, 3), ⟨E k ⟩ = 1 4 Re{ρu ̇iu ̇i}, ⟨E p ⟩ = 1 4 Re{σ ij ε ij } (7.24)
where, Re{■} is the real part operator, u ̇i is the particle velocity of the waveguide along direction x θ i ; the overbar "-" denotes the complex conjugate.

Numerical algorithm of solving the complex DE

The task of solving the complex DE, Eq. (7.20) is extremely complex from mathematical view. Indeed, it is a multivariate transcendental equation given that, at a specified propagation angle θ 0 , for a solution pair (ω, k * ) there are actually three real variables (ω, k r , k i ) to be identified because of k * = k r + ik i . Thus, a certain dimensionality reduction is necessary in order to solve the ternaryvariable equation. Note that the complex stiffness matrix C * (ω) is frequency dependent as implied by Eq. (7.4), thus, ω should be considered as another independent variable in addition to θ during 7.3. DATA-DRIVEN STRUCTURAL PARAMETERS ESTIMATION studied in Chapter 6, at frequencies ω 1 , ω 2 , ω 3 , which is an easy task. This strategy is particularly effective for tracing fundamental modes' curves, i.e. S0, A0 and SH0 modes, since the proximity of the complex case curve to the real case one in ω -k r plane guarantees the success of the tracing process, as illustrated in Figure 7.3. It is worth noting that the extrapolation equation k ′ r,n in Figure 7.3, which is the real part of k ′ * n , can be applied to non-constant frequency increment given that the frequency increment obtained from the real case solution may not be equal to the one of complex case.

The above elaboration is now implemented to be a two-step dGMM algorithm of solving the complex DE as presented in

Data-driven structural parameters estimation

The key to success of predicting attenuation using the developed algorithm in Figure 7.4 relies on the accuracy of the viscoelastic loss factors that are involved in the damping models in Eqs. (7.1), (7.2) and (7.3). These parameters can be obtained from the material manufacturer or from standard mechanical test data [START_REF] Boiko | Measurement method of complex viscoelastic material properties[END_REF][START_REF] Montalvão | Experimental measurement of the complex young's modulus on a cfrp laminate considering the constant hysteretic damping model[END_REF]. However, in most cases, both ways are not available, especially for

Identification of attenuation coefficient

For circular-like crested guided waves propagating in composite plates, geometric spreading and material damping are two main factors of wave amplitude attenuation [START_REF] Gresil | Prediction of attenuated guided waves propagation in carbon fiber composites using rayleigh damping model[END_REF]. The former is responsible for the near-field propagation (usually less than three wavelengths) and the latter mainly for far field propagation [START_REF] Gresil | Prediction of attenuated guided waves propagation in carbon fiber composites using rayleigh damping model[END_REF][START_REF] Ramadas | Three-dimensional modeling of lamb wave attenuation due to material and geometry in composite laminates[END_REF]. In Eq. (7.6), the effect of geometric spreading is not accounted since it is a solution of the plate's free vibration [START_REF] Guo | Dichotomy property of dispersion equation of guided waves propagating in anisotropic composite plates[END_REF], in other words, far field condition is fulfilled. As such, the displacement field of Eq. (7.6) for a general layer l i can be extended to the whole laminate system, as 7.3. DATA-DRIVEN STRUCTURAL PARAMETERS ESTIMATION expressed in Eq. (7.25).

u i (x 1 , x 3 , t) = u ˆi(x 3 )e i(k * x 1 -ωt) (i = 1, 2, 3) (7.25)
where, u ˆi(x 3 ) is the through thickness wave amplitude at the excitation source. Substituting k * = k r + ik i into Eq. (7.25) leads to Eq. (7.26).

u i (x 1 , x 3 , t) = u ˆi(x 3 )e -k i x 1 e i(krx 1 -ωt) (i = 1, 2, 3) (7.26)
Comparing the two equations, the wave amplitude decays exponentially over propagation distance

x 1 with decaying ratio k i , which allows to interpret the physical essence of the imaginary part of the complex wavenumber, i.e. being the attenuation coefficient. Besides, Eq. (7.26) also demonstrates that all layers share the same attenuation coefficient3 since the decaying term e -k i x 1 is independent of the thickness direction coordinate x 3 , and it can be measured by the piezoelectric transducers (PZTs) surface mounted on the laminate. Explicitly, making x 3 = 0 and taking logarithm for both sides of Eq. (7.26) but omitting the phase term e i(krx 1 -ωt) , a linear formulation is yielded in Eq. (7.27).

Log [u i (x 1 , 0)] = -k i x 1 + Log [u ˆi(0)] (i = 1, 2, 3) (7.27)
where, u i (x 1 , 0) represents the wave packet amplitude recorded by PZT that is surface mounted at distance x 1 from the excitation source located at 0. Based on Eq. (7.27), the attenuation coefficient k i can be identified from recorded wave signals by linearly regressing logarithmic amplitude of wave packets versus propagation distance x 1 , from which the negative of the slope of the regressed line is just the attenuation coefficient.

The linear regression (LR) method is also used to identify energy velocity from the recorded wave signals since, at a specified frequency and along a fixed direction, a certain mode's wave packet propagates with constant velocity value. The detailed manipulation will be presented in the case study sections.

Estimation of viscoelastic loss factors

The experimentally identified attenuation coefficients can be used to estimate the viscoelastic loss factors which are not available in most cases. To this end, the attenuation coefficient k i is expressed

γ ˆ= arg min γ m ∑ l=1 [ K(γ, ω l ) -k ˆi,l ] 2 (7.29)
where, the series pair

( ω l , k ˆi,l ) l=1,...,m
is the experimentally identified attenuation coefficients at various frequencies, and γ ˆis the estimated loss factors for a certain damping model.

It is worth noting that the function k i = K(γ, ω) in Eq. (7.28) should represent the same branch of dispersion curves as the experimentally identified one, which is achieved through the single mode curve tracing technique illustrated in Figure 7.3. In some cases, due to the inaccurate elastic moduli or the existence of uncertainties, the computed energy velocities do not match well with the identified ones. To solve this issue, the same updating process is applied to the experimentally identified energy velocities to calibrate the inaccurate elastic moduli. For brevity, detailed formulation is not presented here but an example is provided in the IFS case study.

Overview of the attenuation prediction method

Once all the required material parameters are available, the numerical dGMM algorithm in Fig- In this section, two cases of fan cowl structure (FCS), either unmounted or mounted on an instrumented A380 plane, are employed to validate the proposed guided wave attenuation prediction method. The actual FCS is shown in Figure 7.1(a). This structure is made up of a four-layered carbon epoxy composite plate with stacking sequence [0/-45/+45/0]. Its elastic material properties of each layer can be obtained from Fendzi's PhD thesis [3] and listed in Table 7.1 for convenience, but the viscoelastic properties are not available. Thus, the model updating process introduced previously will be adopted to estimate them. Obtained results are listed in Table 7.1 for HR and KV models beforehand. to R6, according to their geometrical characteristics, as shown in Figure 7.6(b). By doing so, each region can be considered at once and signals are processed with great convenience. The 13 sensors in the unmounted FCS constitute only one region and are thus not marked for brevity. Furthermore, sensors of each region in both cases are intentionally divided into three groups during signal processing according to their alignment pattern, i.e. according to their direction with respect to the ply sequence.

The group information of the unmounted FCS is illustrated in Figure 7.6(a). For the mounted FCS, in each region, Group 1 includes all sensors and Group 2 and 3 contain sensors roughly aligned along the upper and lower horizontal line, respectively. For example, in R1, Group 1 includes sensors from 1 to 9, and Group 2 and 3 includes sensors 1,2,4,6,8 and 3,5,7,9, respectively. The accurate fiber orientation of each layer in the plate is not available but the horizontal direction of the plate is assumed to be the fiber orientation of the 0 • layer because this direction leads to the best match between theoretical predictions and experimental measurements for both energy velocity and attenuation coefficient as will be shown later. enhance signal to noise ratio [START_REF] Yu | Advanced signal processing for enhanced damage detection with piezoelectric wafer active sensors[END_REF]. Note that the frequency sweep testing is required for understanding the mechanism of wave attenuation within an interested frequency range and it is different from a single sweet spot frequency testing which is desired for damage detection [START_REF] Agrahari | A refined lamb wave time-reversal method with enhanced sensitivity for damage detection in isotropic plates[END_REF]. Besides, only the S0 mode wave packet is discernible that makes the subsequent identification of energy velocity and attenuation coefficient only focus on S0 mode. values of energy velocity in the six regions are larger than 0.7 whereas only nine out of fifteen for attenuation coefficient; 2) energy velocities tend to be consistent but the distribution of attenuation coefficients are more variable; 3) the identified energy velocities in different groups of the six regions agree well with the reference value. However there exists more discrepancy between the identified attenuation coefficients and the reference one but the same order of magnitude remained. The quality of energy velocity identification is much better than the quality of the attenuation coefficient which may be interpreted as the attenuation coefficient being much smaller than the propagating wavenumber

(|k i | ≪ |k r |)
and thus much more sensitive to experimental noise. Note that in Figure 7.11 several bars are not shown because of their unphysical meanings such as too large or too small energy velocity or negative attenuation coefficient, which may be attributed to the superposed S0 mode's wave packet by other modes, e.g. A0 or the reflected modes.

Identified dispersion curves of energy velocity and attenuation coefficient

By applying the LR method to each frequency tested for the unmounted FCS, the dispersion curves of energy velocity and attenuation coefficient are formed in Figure 7.12(a) and (b), respectively. Note that in Figure 7.12(a) only the identified curve of Group 1 is presented due to the closing curves amongst the three groups, whereas in Figure 7.12(b) only the identified curves of Group 2 and 3 are shown thanks to the anisotropic effect in Group 1 which has been declared in the previous subsection.

Additionally, the curves of correlation coefficients R 2 corresponding to Figure 7.12(a) and (b) are presented in Figure 7.12(c) to assess the quality of the identified dispersion curves. Obviously, the identified S0 mode's energy velocity curve is accurate because of its greater correlation coefficients (all larger than 0.95). Although the identified attenuation curves (Figure 7.12(b)) are not as good as the energy velocity curve (Figure 7.12(a)), we can still get some meaningful findings: 1) in the frequency range 75-150 kHz, the identified attenuation coefficients are reliable enough given that the corresponding R 2 values are larger than 0.7 except the initial points (50-70 kHz); 2) both curves present the same trend, i.e. attenuation increases with frequency firstly and then decreases; 3) the identified attenuation coefficients for both groups are equal at approximately 95 kHz, and the lower and upper bounds of the two groups are intersected in the vicinity of 95 kHz.

Estimation of viscoelastic material properties of the unmounted FCS

The experimentally identified S0 mode attenuation coefficients in Figure 7.12(b) are further employed to estimate viscoelastic material properties via the least square method formulized in Eq. (7.29).

When specifying damping models, only HR and KV models are used considering that BT model is mainly used for highly damped structures whereas FCS is slightly damped because attenuation coefficient is no more than 2 Np/m in the interested frequency range according to Figure 7.12(b). The estimated viscoelastic properties of the two models are listed in to a different value, the loss factors of KV model will change correspondingly (but the ratio γ KV /f c keeps unchanged) [12].

Once all the required material parameters in Table 7.1 have been obtained, they are used as inputs to the developed algorithm exhibited in Figure 7.4. After computing, the dispersion curves of energy velocity and attenuation coefficient for the two damping models are depicted in Figure 7.13(a) and (b), respectively, in which the experimentally identified data points in Figure 7.12(a) and (b) are also presented for comparison. [3] and are listed in Table 7.4 and 7.5 for convenience. Table 7.6 lists the viscoelastic properties of the two materials corresponding to the specific damping models, which are obtained through model updating process (see Section 7.5.3) since the viscoelastic information is not available from this reference. The original values from [3] that lead to a great agreement to the experimental data of the unmounted IFS. 2 The updated values that result in the best match with the experimental data of the mounted IFS. match between the theoretical predictions and experimental measurements for both energy velocity and attenuation coefficient. The signal acquisition process was similar to the experiments on FCS and thus the detailed experimental setup is not shown for brevity. The frequency series in the unmounted IFS experiment were swept from 50 kHz to 150 kHz at 5 kHz increment but there was only 100 kHz available for the mounted IFS experiment. Time averaging and wavelet denoising strategies were also adopted for signal processing. .17(a) presents the comparison of the identified energy velocity via LR method between the mounted and unmounted IFS at 100 kHz as this frequency is unique for the mounted case. This subfigure shows that both cases produce close energy velocities and simultaneously the higher R 2

CASE STUDY ON AN UNMOUNTED AND MOUNTED IFS OF A380 PLANE

values larger than 0.8 are obtained except the Group 1 of the unmounted IFS. Thus, Group 2 and 3 of the unmounted IFS will be adopted to identify the dispersion curves of energy velocity in the next subsection.

A similar comparison of the identified attenuation coefficient is illustrated in Figure 7.17(b). It

shows that overall, the mounted IFS returns about half the attenuation of its unmounted counterpart.

This could be attributed to different environmental conditions in both cases, and especially to the fact that when IFS is mounted on an A380 plane the structure is stressed which can influence guided waves propagation properties [START_REF] Gandhi | Acoustoelastic lamb wave propagation in biaxially stressed plates[END_REF]. Moreover, the attenuation coefficient of Group 2 is not equal to the one of Group 3 for both cases and the discrepancy in the mounted IFS is more salient, even though Group 2 and 3 have the same propagation direction. This phenomenon can be explained by the fact that the imaginary part of the wavenumber, i.e. the attenuation coefficient, is much smaller than the real part and thus much more prone to noise error. Since the correlation coefficients of Group 2 and 3 of the unmounted IFS are larger than 0.8, the two groups will be selected to identify dispersion curves of attenuation coefficient in the next subsection. To further explore directional dependence of the S0 mode energy velocity, the guided wave signals in Group 1 of the mounted IFS were processed to generate a polar plot of energy velocity, as depicted in Figure 7.18, since this group possess multiple wave propagation directions. Besides, the theoretical curves predicted via the developed dGMM method are also presented in Figure 7.18 for comparison purpose. The curves of dGMM prediction 1 and dGMM prediction 2 are created by using the (E 7.4, respectively.

Identified dispersion curves of energy velocity and attenuation coefficient

The identified dispersion curves of energy velocity and attenuation coefficient of the unmounted IFS are generated in

Estimation of viscoelastic material properties of the unmounted IFS

With the experimentally identified attenuation coefficients in Figure 7.19(b), the viscoelastic material properties of the unmounted IFS can be estimated according to the model updating process formulized in Eq. (7.29). Before performing this process, one has to note that the attenuation of the unmounted IFS is larger than the one of FCS according to Figure 7.19(b), thus BT model should be applied to IFS in a certain form considering that BT model is mainly used for highly damped structures [START_REF] Mastroddi | Time-and frequency-domain linear viscoelastic modeling of highly damped aerospace structures[END_REF], and the unmounted IFS exactly belongs to this case. Furthermore, the IFS is a sandwich type structure composed by two inhomogeneous materials with the aluminum alloy honeycomb core and the carbon epoxy skin layers. Therefore, the core layer and the skin layers can be modeled by different damping models. By following the convention in FCS, the carbon epoxy layers in IFS are modeled by HR or KV models, thus BT model is naturally applied to the aluminum alloy core layer, which finally results in two combinations of damping models, HR+BT and KV+BT models.

By taking the identified attenuation coefficients in and (f). This behavior is a unique property of the damped wave guide [72].

To study the influence of attenuation on phase velocity, the attenuation curves of S0, A0 and SH0 modes in To study the anisotropic effect of phase velocity and attenuation, the polar plots of both quantities are depicted in Finally, the 3D surface diagrams of the three basic modes are predicted in Figure 7.24 to represent the frequency-spatial spectra of phase velocity and attenuation. In Figure 7.24(a)(c)(e), the cylindrical surfaces of S0 and SH0 modes and the paraboloid of revolution of A0 mode signify the quasi-isotropic property of phase velocity. However, the manifold shapes of attenuation in Figure 7. The traditional curve in Figure 7.23 can be reconstructed from Figure 7.24. For instance, the S0 the intersecting line of the plane f = 200 kHz to the surface of HR+BT model in Figure 7.24(f). For brevity, these intersecting lines are not depicted in Figure 7.24. Thus, the frequency-spatial spectra of guided waves will convey more information than the traditional dispersion curves, and it will play a more important role in guided waves based SHM, especially for anisotropic viscoelastic materials.

Discussion

Although this chapter only presents the comparison results of S0 mode between the experimental and predicted data, the developed dGMM method can be generalized to other guided wave modes such as A0 and SH0 modes if these modes' information is experimentally identifiable such that the identified data can be integrated to update the viscoelastic material properties. However, simultaneously identifying S0 and A0 modes usually requires special setup of transducers from the hardware viewpoint that have flexible polarity directions, dual PZT for example [START_REF] Ono | Attenuation of lamb waves in cfrp plates[END_REF], to sufficiently excite both fundamental modes. Besides, from the software viewpoint, any efficient decomposition algorithms are required that can separate coupled S0 and A0 wave packets in the signal processing level [93]. To acquire the SH0 wave signal, special SH wave transducers need to be utilized [START_REF] Miao | Shear horizontal wave transducers for structural health monitoring and nondestructive testing: A review[END_REF].

This chapter takes the damping effect of the elastic moduli into consideration. In some references [79,[START_REF] Ledi | An identification method for frequency dependent material properties of viscoelastic sandwich structures[END_REF], the complex stiffness tensor is employed, i.e.

C * ij (ω) = C ′ ij (ω) -iC ′′ ij (ω)
, to represent the viscoelasticity of composite materials by specifying a certain damping model for the imaginary part C ′′ ij (ω). However, this utilization increases the number of variables for the model updating process, thus demanding advanced optimization algorithms [START_REF] Zhao | Reconstruction of the nine stiffness coefficients of composites using a laser generation based imaging method[END_REF].

In literature, KV model is increasingly adopted to compute the 3D dispersion curves in (f, k r , k i ) domain no matter which numerical method is used [START_REF] Dahmen | Optimization of coupled lamb wave parameters for defect detection in anisotropic composite three-layer with kelvin-voigt viscoelasticity using legendre polynomial method[END_REF]79]. In contrast with this tendency, the two case studies in this chapter demonstrate that the HR involved models predict more accurate attenuation than the KV involved models in comparison with the experimental data. This fact reveals that choosing a damping model to represent the viscoelasticity of composite materials is application dependent.

