Alma Mater Studiorum - Università di Bologna Dottorato di Ricerca in Ingegneria Strutturale ed Idraulica Ciclo XXV Settore Concorsuale di aﬀerenza: 08/B2 Settore Scientiﬁco disciplinare: ICAR/08 Numerical methods for the dispersion analysis of Guided Waves Presentata da: Matteo Mazzotti Coordinatore Dottorato Relatore Prof. Erasmo Viola Prof. Erasmo Viola Correlatori Dott. Alessandro Marzani Dott. Ivan Bartoli Esame ﬁnale anno 2013 To my family Abstract The use of guided ultrasonic waves (GUW) has increased considerably in the ﬁelds of non-destructive (NDE) testing and structural health monitoring (SHM) due to their ability to perform long range inspections, to probe hidden areas as well as to provide a complete monitoring of the entire waveguide. Guided waves can be fully exploited only once their dispersive properties are known for the given waveguide. In this context, well stated analytical and numerical methods are represented by the Matrix family methods and the Semi Analytical Finite Element (SAFE) methods. However, while the former are limited to simple geometries of ﬁnite or inﬁnite extent, the latter can model arbitrary cross-section waveguides of ﬁnite domain only. This thesis is aimed at developing three diﬀerent numerical methods for modelling wave propagation in complex translational invariant systems. First, a classical SAFE formulation for viscoelastic waveguides is extended to account for a three dimensional translational invariant static prestress state. The eﬀect of prestress, residual stress and applied loads on the dispersion properties of the guided waves is shown. Next, a two-and-a-half Boundary Element Method (2.5D BEM) for the dispersion analysis of damped guided waves in waveguides and cavities of arbitrary cross-section is proposed. The attenuation dispersive spectrum due to material damping and geometrical spreading of cavities with arbitrary shape is shown for the ﬁrst time. Finally, a coupled SAFE-2.5D BEM framework is developed to study the dispersion characteristics of waves in viscoelastic waveguides of arbitrary geometry embedded in inﬁnite solid or liquid media. Dispersion of leaky and non- leaky guided waves in terms of speed and attenuation, as well as the radiated waveﬁelds, can be computed. The results obtained in this thesis can be helpful for the design of both actuation and sensing systems in practical application, as well as to tune experimental setup. Contents Contents v List of Figures ix Nomenclature xvii 1 Introduction 1 1.1 Sommario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Research motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.3 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Wave propagation in prestressed waveguides: SAFE method 7 2.1 Sommario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Introduction and literature review . . . . . . . . . . . . . . . . . . . . . 9 2.3 Wave equation in linearized incremental form . . . . . . . . . . . . . . . 11 2.3.1 Linearized strain-displacement relations . . . . . . . . . . . . . . 13 2.3.2 Linearized stress-strain relations . . . . . . . . . . . . . . . . . . 15 2.3.3 Linearized incremental equilibrium equations . . . . . . . . . . . 16 2.4 Equations in the wavenumber-frequency domain . . . . . . . . . . . . . 19 2.5 Domain discretization using semi-isoparametric ﬁnite elements . . . . . 21 2.6 Dispersion analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.6.1 General solutions for lossy and lossless materials . . . . . . . . . 28 2.6.2 Dispersive parameters . . . . . . . . . . . . . . . . . . . . . . . . 31 2.6.2.1 Real wavenumber . . . . . . . . . . . . . . . . . . . . . 31 2.6.2.2 Phase velocity . . . . . . . . . . . . . . . . . . . . . . . 32 2.6.2.3 Attenuation . . . . . . . . . . . . . . . . . . . . . . . . 32 2.6.2.4 Group velocity . . . . . . . . . . . . . . . . . . . . . . . 33 2.6.2.5 Energy velocity 34 . . . . . . . . . . . . . . . . . . . . . . v CONTENTS 2.7 Numerical applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.7.1 Viscoelastic rail under thermal-induced axial stress . . . . . . . . 37 2.7.2 Guided waves propagation in a new roll-straightened viscoelastic rail . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Pipe under initial pressure loading . . . . . . . . . . . . . . . . . 47 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.7.3 2.8 3 Wave propagation in bounded and unbounded waveguides: 2.5D Boundary Element Method 55 3.1 Sommario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.2 Introduction and literature review . . . . . . . . . . . . . . . . . . . . . 57 3.3 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.4 2.5D integral representation theorem . . . . . . . . . . . . . . . . . . . . 61 3.5 Green’s functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.6 Regularized 2.5D boundary integral equation . . . . . . . . . . . . . . . 70 3.6.1 Limiting process . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.6.2 Regularization procedure . . . . . . . . . . . . . . . . . . . . . . 71 3.7 Boundary discretization using semi-isoparametric boundary elements . . 76 3.8 Nonlinear eigenvalue problem . . . . . . . . . . . . . . . . . . . . . . . . 80 3.8.1 Contour integral method . . . . . . . . . . . . . . . . . . . . . . . 81 3.8.2 Deﬁnition of the integral path and permissible Riemann sheets . 82 Dispersion characteristics extraction . . . . . . . . . . . . . . . . . . . . 85 3.9 3.10 Numerical analyses of bounded waveguides . . . . . . . . . . . . . . . . 88 3.10.1 Standard BS11-113A rail . . . . . . . . . . . . . . . . . . . . . . 89 3.10.2 Square bar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 3.11 Surface waves along cavities of arbitrary cross-section . . . . . . . . . . 92 3.11.1 Circular cavity in viscoelastic full-space . . . . . . . . . . . . . . 94 3.11.2 Square cavity in a viscoelastic full-space . . . . . . . . . . . . . . 97 3.12 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4 Leaky Guided Waves in waveguides embedded in solid media: coupled SAFE-2.5D BEM formulation 105 4.1 Sommario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.2 Introduction and literature review . . . . . . . . . . . . . . . . . . . . . 107 4.3 Wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 4.4 SAFE model of the embedded waveguide . . . . . . . . . . . . . . . . . . 114 vi CONTENTS 4.5 BEM model of the surrounding medium . . . . . . . . . . . . . . . . . . 116 4.5.1 Regularized 2.5D boundary integral equation . . . . . . . . . . . 116 4.5.2 Boundary element discretization . . . . . . . . . . . . . . . . . . 120 4.5.3 Evaluation of weakly singular integrals . . . . . . . . . . . . . . . 122 4.6 SAFE-BE coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 4.7 Dispersion analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 4.8 4.7.1 Single-valued deﬁnition of the dynamic stiﬀness matrix . . . . . . 127 4.7.2 Dispersion characteristics extraction . . . . . . . . . . . . . . . . 132 Numerical applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 4.8.1 Elastic steel bar of circular cross section embedded in elastic concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 4.8.2 Viscoelastic steel bar of circular cross section embedded in viscoelastic grout . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 4.8.3 Viscoelastic steel bar of square cross section embedded in viscoelastic grout . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 4.8.4 Viscoelastic steel HP200 beam embedded in viscoelastic soil . . . 144 4.8.5 Rectangular HSS40 20 2 viscoelastic steel tube embedded in viscoelastic grout . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 4.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 5 Leaky Guided Waves in waveguides immersed in perfect ﬂuids: coupled SAFE-2.5D BEM formulation 155 5.1 Sommario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 5.2 Introduction and literature review . . . . . . . . . . . . . . . . . . . . . 157 5.3 Discretized wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . 159 5.3.1 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . 159 5.3.2 SAFE model of the waveguide 5.3.3 2.5D BEM model of the ﬂuid domain 5.3.4 Boundary element discretization . . . . . . . . . . . . . . . . . . 165 5.3.5 Non-uniqueness problem . . . . . . . . . . . . . . . . . . . . . . . 165 . . . . . . . . . . . . . . . . . . . 159 . . . . . . . . . . . . . . . 