Conclusion

This chapter puts forward a dGMM method to predict the frequency-spatial attenuation of guided waves in anisotropic viscoelastic composite laminates by integrating the damping models (HR, KV and BT) into the conventional pure elastic GMM. To efficiently solve the nonlinear transcendental DE, a two-step numerical algorithm is developed that combines dGMM method, 2D MRCM and the curve tracing technique. The proposed attenuation prediction approach is theoretically validated by comparing it with the SAFE method.

CONCLUSION

The energy velocities and attenuation coefficients at various frequencies are identified from recorded wave signals via LR method which can not only process multiple sensing pairs necessarily for monitoring a practical aeronautic structure but also output statistical information such as the confidence interval and correlation coefficient R 2 to assess the quality of the identified parameters. With the experimental attenuation data, the unknown viscoelastic material properties are nondestructively estimated in the least square sense to achieve the best agreement of prediction to experimental data.

Experiments on two physically different structures, the homogeneous FCS and the inhomogeneous IFS with each one unmounted or mounted on an instrumented A380 plane, were carried out to experimentally validate the proposed dGMM approach. FCS is a slightly damped structure and IFS is highly damped. The existence of stress in the mounted case can greatly change the property of attenuation as compared to the unmounted case. To consider the inhomogeneous characteristic of IFS, its core layer and skin layers are modeled by different damping models, thus two combinations of damping models (HR+BT and KV+BT) come into being. Both case studies demonstrate that the HR involved models predict more accurate attenuation than the KV involved models, and the spatial distribution of attenuation holds anisotropic property despite the quasi-isotropic stacking layups. In both structures, A0 mode undergoes much attenuation than S0 mode, thus selecting S0 mode for SHM of similar aeronautic structures is desired if the dispersion aspect is not the determining factor. In summary, using the actual and in-service data of the practical aeronautic structures to validate the proposed attenuation prediction method is not an easy task but makes it extremely valuable for the scientific and industrial communities. In this sense, this chapter escorts the initiative of closing the gap between research and industrial deployment for SHM [START_REF] Cawley | Structural health monitoring: Closing the gap between research and industrial deployment[END_REF].

GENERAL CONCLUSIONS

For the examples of quasi-isotropic and metallic-composite laminates, the computed dispersion curves via HMS are compared to experimental data points extracted from a classical reference, thus validating the effectiveness of HMS.

To solve the lower computational efficiency issue of the standard GMM, the optimized GMM is proposed, which simultaneously takes advantage of the symmetry condition induced global matrix, parallel computing and sparse matrix techniques. The GMM framework has proved its stability in a benchmark test based on the dataset of Open Guided Waves. This test shows the large f d problem of TMM. Thus, when the dispersion curves of a composite laminate at larger frequency values are interested, adopting GMM for computation is necessary. The good agreement between GMM computation and the experimental data of Open Guided Waves validates the effectiveness and feasibility of GMM.

The optimized GMM is successfully applied to an aerospace composite plate having 400 layers, which has the largest number of layers reported so far, to compute its phase velocity dispersion curves. This structure is a component of the booster pressure vessels of the rocket Ariane 6 under development.

Due to the complexity of this structure, the dispersion curves experience heavy veering effect in a small region that will cause trouble when tracing the correct loci of dispersion solutions. To study this effect, the mode shape and MAC value tomography of displacement and stress are analyzed in the regions of veering effect. Analysis results show that within these regions, the dispersion solutions loci belonging to the same symmetry kind can be arbitrarily close but never cross, and this process is accompanied by a rapid exchange of their mode shapes. In comparison with stress mode shapes, displacement mode shapes are a desired indicator to distinguish the branches of dispersion diagram in the veering effect regions. An encouraging result is reported that the optimized GMM can complete computations of the 400-layered plate within half an hour in a standard workstation. This fact breaks down the common sense that GMM framework cannot be applied to laminates having a large number of layers and proves the ability of the optimized GMM in application of aerospace composite materials which usually possess many laminae.

It is furthermore demonstrated that dispersion equations built with PWSA for a single layer plate and with TMM and GMM for a multi-layered plate system hold the dichotomy property in the case of pure elastic materials. This property states that these a priori complex-valued equations collapse to pure real or imaginary valued equations, depending on the behavior of some intermediate param-8.2. PERSPECTIVES components of A380 plane but physically different. For each structure, the dataset of mounted and unmouted cases are analyzed. The first one is a fan cowl structure (FCS) having quasi-isotropic property and the second one an inner fixed structure (IFS) being sandwich type. Analysis shows that FCS is a slightly damped structure and IFS is highly damped. The existence of stress in the mounted case can greatly change the property of attenuation as compared to the unmounted case.

Two combinations of damping models (HR+BT and KV+BT) are used to consider the inhomogeneous characteristic of IFS in the first time, for which the aluminum core layer and carbon epoxy skin layers are modeled by different damping models. Both case studies demonstrate that attenuation coefficients holds anisotropic property despite the quasi-isotropic stacking layups, and A0 mode undergoes much attenuation than S0 mode. Thus, in the later deployment of SHM for similar structures, selecting S0 mode is desired in order to secure enough coverage of wave amplitudes. In summary, using the actual and in-service data of the practical aeronautic structures to validate the proposed attenuation prediction method is not an easy task but makes it extremely valuable for the scientific and industrial communities. In this sense, this paper escorts the initiative of closing the gap between research and industrial deployment for SHM [START_REF] Cawley | Structural health monitoring: Closing the gap between research and industrial deployment[END_REF].

Perspectives

There are considerable works to be done in the future to achieve the application value of the thesis in engineering practice.

As an evolved version of TMM and GMM, the stiffness matrix method (SMM) [START_REF] Rokhlin | Stable recursive algorithm for elastic wave propagation in layered anisotropic media: Stiffness matrix method[END_REF]11] will be implemented to complete the family of matrix-based methods for modelling guided waves propagation in composite laminates. The incompatibility issue in SMM will be checked and how to address this issue will be studied. Benchmark tests can be performed to validate the stability of SMM. Extending it to the viscoelastic case is worthwhile to do and comparing its performance with dGMM shall provide further recommendations for improvement.

More profound viscoelastic damping models should be integrated into dGMM to exactly predict attenuation of complex structures, e.g. the rational model [START_REF] Jm | Viscoelastic vibration toolbox for use with matlab, user's guide version 1.0[END_REF] described in Eq. (8.1). Implementation of this model has the following two fatal issues:

• There is not a unique principle to determine the number of parameters n in Eq. (8.1).

• Even if n is given, it is still difficult to estimate the coefficients α i and β i from experimental attenuation data through the model updating process, unless these coefficients are directly provided.

E * (ω) = E [ 1 + ∑ n i=1 β i (iω) i 1 + ∑ n i=1 α i (iω) i ] (8.1)
To implement the rational model for attenuation prediction, the mechanism of it should be studied firstly. As an alternative, implementing the standard linear solid (SLS) model [START_REF] Pan | Mode selection of guided waves for ultrasonic inspection of gas pipelines with thick coating[END_REF][START_REF] Carcione | P-and s-wave simulation using a cole-cole model to incorporate thermoelastic attenuation and dispersion[END_REF] expressed in Eq. (8.2) is achievable.

E * (ω) = E 1 -iωτ ϵ 1 -iωτ σ (8.2)
where, τ ϵ and τ σ are the creep time and the stress relaxation time, respectively.

Extending the matrix-based methods to release an open source dispersion tool is a promising work.

This tool should include the pre-processing module for easily defining model information (material and structure properties), high efficiency computation module, and user-friendly post-processing module for visualization of results (modal curve tracing and classification, animation of wave mode shapes, etc.). Some basic functionalities have been achieved in the thesis such as displaying dispersion diagram in ξ -v, f -ξ, f -v, f -c g , f -c e , f -k r -k i domains, as well as in polar plot format θ -v and θ -k i domains. Other extension of functionalities can be anticipated, for instance, computing the steering angle between phase velocity and group velocity directions which is a special concept existing in anisotropic composite plates [37]. A comprehensive comparison between the released dispersion tool with discrete methods, e.g. SAFE, should be conducted for validation purpose.

Some advanced signal processing methods will be adopted to replace the linear regression based attenuation identification method. The advanced methods are good candidates for experimentally identifying attenuation coefficients, for example, the matrix pencil method [93] and the estimation of signal parameters via rotational invariance techniques (ESPRIT) [START_REF] Margerit | High resolution wavenumber analysis (hrwa) for the mechanical characterisation of viscoelastic beams[END_REF]78]. These methods can simultaneously identify attenuation coefficient and propagating wavenumber. With more information identified, the model parameters can be estimated more precisely through model updating process, and thus producing more accurate predictions.

Due to the existence of uncertainties, the predicted energy velocity and attenuation coefficient do 8.2. PERSPECTIVES not always match well simultaneously with the respective experimental data (see Figure 7.20). In this situation, the model updating process should yet involve in the identified energy velocity and the elastic constants. That finally leads to the multiple objective optimization problem as presented in Eq. ( 8.3). To conduct this work, some advanced optimization algorithms will be utilized [START_REF] Zhao | Reconstruction of the nine stiffness coefficients of composites using a laser generation based imaging method[END_REF][START_REF] Chen | High-resolution lamb waves dispersion curves estimation and elastic property inversion[END_REF]. number and so on. Indeed, when designing a sensor network, one faces two practical issues: the first one is to get sure that all the structure under study will be covered by guided waves and thus that any potential damage position can be inspected [START_REF] Mitra | Guided wave based structural health monitoring: A review[END_REF]; the second one is to get sure that guided waves with enough amplitude will be reflected from the damage toward piezoelectric elements [START_REF] Su | Guided lamb waves for identification of damage in composite structures: A review[END_REF].

γ ˆ= arg min γ,E i ,G ij ,ν ij ⎧ ⎨ ⎩ ∑ m l=1 [ K(γ, ω l ) -k ˆi,l ] 2 ∑ m l=1 [C e (γ, ω l ) -c ˆe,l ] 2 ⎫ ⎬ ⎭ ( 8
When designing sensor networks, the distance between sensors can be appropriately larger along the orientations of lower attenuation; but along the orientations of larger attenuation, sensors should be placed densely as much as possible to guarantee enough coverage of wave fields, thus increasing the number of sensors. The future work on optimal sensor network deployment procedures will take these factors into account. The attenuation prediction method developed in the thesis paves the way to perform this work. A preliminary idea is stated here. The maximum coverage of wave fields in a composite plate can be designed as the objective function [START_REF] Ghyabi | A simplified framework for prediction of sensor network coverage in real-time structural health monitoring of plate-like structures[END_REF], with the optimization variables of sensor locations and number [START_REF] Yang | Sensor placement for structural health monitoring using hybrid optimization algorithm based on sensor distribution index and fe grids[END_REF]. The predicted spatially dependent attenuation will be designed as restriction conditions of the optimization problem which restricts that at each sensing path of interest, the attenuated wave amplitude should be higher than a preset threshold value in order to keep a sufficient signal to noise ratio (SNR).

Last but not least, improving the traditional damage localization methods such as delay-andsum [START_REF] Michaels | Damage localization in inhomogeneous plates using a sparse array of ultrasonic transducers[END_REF], RAPID [START_REF] Rébillat | Damage localization in geometrically complex aeronautic structures using canonical polyadic decomposition of lamb wave difference signal tensors[END_REF] and Excitelet [START_REF] Quaegebeur | Dispersion-based imaging for structural health monitoring using sparse and compact arrays[END_REF] by taking the viscoelasticity of composite materials into consideration is always of prime importance. Among these methods, temporal or frequency information is necessary, for instance, time of flight (ToF) for delay-and-sum [START_REF] Michaels | Detection, localization and characterization of damage in plates with an in situ array of spatially distributed ultrasonic sensors[END_REF] and wavenumber analysis for

Excitelet [START_REF] Quaegebeur | Correlation-based imaging technique for fatigue monitoring of riveted lap-joint structure[END_REF]. The temporal information can be obtained from the prediction of phase velocity and group velocity dispersion diagrams [START_REF] Zeng | Determination of lamb wave phase velocity dispersion using time-frequency analysis[END_REF], and the frequency information required for wavenumber

(2)

Bisection method Phase change method

(2,3)

(1,3) (1) (1)
(1)

Comments on the numerical labels:

(1) The intermediate parameters keep constants for a fixed 𝑣 0 , which maximizes the computational efficiency of both methods.

(2) The intermediate parameters are computed repeatedly for different 𝑣, which slows down the computational efficiency of both methods.

B.4 Conclusion

En résumé, l'utilisation des données réelles et en service des structures aéronautiques réelles pour valider la méthode de prédiction d'atténuation proposée n'est pas une tâche facile mais la rend extrêmement précieuse pour les communautés académiques et industrielles. En ce sens, cette thèse contribue à combler le fossé entre la recherche et le déploiement industriel des technologies de SHM [START_REF] Cawley | Structural health monitoring: Closing the gap between research and industrial deployment[END_REF]. Abstract : Anisotropic composite plates made up of viscoelastic composite materials are increasingly used in aeronautic industry and structural health monitoring strategies based on guided waves appear as a very promising tool to monitor their health state. In that context, this work is dedicated to the study of guided waves propagation and attenuation in such composite plates. Classical equations of guided waves propagating in a single layer elastic plate are first reviewed based on the three-dimensional (3D) elasticity theory and form the foundation of the partial wave superposition approach (PWSA). In the literature, this approach has been extended to the transfer matrix method (TMM) and global matrix method (GMM) allowing to express the wave propagation characteristics in elastic composite plates. Due to anisotropy, matrix incompatibility issues are encountered when dealing with arbitrarily orientated composite plates for TMM and GMM. An original hybrid matrix strategy (HMS) is proposed to alleviate this issue. It is furthermore demonstrated that dispersion equations built with PWSA for a single layer plate and with TMM and GMM for a multi-layered plate system possess a dichotomy property in the case of pure elastic materials. This property states that these a priori complex-valued equations collapse to pure real or imaginary valued equations, depending on the behavior of some intermediate parameters. This is extremely helpful for overcoming numerical instabilities encountered during the equation solving process. The elastic case is then extended to the viscoelastic case by introducing viscoelastic damping models (Hysteretic, Kelvin-Voigt and Biot models) in composite materials dynamical behavior. This forms the damped GMM (dGMM) proposed here to deal with composite plates made up of arbitrarily orientated anisotropic viscoelastic composite materials. Two industrial case studies, a fan cowl structure (FCS) and an inner fixed structure (IFS) both which are components of an A380 nacelle, are employed to experimentally validate the dGMM by comparing the accuracy of dGMM predictions in terms of velocities and attenuation to experimental measurements.

Keywords : Structural health monitoring, Guided waves, Aeronautic and aerospace composite materials, Anisotropy, Elasticity and viscoelasticity, Composite laminate, Partial wave superposition approach, Transfer matrix method, Global matrix method, Hybrid matrix strategy, Dichotomy property, Damping models, Dispersion equations and curves, Component of A380 nacelle.

  -clés : Surveillance de la santé structurelle, Ondes guidées, Equations et courbes de dispersion, Stratifiés composites anisotropes, Approche par superposition d'ondes partielles, Méthodes de transfert et de matrice globale, Propriété de dichotomie, Modèles d'amortissement viscoélastique, Composant de nacelle A380.

1 Introduction 1 . 1 CONTENTS 3 . 2 . 3 105 4. 9

 1113231059 Background of structural health monitoring . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The history of elastic waves in homogeneous isotropic media . . . . . . . . . . . . . . . 1.3 Modeling methods of guided waves in composite structures . . . . . . . . . . . . . . . 1.3.1 The origin of matrix-based methods for modeling guided waves propagation in layered structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 The modern matrix-based methods . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Finite element methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4 Other methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Organization of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dispersion equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Mode shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Dispersion equations of coupled Lamb and SH waves in monoclinic material . . . . . . 3.3.1 Christoffel equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Displacement and stress fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Dispersion equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Dispersion equations of decoupled Lamb and SH waves in orthotropic material . . . . 3.4.1 Christoffel equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Displacement and stress fields of Lamb wave . . . . . . . . . . . . . . . . . . . 3.4.3 Dispersion equations of Lamb wave . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.4 Displacement and stress fields of SH wave . . . . . . . . . . . . . . . . . . . . . 3.4.5 Dispersion relation of SH wave . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Derivation of the classical Rayleigh-Lamb equation via PWSA . . . . . . . . . . . . . . 3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Guided wave propagation in a multi-layered plate: transfer matrix method 4.1 The multi-layered composite plate model . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Local lamina matrices along the wave propagation direction . . . . . . . . . . . . . . . 4.2.1 Local lamina matrices of coupled Lamb and SH waves . . . . . . . . . . . . . . 4.2.2 Local lamina matrices of pure Lamb waves . . . . . . . . . . . . . . . . . . . . 4.2.3 Local lamina matrices of pure SH waves . . . . . . . . . . . . . . . . . . . . . . 4.3 Transfer matrix method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Local transfer matrix in a lamina . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 The monodromy transfer matrix of the laminate system . . . . . . . . . . . . . 4.3.3 Dispersion equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.3.4 Wave mode shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CONTENTS B.2 Etat de l'art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2.1 Les méthodes matricielles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2.1.1 Méthode de la matrice de transfert . . . . . . . . . . . . . . . . . . . . B.2.1.2 Méthode matricielle globale . . . . . . . . . . . . . . . . . . . . . . . . B.2.1.3 Méthode de la matrice de rigidité . . . . . . . . . . . . . . . . . . . . B.2.2 Méthodes basées sur les éléments finis . . . . . . . . . . . . . . . . . . . . . . . B.2.3 Autres méthodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2.4 Atténuation des ondes guidées . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3 Apport de la thèse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3.1 Chapitre 2 -Aspects fondamentaux de la propagation des ondes dans les solides B.3.2 Chapitre 3 -Propagation des ondes guidées dans une plaque monocouche . . .B.3.3 Chapitre 4 -Propagation des ondes guidées dans une plaque multicouche : méthode de la matrice de transfert . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3.4 Chapitre 5 -Propagation des ondes guidées dans une plaque multicouche : méthode matricielle globale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3.5 Chapitre 6 -Propriété de dichotomie de l'équation de dispersion des ondes guidées se propageant dans des plaques composites anisotropes . . . . . . . . . B.3.6 Chapitre 7 -Prédiction de l'atténuation des ondes guidées se propageant dans les stratifiés composites viscoélastiques anisotropes en fonction de la fréquence et de la direction de propagation . . . . . . . . . . . . . . . . . . . . . . . . . . B.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .List of Tables2.1 The characteristics of stiffness matrix rotation . . . . . . . . . . . . . . . . . . . . . . . 4.1 Four classes of composite laminates and the wave type dependence on the propagation angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Material properties of a transversely isotropic lamina . . . . . . . . . . . . . . . . . . . 4.3 Material properties of a carbon-fiber epoxy lamina . . . . . . . . . . . . . . . . . . . . 4.4 Material properties of aluminum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Material properties of glass fiber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Material properties of composite laminate of Open Guided Waves . . . . . . . . . . . . 5.2 The computational time of the 400-layered laminate using the optimized GMM (unit: seconds) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 The composite materials that automatically exclude the case ∆ > 0. . . . . . . . . . . 6.2 The composite materials that invoke the case ∆ > 0 in a limited phase velocity range. 6.3 The ranges of phase velocity v [m/s] that correspond to the four cases. . . . . . . . . . 6.4 The presence or absence of large f d problem of TMM in classical references. . . . . . . 6.5 The computational time of the 50-layered plate (unit: seconds). . . . . . . . . . . . . . 7.1 Elastic and viscoelastic material properties of composite lamina of the unmounted FCS. 7.2 Identified energy velocities from Figure 7.9(a). R 2 is the correlation coefficient. . . . . 7.3 Identified attenuation coefficient from Figure 7.9(b). R 2 is the correlation coefficient. . LIST OF TABLES 7.4 Elastic material properties of the carbon epoxy ply in the unmounted IFS. . . . . . . . 7.5 Elastic material properties of the aluminum core layer in the unmounted IFS. . . . . . 7.6 Viscoelastic material properties of the carbon epoxy ply and aluminum core in the unmounted IFS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1 Caractérisation des matrices de rigidité avant et après rotation . . . . . . . . . . . . . B.2 Classes de stratifiés composites et la dépendance du type d'onde à l'angle de propagation.242 B.3 Le temps de calcul du stratifié à 400 couches en utilisant le GMM optimisé (unité : secondes) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of Figures 1.1 SHM procedures of aeronautic structures to prevent accidents [1]. . . . . . . . . . . . . 1.2 Increasing usage of composite materials in aircraft structures over the past 40 years [2]. 1.3 Picture of a nacelle [3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Guided waves based SHM system [4]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Dispersion and directivity properties of guided waves velocity [4]. . . . . . . . . . . . . 1.6 Elastic waves in solid media [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Symmetric and anti-symmetric mode shapes of Lamb waves [4]. . . . . . . . . . . . . . 1.8 Labeling system for multilayered plate [6]. . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 SAFE model of wave propagation, (a) discretization along thickness direction, (b) threenode element and degrees of freedom of each node [7]. . . . . . . . . . . . . . . . . . . 2.1 The stress state of a cubic element of a solid material. . . . . . . . . . . . . . . . . . . 2.2 The schematic diagram of a composite lamina. . . . . . . . . . . . . . . . . . . . . . . 2.3 The example of Voigt notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 The schematic diagram of bulk wave propagation. . . . . . . . . . . . . . . . . . . . . 2.5 Elementary bulk waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Wave propagation model of a single-layered plate showing (a) boundary condition, (b) symmetric condition and (c) anti-symmetric condition. . . . . . . . . . . . . . . . . . . 3.2 The polynomial coefficients about α in Eq. (3.9). . . . . . . . . . . . . . . . . . . . . . LIST OF FIGURES 4.1 A three-layered composite plate model [0/θ/90]. The wave propagation direction is along the fiber direction of the middle layer. . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Guided wave propagation model in a n-layered composite laminate. . . . . . . . . . . . 4.3 Pant's pseudo correction measure to deal with the incompatibility issue [8]. . . . . . . 4.4 Barazanchy's vector orthogonality analysis to deal with the incompatibility issue [9]. . 4.5 Conceptual diagram of determining the wave type existing in the cross-ply laminate [(0/90) s ] 2 at wave propagation angle θ = 0 • . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Dispersion curves of pure Lamb and SH waves of the cross-ply laminate when θ = 0 • . 4.7 Conceptual diagram of determining the wave type existing in the cross-ply laminate [(0/90) s ] 2 at wave propagation angle θ = 45 • . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Dispersion curves of the coupled Lamb and SH waves of the cross-ply laminate at θ = 45 • .Displacement mode shapes of the cross-ply laminate for θ = 0 • and 45 • at f = 400 kHz. 4.10 Stress mode shapes of the cross-ply laminate for θ = 0 • and 45 • at f = 400 kHz. . . . 4.11 Conceptual diagram of determining the wave type existing in the quasi-isotropic laminate [0/90/ + 45/ -45] s at the wave propagation angle (a) θ = 0 • , (b) θ = 30 • , (c) θ = 45 • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.12 Dispersion curves of the hybrid or coupled Lamb and SH waves of the quasi-isotropic laminate. The experimental data points are extracted from [8]. . . . . . . . . . . . . . 4.13 Polar plot of wave velocities. The experimental data points are extracted from [8]. . . 4.14 Conceptual diagram of Pant's pseudo correction to deal with the incompatibility issue of the quasi-isotropic laminate [0/90/ + 45/ -45] s when θ = 0 • . . . . . . . . . . . . . . 4.15 Conceptual diagram of determining the wave type existing in the metallic-composite laminate [Al/0/90/Al/90/0/Al] at the wave propagation angle (a) θ = 0 • , (b) θ = 45 • and (c) θ = 45 • for Pant's pseudo correction. . . . . . . . . . . . . . . . . . . . . . . . 4.16 Dispersion curves of the hybrid or pure Lamb and SH waves of the metallic-composite laminate. The experimental data points are extracted from [8]. . . . . . . . . . . . . . 4.17 Polar plot of wave velocities. The experimental data points are extracted from [8]. . . LIST OF FIGURES 5.1 Dispersion curves of Open Guided Waves at the propagation angle θ = 0 • computed via TMM and GMM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Dispersion curves of Open Guided Waves at the propagation angle θ = 0 • computed via GMM and ESM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Polar plot of Open Guided Waves plate at frequency=260 kHz. . . . . . . . . . . . . . 5.4 Equations of ESM to compute the effective stiffness matrix [10]. . . . . . . . . . . . . . 5.5 A symmetric laminate showing only the upper half layers. In the SC ς X,lm bot and ς X,lm mid , the superscript X = s and a represent the symmetric and anti-symmetric condition, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Dispersion curves of the 400-layered aerospace laminate at θ = 0 • direction. . . . . . . 5.7 MAC value tomography of displacement mode shapes in the region R1 of symmetric modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Displacement mode shapes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9 MAC value tomography of stress mode shapes in the region R1 of symmetric modes. . 5.10 Stress mode shapes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 A summary of the solving strategy adopted by Dispersion Calculator [11]. . . . . . . . 6.2 The complex-to-real strategy for solving Rayleigh-Lamb equation [12]. . . . . . . . . . 6.3 The typical ∆ -v relation showing (a) the absence of ∆ > 0 and (b) the presence of ∆ > 0 in a narrow phase velocity range. . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Using bisection method to solve the equation D a (v 0 = 9000, ξ) = 0. . . . . . . . . . . . 6.5 Using phase change method to solve the equation D a (v 0 = 9000, ξ) = 0. . . . . . . . . 6.6 Using bisection method to solve the equation D a (v 0 = 5000, ξ) = 0. . . . . . . . . . . . 6.7 The schematic diagram of phase change method for case 2. . . . . . . . . . . . . . . . 6.8 Using phase change method to solve the equation D a (v 0 = 5000, ξ) = 0. . . . . . . . . 6.9 Using bisection and phase change methods to solve the equation D a (v, ξ 0 = 1750) = 0. The small width of case 3 is caused by the degree of anisotropy. . . . . . . . . . . . . . LIST OF FIGURES 6.10 Dispersion curves of the monoclinic lamina. . . . . . . . . . . . . . . . . . . . . . . . . 6.11 Concluding figures: (a) the strategy to determine the dichotomy property of dispersion equation, (b) the computational efficiency of solving methods adapted to dichotomy property. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.12 Dispersion curves propagating at 0 • for the [0] 50 unidirectional laminate. . . . . . . . . 6.13 Phase angle curve of D G (v, ξ 0 = 3000) = 0 for the [0] 50 unidirectional laminate. . . . . 6.14 Dispersion curves in frequency-wavenumber domain propagating at 45 • of the singlelayered unidirectional composite plate. . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.15 Dichotomy property of dispersion equation at v 0 = 1000 m/s that invokes ∆ > 0. . . . 6.16 Function curves of D X (v 0 , ξ) = 0 at v 0 = 1000 m/s of the Open Guided Waves. . . . . 6.17 The condition number of matrices of the Open Guided Waves at v 0 = 1000 m/s for (a) the monodromy transfer matrix T mono and (b) the global matrix D G . . . . . . . . . . 6.18 Function curve of the symmetric modes of dispersion equation D s (v 0 , ξ) for a triclinic material at v 0 = 4000 m/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 The studied aircraft nacelles of an A380 plane for (a) fan cowl structure and (b) inner fixed structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Schematical diagram of a n-layered aeronautic composite laminate. . . . . . . . . . . . 7.3 The schematic diagram of the curve tracing technique. . . . . . . . . . . . . . . . . . . 7.4 The flowchart of the two-step dGMM algorithm for solving the complex DE. . . . . . 7.5 Overview of the proposed method to predict guided waves attenuation. . . . . . . . . . 7.6 The PZT deployments of FCS for (a) the unmounted case and (b) the mounted case on an instrumented A380 plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 The experimental setup of the unmounted FCS. . . . . . . . . . . . . . . . . . . . . . . 7.8 Typical guided wave signals of the unmounted FCS at 100 kHz. . . . . . . . . . . . . . 7.9 Example of identifying (a) energy velocity and (b) attenuation coefficient via LR method at 100 kHz for the unmounted FCS. Both figures share the same legend and α is used to give the confidence level with 100(1 -α)%. . . . . . . . . . . . . . . . . . . . . . . . LIST OF FIGURES 7.10 Polar plot of S0 mode of (a) energy velocity [m/s] and (b) attenuation coefficient [Np/m] at 100 kHz for the unmounted FCS. Both subfigures share the same legend. . . . . . . 7.11 Bar chart to display regression results of (a) energy velocity and (b) attenuation coefficient via LR method for the mounted FCS at 100 kHz. The horizontal line represents the reference value of the unmounted FCS, and the red triangular arrows denote that the marked bars correspond to a R 2 larger than 0.7. . . . . . . . . . . . . . . . . . . . 7.12 The identified dispersion curves of (a) energy velocity, (b) attenuation coefficient and (c) correlation coefficient. These subfigures relate to the S0 mode propagating at 0 • direction of the unmounted FCS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.13 Comparison between the computed and experimentally identified dispersion curves of (a) energy velocity and (b) attenuation coefficient for the unmounted FCS at θ = 0 • . . 7.14 For the unmounted FCS, HR model's prediction of (a) dispersion curve of phase velocity at θ = 180 • , (b) dispersion curve of attenuation coefficient at θ = 180 • , (c) polar plot of phase velocity [m/s] at f = 500 kHz, (d) polar plot of attenuation coefficient [Np/m] at f = 500 kHz. The open circles in the four subfigures are the solutions computed by SAFE method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.15 For the unmounted FCS, surface plot of phase velocity (left) and attenuation coefficient (right): (a) and (b) S0 mode, (c) and (d) A0 mode, (e) and (f) SH0 mode. The inner and outer surfaces in subfigures (b)(d)(f) correspond to HR and KV models, respectively.196 7.16 The transducers layout of IFS for (a) the unmounted case and (b) the mounted case on an instrumented A380 plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.17 Comparison between the mounted and unmounted IFS for (a) energy velocity and (b) attenuation coefficient at 100 kHz. In the two subfigures, the bars relate to the left y-axis and the lines correlate to the right y-axis. . . . . . . . . . . . . . . . . . . . . . 7.18 Polar plot of S0 mode energy velocity [km/s] in the mounted IFS at 100 kHz. The contour of dGMM prediction 1 and dGMM prediction 2 are computed by using the (E 1 ,E 2 ,E 3 ) values 1 and 2 in Table 7.4, respectively. . . . . . . . . . . . . . . . . . . . 7.19 The identified S0 mode dispersion curves of (a) energy velocity and (b) attenuation coefficient for the unmounted IFS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7. 20

 20 Comparison between the computed and experimentally identified dispersion curves of (a) energy velocity and (b) attenuation coefficient for the unmounted IFS at θ = 0 • . . 7.21 The computed group velocity via the pure elastic GMM for the unmounted IFS at θ = 0 • .202 7.22 For the unmounted IFS, the predicted dispersion curves at θ = 90 • for HR+BT model (left) and KV+BT model (right): (a) and (b) 3D space of (f, k r , k i ), (c) and (d) projection onto (f, k r ) plane, (e) and (f) projection onto (f, k i ) plane. . . . . . . . . . . . 7.23 For the unmounted IFS, HR+BT model's prediction of (a) dispersion curve of phase velocity at θ = 90 • , (b) dispersion curve of attenuation coefficient at θ = 90 • , (c) polar plot of phase velocity [m/s] at f = 200 kHz, (d) polar plot of attenuation coefficient [Np/m] at f = 200 kHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.24 For the unmounted IFS, surface plot of phase velocity (left) and attenuation coefficient (right): (a) and (b) S0 mode, (c) and (d) A0 mode, (e) and (f) SH0 mode. . . . . . . . B.1 Image d'une nacelle d'Airbus A380 [3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2 Système d'étiquetage pour plaque multicouche [6]. . . . . . . . . . . . . . . . . . . . . B.3 Modèle SAFE de propagation des ondes, (a) discrétisation dans le sens de l'épaisseur, (b) élément à trois noeuds et degrés de liberté de chaque noeud [7]. . . . . . . . . . . . B.4 Courbes de dispersion pour le cas d'étude Open Guided Waves pour un angle de propagation θ = 0 • calculées via la TMM et la GMM. . . . . . . . . . . . . . . . . . . . . . B.5 Courbes de dispersion des ondes guidées ouvertes à l'angle de propagation θ = 0 • calculées via GMM et ESM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.6 Courbes de dispersion du stratifié aérospatial à 400 couches dans la direction θ = 0 • . . B.7 Formes de mode de déplacement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.8 Tomographie à valeur MAC des formes de mode de déplacement dans la région R1 des modes symétriques. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.9 (a) Stratégie à adopter pour déterminer la propriété de dichotomie de l'équation de dispersion. (b) Efficacité de calcul des méthodes de résolution basées sur la propriété de dichotomie. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LIST OF FIGURES B.10 Le nombre de conditionnement des matrices pour le cas Open Guided Waves à v 0 = 1000 m/s pour (a) la matrice de transfert de monodromie T mono (méthode TMM) et (b) la matrice globale D G (méthode GMM). . . . . . . . . . . . . . . . . . . . . . . . . B.11 L'organigramme de l'algorithme dGMM en deux étapes pour résoudre le DE complexe. B.12 Le réseau de PZT pour le FCS: (a) le cas ou la structure n'est pas montée sur l'A380 et (b) le cas ou la structure est montée sur un A380 instrumenté. . . . . . . . . . . . . B.13 Le réseau de PZTs pour l'IFS: (a) le cas ou la structure n'est pas montée sur l'A380 et (b) le cas ou la structure est montée sur un A380 instrumenté. . . . . . . . . . . . . . B.14 Prédiction du modèle HR + BT pour l'IFS non monté: (a) courbe de dispersion de la vitesse de phase à θ = 90 • , (b) courbe de dispersion du coefficient d'atténuation à θ = 90 • , (c) tracé polaire de la vitesse de phase [m/s] à f = 200 kHz, (d) tracé polaire du coefficient d'atténuation [Np/m] à f = 200 kHz. . . . . . . . . . . . . . . . . . . . .

1. 1 .

 1 Figure 1.1 -SHM procedures of aeronautic structures to prevent accidents [1].

Figure 1 . 2 -

 12 Figure 1.2 -Increasing usage of composite materials in aircraft structures over the past 40 years [2].

1. 1 .Figure 1 . 3 -

 113 Figure 1.3 -Picture of a nacelle [3]

Figure 1 . 5 -

 15 Figure 1.5 -Dispersion and directivity properties of guided waves velocity [4].

1. 3 .Figure 1 . 6 -

 316 Figure 1.6 -Elastic waves in solid media [5].

Figure 1 . 7 -

 17 Figure 1.7 -Symmetric and anti-symmetric mode shapes of Lamb waves [4].

1. 3 . 1

 31 The origin of matrix-based methods for modeling guided waves propagation in layered structure Matrix-based methods are mainly applied to multi-layered problems, and they have received a great development since 1950s from theoretical advancements to engineering applications. There are three specific methods including transfer matrix method, global matrix method and stiffness matrix method.

Figure 1 . 8 -

 18 Figure 1.8 -Labeling system for multilayered plate [6].

Figure 2 . 2 -

 22 Figure 2.2 -The schematic diagram of a composite lamina.

c

  1111 c 1122 c 1133 c 1123 c 1131 c 1112 c 2211 c 2222 c 2233 c 2223 c 2231 c 2212 c 3311 c 3322 c 3333 c 3323 c 3331 c 3312 c 2311 c 2322 c 2333 c 2323 c 2331 c 2312 c 3111 c 3122 c 3133 c 3123 c 3131 c 3112 c 1211 c 1222 c 1233 c 1223 c 1231 c 1212

Figure 2 . 4 -

 24 Figure 2.4 -The schematic diagram of bulk wave propagation.

Figure 2 . 5 -

 25 Figure 2.5 -Elementary bulk waves.

Figure 3 . 3 3. 1 .

 331 1(a), v is the phase velocity, 1 U = U 1 , V = U 2 , W = U 3 are the amplitudes of displacements along x 1 , x 2 , x WAVE PROPAGATION MODEL OF A SINGLE LAYER PLATE direction, respectively.

Figure 3 . 1 -

 31 Figure 3.1 -Wave propagation model of a single-layered plate showing (a) boundary condition, (b) symmetric condition and (c) anti-symmetric condition.

Figure 3 .

 3 Figure 3.1(a), the three stresses (σ 33 , σ 23 , σ 13 ) should be zero when x 3 = ±h as formulated in Eq. (3.2).

3. 2 .

 2 DISPERSION EQUATIONS OF COUPLED LAMB AND SH WAVES IN TRICLINIC MATERIAL derivative of displacement has been presented in Eqs. (3.5) and (3.6), respectively, based on Eq. (3.1).