162 5.4 Fluid-structure coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 5.5 Single valued deﬁnition of the dynamic stiﬀness matrix . . . . . . . . . . 168 5.6 Eigenvalue analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 5.7 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 5.7.1 Validation case: elastic titanium bar of circular cross-section immersed in oil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 vii CONTENTS 5.7.2 5.8 Viscoelastic steel bar of square cross-section immersed in water . 174 5.7.3 L-shaped viscoelastic steel bar immersed in water . . . . . . . . . 177 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 6 Conclusions 187 6.1 Sommario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 6.2 Conclusions and future works . . . . . . . . . . . . . . . . . . . . . . . . 188 A List of publications 193 Bibliography 195 viii List of Figures 2.1 Fundamental conﬁgurations for the wave propagation problem in prestressed waveguides. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Finite element mesh used for the dispersion curves extraction in sections 2.7.1 and 2.7.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 23 38 Phase velocity, energy velocity and attenuation for the loaded and unloaded cases. The ﬁrst ﬁve modes m1, m2, m3, m4 and m5 are identiﬁed as in [Bartoli et al., 2006]. . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 41 Percent variations between the loaded and unloaded cases for the axially loaded rail. Thin lines denote positive variations while thick lines denote negative variations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Reconstructed stress patterns for the roller straightened 113A standard proﬁle in [Keller et al., 2003]. . . . . . . . . . . . . . . . . . . . . . . . . 2.6 43 Phase velocity, energy velocity and attenuation for the unloaded rail and the roller straightened rail in [Keller et al., 2003]. . . . . . . . . . . . . . 2.7 42 45 Percent variations between the loaded and unloaded case for the roller straightened rail. Thin lines denote positive variations while thick lines denote negative variations. . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 46 Finite element mesh of 112 nodes and 150 linear triangular elements for the ASME 1-1/2 Schedule 160 pipe. The transversal stress contours are relative to an inner pressure pi = 10 MPa and an outer pressure pe = 5 MPa (case 3). Negative values denote compressive stresses. . . . . . . . 2.9 48 Phase velocity, energy velocity and attenuation for the ASME 1-1/2 Schedule 160 pipe under diﬀerent pressure gradients. . . . . . . . . . . . 50 2.10 Percent variations between the loaded and unloaded cases for the pressurized pipe. Thin lines denote positive variations while thick lines denote negative variations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix 51 LIST OF FIGURES 3.1 Analytical model of the bounded waveguide. . . . . . . . . . . . . . . . . 62 3.2 Exclusion neighborhood used for the limiting process ε → 0. . . . . . . . 71 3.3 Auxiliary domain for a bounded waveguide. . . . . . . . . . . . . . . . . 73 3.4 Semi-isoparametric discretization using mono-dimensional elements with linear shape functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 77 Complex κz -plane with branch points, branch cuts and integration path Γ (κz ). The notation (·, ·) stands for the choice of the signs in Eq. (3.84) for κα and κβ respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.6 Subdivision of the domain Ωb by means of integration cells. . . . . . . . 86 3.7 (a) Boundary element mesh with internal cells subdivision and (b) SAFE mesh of the BS11-113A rail. . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Singular values distribution after 50 frequency steps for the standard 113A rail in Sec. 3.10.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 90 91 Real wavenumber dispersion curves for the viscoelastic steel BS11-113A rail. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 3.10 Phase velocity dispersion curves for the viscoelastic steel BS11-113A rail. 93 3.11 Attenuation dispersion curves for the viscoelastic steel BS11-113A rail. . 94 3.12 Energy velocity dispersion curves for the viscoelastic steel BS11-113A rail. 95 3.13 (a) Boundary element mesh with internal cells subdivision and (b) SAFE mesh of the square bar. . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.14 Singular values distribution after 50 frequency steps for the square bar in Sec. 3.10.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 3.15 Real wavenumber dispersion curves for the viscoelastic steel square bar of 2.0 mm side length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 3.16 Phase velocity dispersion curves for the viscoelastic steel square bar of 2.0 mm side length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 3.17 Attenuation dispersion curves for the viscoelastic steel square bar of 2.0 mm side length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 3.18 Energy velocity dispersion curves for the viscoelastic steel square bar of 2.0 mm side length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 3.19 Dimensionless (a) real axial wavenumbers and (b) attenuations versus dimensionless frequency for ν̃ = 0.3 − i4.5 × 10− 4. The normal modes are identiﬁed as in Boström and Burden [1982]. . . . . . . . . . . . . . . 102 3.20 Dimensionless (a) real axial wavenumbers and (b) attenuations versus dimensionless frequency for ν̃ = 0.3 − i4.5 × 10−4 . . . . . . . . . . . . . . 103 x LIST OF FIGURES 4.1 Analytical model of the embedded waveguide. . . . . . . . . . . . . . . . 113 4.2 Complex κz -plane with bulk wavenumbers, vertical branch cuts and integration path for an external isotropic viscoelastic medium. The signs of κα and κβ on Ω∗ and along Γ (κz ) are determined by imposing the conditions on their imaginary parts as indicated in the diﬀerent regions. 128 4.3 Wave vectors conﬁgurations for the point P 3 of Fig. 4.2. The propagation vector kRe S is oriented along the radiation direction (dashed gray lines), while the attenuation vector kIm S is perpendicular to equiamplitude lines (solid gray lines) and oriented in the direction of maximum decay. Magnitude of displacements is proportional to the thickness of equi-amplitude lines. 4.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Wave vectors conﬁgurations for the point P 2 of Fig. 4.2. The propagation vector kRe S is oriented along the radiation direction (dashed gray lines), while the attenuation vector kIm S is perpendicular to equiamplitude lines (solid gray lines) and oriented in the direction of maximum decay. Magnitude of displacements is proportional to the thickness of equi-amplitude lines. 4.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 Wave vectors conﬁgurations for the point P 1 of Fig. 4.2. The propagation vector kRe S is oriented along the radiation direction (dashed gray lines), while the attenuation vector kIm S is perpendicular to equiamplitude lines (solid gray lines) and oriented in the direction of maximum decay. Magnitude of displacements is proportional to the thickness of equi-amplitude lines. 4.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Complex κz -plane with bulk wavenumbers, vertical branch cuts and integration path for an external isotropic elastic medium. The signs of κα and κβ on Ω∗ and along Γ (κz ) are determined by imposing the conditions on their imaginary parts as indicated in the diﬀerent regions. . . . . . . 132 4.7 SAFE-BEM mesh of the elastic steel bar of circular cross section embedded in elastic concrete. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 4.8 Phase velocity dispersion curves for the elastic steel bar of circular cross section embedded in elastic concrete of Fig. 4.7. Modes are indicated as in Ref. [Castaings and Lowe, 2008]. . . . . . . . . . . . . . . . . . . . . . 136 4.9 Attenuation dispersion curves for the elastic steel bar of circular cross section embedded in elastic concrete of Fig. 4.7. Modes are indicated as in Ref. [Castaings and Lowe, 2008]. . . . . . . . . . . . . . . . . . . . . . 136 xi LIST OF FIGURES 4.10 Energy velocity dispersion curves for the elastic steel bar of circular cross section embedded in elastic concrete of Fig. 4.7. Modes are indicated as in Ref. [Castaings and Lowe, 2008]. . . . . . . . . . . . . . . . . . . . . . 137 4.11 SAFE-BEM mesh of the viscoelastic steel bar of circular cross section embedded in viscoelastic grout. . . . . . . . . . . . . . . . . . . . . . . . 