Figure 3 . 2 -

 32 Figure 3.2 -The polynomial coefficients about α in Eq. (3.9).

material, C 14 ,

 14 C 15 , C 24 , C 25 , C 34 , C 35 , C 46 , C 56 = 0. Thus, the Christoffel equation of triclinic material in Eq. (3.7) changes to Eq. (3.22), though it remains the same form as Eq. (3.7) yet with different elements in the matrix K(α) as shown in Eq. (3.23).

  (3.34) and D a (v, ξ) in Eq. (3.36) are derived into Eqs. (3.37a) and (3.37b) by right of the software Mathematica

  clinic material, C 16 , C 26 , C 36 , C 45 = 0, which makes the Christoffel equation of monoclinic material in Eq. (3.22) change to Eq. (3.40).

3 . 40 )

 340 Naturally, Eq. (3.40) can be separated as Eqs. (3.42) and (3.41). Since Eq. (3.41) only involves in the displacement u 2 (shear horizontal direction), the wave derived from Eq. (3.41) is termed as SH wave. Correspondingly, the wave derived from Eq. (3.42) is termed as Lamb wave because this equation involves in the displacements u 1 (longitudinal direction) and u 3 (shear vertical direction).

  (3.47)(3.48), substituting them into the boundary condition of Eq. (3.2) (only σ 33 and σ 13 are evaluated) and the symmetric condition of Eq. (3.3) (only u 3 and σ 13 are evaluated) allows to derive the dispersion equation of symmetric modes of Lamb wave, which is presented in Eqs. (3.50)(3.51).

11 β 33 β 13 β 31 for symmetric modes ( 3 . 53 )

 353 To obtain the dispersion equation of anti-symmetric modes, the displacement and stress field equations in Eqs. (3.47)(3.48) are substituted into the boundary condition of Eq. (3.2) (only σ 33 and σ 13 are evaluated) and the anti-symmetric condition of Eq. (3.4) (only u 1 and σ 33 are evaluated). The resulted equations are presented in Eqs. (3.54)(3.55).

  13 β 31 β 11 β 33 for anti-symmetric modes (3.57) 3.4.4 Displacement and stress fields of SH wave Eq. (3.41) characterizes the displacement and stress fields of SH wave, from which the two α's roots can be easily solved as shown in Eq. (3.58) with the indices 5 and 6 given that the indices from 1 to 4 have been occupied by Lamb wave in Eq. (3.44). Accordingly, there are two partial waves to compose SH wave. The displacement u 2 and stress σ 23 are expressed in Eqs. (3.59) and (3.60) respectively based on PWSA.

66 ) 67 )

 6667 Obviously, letting n = 0 in Eq. (3.66) produces the fundamental SH0 mode which possesses a constant phase velocity as written in Eq. (3.67), indicating the non-dispersive characteristic with respect to frequency ω or wavenumber ξ.For each (v, ξ) solution point of a SH mode, the corresponding displacement and stress mode shapes computed via Eqs. (3.59)(3.60) can be also classified into symmetric and anti-symmetric modes via the symmetry condition stated in Eqs. (3.3)(3.4) in which only u 2 and σ 23 are evaluated. Specifically, the symmetric mode corresponds to the condition σ 23 | x 3 =0 = 0 and the anti-symmetric mode makes the requirement u 2 | x 3 =0 = 0, where x 3 = 0 indicates the midplane. It should be noted that although Eqs. (3.53)(3.57)(3.65) are derived through the stiffness matrix of orthotropic material, they are still compatible with transversely isotropic and isotropic materials given that the stiffness matrix of the two materials belong to the stiffness matrix set of orthotropic materials, i.e. C TI , C I ∈ C O (see Section 2.2.4 and 2.3.3). Thus, one just needs to substitute the corresponding elements in the stiffness matrix of transversely isotropic or isotropic material into Eqs. (3.53)(3.57)(3.65)

  (3.53)(3.57) which are based on PWSA to complete the wave propagation theory. When it comes to Rayleigh-Lamb equation, it belongs to the isotropic material case. Thus, all the derivations in section 3.4.2 should be replaced by the parameters of isotropic material. The derivation starts from Eq.(3.44). Thus, the polynomial coefficients A 4 , A 2 , A 0 in Eq.(3.45) are now computed by using the stiffness coefficients of the isotropic material (see Eq. (2.14)) as follows.

  velocity of longitudinal and transverse modes, respectively. Substitute Eq. (3.70) into Eq. (3.46) to get W L and W T .

  Substitute Eq. (3.74) into Eqs. (3.53) and (3.57) to retrieve the classical Rayleigh-Lamb equation.

Figure 4 . 1 -

 41 Figure 4.1 -A three-layered composite plate model [0/θ/90]. The wave propagation direction is along the fiber direction of the middle layer.

  Figure 4.1(b). For an interested wave propagation direction, it may coincide with the principal direction of a certain lamina (see the middle layer in Figure 4.1(b)), but it may correspond to the off-principal direction of another lamina (see the top layer in Figure 4.1(b)) due to the varying stacking angles of layups. For each lamina, its stiffness matrix keeps the orthotropic type, i.e. C O , in the principal and minor-principal directions according to the Characteristic 2 of Section 2.3.3. However, for a lamina whose principal direction does not coincide with the wave propagation direction, its C O should be rotated to the wave propagation direction with angle θ, denoted as C ′ O . According to the Characteristic 1 of Section 2.3.3, it can be easily inferred that C ′ O ∈ C M .

Figure 4 .

 4 2(a). The profile section of the laminate along the wave propagation direction is shown in Figure 4.2(b). For a general layer l k , its rotation angle becomes (θ -ϕ k ), thus the rotated stiffness matrix of this layer becomes

  (a) The 𝑛-layered laminate model with stacking angles of layups 𝜙 1 /𝜙 2 / ⋯/𝜙 𝑛 The profile section of the laminate in the wave propagation direction

Figure 4 . 2 -

 42 Figure 4.2 -Guided wave propagation model in a n-layered composite laminate.

17

 17

  )Eqs.(4.14) and (4.16) can be concatenated into a single equation, see the matrix form in Eq.(4.18)

  2.1) (p = 6), for the pure Lamb waves in Section (4.2.2) (p = 4) and for the pure SH waves in Section (4.2.3) (p = 2).

4. 3 . 2

 32 The monodromy transfer matrix of the laminate systemThe local transfer matrix T l k in Eq. (4.29) can be synthesized to yield the monodromy transfer matrix of the laminate. Primarily, each layer's local transfer matrix should have the same dimension, i.e. T l k ∈ C p×p for k = 1, . . . , n, otherwise the incompatibility issue of matrix dimension just occurs and it will be discussed in Section 4.4. By virtue of Eq. (4.28), the field variables at the top and bottom sides of each layer can be linked based on the following transferring rule.

1 .

 1 The coupled Lamb and SH waves for T mono ∈ C 6×6 When T mono ∈ C 6×6 , Eq. (4.33) can be expanded in Eq. (4.35), which characterizes the field variables regarding to the coupled Lamb and SH waves.

  characterizes the pure SH waves. Expand Eq. (4.33) to derive the dispersion equation of the pure SH waves, as presented in Eq. (4.43).

8 : end if Algorithm 2

 82 Process the coupled Lamb and SH waves via TMM Input: Each layer's material properties (C k , ρ k , d k , ϕ k ) for (k = 1, . .

4. 4

 4 Hybrid matrix strategy 4.4.1 The incompatibility issue of matrix dimension of the local transfer matrices In Eq. (4.34), there is a requirement that all layers' local transfer matrices T l k (k = 1, . .. , n)should have the same dimension in order to produce the monodromy transfer matrix T mono ∈ C p×p , otherwise an incompatibility issue of the local transfer matrices in terms of their matrix dimension 4.4. HYBRID MATRIX STRATEGY is encountered. Physically, this issue corresponds to the case that the interested wave propagation direction is along the principal direction of one layer (producing the decoupled Lamb and SH waves in this layer) but along the off-principal direction of another layer (producing the coupled Lamb and SH

Figure 4 . 3 -

 43 Figure 4.3 -Pant's pseudo correction measure to deal with the incompatibility issue [8].

  Process the coupled Lamb and SH waves.

Figure 4 . 6 .

 46 Figure 4.6. For another wave propagation angle θ = 90 • , a similar diagram than Figure 4.5 can be generated to determine the accessible wave type existing in the cross-ply laminate at the specified angle θ = 90 • . Due to the reciprocal condition between θ = 0 • and θ = 90 • for the cross-ply stacking sequence [(0/90) s ] 2 , it is easy to infer that the rotated C ′ i ∈ C O (i = 1, . . . , 8) for θ = 90 • . Thus, the accessible

Figure 4 . 5 -Figure 4 . 6 -

 4546 Figure 4.5 -Conceptual diagram of determining the wave type existing in the cross-ply laminate [(0/90) s ] 2 at wave propagation angle θ = 0 • .

4. 6 . 2

 62 The coupled Lamb and SH waves when θ = 45 • When we consider θ = 45 • , the conceptual diagram of determining the plate's wave type is illustrated in Figure 4.7. It shows that each layer's stiffness matrix after rotation changes to C ′ i ∈ 4.6. NUMERICAL EXAMPLE ON A CROSS-PLY LAMINATE C M (i = 1, . . . , 8). This outcome immediately leads to the coupled Lamb and SH waves according to the Branch 1 of Algorithm 1, and Algorithm 2 should be adopted to compute the dispersion curves. After running Algorithm 2, the resulted dispersion curves of the coupled Lamb and SH waves are depicted in Figure 4.8. 𝑖 , 𝜃 𝑖 }

Figure 4 . 7 -

 47 Figure 4.7 -Conceptual diagram of determining the wave type existing in the cross-ply laminate [(0/90) s ] 2 at wave propagation angle θ = 45 • .

Figure 4 . 8 -Figure 4 . 9 -Figure 4 . 10 -

 4849410 Figure 4.8 -Dispersion curves of the coupled Lamb and SH waves of the cross-ply laminate at θ = 45 • .

Figures 4 .

 4 Figures 4.6and 4.8 to compute their displacement mode shapes (see Figure 4.9) and stress mode shapes (see Figure 4.10).

Figure 4 . 6 .

 46 Specifically, Figure 4.9(a) corresponds to the point A of Figure 4.6 that belongs to the pure Lamb modes. Thus, u 2 component vanished in a pure Lamb mode. Likewise, Figure 4.9(c) corresponds to the point C of Figure 4.6 belonging to the pure SH modes and thus the u 1 and u 3 components vanished in a pure SH mode.

Figure 4 .

 4 Figure 4.9(d), (e), (f) corresponds to the point A, B, C of Figure 4.8, respectively, which belong to the coupled Lamb and SH waves. The three displacement components are coexisting in Figure 4.9(d)(e)(f). For the coupled modes, the mode order is named by following the terminology of

Figure 4 .

 4 10, the zero-valued stress mode shapes at the top-and bottom-most surfaces can be observed thus demonstrating the traction-free boundary condition considered in Eq. (4.36). Besides, the continuous curves of the displacement and stress mode shapes reveal that the continuity condition postulated during deriving the TMM framework (see Eq. (4.31)) is legitimate. In Figures 4.9 and 4.10, the dotted horizontal line represents the midplane of the plate, which is mentioned here to validate the symmetry condition (see Figure 3.1(b)(c) and Eqs. (3.3)(3.4) for the single-layered system, where the midplane is represented by x 3 = 0). As a consequence, the dispersion curves in Figures 4.6 and 4.8 are further classified into symmetric and anti-symmetric modes.

4. 7 . 1

 71 The hybrid Lamb and SH waves when θ = 0 • , 45 • or 90•

Figure 4 .

 4 Figure 4.11(a) illustrates the process of determining the wave type existing in the laminate at the current propagation angle θ = 0 • . It is clear that the stiffness matrices of some layers after rotation (specifically, the 1 st , 2 nd , 7 th , 8 th layer) belong to the set C O but another layers (from the 3 rd to the 6 th layer) belong to the set C M . For the rotated C ′ j ∈ C O , there exists two kinds of local transfer matrix T l j 4×4 and T l j 2×2 according to the Section 4.2.2 and 4.2.3, which correspond to the pure Lamb and SH waves in this layer. But for another rotated C ′ i ∈ C M , it has only the local transfer matrix T l i 6×6 according to the Section 4.2.1, which corresponds to the coupled Lamb and SH waves in this layer.

4. 7 .)

 7 NUMERICAL EXAMPLE ON A QUASI-ISOTROPIC LAMINATE the hybrid Lamb and SH waves are obtained and the dispersion curves are depicted in Figure 4.12(a) which corresponds to the current propagation angle θ = 0 • . HMS based on Eqs. (4.57)(4.59)--------------------→T ˜lj 6×6 The incompatibility issue is still occurring at θ = ±45 • and 90 • due to the stacking angles of layups [0/90/ + 45/ -45] s , and it can be qualitatively analyzed by making the same kind of conceptual diagram as the angle θ = 0 • in Figure 4.11(a). To avoid redundancy, we only illustrate the angle θ = 45 • in Figure 4.11(c). It shows that, after rotation, the stiffness matrices of the 3 rd to the 6 th layer keep the orthotropic type, i.e. C ′ j ∈ C O (j = 3, 4, 5, 6), which is conjugated with Figure 4.11(a) in the graphical sense. Thus, the HMS should be applied to these layers. After running Algorithm 1 and 5, the dispersion curves of the hybrid Lamb and SH waves are depicted in Figure 4.12(c) and (d) for θ = 45 • and θ = 90 • , respectively.

4. 7 . 2

 72 The coupled Lamb and SH waves when θ = 30• When the wave direction of interest does not coincide with any principal axis of all layers, i.e. θ / ∈ {0 • , 45 • , 90 • , -45 • }, the rotated C ′ i of all layers shall belong to the set C M , which gives rise to the unified local transfer matrix T l i 6×6 between all layers according to the Section 4.2.1. Here, we take a typical angle θ = 30 • to present the process of determining the coupled type of Lamb and SH waves as illustrated in Figure 4.11(b). For the coupled case, Algorithm 2 should be adopted to compute the dispersion solutions according to the result provided by Algorithm 1. After computation, the dispersion curves of the coupled Lamb and SH waves are depicted in Figure 4.12(b) for the current angle θ = 30 • .

Figure 4 .

 4 13 for f d = 0.1 MHz • mm and f d = 0.4 MHz • mm, where f d represents the product of frequency f and plate's total thickness d. In this figure, the eight singular angles are indicated, they are θ ∈ {0 • , 45 • , 90 • , 135 • , 180 • , 225 • , 270 • , 315 • }.

4. 7 .

 7 NUMERICAL EXAMPLE ON A QUASI-ISOTROPIC LAMINATE

Figure 4 . 11 -

 411 Figure 4.11 -Conceptual diagram of determining the wave type existing in the quasi-isotropic laminate [0/90/ + 45/ -45] s at the wave propagation angle (a) θ = 0 • , (b) θ = 30 • , (c) θ = 45 • .

Figure 4 . 12 -

 412 Figure 4.12 -Dispersion curves of the hybrid or coupled Lamb and SH waves of the quasi-isotropic laminate. The experimental data points are extracted from [8].

Figure 4 . 13 -

 413 Figure 4.13 -Polar plot of wave velocities. The experimental data points are extracted from [8].

Figure 4 .

 4 Figure 4.11(a). The core idea of this measure is detailed in the right-most two columns of Figure 4.14. By additionally rotating the C ′ j ∈ C O (j = 1, 2, 7, 8) with a minor angle ∆θ = -1 • , all the rotated stiffness matrices now belong to the set C M that finally produces the coupled Lamb and SH waves.

Figure 4 . 14 -

 414 Figure 4.14 -Conceptual diagram of Pant's pseudo correction to deal with the incompatibility issue of the quasi-isotropic laminate [0/90/ + 45/ -45] s when θ = 0 • .

  layers and four glass fiber layers with stacking angles of layups [Al/0/90/Al/90/0/Al]. The material 4.8. NUMERICAL EXAMPLE ON A METALLIC-COMPOSITE LAMINATE

Figure 4 .

 4 Figure 4.15(a) illustrates the process of determining wave type when one principle-axis direction θ = 0 • is concerned. At this direction, the rotated stiffness matrices of aluminum layers remain the isotropic property C ′ i ∈ C I (i = 1, 4, 7), and the rotated stiffness matrices of glass fiber layers keep the orthotropic property C ′ j ∈ C O (j = 2, 3, 5, 6). According to the classification C I ⊆ C O defined in the Characteristic 3 of Section 2.3.3, we can also classify the C ′ i ∈ C O (i = 1, 4, 7). As such, all the rotated stiffness matrices now belong to the orthotropic type that finally produces the pure Lamb and SH waves. For another principle-axis direction θ = 90 • , the same outcome of the pure case is anticipated. The dispersion curves of the pure Lamb and SH waves at the two principle-axes directions are then retrieved by running Algorithm 3 and 4 and are depicted in Figure 4.16(a)(e). The good agreement between the computed curves and the experimental data points demonstrates the high accuracy of the TMM framework in processing the metallic-composite laminate.

Figure 4 . 15 -Figure 4 . 16 -

 415416 Figure 4.15 -Conceptual diagram of determining the wave type existing in the metallic-composite laminate [Al/0/90/Al/90/0/Al] at the wave propagation angle (a) θ = 0 • , (b) θ = 45 • and (c) θ = 45 • for Pant's pseudo correction.

Figure 4 . 17 -

 417 Figure 4.17 -Polar plot of wave velocities. The experimental data points are extracted from [8].

Figure 4 .

 4 15(b) illustrates the process of determining wave type when θ = 45 • . At this direction, the stiffness matrices of aluminum layers are rotated to C ′ i ∈ C I ⊆ C O (i = 1, 4, 7) due to the rotation invariance property of isotropic material, and the stiffness matrices of the glass fiber layers are rotated to C ′ j ∈ C M (j = 2, 3, 5, 6). The inconformity of stiffness matrix classes between aluminum and glass fiber layers at the current angle θ = 45 • makes the requirement that only the hybrid local transfer matrices of the aluminum layers T ˜li 6×6 (i = 1, 4, 7) are compatible with the local transfer matrices of the glass fiber layers T l j 6×6 (j = 2, 3, 5, 6) as illustrated in Figure 4.15(b). Correspondingly, the hybrid type of Lamb and SH waves is determined. For another two angles θ = 20 • and 70 • , the same hybrid wave type can be analyzed. After running Algorithm 5, the dispersion curves at the three angles θ = 20 • , 45 • , 70 • are retrieved in Figure 4.16(b), (c), (d), respectively. The reason of exhibiting the dispersion curves of θ = 20 • and 70 • in Figure4.16 is that the experimental data of the two angles are available from[8]. The good agreement between the computed curves and the experimental data points validates the feasibility of the HMS in processing the metallic-composite laminate.In the quasi-isotropic laminate [0/90/ + 45/ -45] s , there are only eight singular angles θ ∈ {0 • , 45 • , 90 • , 135 • , 180 • , 225 • , 270 • , 315 • } as indicated in Figure 4.13. However, in the metallic-composite 4.9. CONCLUSION laminate, any propagation angle is singular except θ ∈ {0 • , 90• , 180 • , 270 • }. This is the major difference between the polar plot of the quasi-isotropic laminate in Figure4.13 and the polar plot of the metallic-composite laminate in Figure4.17

  .

4. 8 . 3

 83 Superiority of HMS over Pant's pseudo correction measure

Figure 4 .

 4 Figure 4.15(c) illustrates the ineffectiveness of Pant's pseudo correction measure during processing the incompatibility issue of metallic-composite laminate. It shows that the additionally rotated stiffness matrices of aluminum layers still remains isotropic property, C ′′ i ∈ C I (i = 1, 4, 7), due to the rotation invariance of isotropic material. In consequence, the two kinds of local transfer matrices of aluminum layers ( T l i 4×4 , T l i 2×2

Algorithm 1

 1 is the quintessence of this chapter. It judges which wave type exists in the currently studied composite laminate at the interested propagation angle. The related variables, matrices and equations corresponding to each wave type have been programmatically interpreted in Algorithm 2, 4.9. CONCLUSION 3, 4 and 5. These algorithms are controlled by Algorithm 1. With the smooth cooperation between Algorithm 1 and Algorithms 2-5, the correct wave characteristics can be computed including the dispersion relation (phase and group velocities), wave mode shapes and polar plots of these quantities. Numerical examples on three commonly used composite laminates validates the effectiveness of the standard TMM and the HMS. TMM framework is accurate enough when comparing to experimental data. The HMS is also superior than Pant's pseudo correction measure for dealing with the wave propagation problem in metallic-composite plate.Chapter 5

Figure 4 .

 4 2(b), it corresponds to the following equation. σ l 1 top = 0 and σ ln bot = 0 (5.4) 5.1. THE STANDARD GLOBAL MATRIX METHOD Substituting Eq. (5.1b) into Eq. (5.4) can lead to Eq. (5.5).

Figure 4 .

 4 2(b) and combining Eq. (5.5), all the resulted equations can be regularly organized to a global linear homogeneous equation set, Eq. (5.6), which requires the vanished determinant of the coefficients matrix, D G (v, ξ) ∈ C np×np , finally generating the implicit dispersion equation of guided waves propagating in a laminate as presented in Eq. (5.7).

Figure 5 . 1 - 123 5. 2 .

 511232 Figure 5.1 -Dispersion curves of Open Guided Waves at the propagation angle θ = 0 • computed via TMM and GMM.

Figure 5 . 2 -

 52 Figure 5.2 -Dispersion curves of Open Guided Waves at the propagation angle θ = 0 • computed via GMM and ESM.

Figure 5 . 3 -

 53 Figure 5.3 -Polar plot of Open Guided Waves plate at frequency=260 kHz.

Figure 5 . 4 -

 54 Figure 5.4 -Equations of ESM to compute the effective stiffness matrix [10].

5. 3 .

 3 THE OPTIMIZED GLOBAL MATRIX METHOD fact leads to that the guided waves can be always separated into symmetric and anti-symmetric modes, regardless of propagation direction. If the symmetry condition (SC) can be embedded into the global matrix ahead, the size of the global matrix D G (v, ξ) can be halved to reduce computational burden. Specifically, the SC in Eqs.(3.3) and(3.4) now has to be evaluated at the midplane of a symmetric laminate, as follows.

Figure 5 . 5 -

 55 Figure 5.5 -A symmetric laminate showing only the upper half layers. In the SC ςX,lm bot and ς X,lm mid , the superscript X = s and a represent the symmetric and anti-symmetric condition, respectively.

Figure 5 .

 5 5(a). Substituting the specific terms from Eqs. (3.29) and (3.30) into Eqs. (5.10) and

Figure 5 .

 5 Figure 5.5(b). In this case, Λ lm (d m ) in Eq. (5.12) should be replaced by Λ lm (d m /2), leading to the following compact form of SC.

  0 and D(v, ξ

Figure 5 . 6 -

 56 Figure 5.6 -Dispersion curves of the 400-layered aerospace laminate at θ = 0 • direction.

Figure 5 . 7 -Figure 5 . 8 -Figure 5 . 8 .

 575858 Figure 5.7 -MAC value tomography of displacement mode shapes in the region R1 of symmetric modes.

Figure 5 .

 5 Figure 5.8C, u 1 and u 3 become larger components than u 2 , thus point C should belong to the S0 mode,

Figure 5 .Figure 5 .

 55 Figure 5.7B, the MAC values of the upper-right curve to the turning point B are almost zero. Likewise, in Figure 5.7E, the MAC values of the lower-left curve to the turning point E are nearly zero as well.

Figure 5 . 9 -

 59 Figure 5.9 -MAC value tomography of stress mode shapes in the region R1 of symmetric modes.

Figure 5 . 10 -

 510 Figure 5.10 -Stress mode shapes.

Figure 6 . 1 -

 61 Figure 6.1 -A summary of the solving strategy adopted by Dispersion Calculator [11].

Figure 6 . 2 -

 62 Figure 6.2 -The complex-to-real strategy for solving Rayleigh-Lamb equation [12].

Figure 6 . 3 -

 63 Figure 6.3 -The typical ∆ -v relation showing (a) the absence of ∆ > 0 and (b) the presence of ∆ > 0 in a narrow phase velocity range.

6. 3 . 1 . 3

 313 Case 3: only α 1 , α 3 are purely imaginary numbers. In this case, set α 1 = ia 1 , α 3 = ia 3 , meanwhile according to the Property 2 and 3, set β 21 = ip 21 , β 31 = ip 31 , β 23 = ip 23 , β 33 = ip 33 , B 5 = iP 5 , where a 1 , a 3 , p 21 , p 31 , p 23 , p 33 , P 5 are real numbers. With the new substitutions, B 1 , B 2 , B 3 in Eq. (6.9) can be deduced as follows. B 1 = β 11 (β 25 ip 33 -ip 23 β 35 ) = iβ 11 (β 25 p 33 -p 23 β 35 ) = iP 1 ⇒ an imag. number B 2 = β 13 (ip 21 β 35 -β 25 ip 31 ) = iβ 13 (p 21 β 35 -β 25 p 31 ) = iP 2 ⇒ an imag. number B 3 = β 15 (ip 23 ip 31 -ip 21 ip 33 ) = β 15 (-p 23 p 31 + p 21 p 33 ) ⇒ a real number (6.13)

6. 3 .Figure 6 . 4 -

 364 Figure 6.4 -Using bisection method to solve the equation D a (v 0 = 9000, ξ) = 0.

  4(b), if moving point 1 to 2 along the curve, there should have a sudden 180 • phase change in Figure 6.5(a) when crossing the accurate solution ξ 1 = 538.95. Thus, like the bisection method, one can consecutively monitor the occurrence of 180 • phase change to search ξ solutions.

Figure 6 .Figure 6 . 5 -

 665 Figure 6.5(b) intuitively shows the continuous curves of phase angle changing with ξ. Employing the phase change method, all the ξ solutions can be obtained as shown in this figure, which are the same as the solutions via bisection method marked in Figure 6.4(b).

Figure 6 Figure 6 . 6 -

 666 Figure 6.6(a).

Figure 6 . 7 -Figure 6 . 8 -

 6768 Figure 6.7 -The schematic diagram of phase change method for case 2.

Figure 6 .

 6 Figure 6.9 shows the typical function curves of implicit D a (v, ξ 0 ) at ξ 0 = 1750 [1/m] in the format of real and imaginary parts as well as phase angle curves. It can be seen that the whole range of v interested is large enough to span all the four cases related with α 1 , α 3 , α 5 . Within each case, only the effective part is plotted, and at the boundary of two different cases, there exists a switch between real and imaginary parts, this occurrence is accompanied by a 90 • phase change in the phase angle curve.

6. 3 .

 3 DICHOTOMY PROPERTY OF DISPERSION EQUATION OF A SINGLE LAYER PLATE of different cases, 90 ° phase change will occur.

Figure 6 . 9 -Figure 6 . 10 -

 69610 Figure 6.9 -Using bisection and phase change methods to solve the equation D a (v, ξ 0 = 1750) = 0.The small width of case 3 is caused by the degree of anisotropy.

6. 3 .

 3 DICHOTOMY PROPERTY OF DISPERSION EQUATION OF A SINGLE LAYER PLATE relation f = ξv/(2π). D(f, ξ) ≜ D(2πf /ξ, ξ) = 0 frequency-wavenumber domain (6.18a) d(v, f ) ≜ D(v, 2πf /v) = 0 frequency-phase velocity domain (6.18b) The newly generated two dispersion equations D(f, ξ) = 0 and d(v, f ) = 0 still hold the dichotomy property because they naturally succeed from the equation D(v, ξ) = 0. Thus, bisection and phase

  𝑣 0 , 𝜉 𝑖 for 𝜉 𝑖 ∈ 𝜉 min , 𝜉 max If 𝑀 𝑅 ≫ 𝑀 𝐼 , ൝ 𝒟 𝑋 𝑅 𝑣, 𝜉 = 0 𝒟 𝑋 𝐼 𝑣, 𝜉 ≡ 0 If 𝑀 𝑅 ≪ 𝑀 𝐼 , 𝒟 𝑎 𝑣, 𝜉 = 0 or 𝒟 𝑠 𝑣, 𝜉 = 0 (enumeration strategy) A single-or multilayered laminate for 𝒟 𝑋 𝑣, 𝜉 = 0 (sampling strategy)CasesFor any case, make sampling and comparison 𝑋 represents 𝑎 or 𝑠 for a single-layered monoclinic lamina, or 𝐺, 𝑇, 𝑆 for a multi-layered laminate respectively representative of GMM, TMM, SMM.

( 3 )Figure 6 . 11 -

 3611 Figure 6.11 -Concluding figures: (a) the strategy to determine the dichotomy property of dispersion equation, (b) the computational efficiency of solving methods adapted to dichotomy property.

  recomputed graph via GMM and CPCM

Figure 7 .Figure 6 . 12 -Figure 6 . 13 -

 7612613 Figure 7. Dispersion curves for 50-layer 1 mm thick laminate using GMM and TMM isotropic; (c) GMM, orthotropic; (d) TMM, orthotropic. (b) graph showing numerical instabilities (from [9]) Figure 6.12 -Dispersion curves propagating at 0 • for the [0] 50 unidirectional laminate.

Figure 6 . 14 -Figure 6 . 15 -

 614615 Figure 6.14 -Dispersion curves in frequency-wavenumber domain propagating at 45 • of the singlelayered unidirectional composite plate.

6. 6 .Figure 6 . 16 -

 6616 Figure 6.16 -Function curves of D X (v 0 , ξ) = 0 at v 0 = 1000 m/s of the Open Guided Waves.

Figure 6 . 17 -

 617 Figure 6.17 -The condition number of matrices of the Open Guided Waves at v 0 = 1000 m/s for (a) the monodromy transfer matrix T mono and (b) the global matrix D G .

Figure 6 . 18 -

 618 Figure 6.18 -Function curve of the symmetric modes of dispersion equation D s (v 0 , ξ) for a triclinic material at v 0 = 4000 m/s.

Figure 7 . 1 -

 71 Figure 7.1 -The studied aircraft nacelles of an A380 plane for (a) fan cowl structure and (b) inner fixed structure.

7. 2

 2 Wave propagation theory in the viscoelastic composite laminate The composite laminate model introduced in Chapter 4 are directly duplicated in Figure 7.2 for problem statement. The intrinsic viscoelastic damping properties of composite materials are taken into account in this chapter that cause attenuation of guided waves during propagation. The damping 7.2. WAVE PROPAGATION THEORY IN THE VISCOELASTIC COMPOSITE LAMINATE models adopted in this chapter are first introduced in order to integrate them into the standard GMM for predicting attenuation coefficients.

Figure 7 . 2 -

 72 Figure 7.2 -Schematical diagram of a n-layered aeronautic composite laminate.

7. 2 .

 2 WAVE PROPAGATION THEORY IN THE VISCOELASTIC COMPOSITE LAMINATE materials. C * (ω, θ) =

  )

7. 2 .

 2 WAVE PROPAGATION THEORY IN THE VISCOELASTIC COMPOSITE LAMINATEσ l 1 | x 3 =0 = 0 and σ ln | x 3 =dn = 0(7.18)

7. 2 .

 2 WAVE PROPAGATION THEORY IN THE VISCOELASTIC COMPOSITE LAMINATE correlated eigenvectors of the global matrix G(ω, k * , θ). Finally, the displacement and stress mode shapes of the laminate can be portrayed along the plate thickness direction through Eqs. (7.6) and (7.7).With the solved dispersion relation between ω and k * at a specified θ 0 , phase velocity c p is computed from Eq.(7.21). If the damping effect is slight, i.e. |k i | ≪ |k r |, group velocity c g can be computed

Figure 7 . 4 .

 74 Step 1 is trivial because it works out the real case DEs through the undamped GMM of Chapter 5. Step 2 is just the realization of the combined dGMM, curve tracing technique and 2D MRCM.

Figure 7 . 3 -

 73 Figure 7.3 -The schematic diagram of the curve tracing technique.

Step 2 Figure 7 . 4 -

 274 Figure 7.4 -The flowchart of the two-step dGMM algorithm for solving the complex DE.

ure 7 .

 7 4 is driven to predict various spectra of attenuation including dispersion curves and spatial attenuation distribution.

Figure 7 .

 7 5 just outlines the overview of the proposed attenuation prediction method in which the three purple boxes summarize the theoretical works established in the previous sections.

Figure 7 . 5 -

 75 Figure 7.5 -Overview of the proposed method to predict guided waves attenuation.

7. 4

 4 Case study on an unmounted and mounted FCS of A380 plane 7.4.1 Experimental setup

Figure 7 . 6 -

 76 Figure 7.6 -The PZT deployments of FCS for (a) the unmounted case and (b) the mounted case on an instrumented A380 plane.

Figure 7 . 7 -

 77 Figure 7.7 -The experimental setup of the unmounted FCS.

Figure 7 .

 7 Figure 7.8 shows several typical guided wave signals in Group 2 of the unmounted FCS in which PZT 1 serves as the actuator and its signal is normalized to the same order of magnitude with the receivers. Evidently, the signal amplitude decreases with the increasing of propagation distance.

Figure 7 . 8 -

 78 Figure 7.8 -Typical guided wave signals of the unmounted FCS at 100 kHz.

Figure 7 . 11 -

 711 Figure 7.11 -Bar chart to display regression results of (a) energy velocity and (b) attenuation coefficient via LR method for the mounted FCS at 100 kHz. The horizontal line represents the reference value of the unmounted FCS, and the red triangular arrows denote that the marked bars correspond to a R 2 larger than 0.7.

Figure 7 . 12 -

 712 Figure 7.12 -The identified dispersion curves of (a) energy velocity, (b) attenuation coefficient and (c) correlation coefficient. These subfigures relate to the S0 mode propagating at 0 • direction of the unmounted FCS.

Figure 7 .

 7 13(a) shows that the computed S0 mode energy velocity curve agrees well with the experimental points, among which the diamond point is the mean value of the bars in Figure7.11(a), representing the mounted FCS. Besides, the two damping models produce the mutually overlapped energy velocity curves for the three basic modes in the shown frequency range, denoting that viscoelastic damping effect does not change the property of energy velocity for the slightly damped FCS.

Figure 7 . 13 -

 713 Figure 7.13 -Comparison between the computed and experimentally identified dispersion curves of (a) energy velocity and (b) attenuation coefficient for the unmounted FCS at θ = 0 • .

Figure 7 .Figure 7 . 14 -

 7714 Figure 7.13(b) shows that the curve of HR model presents the linear trend whereas the curve of KV model expresses the parabola trend, which results in overestimated attenuation prediction in higher frequency range. Furthermore, the upward concavity of the curve of KV model is contrary to the upward convexity of the experimental data points, thus, HR model fits better with the experimental

Figure 7 . 15 -

 715 Figure 7.15 -For the unmounted FCS, surface plot of phase velocity (left) and attenuation coefficient (right): (a) and (b) S0 mode, (c) and (d) A0 mode, (e) and (f) SH0 mode. The inner and outer surfaces in subfigures (b)(d)(f) correspond to HR and KV models, respectively.

Figure 7 . 16 -

 716 Figure 7.16 -The transducers layout of IFS for (a) the unmounted case and (b) the mounted case on an instrumented A380 plane.

7. 5 . 2

 52 Figure 7.17(a) presents the comparison of the identified energy velocity via LR method between

Figure 7 . 17 -

 717 Figure 7.17 -Comparison between the mounted and unmounted IFS for (a) energy velocity and (b) attenuation coefficient at 100 kHz. In the two subfigures, the bars relate to the left y-axis and the lines correlate to the right y-axis.

Figure 7 .

 7 [START_REF] Mitra | Guided wave based structural health monitoring: A review[END_REF] shows that the original (E 1 , E 2 , E 3 ) values 1 yield to an overestimated prediction (the curve of dGMM prediction 1). When these values are updated to values 2, the predicted curve of dGMM prediction 2 matches well with the experimental data points of the mounted IFS, which is the same structure as the unmounted counterpart but undergoing different environmental conditions, i.e. existing stress in the mounted case thus modifying wave propagation properties[START_REF] Gandhi | Acoustoelastic lamb wave propagation in biaxially stressed plates[END_REF]. Figure 7.18 shows that the IFS structure holds quasi-isotropic property in terms of S0 mode energy velocity that is consistent with the symmetric stacking sequence of laminae [0/-45/+45/0/Al/0/+45/-45/0].

Figure 7 . 18 -

 718 Figure 7.18 -Polar plot of S0 mode energy velocity [km/s] in the mounted IFS at 100 kHz. The contour of dGMM prediction 1 and dGMM prediction 2 are computed by using the (E 1 ,E 2 ,E 3 ) values 1 and 2 in Table 7.4, respectively.

Figure 7 .