139 4.12 Phase velocity dispersion curves for the viscoelastic steel circular bar embedded in viscoelastic grout of Fig. 4.11. Modes are indicated as in Ref. [Pavlakovic et al., 2001]. . . . . . . . . . . . . . . . . . . . . . . . . 139 4.13 Attenuation dispersion curves for the viscoelastic steel circular bar embedded in viscoelastic grout of Fig. 4.11. Modes are indicated as in Ref. [Pavlakovic et al., 2001]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 4.14 Energy velocity dispersion curves for the viscoelastic steel circular bar embedded in viscoelastic grout of Fig. 4.11. Modes are indicated as in Ref. [Pavlakovic et al., 2001]. . . . . . . . . . . . . . . . . . . . . . . . . 140 4.15 SAFE-BEM mesh of the viscoelastic steel bar of square cross-section embedded in viscoelastic grout. . . . . . . . . . . . . . . . . . . . . . . . 142 4.16 Phase velocity dispersion curves for the viscoelastic steel square bar embedded in viscoelastic grout of Fig. 4.15. Modes are indicated as in Ref. [Gunawan and Hirose, 2005], where a square bar in vacuum was considered.142 4.17 Attenuation dispersion curves for the viscoelastic steel square bar embedded in viscoelastic grout of Fig. 4.15. Modes are indicated as in Ref. [Gunawan and Hirose, 2005], where a square bar in vacuum was considered.143 4.18 Energy velocity dispersion curves for the viscoelastic steel square bar embedded in viscoelastic grout of Fig. 4.15. Modes are indicated as in Ref. [Gunawan and Hirose, 2005], where a square bar in vacuum was considered. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 4.19 SAFE-BEM mesh of the HP200 viscoelastic steel beam embedded in viscoelastic soil. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 4.20 Phase velocity velocity dispersion curves for the viscoelastic steel HP200 beam embedded in viscoelastic soil of Fig. 4.19. . . . . . . . . . . . . . . 146 4.21 Attenuation dispersion curves for the viscoelastic steel HP200 beam embedded in viscoelastic soil of Fig. 4.19. . . . . . . . . . . . . . . . . . . . 146 4.22 Energy velocity dispersion curves for the viscoelastic steel HP200 beam embedded in viscoelastic soil of Fig. 4.19. . . . . . . . . . . . . . . . . . 147 4.23 Mode shapes and waveﬁeld in soil for (a) mode m2 at 88.38 Hz, (b) mode m2 at 616.16 Hz and (c) mode m2 at 952.02 Hz. . . . . . . . . . . . . 148 xii LIST OF FIGURES 4.24 SAFE-BEM mesh of the embedded HSS40 20 2 rectangular steel tube 150 4.25 Dispersion curves for the viscoelastic HSS40 20 2 steel section embedded in viscoelastic grout of Fig. 4.24. . . . . . . . . . . . . . . . . . . . . 152 5.1 Analytical model of the immersed waveguide. . . . . . . . . . . . . . . . 161 5.2 Real wavenumber dispersion curves for the elastic steel bar of circular cross section immersed in oil. . . . . . . . . . . . . . . . . . . . . . . . . 172 5.3 Phase velocity dispersion curves for the elastic steel bar of circular cross section immersed in oil. . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 5.4 Attenuation dispersion curves for the elastic steel bar of circular cross section immersed in oil. . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 5.5 Energy velocity dispersion curves for the elastic steel bar of circular cross section immersed in oil. . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 5.6 Real wavenumber dispersion curves for the viscoelastic steel square bar immersed in water. Guided modes are named as in Ref. [Gunawan and Hirose, 2005]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 5.7 Phase velocity dispersion curves for the viscoelastic steel square bar immersed in water. Guided modes are named as in Ref. [Gunawan and Hirose, 2005]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 5.8 Attenuation dispersion curves for the viscoelastic steel square bar immersed in water. Guided modes are named as in Ref. [Gunawan and Hirose, 2005]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 5.9 Energy velocity dispersion curves for the viscoelastic steel square bar immersed in water. Guided modes are named as in Ref. [Gunawan and Hirose, 2005]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 5.10 Real wavenumber dispersion curves for the L-shaped viscoelastic steel bar immersed in water. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 5.11 Phase velocity dispersion curves for the L-shaped viscoelastic steel bar immersed in water. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 5.12 Attenuation dispersion curves for the L-shaped viscoelastic steel bar immersed in water. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 5.13 Energy velocity dispersion curves for the L-shaped viscoelastic steel bar immersed in water. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 5.14 Normalized in-plane displacement and pressure ﬁelds for (a) the m1 mode and (b) the h3 mode at 51.2 kHz. . . . . . . . . . . . . . . . . . . 181 xiii LIST OF FIGURES 5.15 Normalized in-plane displacement and pressure ﬁelds for the m1 mode at 77.0 kHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 5.16 Normalized in-plane displacement and pressure ﬁelds for the m2 mode at 9.7 kHz in (a) the near ﬁeld and (b) the far ﬁeld (the normalized scale is the same in (a) and (b)). . . . . . . . . . . . . . . . . . . . . . . . . . 183 5.17 Normalized in-plane displacement and evanescent pressure ﬁelds for (a) the m3 and (b) the m4 mode at 40 kHz. . . . . . . . . . . . . . . . . . . 184 xiv Nomenclature α attenuation σ0, σ Cauchy stress tensors ε, independent linear strains εNL , NL independent nonlinear strains Ξ 1st Piola-Kirchhoﬀ stress tensor ξ natural coordinates for bidimensional elements δij Kronecker’s delta η natural coordinate for monodimensional elements λ, λ̃ Lamé’s ﬁrst constant λ, μ̃ Lamé’s second constant E, e Green-Lagrange strain F deformation gradient f 0 , fc volume forces vectors I identity matrix J Poynting vector kL , κL bulk longitudinal wavevector and wavenumber kS , kS bulk shear wavevector and wavenumber kz , κz axial wavevector and wavenumber kα , κα radial bulk longitudinal wavevector and wavenumbers kβ , κβ radial bulk shear wavevector and wavenumbers n outward normal N, N shape functions S 2nd Piola-Kirchhoﬀ stress tensor xv LIST OF FIGURES s0 , s vectors of independent linear stress components t0 , tc , ts , tb , t tractions vectors tnc nonconservative surface load u0 , u, uc displacements vectors X = [x, y, z]T conﬁguration of the material particle on V x = [x, y]T conﬁguration of the material particle in the x − y plane X = [x , y , z ]T source point on V x = [x , y ]T source point on the x − y plane xc collocation point C, C̃ fourth order tensors of viscoelastic moduli C0 , C, Ct material conﬁgurations Bxy , Bz compatibility matrices K , W , Vc , Vnc energy quantities Lx , Ly , Lz compatibility operators ∇ gradient operator ν Poisson’s ratio ω circular frequency aux Ωaux b , ∂Ωb domain and boundary of the auxiliary problem Ωb , ∂Ωb area and boundary of the solid BEM region on the x − y plane Ωf , ∂Ωf area and boundary of the ﬂuid domain in the x − y plane Ωs , ∂Ωs area and boundary of the waveguide cross-section in the x − y plane ∂p/∂n pressure ﬂux ρ material density ce energy velocity cg group velocity cL , c̃L bulk longitudinal velocities cp phase velocity cS , c̃S bulk shear velocities c∞ adimensional coeﬃcient cij , caux ij , c∞ , c free terms xvi LIST OF FIGURES Du directional derivative in the direction of u E Young’s modulus (1) (2) Hn (·), Hn (·) nth order Hankel function of the ﬁrst and second kind P D , ∂P D /∂n fundamental pressures and ﬂuxes of the 2.5D Helmholtz equation P S , ∂P S /∂n fundamental pressures and ﬂuxes of the 2D Laplace equation p0 , p pressure in the acoustic domain R, r, rc source-receiver distances t time D , TijD Uij fundamental displacement and tractions for the 2.5D elastodynamic problem S , TijS Uij fundamental displacement and tractions for the 2.5 plane strain problem un = u · n normal displacement V , ∂V volume and boundary surface of the waveguide i imaginary unit xvii Chapter 1 Introduction 1.1 Sommario Nel capitolo introduttivo vengono inizialmente descritte in modo sommario le potenzialità dei metodi basati su onde ultrasoniche di tipo guidato (GUW) nei campi delle indagini non distruttive, del monitoraggio strutturale e della caratterizzazione dei materiali. Poichè tali metodi richiedono un’accurata conoscenza dei parametri di dispersione per la speciﬁca guida d’onda oggetto dell’indagine, l’attenzione viene in seguito focalizzata sulle tecniche analitiche e numeriche utilizzate in letteratura per il calcolo delle curve di dispersione, mettendone in luce i principali vantaggi e limitazioni. Sono successivamente descritte le motivazioni che hanno guidato la ricerca ed hanno condotto alla stesura della presente tesi. Inﬁne, vengono brevemente illustrati gli aspetti innovativi introdotti e i principali risultati ottenuti in ogni capitolo. 1.2 Research motivations Guided Ultrasonics Waves (GUW) are recognized as an eﬀective diagnostic tool in the ﬁelds of nondestructive evaluation (NDE) testing, structural health monitoring (SHM) and materials characterization. The basic concept behind guided waves is that a structural component with invariant geometric and mechanical characteristics along one or more dimensions (waveguide), can be used as support to “drive” the wave propagation, thus providing a fundamental means for its inspection. Compared with classical ultrasonic testing techniques, some advantages exist. First, the energy of the wave is carried for long distances over the waveguide, whereas in 1 1. INTRODUCTION standard ultrasonic testings only small areas of the structural component can be investigated at once. As a consequence, operations times are also drastically reduced. The second advantage is that guided waves can provide a complete inspection of the entire waveguide cross-section. Additionally, guided waves have inherent potential to target particular defects. Unlike longitudinal and shear waves used in standard ultrasonic techniques, guided waves are multi modal, i.e. many modes can carry energy at a given frequency. This property allows one to select several modes having the greatest sensitivity with respect to a speciﬁc defect or a mechanical parameter that must be identiﬁed. Another advantage oﬀered by GUW is the possibility to design permanent monitoring systems with relatively small hardware. For the reasons above, the importance of GUW in civil, industrial and medical applications has increased considerably in recent years. For example, guided waves are widely used in the water and oil transportation industry for the detection of defects in pipelines. In the railroad industry they are used to monitoring the conditions of rails, with the aim to prevent failures that can cause disservices or compromise safety. In the aerospace industry, they are largely employed for the quality assessment of adhesively-bonded components. In the civil engineering ﬁeld, guided waves have proven to be eﬀective in the damage detection of bridge cables, inspections of foundation piles, weld inspections and characterization of the material constants in composite structural components. All these applications require an accurate knowledge of the dispersive characteristics of guided waves. The concept of dispersion denotes a variation of the behaviour of guided waves as a function of the frequency, and is a consequence of the interaction of the wave propagation process with the structural geometry. The fundamental dispersive parameters are the phase velocity, the attenuation, group velocity and attenuation. The phase velocity denotes the rate at which the crests of a particular guided mode propagate along the waveguide at a certain frequency. The attenuation expresses instead the amplitude decay per unit of distance traveled. This information is of great importance, especially in leaky systems, where attenuation mostly aﬀects the length of inspection ranges. The group velocity indicates the rate at which packets of waves at inﬁnitely close frequencies move along the waveguide. This feature gives an indication about how much dispersion occurs for a signal generated in a certain frequency range, i.e. how much the shape of the signal is distorted while it propagates along the waveguide. The energy velocity represents a generalization of the energy velocity concept for attenuative systems, and correspond to the rate at which the energy carried by the wave moves along the propagation direction. 2 1.2. RESEARCH MOTIVATIONS In order to eﬃciently exploit guided waves, it is therefore necessary to chose the guided modes that maximize the inspection ranges with high sensitivity with respect to the defect or parameter to be identiﬁed. From the above considerations it clearly appears the need of mathematical tools able to extract dispersive data for waveguides with diﬀerent geometries and materials, as well as to model the interaction of the waveguide with the surrounding environment. To this purpose, diﬀerent analytical and numerical methods are available in literature. The Transfer Matrix Method (TMM) [Haskell, 1953; Thomson, 1950] and the Global Matrix Method (GMM) [Knopoﬀ, 1964; Lowe, 1992; Pavlakovic, 1998] represent the most widely adopted techniques in the context of analytical methods. These methods are able to extract the dispersion curves for plate-like and cylindrical waveguides that are immersed in vacuum or embedded in solid or ﬂuid media. Their capability to handle multilayered waveguides and to provide very accurate solutions makes them very appealing for diﬀerent wave propagation problems involving civil, mechanical and aerospace structures. The Finite Element Methods (FEM) [Chen and Wilcox, 2007; Sorohan et al., 2011] and the Semi-Analytical Finite Element (SAFE) methods [Bartoli et al., 2006; Gavric, 1995; Hayashi et al., 2003; Hladky-Hennion, 1996; Shah et al., 2001] have instead the unique capability to model waveguides of complex geometries and materials, for which theoretical solutions are not available. Moreover, they generally lead to well posed problems, while Matrix Methods may result unstable when the waveguide presents a large number of layers, as in the case of composite laminates. Although the above methods can model a large variety of problems, some situations that are often encountered in practice have not been investigated in the literature. These are, for example, the cases of prestressed viscoelastic waveguides and waveguides of complex geometry and materials that are embedded in solids or immersed in ﬂuids. In this thesis, three diﬀerent numerical methods are presented for the solution of the above mentioned problems. The ﬁrst is an extension of the Semi-Analytical Finite Element (SAFE) method for the extraction of the waves modal properties in viscoelastic prestressed waveguides. The main novelty introduced is the derivation of the wave equation, which is obtained in linearized incremental form within an Updated Lagrangian framework and by considering the inﬂuence of nonconservative loads. A modal formula for the wave energy velocity calculation is also proposed, which is based on the linearized incremental form of the Poynting theorem obtained by manipulating the energy balance principle expressed in material description. The second numerical method developed is a two-and-a-half (2.5D) Boundary Ele- 3 1. INTRODUCTION ment Method (BEM) able to predict the dispersion properties of damped guided waves in waveguides and cavities of arbitrary cross-section. In this formulation, the Cauchy Principal Value integrals and the boundary coeﬃcients are treated by means of a regularization procedure. Unlike the SAFE formulations, where the dispersion analysis consists in solving a linear eigenvalue problem, the dispersive wave equation resulting from the regularized 2.5D BEM is conﬁgured as a nonlinear eigenvalue problem. This problem is solved by means of a recently developed Contour Integral Method. In relation to the singular characteristics and the multivalued feature of the Green functions, the properties of various Riemann sheets are investigated and a contour integration path is proposed, which takes into account the presence of the Sommerfeld branch cuts in the complex plane of the axial wavenumbers. By means of some numerical examples, a comparative analysis between the 2.5D BEM and the SAFE is performed, while some new results are obtained concerning the dispersive properties of surface guided waves along cavities of diﬀerent geometries. The third method proposed is a coupled SAFE-2.5D BEM approach for the dispersion analysis of leaky guided waves in viscoelastic waveguides of arbitrary cross-section that are embedded in viscoelastic isotropic media. So far, leaky guided waves have been essentially investigated for waveguides of simple geometries by means of analytical methods. Few studies have been proposed in literature in which are modeled using diﬀerent approaches, such as absorbing regions [Castaings and Lowe, 2008; Fan et al., 2008], inﬁnite elements [Jia et al., 2011] or Perfectly Matched Layers [Treyssède et al., 2012]. However, all the numerical methods above present some approximations in the description of the radiated waveﬁeld, and the problem of how correctly model leaky guided waves in complex structures is still challenging. In the proposed formulation, the energy radiation due to leakage of bulk waves is introduced in the SAFE model by converting the BEM impedance matrix into an equivalent dynamic stiﬀness matrix, which is manipulated as a single, wavenumber and frequency dependent, ﬁnite element of inﬁnite extension. Due to singular characteristics of leaky modes, additional conditions are introduced in the Green functions in order to satisfy the Snell-Descartes law at the SAFE-BEM interface. The coupled SAFE-2.5D BEM formulation is also presented for waveguides immersed in ﬂuids, in which the solution in the ﬂuid region is assumed to satisfy the 2.5D Helmholtz equation. The results obtained in this thesis and can be helpful for the design of both actuation and sensing systems in practical application, as well as to tune experimental setup. 