 7 19(a) and (b), respectively. Note that the corresponding correlation coefficient curves are not presented here because all values are larger than 0.8 which demonstrates reliable results of dispersion curves identification. Observing Figure7.19(a), the energy velocity curves of both groups basically overlap to each other, whereas in Figure7.19(b), Group 3 brings a lower attenuation than Group 2, but both curves keep the same tendency as the unmounted FCS, as seen 7.5. CASE STUDY ON AN UNMOUNTED AND MOUNTED IFS OF A380 PLANE in Figure 7.12(b).

Figure 7 .

 7 19(b) reveals that for practical aeronautic composite structures, the attenuation mechanism induced by material damping is more complex than for a unidirectional CFRP composite plate[79]. The lower and upper bounds in Figure7.19(b) represent the confidence intervals which are extracted from LR method under 95% confidence level.

Figure 7 . 19 -

 719 Figure 7.19 -The identified S0 mode dispersion curves of (a) energy velocity and (b) attenuation coefficient for the unmounted IFS.

Figure 7 .

 7 19(b) as the training data to the model updating process, the estimated viscoelastic material properties of the two combinations of damping models are obtained in Table7.6. It shows that the parameters of BT model in the aluminum alloy core 7.5. CASE STUDY ON AN UNMOUNTED AND MOUNTED IFS OF A380 PLANE layer are larger than the parameters of HR or KV model in the carbon epoxy layers, thus manifesting that the large attenuation of IFS is controlled by the highly damped aluminum alloy honeycomb core layer.With the obtained viscoelastic material properties of the two combinations of damping models in Table7.6, the energy velocity and attenuation curves are theoretically computed via the developed dGMM as depicted in Figure7.[START_REF] Miao | Shear horizontal wave transducers for structural health monitoring and nondestructive testing: A review[END_REF](a) and (b), respectively, in which the experimentally identified data points in Figure 7.19(a) and (b) are also presented for comparison, along with the diamond points in Figure 7.20(a) and (b) being the mean values of the mounted IFS in Figure 7.17

  (a) and (b), respectively. We firstly analyze Figure 7.20(b). It shows that the linear trend curve of HR+BT model in the frequency range 0-150 kHz fits better with the experimental data than the parabola trend curve of KV+BT model in 0-200 kHz. Since the concavity of KV+BT model's curve is reversed to the trend of the experimental data points, KV+BT model predicts underestimated and overestimated attenuation in the lower and higher frequency range, respectively. For HR+BT model, its linearized prediction represents the average effect to the experimental data in the least square sense.

Figure 7 . 20 -

 720 Figure 7.20 -Comparison between the computed and experimentally identified dispersion curves of (a) energy velocity and (b) attenuation coefficient for the unmounted IFS at θ = 0 • .

Figure 7 . 21 -

 721 Figure 7.21 -The computed group velocity via the pure elastic GMM for the unmounted IFS at θ = 0 • .

Figure 7 .

 7 22(e) are separately drawn in Figure7.23(b), then superimposed on the phase velocity curves in color code as illustrated in Figure7.23(a). This subfigure shows that there is a step in the S0 mode phase velocity curve around 200 kHz, which is caused by the fluctuation of the S0 mode attenuation curve around 200 kHz in Figure7.23(b). Besides, S0 mode holds a slight attenuation in comparison with SH0 and A0 modes. This could be the reason that only S0 mode information has been identified experimentally (see Figure7.19) as SH0 and A0 modes have been rapidly attenuated. In short, Figure7.23(a)(b) manifest that the phase velocity of guided waves in damped sandwich structure is characterized simultaneously by frequency (dispersion property) and attenuation (viscoelastic damping).

Figure 7 .

 7 23(c) and (d) at the same frequency f = 200 kHz.

Figure 7 .

 7 23(c) displaysthe quasi-isotropic property of phase velocity of the three basic modes. Nevertheless, pure anisotropic behavior of attenuation is observed in Figure7.23(d) for S0 and A0 modes, both of which hold the largest and smallest attenuation in the principal (0 • and 180 • ) and minor (90 • and 270 • ) directions, respectively. As for SH0 mode's attenuation, its anisotropic degree is slight along the omnidirectional propagation angles. In other words, Figure7.23(c)(d) give the evidence that despite the quasi-isotropic property, attenuation of guided waves in damped sandwich structure still possesses anisotropic effect.

  Figure 7.24(a)(c)(e), KV+BT model's predictions are not presented since it yields almost the same

Figure 7 . 22 -

 722 Figure 7.22 -For the unmounted IFS, the predicted dispersion curves at θ = 90 • for HR+BT model (left) and KV+BT model (right): (a) and (b) 3D space of (f, k r , k i ), (c) and (d) projection onto (f, k r ) plane, (e) and (f) projection onto (f, k i ) plane.

Figure 7 . 23 -

 723 Figure 7.23 -For the unmounted IFS, HR+BT model's prediction of (a) dispersion curve of phase velocity at θ = 90 • , (b) dispersion curve of attenuation coefficient at θ = 90 • , (c) polar plot of phase velocity [m/s] at f = 200 kHz, (d) polar plot of attenuation coefficient [Np/m] at f = 200 kHz.

7. 5 .Figure 7 . 24 -

 5724 Figure 7.24 -For the unmounted IFS, surface plot of phase velocity (left) and attenuation coefficient (right): (a) and (b) S0 mode, (c) and (d) A0 mode, (e) and (f) SH0 mode.

  longueur d'onde et de la vitesse des ondes avec les modes de propagation et la fréquence[START_REF] Wilcox | The effect of dispersion on long-range inspection using ultrasonic guided waves[END_REF]. La seconde caractéristique est due à l'anisotropie inhérente des matériaux composites aéronautiques qui entraîne une dépendance des propriétés des ondes guidées avec leur direction de propagation[START_REF] Lammering | Lamb-wave based structural health monitoring in polymer composites[END_REF].La troisième caractéristique est l'atténuation des ondes guidées qui est généralement causée par la viscoélasticité des matériaux composites[START_REF] Schubert | On attenuation and measurement of lamb waves in viscoelastic composites[END_REF][START_REF] Asamene | Influence of attenuation on acoustic emission signals in carbon fiber reinforced polymer panels[END_REF].

Figure B. 1 -

 1 Figure B.1 -Image d'une nacelle d'Airbus A380 [3].

Figure B. 2 -Forme

 2 Figure B.2 -Système d'étiquetage pour plaque multicouche [6].

Figure B. 4 -Figure B. 6 -Figure B. 7 -Figure B. 8 -

 4678 Figure B.4 -Courbes de dispersion pour le cas d'étude Open Guided Waves pour un angle de propagation θ = 0 • calculées via la TMM et la GMM.

Figure B. 9 .

 9 Figure B.9. Cette propriété est extrêmement utile pour surmonter les instabilités numériques rencontrées lors du processus d'obtention des équations de dispersion. Le cas des grands produits f × d rencontré par la TMM est exploré quantitativement en utilisant la propriété de dichotomie. L'analyse montre que le problème des grands produits f × d rencontré par la TMM est causé par un mauvais conditionnement de la matrice de transfert de monodromie (voir la Figure B.10).

  for 𝜉 𝑖 ∈ 𝜉 min , 𝜉 max If 𝑀 𝑅 ≫ 𝑀 𝐼 , monoclinic lamina for 𝒟 𝑎 𝑣, 𝜉 = 0 or 𝒟 𝑠 𝑣, 𝜉 = 0 (enumeration strategy)A single-or multilayered laminate for 𝒟 𝑋 𝑣, 𝜉 = 0 (sampling strategy)CasesFor any case, make sampling and comparison 𝑋 represents 𝑎 or 𝑠 for a single-layered monoclinic lamina, or 𝐺, 𝑇, 𝑆 for a multi-layered laminate respectively representative of GMM, TMM, SMM.know-why know-how

( 3 )

 3 Figure B.9 -(a) Stratégie à adopter pour déterminer la propriété de dichotomie de l'équation de dispersion. (b) Efficacité de calcul des méthodes de résolution basées sur la propriété de dichotomie.

Figure B. 10 - 3 )Step 2 Figure B. 11 2 (

 1032112 Figure B.10 -Le nombre de conditionnement des matrices pour le cas Open Guided Waves à v 0 = 1000 m/s pour (a) la matrice de transfert de monodromie T mono (méthode TMM) et (b) la matrice globale D G (méthode GMM).

Figure B. 12 -

 12 Figure B.12 -Le réseau de PZT pour le FCS: (a) le cas ou la structure n'est pas montée sur l'A380 et (b) le cas ou la structure est montée sur un A380 instrumenté.

Figure B. 13 -

 13 Figure B.13 -Le réseau de PZTs pour l'IFS: (a) le cas ou la structure n'est pas montée sur l'A380 et (b) le cas ou la structure est montée sur un A380 instrumenté.

Figure B. 14 -

 14 Figure B.14 -Prédiction du modèle HR + BT pour l'IFS non monté: (a) courbe de dispersion de la vitesse de phase à θ = 90 • , (b) courbe de dispersion du coefficient d'atténuation à θ = 90 • , (c) tracé polaire de la vitesse de phase [m/s] à f = 200 kHz, (d) tracé polaire du coefficient d'atténuation [Np/m] à f = 200 kHz.

  Shuanglin GUO Contribution to the study of guided waves propagation and attenuation in anisotropic composite laminates made up of viscoelastic composite materials : Application to A380 mounted nacelle parts Résumé : Les plaques composites anisotropes constituées de matériaux composites viscoélastiques sont de plus en plus utilisées dans l'industrie aéronautique et les stratégies basées sur les ondes guidées apparaissent comme un outil très prometteur pour surveiller leur état de santé. Dans ce contexte, ce travail est dédié à l'étude de la propagation et de l'atténuation des ondes guidées dans de telles plaques composites. Les équations classiques des ondes guidées se propageant dans une plaque élastique monocouche sont d'abord passées en revue sur la base de la théorie de l'élasticité tridimensionnelle (3D) et constituent le fondement de l'approche par superposition d'ondes partielles (PWSA). Dans la littérature, cette approche a été étendue à la méthode des matrices de transfert (TMM) et à la méthode des matrices globales (GMM) permettant d'exprimer les caractéristiques de propagation des ondes dans les plaques composites élastiques. En raison de l'anisotropie, des problèmes d'incompatibilité matricielle sont rencontrés lors du traitement de plaques composites arbitrairement orientées pour la TMM et la GMM. Une stratégie originale basée sur des matrices hybrides (HMS) est proposée pour pallier ce problème. Il est en outre démontré que les équations de dispersion construites avec PWSA pour une plaque monocouche et avec TMM et GMM pour un système de plaques multicouches possèdent une propriété de dichotomie dans le cas de matériaux élastiques purs. Cette propriété indique que ces équations a priori à valeurs complexes se réduisent en fait à des équations à valeurs réelles ou imaginaires pures, en fonction du comportement de certains paramètres intermédiaires. Cette propriété est extrêmement utile pour surmonter les instabilités numériques rencontrées au cours du processus de résolution des équations. Le cas élastique est ensuite étendu au cas viscoélastique en introduisant des modèles d'amortissement viscoélastiques (modèles d'hystérésis, de Kelvin-Voigt et de Biot) dans le comportement dynamique des matériaux composites. Ceci forme une méthode appélée « damped GMM » (dGMM) proposé ici pour traiter les plaques composites constituées de matériaux composites viscoélastiques anisotropes orientés arbitrairement. Deux études de cas industrielles, une structure de capot de soufflante (FCS) et une structure fixe intérieure (IFS), toutes deux composants d'une nacelle d'A380, sont utilisées pour valider expérimentalement la méthode dGMM en comparant la précision des prédictions par dGMM en termes de vitesses et d'atténuation aux mesures expérimentales. Mots clés : Surveillance de la santé structurelle, Ondes guidées, Matériaux composites aéronautiques et aérospatiaux, Anisotropie, Élasticité et viscoélasticité, Stratifié composite, Approche de superposition d'ondes partielles, Méthode de matrice de transfert, Méthode de matrice globale, Stratégie de matrice hybride, Propriété de dichotomie, Modèles d'amortissement, Équations de dispersion et courbes, Composant de nacelle A380.

  1.3. MODELING METHODS OF GUIDED WAVES IN COMPOSITE STRUCTURESwave problem, formed the well-known Lamb wave, and his solutions led to two types of wave modes including symmetric (longitudinal) and anti-symmetric (transverse) ones but they are coupled to each other during propagation. One important characteristic of Lamb waves is the dispersion, i.e. phase velocity of a specific mode varies with frequency[26]. Lamb's pioneering research, at present, laid the foundation of non-destructive evaluation (NDE) based on Lamb waves techniques[START_REF] Su | Guided lamb waves for identification of damage in composite structures: A review[END_REF]. In 1924, Stoneley wave was discovered by Robert Stoneley who explored the wave behavior propagation along the interface of two solid semi-space[START_REF] Stoneley | Elastic waves at the surface of separation of two solids[END_REF]. In 1947, when Scholte studied the Stoneley's problem, he replaced one solid medium with water and afterwards discovered a new wave type, now named

Scholte wave, which possesses the property of a lower speed than the bulk wave speeds of both the solid and the water. Scholte's work mainly focused on the wave propagation without energy leaking into neighboring solids but later works by Pilant in 1979 completed the works of wave leaking energy

  1 in Cartesian coordinate system O -x 1 x 2 x 3 . In this figure, the labelled quantities (σ 11 , σ 22 , σ 33 ) are the normal stresses, and (σ 12 , σ 21 , σ 13 , σ 31 , σ 23 , σ 32 ) are the shear stresses. Due to the reciprocal condition, i.e. σ 12 = σ 21 , σ 13 = σ 31 , σ 23 = σ 32 , the nine variables of

	σ =	[

stresses reduce to six, that form the stress vector σ. Correspondingly, there are six variables of strain (ε 11 , ε 22 , ε 33 , ε 23 , ε 13 , ε 12 ) that form the strain vector ε. For a typical 3D elastic problem, there are three displacement variables (u 1 , u 2 , u 3 ) that form the displacement vector u. The relation between

2.1. BASIC EQUATIONS OF 3D ELASTICITY

stress and strain is characterized by the general Hookes' law as formalized in Eq. (2.4). σ 11 , σ 22 , σ 33 , σ 23 , σ 13 , σ 12

  C 11 C 12 C 13 C 14 C 15 C 16 C 12 C 22 C 23 C 24 C 25 C 26 C 13 C 23 C 33 C 34 C 35 C 36 C 14 C 24 C 34 C 44 C 45 C 46 C 15 C 25 C 35 C 45 C 55 C 56 C 16 C 26 C 36 C 46 C 56 C 66

	2.3. STIFFNESS MATRIX ROTATION
		𝑖𝑗 or 𝑘𝑙	𝑝 or 𝑞	Take 𝑐 1231 as the example
		11 22	1 2	1231⇒"12"+"31"
		33	3	6	5
		23 or 32	4
		13 or 31 12 or 21	5 6	𝑐 1231 ⟺ 𝐶 65
		Figure 2.3 -The example of Voigt notation.
		⎡
	C =	⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

  C 11 cos 4 θ + C 22 sin 4 θ + (2C 12 + 4C 66 ) sin 2 θ cos 2 θ, C ′ 12 = (C 11 + C 22 -4C 66 ) sin 2 θ cos 2 θ + C 12 (sin 4 θ + cos 4 θ), C ′ 13 = C 13 cos 2 θ + C 23 sin 2 θ, C ′ 16 = (C 12 + 2C 66 -C 11 ) sin θ cos 3 θ + (C 22 -C 12 -2C 66 ) cos θ sin 3 θ, C ′ 22 = C 11 sin 4 θ + C 22 cos 4 θ + (2C 12 + 4C 66 ) sin 2 θ cos 2 θ, C ′ 23 = C 23 cos 2 θ + C 13 sin 2 θ, C ′ 26 = (C 12 + 2C 66 -C 11 ) cos θ sin 3 θ + (C 22 -C 12 -2C 66 ) sin θ cos 3 θ, C 55 cos 2 θ + C 44 sin 2 θ, C ′ 66 = C 66 (sin 4 θ + cos 4 θ) + (C 11 + C 22 -2C 12 -2C 66 ) sin 2 θ cos 2 θ.

	2.3. STIFFNESS MATRIX ROTATION
	C ′ 33 = C 33 , C ′ 36 = (C 23 -C 13 ) sin θ cos θ, C ′ 44 = C 44 cos 2 θ + C 55 sin 2 θ, C ′ 45 = (C 44 -C 55 ) sin θ cos θ, C ′ 55 = (2.19)

  in which the four constants C 13 , C 33 , C 44 , C 66 are now replaced byC 13 = C 12 , C 33 = C 22 , C 44 = C 22 -C 23 2, C 66 = C 55 because there are only five independent constants in C TI as indicated in Eq.(2.11). After substitution, one can know that the resulting C ′ TI keeps the same form as C ′ O in Eq.(2.18). This is the reason that we classify C TI ∈ C O in Section 2.2.4.

	The elements of C I in Eq. (2.13) are further substituted into Eq. (2.19) to derive the rotated
	stiffness matrix of the isotropic material C ′ I as concluded in Eq. (2.21), which shows that C ′ I =C I .
	This outcome conforms with the property of isotropic materials that has the highest symmetry such
	that its mechanical properties do not change with wave propagation direction.