4 1.3. OUTLINE OF THE THESIS 1.3 Outline of the thesis The thesis is organized as follows. In Chapter 3, an extension to the Semi-Analytical Finite Element (SAFE) method is proposed in order to include the eﬀect of a general state of initial stress on the dispersive behavior of damped guided waves. The wave equation is derived in linearized incremental form within an Updated Lagrangian framework. A modal formula for the wave energy velocity calculation is proposed, which is based on the linearized incremental form of the Poynting theorem in material description. New results not available in literature are discussed, which can be helpful in guided wave testing of loaded rails and pressurized pipelines. In Chapter 3, a 2.5D Boundary Element formulation is developed to predict the dispersion properties of damped guided waves in waveguides and cavities of arbitrary cross-section. A regularization procedure is described to treat Cauchy Principal Value Integrals and boundary coeﬃcients, while the resulting nonlinear eigenvalue problem is solved by using a recently developed Contour Integral Method. A Riemann surface analysis is also presented, and a contour integration path is described for the elastic and viscoelastic cases. The method is ﬁrst validated against the SAFE method, while new results are discussed for cavities of diﬀerent geometries. Chapter 4 is dedicated to the study of leaky guided waves in viscoelastic waveguides of arbitrary cross-section embedded in viscoelastic media. The problem is solved by using a coupled SAFE-2.5D approach, in which the SAFE is used to model the embedded waveguide and the 2.5D BEM to represent the impedance of the surrounding medium. A single-valued analysis is presented for the resolvent stiﬀness operator, which is based on supplementary interface conditions introduced via the Snell-Descartes law. The proposed method is ﬁrst validated against some results available in literature for simple geometries, while some new applications for complex geometries are proposed for the ﬁrst time. Chapter 5 describes a coupled SAFE-2.5D BEM model for the computation of the dispersion properties of leaky guided waves in waveguides immersed in ideal ﬂuids. As in Chapter 3, a regularization procedure is adopted for the desingularization of the boundary integrals. To improve the numerical stability of the external Helmholtz problem, the so called CHIEF method is also implemented. The results obtained using the proposed procedure are ﬁrst compared with those given by the GMM method. New results not available in literature are ﬁnally presented. Finally, in Chapter 5 some brief conclusions are presented, with emphasis on the 5 1. INTRODUCTION new contributions given in this study. 6 Chapter 2 Wave propagation in prestressed waveguides: SAFE method 2.1 Sommario Un’estensione del metodo semi-analitico agli elementi ﬁniti (SAFE method) viene proposta al ﬁne di studiare l’eﬀetto di uno stato pluriassiale di pretensione o predeformazione sul comportamento dispersivo di onde guidate che si propagano in guide d’onda dissipative. L’equazione del moto viene ricavata in un sistema di riferimento Lagrangiano aggiornato, nel quale la conﬁgurazione di pretensione viene assunta come conﬁgurazione di riferimento. Poichè in applicazioni pratiche le deformazioni indotte nelle guide d’onda risultano di alcuni ordini di grandezza inferiori a quelle prodotte dai normali carichi di servizio, lo stato di deformazione iniziale può considerarsi ﬁnito in rapporto a quello generato dall’onda anche se la guida possiede, in questo stato, una riserva elastica. In conformità a queste ipotesi, le equazioni di congruenza, costitutive e di equilibrio sono ricavate in forma incrementale linearizzata, includendo l’eﬀetto di carichi di tipo non conservativo. L’equazione d’onda per il sistema semi-discretizzato conduce ad un problema polinomiale agli autovalori, dal quale i numeri d’onda e le associate forme modali vengono estratti per diverse ﬁssate frequenze. Il set di soluzioni calcolato viene successivamente impiegato nell’estrazione dei parametri di dispersione: velocità di fase, attenuazione e velocità di gruppo. Mentre i primi due parametri possono essere estratti direttamente dal set di soluzioni calcolate, la velocità di gruppo richiede un’ulteriore elaborazione dei risultati. Una formula per il calcolo della velocità di gruppo è stata presentata e validata in letteratura per soli stati tensionali iniziali di tipo monoas- 7 2. WAVE PROPAGATION IN PRESTRESSED WAVEGUIDES: SAFE METHOD siale. Nel presente studio questa formula viene pertanto estesa a stati di pretensione o predeformazione di tipo pluriassiale. Tuttavia, il concetto di velocità di gruppo perde signiﬁcato ﬁsico nel caso di guide d’onda dissipative, essendo sostituito dal più generale concetto di velocità dell’energia. Poichè i modi attenuati sono di notevole interesse in ambito teorico ed applicato, viene proposta una formula modale per il calcolo della velocità dell’energia. Tale formula viene derivata in forza al teorema di Umov-Poynting stabilendo una legge di bilancio dell’energia in forma incrementale linearizzata. La formulazione proposta viene dapprima validata comparando i risultati ottenuti con due casi noti in letteratura, una barra a sezione circolare ed un binario soggetto a variazioni termiche uniformi. Nuovi casi studio vengono proposti, riguardanti l’eﬀetto delle tensioni residue derivanti dai processi di produzione dei binari e l’eﬀetto di una pressione iniziale di tipo idrostatico sulla propagazione di onde guidate in condotte in mezzi ﬂuidi. In tutti i casi, l’eﬀetto dovuto allo stato di pretensione iniziale risulta maggiormente evidente alle basse frequenze, dove il fenomeno di propagazione risulta più sensibile alle variazioni di rigidezza geometrica della guida. Ad alte frequenze il moto risulta quasi totalmente dominato dalla rigidezza meccanica della guida e l’eﬀetto della pretensione diventa sostanzialmente trascurabile. 8 2.2. INTRODUCTION AND LITERATURE REVIEW 2.2 Introduction and literature review A ﬁrst rigorous mathematical treatment of wave propagation problems in solids with a predeformation or a prestress state has been provided by Biot [1957, 1940, 1965] and Hayes [1963]. Through the years, the problem has been subjected to an intensive research. Williams and Malvern [1969] used the harmonic analysis to get the phasevelocity dispersion curves for prestressed circular rods, ﬂat plates and unbounded mediums considering both strain-rate-independent and strain-rate-dependent constitutive equations. The eﬀect of tensile and compressive axial loads on the dispersive characteristic of elastic waves propagating in submerged beams was investigated by Cook and Holmes [1981]. More recently, Bhaskar [2003] studied the dispersion relations for propagative and evanescent modes with bending-torsion coupling, while Tanuma and Man [2006] considered Rayleigh waves propagating along the free surface of a prestressed anisotropic media, deriving a ﬁrst-order perturbation formula for the phase velocity shift of Rayleigh waves from its comparative isotropic value. Frikha et al. [2011] have demonstrated that the eﬀect of a compressive or tensile axial load on the elastic wave propagation in helical beams is signiﬁcant for the four propagating modes in a low-frequency range. The wave propagation problem in waveguide-like structures has been investigated in the literature using diﬀerent mathematical approaches. In their work, Chen and Wilcox [2007] proposed a three-dimensional ﬁnite element based procedure to predict the eﬀect of axial load on the dispersive properties of guided waves in elastic waveguides of arbitrary cross section such rods, plates and rails, validating the method at low frequencies by using analytical formulae for low order theories. Osetrov et al. [2000] applied the Transfer Matrix Method (TMM) to study Surface Acoustic Waves (SAW) propagating in anisotropic and hyperelastic layered systems under residual stress, including also changes in density, modiﬁcation of the elastic stiﬀness tensor by residual strain and third-order stiﬀness constants. Lematre et al. [2006] applied matrix methods to predict Lamb, Shear Horizontal (SH) and SAW propagation in piezoelectric plates subjected to diﬀerent stress proﬁles and to calculate the acoustoelastic eﬀect on Lamb wave propagation in stressed thin-ﬁlms as well as in multilayered heterostructures under biaxial residual stresses. The prediction of dispersive characteristics of waves traveling along waveguides of arbitrary cross section represents a computationally expensive problem, especially when dispersive data is required at high frequencies. For waveguides of arbitrary but constant cross section the Semi Analytical Finite Element (SAFE) technique represents a very 9 2. WAVE PROPAGATION IN PRESTRESSED WAVEGUIDES: SAFE METHOD eﬃcient tool, since it allows to discretize the waveguide cross section only, reducing drastically the dimension of the problem [Bartoli et al., 2006; Mu and Rose, 2008; Treyssède, 2008]. To date, Semi Analytical Finite Element (SAFE) formulations were predominantly exploited for axially-loaded waveguides of linear elastic materials only [Loveday, 2009]. In this work, Loveday included the eﬀect of the axial load, resulting in a additional geometric stiﬀness matrix proportional to the mass matrix through the ratio between the axial stress and mass density. At low frequencies numerical results were shown to be in good agreement with those predicted by the Euler-Bernoulli beam theory. This extension has been used subsequently to evaluate the inﬂuence of axial load changes in rails by using sensitivity analysis and phase shift [Loveday and Wilcox, 2010] as well to support the development of a prototype aimed at predicting incipient buckling in Continuously Welded Rails (CWR) [Bartoli et al., 2010]. Experimental validations of the various formulations proposed in the literature can be found in diﬀerent works. For instance, in their work, Chaki and Bourse [2009] applied simpliﬁed acoustoelastic formulations to calibrate a guided ultrasonic wave procedure for monitoring the stress level in seven-wire steel strands while Shen et al. [2008] used guided waves to localize defects in pipes bearing high pressure gases. Since the use of guided waves for long range inspection applications is increasing, a further development of the SAFE formulation is necessary to extend it beyond the case of mono-axial prestress states. To this aim, the study presented in this chapter generalizes the SAFE formulations to viscoelastic waveguides subject to a three-dimensional state of prestress. The present extension allows thus to predict the eﬀect of prestress on the guided waves group and energy velocity as well as the wave attenuation. In this context, Caviglia and Morro [1992, 1998] provided a rigorous mathematical treatment of the energy ﬂux and dissipation of waves traveling in prestressed anisotropic viscoelastic solids. In their work, Degtyar and Rokhlin [1998] used a energy velocity formula to investigate the reﬂection/refraction problem for elastic wave propagation through a plane interface between two anisotropic stressed solids and between a ﬂuid and a stressed anisotropic solid with arbitrary propagation directions and arbitrary incident wave type. The present Chapter is organized in the following manner: the equilibrium equations of the incremental linearized theory are ﬁrst reviewed including the general state of prestress, the viscoelastic properties of the material and the eﬀect of nonconservative forces. The discretized system governing the wave propagation problem is then derived via application of the SAFE method. The group velocity formula proposed by Loveday 10 2.3. WAVE EQUATION IN LINEARIZED INCREMENTAL FORM [2009] is updated to account for the new stiﬀness operators without including the viscoelastic eﬀect, which is taken into account in the energy velocity formula derived from the energy balance principle recasted in incremental form. The scheme developed is suﬃciently general to cover also prestressed waveguides of viscoelastic anisotropic materials. 2.3 Wave equation in linearized incremental form The incremental equation of motion is derived in the Lagrangian framework depicted in Fig. 2.1 where C0 is a stress-free initial conﬁguration in which the waveguide is not subjected to any static or dynamic loading process. The generic material particle is individuated in C0 by the position vector X0 = [x0 , y 0 , z 0 ]T . If a static load is applied to the stress free conﬁguration C0 , the particle X0 moves T by a quantity u0 (X) = u0x , u0y , u0z and takes place in the conﬁguration C, which is indicated as the prestressed conﬁguration. The volume and the boundary surface of the waveguide in C conﬁguration are denoted with V and ∂V , respectively. The general particle at X = X0 + u0 = [x, y, z]T in the prestressed conﬁguration is subjected to a stress ﬁeld denoted by the Cauchy stress tensor σ0 (X), which is assumed to satisfy the static equilibrium conditions with the external applied body and surface forces, denoted by f 0 and t0 , respectively. The ﬁnal conﬁguration of the waveguide is denoted by Ct and is considered due to a displacement ﬁeld resulting from the application of a dynamic pulse to the prestressed conﬁguration. The current conﬁguration vector at time t is given by Xt (u) = X + u = [xt , y t , z t ]T and results from the superimposition of the (small) incremental timedependent displacement ﬁeld u = [ux , uy , uz ]T due to the mechanical waves on the prestressed conﬁguration X. The equilibrium equations in incremental form can be obtained by following diﬀerent approaches. Based on the coordinate systems chosen to describe the behavior of the body whose motion is under consideration, relevant quantities, such as deformations, constitutive relations and stresses can be described in terms of where the body was before any deformation due to externally applied loads or where it is during deformation; the former is called a material description, and the latter is called a spatial description [Bonet and Wood, 2008]. Alternatively, these are often referred to as Lagrangian and Eulerian descriptions respectively. Therefore, a material description refers to the behavior of a material particle, whereas a spatial description refers to the behavior at a spatial position. If the deformation state in the current conﬁguration is described with 11 2. WAVE PROPAGATION IN PRESTRESSED WAVEGUIDES: SAFE METHOD respect to a coordinate system that does not correspond to the stress-free conﬁguration, one refer in this case to the Updated Lagrangian description. Although the Total Lagrangian (TL) description is widely used in the context of nonlinear solid mechanics, for the purpose of this study the Updated Lagrangian (UL) formulation results more convenient. According to the UL description, the C conﬁguration is taken as reference and it can be computed from C0 considering the initial static displacement u0 , which is assumed to be known, for example, from previous static analysis. Using this approach, the initial static displacement ﬁeld u0 is accounted implicitly in the SAFE mesh that is used to discretize the cross-section of the waveguide in C, thus without the need to include the static terms in the equilibrium equations. The TL description obviously still remains of general validity although the nonlinear compatibility relations would include in this case some additional high-order terms in u0 , leading to more complicated equations [Bathe, 1996]. However, when deformations are superimposed on ﬁnite strains, the prestressed state is generally assumed identical or at most slightly deviated from the unstressed state and the TL and UL formulations can