  -iξα 1 h β 12 e -iξα 2 h β 13 e -iξα 3 h β 14 e -iξα 4 h β 15 e -iξα 5 h β 16 e -iξα 6 h β 21 e -iξα 1 h β 22 e -iξα 2 h β 23 e -iξα 3 h β 24 e -iξα 4 h β 25 e -iξα 5 h β 26 e -iξα 6 h β 31 e -iξα 1 h β 32 e -iξα 2 h β 33 e -iξα 3 h β 34 e -iξα 4 h β 35 e -iξα 5 h β 36 e -iξα 6 h β 11 e iξα 1 h β 12 e iξα 2 h β 13 e iξα 3 h β 14 e iξα 4 h β 15 e iξα 5 h β 16 e iξα 6 h β 21 e iξα 1 h β 22 e iξα 2 h β 23 e iξα 3 h β 24 e iξα 4 h β 25 e iξα 5 h β 26 e iξα 6 h β 31 e iξα 1 h β 32 e iξα 2 h β 33 e iξα 3 h β 34 e iξα 4 h β 35 e iξα 5 h β 36 e iξα 6 h

	3.2. DISPERSION EQUATIONS OF COUPLED LAMB AND SH WAVES IN
	TRICLINIC MATERIAL
	modes in implicit 2 form.
	⎡ β 11 e
	⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
	.14)
	3.2.3 Dispersion equations
	Substituting Eq. (3.13) into Eq. (3.2) to apply the BC, the following linear homogeneous equation
	set D(v, ξ)η = 0 is obtained, where D(v, ξ) is the coefficient matrix in Eq. (3.15) depending on phase
	velocity v and wavenumber ξ. In order to produce non-trivial solution of η, D(v, ξ) should vanish as
	presented in Eq. (3.16), which is the dispersion equation for coupled symmetric and anti-symmetric

11 e iξα 1 h β 12 e iξα 2 h β 13 e iξα 3 h β 14 e iξα 4 h β 15 e iξα 5 h β 16 e iξα 6 h β 21 e iξα 1 h β 22 e iξα 2 h β 23 e iξα 3 h β 24 e iξα 4 h β 25 e iξα 5 h β 26 e iξα 6 h β 31 e iξα 1 h β 32 e iξα 2 h β 33 e iξα 3 h β 34 e iξα 4 h β 35 e iξα 5 h β 36 e iξα 6 h

							.2)
	is redundant. Thus, the BC at x 3 = h in Eq. (3.2) and the SC in Eq. (3.3) are employed to derive the
	dispersion equation of symmetric modes. Substituting Eqs. (3.11) and (3.13) into the two equations,
	meanwhile reorganizing the resulted equation set to matrix form with the participation factors η being
	the unknown vector, one obtains Eqs. (3.17) and (3.18).
	⎡	W 1	W 2	W 3	W 4	W 5	W 6
	⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎢ ⎢ β	β 21 β 31	β 22 β 32	β 23 β 33	β 24 β 34	β 25 β 35	β 26 β 36

  iξα 1 h β 12 e iξα 2 h β 13 e iξα 3 h β 14 e iξα 4 h β 15 e iξα 5 h β 16 e iξα 6 h β 21 e iξα 1 h β 22 e iξα 2 h β 23 e iξα 3 h β 24 e iξα 4 h β 25 e iξα 5 h β 26 e iξα 6 h β 31 e iξα 1 h β 32 e iξα 2 h β 33 e iξα 3 h β 34 e iξα 4 h β 35 e iξα 5 h β 36 e iξα 6 h

						6
	β 11	β 12	β 13	β 14	β 15	β 16
	β 11 e

  -iξα 5 h α 6 e -iξα 6 h α 5 e iξα 5 h α 6 e iξα 6 h

6

h η 6 = 0 α 5 e +iξα 5 h η 5 + α 6 e +iξα 6 h η 6 = 0 (3.61) ORTHOTROPIC MATERIAL Rewrite Eq. (3.61) as the matrix form as follows, [ α 5 e

  .72) 3.6. CONCLUSION Substitute Eqs. (3.71) and (3.72) into Eq. (3.49) to compute β 11 , β 31 , β 13 , β 33 .

  12 β 13 β 14 β 15 β 16 β 21 β 22 β 23 β 24 β 25 β 26 β 31 β 32 β 33 β 34 β 35 β 36

  12 β 13 β 14 β 15 β 16 β 21 β 22 β 23 β 24 β 25 β 26 β 31 β 32 β 33 β 34 β 35 β 36

⎤ ⎥ ⎦

  T 11 T 12 T 13 T 14 T 15 T 16 T 21 T 22 T 23 T 24 T 25 T 26 T 31 T 32 T 33 T 34 T 35 T 36 T 41 T 42 T 43 T 44 T 45 T 46 T 51 T 52 T 53 T 54 T 55 T 56 T 61 T 62 T 63 T 64 T 65 T 66

  T 11 T 12 T 13 T 14 T 15 T 16 T 21 T 22 T 23 T 24 T 25 T 26 T 31 T 32 T 33 T 34 T 35 T 36 T 41 T 42 T 43 T 44 T 45 T 46 T 51 T 52 T 53 T 54 T 55 T 56 T 61 T 62 T 63 T 64 T 65 T 66

  it characterizes the pure Lamb waves. The corresponding dispersion equation of the pure Lamb waves can be derived by expanding Eq. (4.33) into Eq. (4.40). T 11 T 12 T 13 T 14 T 21 T 22 T 23 T 24 T 31 T 32 T 33 T 34 T 41 T 42 T 43 T 44

	⎡	u 1	⎤ ln		⎡
	⎢ ⎢ ⎢ ⎣ σ 33 u 3	⎥ ⎥ ⎥ ⎦	=	⎢ ⎢ ⎢ ⎣
	σ 13	bot

  T 13 T 14 T 21 T 22 T 23 T 24 T 31 T 32 T 33 T 34 T 41 T 42 T 43 T 44

  Rewrite Eq. (4.54) as the matrix form in Eq. (4.55) and the compact form in Eq. (4.56).

	4.4. HYBRID MATRIX STRATEGY
	⎡ σ 33	⎤	⎡ β 11 β 12 β 13 β 14 0	0
	⎢ ⎣ σ 23 σ 13	⎥ ⎦ =	⎢ ⎣ β 31 β 32 β 33 β 34 0 0 0 0 0 β 25 β 26 0
				(4.54)

1 -vt)

Table 4 .

 4 1 -Four classes of composite laminates and the wave type dependence on the propagation angle.

	Laminate class	Stacking angles	Propagation angle θ	Coupled waves	Pure waves	Hybrid waves
	Unidirectional 1

Table 4 .

 4

	2 -Material properties of a transversely isotropic lamina
	Density E 1608 [ kg/m 3 ] 172	9.8	6.1	0.37 0.55 0.2 mm	[(0/90) s ] 2
		⎡ 178.2 8.347 8.347	0	0	0	⎤
	C =	⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣	14.44 8.119 14.44 sym	0 0 3.161 0 0 0 6.1 0 0 0 0	⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥	GPa	(4.60)
						6.1

1 [GPa] E 2 [GPa] G 12 [GPa]

ν 12 ν 23 Ply thickness Stacking angles 4.6.1 The pure Lamb and SH waves when θ = 0 •

  Rot{𝐂 𝑖 , 𝜃 𝑖 }

	𝐂 1 ∈ 𝒞 𝒪	𝐂 1 ′ ∈ 𝒞 𝒪	𝐓 4×4 𝑙 1	𝐓 2×2 𝑙 1
	𝐂 2 ∈ 𝒞 𝒪	𝐂 2 ′ ∈ 𝒞 𝒪	𝐓 4×4 𝑙 2	𝐓 2×2 𝑙 2
	𝐂 3 ∈ 𝒞 𝒪	𝐂 3 ′ ∈ 𝒞 𝒪	𝐓 4×4 𝑙 3	𝐓 2×2 𝑙 3
	𝐂 4 ∈ 𝒞 𝒪	𝐂 4 ′ ∈ 𝒞 𝒪	𝐓 4×4 𝑙 4	𝐓 2×2 𝑙 4
	𝐂 5 ∈ 𝒞 𝒪	𝐂 5 ′ ∈ 𝒞 𝒪	𝐓 4×4 𝑙 5	𝐓 2×2 𝑙 5
	𝐂 6 ∈ 𝒞 𝒪	𝐂 6 ′ ∈ 𝒞 𝒪	𝐓 4×4 𝑙 6	𝐓 2×2 𝑙 6
	𝐂 7 ∈ 𝒞 𝒪	𝐂 7 ′ ∈ 𝒞 𝒪	𝐓 4×4 𝑙 7	𝐓 2×2 𝑙 7
	𝐂 8 ∈ 𝒞 𝒪	𝐂 8 ′ ∈ 𝒞 𝒪	𝐓 4×4 𝑙 8	𝐓 2×2 𝑙 8
			𝐓 4×4 mono	𝐓 2×2 mono
		Accessible wave type:	Pure Lamb and SH waves

  The laminate consists of eight unidirectional composite laminae with stacking angles of layups [0/90/ + 45/-45] s , indicating the quasi-isotropic property of this plate. Each lamina is made up of carbon-fiber epoxy prepreg having transversely isotropic material properties as listed in Table4.3, which is cited from[8]. This material is commonly used in the aeronautic and aerospace industry. The material properties are further used to compute the stiffness matrix in Eq. (4.61) along its principal direction 4.7. NUMERICAL EXAMPLE ON A QUASI-ISOTROPIC LAMINATE based on Eq. (2.12).

			Table 4.3 -Material properties of a carbon-fiber epoxy lamina
	Density	[ kg/m 3 ]	E 1 [GPa] E 2 [GPa] G 12 [GPa] ν 12 ν 23 Ply thickness Stacking angles
	1650		143	9.1	4.8	0.3 0.3 0.17 mm	[0/90/ + 45/ -45] s
				⎡ 145.38 3.96 3.96	0	0	0
			C =	⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣	10.11 3.11 10.11 0 0 3.5 0 0 0 sym 4.8 0 0 0 0
							4.8

Table 4 .

 4

		Table 4.4 -Material properties of aluminum
	Density	[ kg/m 3 ]	E [GPa]	ν	Ply thickness [mm]
	2780		73.1	0.33	0.33

• . 5 -Material properties of glass fiber Density (

solutions cannot be computed via Pant's pseudo correction for metallic-composite laminate (c)

	𝐂 𝑖 type along its principal direction	Stacking angles 𝜙 𝑖	Rotated stiff. matrix 𝐂 𝑖 ′ = Rot{𝐂 𝑖 , 𝜃 𝑖 } Wave propagation angle 𝜃 = 45 Rotation angle 𝜃 𝑖 = 𝜃 -𝜙 𝑖 °Local transfer matrix 𝐓 𝑝×𝑝 𝑙 𝑖 For 𝐂 𝑖 𝐂 𝑖 ′′ = Rot{𝐂 𝑖 ′ , -1 °} ′ ∈ 𝒞 ℐ ⊆ 𝒞 𝒪 , compute Pant's pseudo correction
				𝐂 1 ′′ ∈ 𝒞 ℐ	𝐓 2×2 𝑙 1
				𝐂 2 ′ ∈ 𝒞 ℳ
				𝐂 3 ′ ∈ 𝒞 ℳ	𝐓 6×6 𝑙 3
				𝐂 4 ′′ ∈ 𝒞 ℐ	𝐓 4×4 𝑙 4 , 𝐓 2×2 𝑙 4
				𝐂 5 ′ ∈ 𝒞 ℳ	𝐓 6×6 𝑙 5
				𝐂 6 ′ ∈ 𝒞 ℳ	𝐓 6×6 𝑙 6
				𝐂 7 ′′ ∈ 𝒞 ℐ	𝐓 4×4 𝑙 7 , 𝐓 2×2 𝑙 7
		No effective monodromy transfer matrix 𝐓 𝑝×𝑝 mono	Incompatible local transfer matrices 𝐓 𝑝×𝑝 𝑙 𝑖
		Wave

Table 5 .

 5 1 -Material properties of composite laminate of Open Guided Waves

	Density	[ kg/m 3 ]	Ply thickness [mm] Stacking angles
	1571		0.125		[45/0/ -45/90/ -45/0/45/90] s
				⎡ 130 6.1 6.1	0	0	0	⎤
		C O =	⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣	11.2 5.2 11.2 0 0 3.0 0 0 0 sym 4.2 0 0 0 0	⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
						4.2

  6 β 21 β 22 β 23 β 24 β 25 β 26 β 31 β 32 β 33 β 34 β 35 β 36

Table 5 .

 5

	2 -The computational time of the 400-layered laminate using the optimized GMM (unit:
	seconds)
	Bisection method		Phase change method
	sweep v sweep f sum	sweep v sweep f sum
	899 s	462 s	1361 s	871 s	469 s	1340 s
	780 a	454 a	1234 a	752 a	453 a	1205 a

  In this case, set α 1 = ia 1 . According to the Property 2 and 3, β 21 , β 31 , B 5 become purely imaginary numbers, so one can set them as β 21 = ip 21 , β 31 = ip 31 , B 5 = iP 5 , here a 1 , p 21 , p 31 , P 5 are real numbers.In that way, B 1 , B 2 , B 3 in Eq. (6.9) can be deduced as follows. B 1 = β 11 (β 25 β 33 -β 23 β 35 ) ⇒ a real number B 2 = β 13 (ip 21 β 35 -β 25 ip 31 ) = iβ 13 (p 21 β 35 -β 25 p 31 ) = iP 2 ⇒ an imag. number B 3 = β 15 (β 23 ip 31 -ip 21 β 33 ) = iβ 15 (β 23 p 31 -p 21 β 33 ) = iP 3 ⇒ an imag. number

	6.3. DICHOTOMY PROPERTY OF DISPERSION EQUATION OF A SINGLE
	LAYER PLATE
	equation easy to be solved via bisection method. This phenomenon casts the essence of dichotomy
	property.
	6.3.1.2 Case 2: only α 1 is a purely imaginary number.

(6.11)

where, P 2 , P 3 are definitely real numbers.

  ξa 3 h) sin(ξα 5 h)]P 5 ⇒ a purely imag. quantity Case 4: all α 1 , α 3 , α 5 are purely imaginary numbers.In this case, set α 1 = ia 1 , α 3 = ia 3 , α 5 = ia 5 , according to the Property 2 and 3, continuously setβ 21 = ip21 , β 31 = ip 31 , β 23 = ip 23 , β 33 = ip 33 , β 25 = ip 25 , β 35 = ip 35 , B 5 = iP 5 , where, a 1 , a 3 , a 5 , p 21 , p 31 , p 23 , p 33 , p 25 , p 35 , P 5 are real numbers. Substitute these new settings into Eq. (6.9) to deduce as follows. B 1 = β 11 (ip 25 ip 33 -ip 23 ip 35 ) = β 11 (-p 25 p 33 + p 23 p 35 ) ⇒ a real number B 2 = β 13 (ip 21 ip 35 -ip 25 ip 31 ) = β 13 (-p 21 p 35 + p 25 p

	(6.14)
	6.3.1.4

31 ) ⇒ a real number B 3 = β 15 (ip 23 ip 31 -ip 21 ip 33 ) = β 15 (-p 23 p 31 + p 21 p 33 ) ⇒ a real number (6.15)

Table 6 .

 6 [START_REF] Su | Identification of damage using Lamb waves: from fundamentals to applications[END_REF] -The computational time of the 50-layered plate (unit: seconds).

	Bisection method		Phase change method
	sweep v sweep ξ sum	sweep v sweep ξ sum
	6437 s	3052 s	9489 s	6729 s	3071 s	9800 s
	2284 p	1058 p	3342 p	2338 p	1052 p	3390 p

s Serial computing technique is applied. p Parallel computing technique is applied.

  1 , E 2 , E 3 , and three shear moduli G 23 , G 31 , G 12 , as well as three Poisson's ratios ν 12 , ν 13 , ν 23 . Since the viscoelastic property of composite materials is considered in this chapter, a given damping model selected among Eqs. (7.1), (7.2) or(7.3) can be applied to the six elastic moduli, thus producing frequency dependent complex moduli[START_REF] Manconi | Estimation of the loss factor of viscoelastic laminated panels from finite element analysis[END_REF]

  12 β 13 β 14 β 15 β 16 β 21 β 22 β 23 β 24 β 25 β 26 β 31 β 32 β 33 β 34 β 35 β 36

	⎤
	⎥ ⎦	(7.9)

  2 , η 3 , η 4 , η 5 , η 6 ] T (7.11)7.2. WAVE PROPAGATION THEORY IN THE VISCOELASTIC COMPOSITE LAMINATEIn Eqs. (7.8)(7.9)(7.10)(7.11), α j , V j and W j with (j = 1, . . . , 6) are computed from the (complex) Christoffel equation, Eq. (7.12), i.e. the non-trivial vector [U, V, W ] T requires the singularity of the coefficient matrix in Eq. (7.12), which further leads to a cubic equation in terms of α 2 as presented in Eq.(7.13).

	⎡ ⎢ ⎣ C * 11 -ρv 2 + C * 55 α 2 C * 16 + C * 45 α 2 (C * 13 + C * 55 )α	C * 16 + C * 45 α 2 66 -ρv 2 + C * C * 44 α 2 (C * 36 + C * 45 )α	(C * 13 + C * 55 )α (C * 36 + C * 45 )α 55 -ρv 2 + C * C * 33 α 2

  7.3. DATA-DRIVEN STRUCTURAL PARAMETERS ESTIMATION as the function of frequency ω with undetermined loss factors γ. = [γ 1 , γ 2 , γ 3 , γ 23 , γ 31 , γ 12 ] in which the six elements are the loss factors of E * 1 , E * 2 , E * 3 , G * 23 , G * 31 , G * 12 , respectively, as defined in Eq. (7.4). When computing, γ is specialized to a certain damping model, as defined in Eqs. (7.1), (7.2) and (7.3), γ HR , γ KV and γ BT . The loss factors γ is then determined in the least square sense by inverting the experimentally identified attenuation coefficients, also known as model updating process (we use here the lsqcurvefit.m MATLAB function):

	k i = K(γ, ω)	(7.28)
	where, γ

Table 7 .

 7 1 -Elastic and viscoelastic material properties of composite lamina of the unmounted FCS. =0.01%, γ 3 =0.01%, γ 23 ,γ 31 ,γ 12 =0.01% KV model γ 1 =3.47%, γ 2 =0.01%, γ 3 =0.012%, γ 23 ,γ 31 ,γ 12 =0.01%, f c = 250 kHz The PZT deployments for both cases are shown in Figure 7.6. There are in total 13 and 43 7.4. CASE STUDY ON AN UNMOUNTED AND MOUNTED FCS OF A380 PLANE sensors surface installed on the unmounted and mounted FCS, respectively. Due to the presence of stiffeners, the 43 sensors in the mounted FCS are partitioned to six regions, marked from R1

	(E 1 , E 2 , E 3 ) [GPa]	(G 23 , G 31 , G 12 ) [GPa]	(ν 12 , ν 13 , ν 23 )	ρ [kg/m 3 ]	Ply thickness [mm]	Stacking sequence
	(60,40,8.1)	(4.8,4.8,4.8)	(0.03,0.3,0.3) 1554	0.28	[0/-45/+45/0]
	HR model	γ 1 =1.74%, γ 2

Table 7

 7 .1. From this table, for both 7.4. CASE STUDY ON AN UNMOUNTED AND MOUNTED FCS OF A380 PLANE models γ 1 is greatly larger than the remaining five parameters, meaning that the damping effect is predominated in the principal fiber direction of composite lamina. Besides, the loss factors of KV model are the relative values to the ones at the characterization frequency f c = 250 kHz. If f c changes

Table 7 .