therefore be confused, i.e. one can assume X0 ≈ X. This simpliﬁcation cannot be applied when large strains and stresses are involved since it requires the use of appropriate incremental kinematic and constitutive relations [Bathe, 1996; Bažant and Cedolin, 1991; Yang and Kuo, 1994]. Such cases are not considered in this study but are of great importance, especially when the stress level reaches the same order of magnitude of the incremental tangential moduli or, if the body is not thin, when the incremental material moduli shows high anisotropy [Bažant and Cedolin, 1991]. In ﬁnite deformation analysis that use FEM formulations, the Updated Lagrangian description is generally adopted to give a linearization of the equilibrium relations within a Newton method scheme [Bonet and Wood, 2008; Wriggers, 2008]. Following this scheme, the equilibrium conﬁguration corresponding to a ﬁxed load increment is found by subdividing ﬁrst the load increment into diﬀerent load steps and proceeding iteratively by solving a linearized system at each load step until convergence. In reality, since only small pulses are applied on the waveguide, a fully nonlinear system of governing equations is not necessary. In fact, the hypothesis of small incremental loads and small deformations is easily veriﬁed if one observes that in many practical applications waveguides can be treated as slender structures for which magnitudes of strains arising during their service state are generally included in the range of 10−4 ÷10−3 , while guided waves generated by means of ultrasonic equipments generally produce strains in the order of 10−7 [Man, 1998; Rose, 12 2.3.1. LINEARIZED STRAIN-DISPLACEMENT RELATIONS 2004]. This means that typical strains involved in slender structures can be considered “ﬁnite” if compared with ultrasonic strains even if the prestressed conﬁguration posses an elastic reserve. 2.3.1 Linearized strain-displacement relations The geometric nonlinearities associated with the initial stress enter the problem via the kinematic relations in force of the ﬁnite strains assumption. A key quantity in ﬁnite deformation analysis is the deformation gradient F (u), which is involved in all equations relating quantities before deformation to corresponding quantities after (or during) deformation. The deformation gradient tensor enables the relative spatial position of two neighboring particles after deformation to be described in terms of their relative material position before deformation. Denoting with X and X the two position of a material particle in the prestressed and current conﬁguration, respectively, the deformation gradient associated to the particle motion is expressed as ∂Xt (u) ∂X ∂ (X + u (X, t)) = ∂X = I + ∇u, F (u) = (2.1) where I denotes the identity matrix and ∇ (·) denotes the gradient with respect to the prestressed conﬁguration C. A general measure of the deformation in the material description is represented by the Green-Lagrange (GL) strain tensor, which can be expressed in terms of deformation gradient as [Bonet and Wood, 2008; Wriggers, 2008] E (u) = 1 T F F−I , 2 (2.2) or, by using the substitutions in Eq. (2.1), in terms of displacement gradient as E (u) = 1 ∇u + (∇u)T + (∇u)T ∇u . 2 (2.3) The Green-Lagrange can be conveniently decomposed into the sum of two tensors as E (u) = ε (u) + εN L (u), denoting ε (u) and εN L (u) the tensors of the strain components that are linear and nonlinear in the displacements u (X, t), respectively. The linear strain tensor corresponds to the symmetric part of the GL strain tensor and is 13 2. WAVE PROPAGATION IN PRESTRESSED WAVEGUIDES: SAFE METHOD given by ε (u) = sym (E) = 1 ∇u + ∇ (u)T , 2 (2.4) while the tensor of nonlinear strains takes the form εN L (u) = 1 (∇u)T ∇ (u) . 2 (2.5) In view of the Semi-Analytical Finite Element discretization, the independent components of the linear and nonlinear strain tensors in Eqs. (2.4) and (2.5) are collected in the 6 × 1 vector e (u) = (u) + N L (u) , (2.6) (u) = [εxx , εyy , εzz , εyz , εxz , εxy ]T ∂ ∂ ∂ +Ly +Lz u = Lx ∂x ∂y ∂z (2.7) where is the vector of linear strain components and 1 N L (u) = 2 ∂uT ∂u ∂uT ∂u ∂uT ∂u ∂uT ∂u ∂uT ∂u ∂uT ∂u 2 2 2 ∂x ∂x ∂y ∂y ∂z ∂z ∂y ∂z ∂x ∂z ∂x ∂y T (2.8) is the vector of nonlinear strain components. In Eqs. (2.7) and (2.8) the Voigt notation has been used, while the 6 × 3 compatibility operators L i appearing in Eq. (2.7) are deﬁned as [Bartoli et al., 2006] ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ Lx = ⎢ ⎢ ⎢ ⎢ ⎣ 1 0 0 ⎤ ⎡ 0 0 0 ⎥ ⎢ ⎢ 0 1 0 0 0 0 ⎥ ⎥ ⎢ ⎥ ⎢ 0 0 0 0 0 0 ⎥ ⎢ ⎥, Ly = ⎢ ⎥ ⎢ 0 0 1 0 0 0 ⎥ ⎢ ⎢ 0 0 0 0 0 1 ⎥ ⎦ ⎣ 0 1 0 1 0 0 ⎤ ⎡ 0 0 0 ⎥ ⎢ ⎥ ⎢ 0 0 ⎥ ⎢ ⎥ ⎢ 0 0 ⎥ ⎢ ⎥, Lz = ⎢ ⎥ ⎢ 0 1 ⎥ ⎢ ⎥ ⎢ 1 0 ⎦ ⎣ 0 0 ⎤ ⎥ 0 ⎥ ⎥ 1 ⎥ ⎥ ⎥. 0 ⎥ ⎥ 0 ⎥ ⎦ 0 (2.9) Since the strain quantities deﬁned in Eqs. (2.7) and (2.8) are nonlinear expressions in the displacement u (X, t), they will lead to nonlinear governing equations. In force of the assumptions of small applied loads and small displacements, the governing equations can be recasted in a incremental linearized form. Assuming as incremental those quantities associated with the diﬀerence of motion between the current (Ct ) and the prestressed (C) conﬁgurations, the linearized incremental strain-displacement relations 14 2.3.2. LINEARIZED STRESS-STRAIN RELATIONS can be obtained by means a ﬁrst order Taylor series expansion in the neighborhood of the prestressed conﬁguration, which reads where f (X + βu) − f (X) = Du f (X) , (2.10) d f (X + βu) Du f (X) = dβ β=0 (2.11) denotes the directional derivative at X in the direction of the incremental displacement u (X, t). Using Eq. (2.11), the linearizations of Eqs. (2.2) and (2.6) take the form Du E (u) = ε (u) , (2.12) Du e (u) = (u) , (2.13) while the linearizations of the ﬁrst variations of Eqs. (2.2) and (2.6) are expressed as Du δE (u) = δεN L (u) , (2.14) Du δe (u) = δN L (u) , (2.15) in which δ denotes the ﬁrst variation with respect to u. 2.3.2 Linearized stress-strain relations The increment of stress related to any strain increment E (u) results to be small as it depends on the small amplitude waves assumption. From an energetic point of view, the use of the 2nd Piola-Kirchhoﬀ stress tensor S (u) is required as work-conjugate of the GL strain tensor [Bažant and Cedolin, 1991; Bonet and Wood, 2008; Wriggers, 2008]. Because of only small amplitude waves are applied on the initial prestressed conﬁguration, the state of stress in the current conﬁguration will diﬀer slightly from the prestressed state. As a consequence, the 2nd Piola-Kirchhoﬀ stress tensor can be confused with the Cauchy stress tensor σ (u). Making use of Eqs. (2.11) and (2.12), the above statement can be expressed in terms of linearized stress-strain relations as ∂S (u) : Du E (u) Du S (u) = ∂E (u) u=0 (2.16) = C (X) : ε (u) = σ (u) 15 2. WAVE PROPAGATION IN PRESTRESSED WAVEGUIDES: SAFE METHOD where Cijkm (X) = ∂Sij (u) /∂Ekm (u) |u=0 is the 6 × 6 fourth order symmetric tensor of tangential moduli of the material at point X in the prestressed conﬁguration. Since the linearized 2nd Piola-Kirchhoﬀ stress tensor and the Cauchy stress tensor coalesce under the hypothesis of small displacements, their independent components can be uniquely collected in the 6 × 1 vector s (u) = [sxx , syy , szz , syz , sxz , sxy ]T = [σxx , σyy , σzz , σyz , σxz , σxy ]T . (2.17) Using Eqs. (2.7) and (2.17), the stress-strain relation in Eq. (2.16) can be reexpressed as s (u) = C (X) (u) . (2.18) If an isotropic material with linear viscoelastic behaviour is considered, the Boltzmann superposition principle can be used to express the incremental stress in force of the small amplitude waves. The linearized incremental stress vector in Eq. (2.18) can be rewritten in terms of convolution integral as [Christensen, 2010; Lee and Oh, 2005] ∂ u (X, τ ) dτ C (X, t − τ ) s (u) = ∂τ −∞ t (2.19) being now C (X, t − τ ) the fourth order symmetric tensor of relaxation functions and t the current time instant. 