 7 [START_REF] Lammering | Lamb-wave based structural health monitoring in polymer composites[END_REF] -Elastic material properties of the carbon epoxy ply in the unmounted IFS.

	(E 1 , E 2 , E 3 ) [GPa]	(G 23 , G 31 , G 12 ) [GPa]	(ν 12 , ν 13 , ν 23 )	ρ kg/m 3	Ply thickness [mm]
	(51.0,65.7,8.1) 1 (40.0,55.0,8.1) 2	(5.2,5.2,5.2)	(0.02,0.3,0.3)	1554	0.28
	1

Table 7 .

 7 5 -Elastic material properties of the aluminum core layer in the unmounted IFS.

	E [MPa]	G [MPa]	ν	ρ [kg/m 3 ]	Layer thickness [mm]
	866	354	0.2232	67	1.2

There are in total 10 PZTs surface installed on the unmounted and mounted IFS, which are

Table 7 .

 7 6 -Viscoelastic material properties of the carbon epoxy ply and aluminum core in the unmounted IFS. HR model is used for carbon epoxy plies and BT model is applied for aluminum core.2 KV model is used for carbon epoxy plies and BT model is applied for aluminum core.

	Combination	Carbon epoxy ply (HR or KV)				Aluminum (BT)
	of models	γ 1	γ 2	γ 3	γ 23	γ 31	γ 12	f c [kHz] γ E	γ G	ϵ
	HR+BT model 1 13.9% 0.04% 0.02% 5.60% 5.60% 5.60% -	19.9% 19.9% 10
	KV+BT model 2 27.4% 0.01% 0.01% 0.01% 0.01% 0.01% 250	30.0% 30.0% 10

1

  1 , E 2 , E 3 ) values 1 and 2 listed in Table 7.4, respectively. The other material properties listed 7.5. CASE STUDY ON AN UNMOUNTED AND MOUNTED IFS OF A380 PLANE in Table 7.4 and 7.5 are the same.

  .3) Figures 7.14(d) and7.23(d) shows that even if many aeronautic composite plates are designed and manufactured as quasi-isotropic property, attenuation of guided waves presents anisotropy in terms of spatial distribution. This property can affect the layout of sensors on distance, orientation,

  B.1. CONTEXTE G ÉN ÉRAL DE LA TH ÈSEsonores guidées dans les structures minces sont particulièrement efficaces pour surveiller l'état de santé et l'intégrité des structures composites aéronautiques[START_REF] Mitra | Guided wave based structural health monitoring: A review[END_REF]. Ces ondes ultrasonores guidées dans des structures minces ont la particularité de pouvoir se propager sur de relativement grandes distances et peuvent donc couvrir une surface de contrôle importante avec peu d'élements actifs et en un temps plutôt court. Cet avantage se traduit en pratique par une réduction de la main d'oeuvre et des équipements nécessaires pour effectuer la surveillance de ces structures[START_REF] Su | Guided lamb waves for identification of damage in composite structures: A review[END_REF].Lors du choix des ondes guidées comme outil d'inspection des structures composites aéronautiques, il est en préambule nécessaire de souligner leur aspect multi-modal: à une fréquence d'excitation donnée plusieurs types d'ondes (aussi appelés "modes") peuvent se propager simultanément avec des longueurs d'ondes et des vitesses différentes. Trois caractéristiques de ces ondes doivent en particulier être considérées avec attention pour bien comprendre la façon dont elles de propagent dans les matériaux

d'intérêt. La première caractéristique est leur comportement dispersif qui décrit l'évolution de la

Table B .

 B 2 -Classes de stratifiés composites et la dépendance du type d'onde à l'angle de propagation. -45] s θ ∈ {0• , 45 • , 90 • , -45 • } √ θ / ∈ {0 • , 45 • , 90 • , -45 • } √ Composite métallique [Al/0/90/Al/90/0/Al] 2 θ ∈ {0 • , 90 • } Remarque relative à la classe des stratifiés unidirectionnels : étant donné que les variables de champ (déplacement et contrainte) sont considérées comme continues à travers l'interface des couches adjacentes (la condition d'interface parfaite est supposée), le stratifié unidirectionnel [0]8 est mathématiquement équivalent à la plaque monocouche [0]1 ayant la même épaisseur que la plaque [0]8. 2 Al désigne la couche d'aluminium qui est un matériau isotrope.

	Classe de stratifié Unidirectionnel 1 Plis croisés	Angles d'empilage [0] 8 [(0/90) s ] 2	Angle de propagation θ θ ∈ {0 • , 90 • } θ / ∈ {0 • , 90 • } θ ∈ {0 • , 90 • } θ / ∈ {0 • , 90 • }	Cas couplé √ √	Cas découplé √ √	Cas hybride
	Quasi-isotrope	[0/90/ + 45/ √
			θ / ∈ {0 • , 90 • }			√

1

Table B .

 B

	Méthode de bissection	Méthode de changement de phase
	sweep v sweep f sum	sweep v sweep f sum
	899 s	462 s	1361 s	871 s	469 s	1340 s
	780 a	454 a	1234 a	752 a	453 a	1205 a

3 -Le temps de calcul du stratifié à 400 couches en utilisant le GMM optimisé (unité : secondes)

s Le temps de calcul correspond aux modes symétriques. a Le temps de calcul correspond aux modes antisymétriques.

ACKNOWLEDGEMENTS

1.4. ORGANIZATION OF THE THESIS

Although θ = 0 • does not mean an effective rotation, it is still applied here for notation homogeneity purpose.

The media admissible for bulk wave propagation should be an unbounded solid space, however the unidirectional composite plate shown in Figure2.4 is only for illustration purpose.

Elastic material makes the real-valued stiffness matrix and is considered in Chapters 3, 4, 5, 6 of the thesis. In contrast, viscoelastic material leads to the complex-valued stiffness matrix and is studied in Chapter 7.

In somewhere of the thesis, phase velocity is signified by cp.

In the thesis, the term implicit means that the dispersion equation is defined in matrix-determinant form.

In this thesis, the term explicit means that the dispersion equation is simplified from the original one defined in matrix-determinant form.

4.4. HYBRID MATRIX STRATEGY

GMM framework is studied in Chapter 5.

This item is used to generate Figure6.3(a). Table6.2 -The composite materials that invoke the case ∆ > 0 in a limited phase velocity range.

This item is used to generate Figure 6.3(b).

In the author's opinion, the phase change method is evolved from the convergence criteria of Disperse that checks the included phase angle of the complex-valued dispersion function being greater than 90 • for real solutions of dispersion equation[START_REF] Lowe | Plate waves for the ndt of diffusion bonded titanium[END_REF].

SMM is not considered in the thesis according to the statement in Section 1.3.1, but SMM still holds dichotomy property after validation.

For notation homogeneity and for readability purposes, the frequency dependency is also shown for the HR model even if it is by definition frequency-independent.

It is possible to consider the complex frequency with real wavenumber. But this case does not relate to the wave propagation problem at hand[6].

This property conforms with the Snell's law[2] which requires that all the partial waves keep the same (complex) wavenumber in the interface between adjacent laminae along the wave propagation direction.

8.3. PUBLICATIONS RELATED TO THE THESIS

Acknowledgements

ACKNOWLEDGEMENTS

My research works were funded by the China Scholarship Council (CSC). My special appreciation goes to CSC.

Finally, I would like to thank my family. The achievement of this thesis would not have been possible if without your support and understanding.

Chapter 7

Prediction of frequency and spatially dependent attenuation of guided waves propagating in anisotropic viscoelastic composite laminates

The industrial application of guided waves-based structural health monitoring techniques is highly tributary of the number and placement of the active sensor elements. Yet, the optimal sensorization of a structure relies on the decrease in amplitude of guided waves over propagation distance. A reliable prediction of amplitude attenuation of guided waves is still a challenge especially for anisotropic viscoelastic composite materials which exhibits complex changes of attenuation with propagation direction, and thus a spatial dependency of attenuation. In Chapter 5, GMM is applied to the pure elastic case. In this chapter, the damped global matrix method (dGMM), having stable and efficient merits, is developed to predict the frequency and spatially dependent attenuation of waves propagating in anisotropic viscoelastic composite materials. dGMM integrates three damping models (Hysteretic, Kelvin-Voigt and Biot models) into the conventional undamped GMM of Chapter 5 to consider viscoelasticity of composite laminates. The proposed dGMM is presented and then validated by numerical comparison with the semi-analytical finite element method. In addition, two industrial case studies, parts of an A380 nacelle at scale one, are employed to experimentally validate the proposed attenuation prediction method. The first one is a fan cowl structure and the second one an inner fixed structure, both either unmounted or mounted on an actual instrumented A380 plane. This makes the validation extremely valuable for both the scientific and industrial communities. The proposed

WAVE PROPAGATION THEORY IN THE VISCOELASTIC COMPOSITE LAMINATE

the solving process. Specifically, at a specified propagation angle θ 0 , ω is independently fixed as a constant ω 0 , by doing so, the original quaternary-variable equation D(ω, k r , k i , θ) = 0 is now transited to a dual-variable equation D(k r , k i ; ω 0 , θ 0 ) = 0. This equation can be solved by applying a dual variable searching method [6]. In the thesis, a recently developed method named 2D MRCM [START_REF] Zhu | A numerical algorithm to solve multivariate transcendental equation sets in complex domain and its application in wave dispersion curve characterization[END_REF] is employed which mathematically transforms the dual-variable root-finding process to search the global minimal modulus of the characteristic function |D(k r , k i )| by checking the module ratio convergence.

The interested readers can refer to Zhu's works [START_REF] Zhu | A numerical algorithm to solve multivariate transcendental equation sets in complex domain and its application in wave dispersion curve characterization[END_REF][START_REF] Zhu | Accurate characterization of 3d dispersion curves and mode shapes of waves propagating in generally anisotropic viscoelastic/elastic plates[END_REF] for more information. To get the full-frequency spectrum of k r and k i , the above solving process is repeated by changing ω 0 , which reveals the dispersion property of both quantities. Alternatively, by changing θ 0 , the resulted diagrams are the polar plot of k r and k i , which just shows the anisotropic property of guided waves propagating in anisotropic composite laminates, viz. the spatial effect.

It is well known that there are multiple solutions (k r , k i ) at a stationary pair (ω 0 , θ 0 ), which represents multiple guided wave modes propagation at the specified frequency ω 0 . 2D MRCM can search all the solutions by scanning the structured grids as long as the step length of scanning element is small enough. Scanning the whole space of interest is required to generate the multi-modal 3D dispersion curves (k r , k i , ω), known as the forward procedure, but is time-consuming. If only a single mode's dispersion curve is desired, e.g. S0 or A0 mode, a single curve tracing technique can accelerate the searching efficiency since tracing the trajectory of a single mode's solution is just a fractional workload of searching the full 3D spectral lines. In addition, in the following inverse procedure of model updating process, the traced single mode's dispersion curve can be immediately contrasted to the counterpart that is experimentally measured, thereof avoiding the branch identification and mode matching problem [52]. To this end, a quadratic extrapolation-based single mode curve tracing technique is developed, as schematically illustrated in Figure 7 The LR method introduced previously is used to identify energy velocity at a given frequency 100 kHz in the unmounted FCS experiments, as illustrated in Figure 7.9(a). The regressed lines for the three groups are plotted based on the scatter points representing the time of arrival (ToA) of S0 mode's wave packet versus sensing distance, from which energy velocity is identified from the inverse of the slope of the regressed line. The regression results are listed in Table 7.2, which shows the consistent energy velocities identified amongst the three sensor groups and meanwhile the higher correlation coefficients R 2 (0.99). Attenuation coefficient can also be identified by using LR method based on Eq. (7.27), as illustrated in Figure 7.9(b), in which some points of Group 1 are overlapped by the points of Group 2 and 3 because the sensors in Group 2 and 3 are contained in Group 1. The results extracted from Figure 7.9(b) are listed in Table 7.3, from which the R 2 in Group 1 (0.39) is greatly lower than the one of Group 2 (0.92) and Group 3 (0.79). This result is consistent with the dispersed scatter points of Group 1 in Figure 7.9(b). More fundamentally, the lower R 2 in Group 1 is caused by the anisotropic effect of composite materials since the sensors in Group 1 yield multiple directions of propagation paths whereas Group 2 and 3 only orient at the horizontal direction, thus a larger R 2 value is guaranteed respectively for the two groups.

CASE STUDY ON AN UNMOUNTED AND MOUNTED FCS OF A380 PLANE

Attenuation coefficient prediction for the unmounted FCS

By using the elastic and viscoelastic parameters of HR model in Table 7.1, the attenuation coefficient can be predicted in two ways. One is the traditional dispersion curve in a wide frequency range at a fixed propagation direction as depicted in respectively. Attenuation of the SH0 mode is less variable, the smallest value appearing at 30 • in the first quadrant. Note that the open circles in Figure 7.14 are the solutions of SAFE method [START_REF] Bartoli | Modeling wave propagation in damped waveguides of arbitrary cross-section[END_REF]79] that is used here for comparison and verification with the proposed dGMM approach. For SAFE computation in this instance, five 1D quadratic elements per layer are used to mesh the through thickness section of the four-layered [0/-45/+45/0] composite laminate, which guarantees its convergence as proved in

Chapter 8

General conclusions and perspectives

General conclusions

The thesis has presented the properties of guided waves propagating in elastic and viscoelastic composite laminates through theoretical derivation, numerical illustration and experimental validation.

The focus of the thesis is placed on how to establish dispersion equations via matrix-based methods, and efficiently and stably solve dispersion equations in pure elastic and viscoelastic cases. The following conclusions and original findings are outlined.

The PWSA introduced in Chapter 3 is extended to the TMM to study the property of guided waves propagating in a multi-layered composite laminate. It combines the traction free boundary condition and perfect continuity condition. The later is formed using the transferring rule to link the field variables between adjacent layers and yields the monodromy transfer matrix of the whole plate system for applying the boundary condition. In accordance with different material types in each layer of the composite laminate, there exists three cases of guided waves, namely, the coupled Lamb and SH waves, the separated Lamb and SH waves, and the hybrid case which means that in some layers of the laminate the two wave types are coupled to each other but in another layers decoupled. The hybrid case is generally existing in a quasi-isotropic composite plate and metallic-composite plate for which metallic layers are isotropic but composite laminae are anisotropic. The standard TMM can deal with the coupled and decoupled cases, but for the hybrid case, an original HMS is proposed to address the local transfer matrix incompatibility issue. Three numerical examples are used to illustrate the three cases of guided waves coupling issues. Each example corresponds to a class of composite plates, i.e. cross-ply laminate, quasi-isotropic laminate and metallic-composite laminate.

GENERAL CONCLUSIONS

eters. This property is present in composite materials (excluding triclinic ones of few specific case) of propagating and evanescent waves, in wavenumber-phase velocity domain, frequency-wavenumber domain and frequency-phase velocity domain. For the single layer plate, dichotomy property can be analyzed through enumeration strategy that strictly follows the evolution of intermediate parameters'

properties. For the multi-layered plate system, a simple sampling strategy is proposed to numerically evaluate dichotomy property. This property is extremely helpful for overcoming numerical instabilities encountered during the equation solving process. The large f d problem of TMM is quantitatively explored by using the methodology of studying dichotomy property. Analysis shows that the large f d problem of TMM is caused by the vanishing of effective part between real and imaginary parts of dispersion equation built with TMM due to the poor conditioning of the monodromy transfer matrix.

Considering the inherent viscoelasticity of composite materials, the damping effect of dispersion curves are investigated. Three damping models (Hysteretic, Kelvin-Voigt and Biot models) are integrated into the standard GMM to form an extended version, the damped GMM (dGMM). In viscoelastic case, the complex-valued dispersion equations built with dGMM have the solution pair of real frequency and complex wavenumber, and the attenuation coefficients of guided waves can be extracted from the imaginary part of the complex wavenumber. Due to the complexity of the viscoelastic case, solving the multivariate transcendental dispersion equation is mathematically intractable. To address this issue, a two-step root-finding approach is originally proposed that integrates the undamped GMM, 2D Module Ratio Convergence Method (2D MRCM) and curve tracing technique. With this approach, the 3D dispersion curves in real frequency-complex wavenumber domain can be computed.

The classical SAFE method is adopted to validate the effectiveness of the proposed methodology.

In addition to the numerical computation approach, an identification method of attenuation coefficients is originally proposed that utilizes linear regression technique and is based on the propagation nature of wave displacement fields. With this method, attenuation coefficients can be experimentally identified from wave signals acquired in a pitch-catch sensor network. Considering that viscoelastic parameters of damping models are generally not available in practice, the model updating process is adopted to estimate these parameters in the least square sense that fits the experimentally identified attenuation coefficients with the dGMM such that there is a good agreement between theoretical computation and experimental identification.

The entire framework developed in the thesis is applied to two engineering structures, both are 8.3. PUBLICATIONS RELATED TO THE THESIS analysis is the output of the dispersion solutions in frequency-wavenumber domain [START_REF] Mei | Characterization of multilayer delaminations in composites using wavenumber analysis: numerical and experimental studies[END_REF]. In anisotropic viscoelastic plates, the existence of wave attenuation can change properties of temporal and frequencyrelated quantities. Thus, improvement can be made by bridging the gap between the temporal-or frequency-based damage localization methods and the viscoelastic wave propagation models studied in the thesis. This leaves the ongoing works.

Publications related to the thesis

This section groups together the various publications that are related to the thesis presented here.

Journal papers For the Property 2, set α r = ia r , here a r is a real number, and substitute it into Eq. (3.27) to make the following derivations.

⇒ a purely imaginary number In this case, set W r = iw r , here w r is a real number, but keep V r unchanged because it is already a real number. Substitute ia r , V r , iw r into Eq. (3.31) to make further derivations. Cette thématique de recherche est à l'étude depuis le début des année 1970 et en un demi-siècle, les techniques de SHM ont progessé vers leur maturité en jouant un rôle important dans l'évaluation de l'intégrité et de la durabilité des structures [START_REF] Su | Identification of damage using Lamb waves: from fundamentals to applications[END_REF]. Grâce à des techniques de SHM efficaces, continues et automatisées, il est en théorie possible d'identifier les dommages structurels à un stade précoce afin d'éviter de nouvelles défaillances, générant ainsi d'énormes avantages économiques et ainsi que la sauvegarde de vies humaines [START_REF] Sohn | A review of structural health monitoring literature: 1996-2001[END_REF].

Parmi tous les types de techniques de SHM existantes, les stratégies basées sur les ondes ultra-