2.3.3 Linearized incremental equilibrium equations The equilibrium of the waveguide in incremental form is obtained by subtracting from the linearized equilibrium equations in the conﬁguration Ct those written in the conﬁguration C. The equilibrium equations for both conﬁgurations can be obtained via application of the Hamilton’s variational principle t2 δH (u, δu) = δ K − W + Vc + Vnc dt = 0 (2.20) t1 where K denotes the kinetic energy of the waveguide, W accounts for the stored stored elastic energy and the dissipated energy, Vc is the work done by the external conservative volume and surface forces and δVnc is the nonconservative virtual work done by external deformation-dependent loads. The various energetic terms at a generic point Xt in the conﬁguration C t can be 16 2.3.3. LINEARIZED INCREMENTAL EQUILIBRIUM EQUATIONS expressed with respect to the conﬁguration C as follows K (u)|Ct W (u)|Ct Vc (u)|Ct δVnc (u)|Ct 1 = ρ (X) u̇2 dv, 2 V 1 = ( (u))T s (u) dv, 2 V 0 T = u fc (X) + fc (X, t) dv + uT t0c (X) + tc (X, t) da, ∂V V T 0 δu tnc (u) + tnc (u) da, (2.21) (2.22) (2.23) (2.24) ∂V where fc0 (X) and fc (X, t) are the vectors of initial and incremental conservative volume loads, respectively, t0c (X) and tc (X, t) denote the initial and incremental conservative traction loads, t0nc (u) is the vector of nonconservative traction loads in the prestressed conﬁguration and tnc (u) stands for a small displacement-dependent increment of the nonconservative traction loads. It should be remarked that the nonconservative external virtual work must be evaluated at the current conﬁguration Xt (u), which is unknown. Therefore, the spatial description should be used rigorously instead of the material description. However, if the increment in magnitude of the load is suﬃciently small, the integration of the current load intensity can be performed with good accuracy over the surface of the prestressed conﬁguration ∂V [Bathe, 1996]. The linearized variations of Eqs. (2.21)-(2.23) take the following representations Du δK (u, δu) = Du δW (u, δu) = δu̇T ρ (X) u̇dv, V V = (2.25) T T Du (δe (u)) s (u) |u=0 + (δe (u)) |u=0 Du s (u) dv (δN L (u))T s0 (X) + (δ (u))T s (u) dv V (δN L (u))T s0 (X) dv t ∂ (u (X, τ )) dtdv, (δ (u))T C ((X) , t − τ ) + ∂τ V −∞ T δu fc (X, t) dv + δuT tc (X, t) da, Du δVc (u, δu) = = V V (2.26) (2.27) ∂V in which s0 (X) = [s0xx , s0yy , s0zz , s0yz , s0xz , s0xy ]T is the vector collecting the independent components of the Cauchy stress tensor σ0 (X) in the prestressed conﬁguration. It is 17 2. WAVE PROPAGATION IN PRESTRESSED WAVEGUIDES: SAFE METHOD noted that Eq. (2.19) for the time-dependent stress-strain relations has been used in Eq. (2.26). In the rest of this chapter, it is assumed that the nonconservative forces applied to the system are of pressure type only, with no friction between the solid-ﬂuid interfaces. In this case one can recognize that t0nc (u) = −p0 n (u) , (2.28) tnc (u) = −pn (u) , (2.29) where p0 and p are, respectively, the hydrostatic pressure acting on the waveguide in the prestressed conﬁguration and the incremental pressures applied at the boundary, while n (u (X, t)) is the outward normal at the point X of the boundary surface ∂V . It is noted that, since t0nc (u) and tnc (u) must represent a traction, the notation used in Eqs. (2.28) and ((2.29)) implies that the pressure is positive in compression. If the ﬂuid-structure interaction is neglected, then the term tnc (u) on the right hand side of Eq. (2.24) vanishes, since in this case the magnitude of the pressure does not depend upon the deformation but only on the load direction pn (u). As a consequence, the linearized external virtual work can be reexpressed as Du δVnc (u, δu) = Du δVp (u, δu) = where Du n (u) = ∂n(u) ∂u u ∂Ωs −p0 δuT Du n (u) da (2.30) denotes the linearized change of orientation of the outward normal at X due to the displacement u (X, t) at the same point. In order to obtain the linearized incremental form of the equation of motion, Eq. (2.20) can be ﬁrst substituted into Eq. (2.10) to give Du δH (u, δu) = 0, (2.31) which expresses that the directional derivative of the ﬁrst variation of the Hamilton’s functional must vanish for any given small displacement u (X, t) applied at point X on the prestressed conﬁguration must vanish. Making use of Eqs. (2.25), (2.26), (2.27) and (2.30), after some algebra the ﬁrst variation of the Hamiltonian action in linearized 18 2.4. EQUATIONS IN THE WAVENUMBER-FREQUENCY DOMAIN incremental form is obtained t2 T 0 T T − (δu) ρ (X) ü − δN L (u) s (X) + (δu) fc (X, t) dvdt Du δH (u, δu) = t1 V t2 t T ∂ u (X, τ ) dτ dvdt δ (u) C (X, t − τ ) − ∂τ t1 V −∞ t2 + (δu)T tc (X, t) dadt − t1 t2 t1 ∂V ∂V (δu)T p0 ∂n (u) udadt ∂u =0 (2.32) Eq. (2.32) represents the basic system governing the dynamic of small oscillations of a three dimensional viscoelastic body subjected to an initial generic stress ﬁeld. 2.4 Equations in the wavenumber-frequency domain Given the longitudinal invariance, or periodicity, of both material and geometric characteristics of the waveguide in direction z and considering a wavenumber-frequency dependence of the form exp [i (κz z − ωt)] (2.33) where κz denotes the wavenumber in the direction of propagation, ω is the angular frequency and i is the imaginary unit, any scalar or vectorial ﬁeld can be contracted from the space-time domain to the wavenumber-frequency domain using the Fourier transforms f (z, ω) = F f (z, t) (ω) = f (κz , t) = F f (z, t) (κz ) = +∞ f (z, t) exp (−iωt)dt, (2.34) f (z, t) exp (−iκz z)dz. (2.35) −∞ +∞ −∞ Important consequences of this transformation convention concern the direction of positive wave propagation and decay and the location of poles for the dynamic system under consideration [Kausel, 2006]. These, in turn, relate to the principles of radiation and boundedness at inﬁnity, which will be addressed in the next chapters. Since the Fourier transforms act only on the t and z dependent ﬁelds, each wavenumber κz (ω) (or, conversely, each angular frequency ω (κz )) is projected on the x − y plane and the 19 2. WAVE PROPAGATION IN PRESTRESSED WAVEGUIDES: SAFE METHOD corresponding waveform propagating in the z-direction can be captured in the x − y plane by an in-plane mesh of the waveguide cross section. The stress-strain relation in Eq. (2.19) can be contracted from the time to the frequency domain, yielding to the well known relation [Christensen, 2010] Du s (X, ω) = +∞ t −∞ −∞ C (X, t − τ ) ∂ (u (X, t)) exp (−iωt) dτ dt ∂τ (2.36) = C̃ (X, ω) (X, ω) , which states that the incremental stress relative to small deformations can be obtained in the frequency domain as in a linear elastic analysis, providing only the substitution of the real tensor of elastic moduli with the tensor of relaxation functions complex C̃ (ω) = Re[C̃ (ω)] + iIm[C̃ (ω)], where Re C̃[ω ] is the so-called tensor of storage moduli and Im[C̃ (ω)] denotes the tensor of loss moduli. The versatility of the ﬁnite element formulation allows considering several types of visco-elastic rheological models, by simply assuming the opportune complex moduli matrix C̃ (ω). Generally, two diﬀerent models are used in the literature to describe absorbing media. One of them, the Maxwell rheological model, expresses the dynamic behavior of the hysteretic stress-strain relationship Re C̃ (ω) = D, Im C̃ (ω) = −iη, (2.37) where D is the well known tensor of elastic moduli while η is the viscosity tensor. Compared to a non-absorbing propagation model, the only modiﬁcation is that the visco-elastic tensor becomes complex. In contrast, the Kelvin-Voigt model assumes a linear dependence of Im[C̃ (ω)] on the frequency: Re C̃ (ω) = D, Im C̃ (ω) = −iωη. (2.38) In Eqs. (2.37) and (2.38), a negative loss modulus is considered according to the harmonic deﬁnition of the displacement ﬁeld given in Eq. (2.33). In fact, depending on the sign of the temporal term (iωt), the sign of the loss modulus can assume positive or negative value. When used in the equation of motion, the eﬀect of the Kelvin-Voigt model, bringing out the frequency dependence of the tensor, requires the imaginary part of the visco-elastic tensor to be recalculated at each frequency. The impact of both models has been thoroughly investigated in [Neau, 2003] and [Rose, 2004]. It appears that the attenuation is proportional to the frequency times the imaginary part of 20 2.5. DOMAIN DISCRETIZATION USING SEMI-ISOPARAMETRIC FINITE ELEMENTS the viscoelastic tensor and, being the loss per unit distance traveled, is a linear function of the frequency in the case of the hysteretic model and a quadratic function of the frequency in the case of the Kelvin-Voigt model. While in the case of hysteretic damping the complex part of the viscoelastic tensor usually is given without any reference to the frequency value for which the tensor itself is obtained, it is important to remind that in the case of Kelvin-Voigt model, such frequency has to be speciﬁed. Since the study of the diﬀerent behaviours of guided waves under diﬀerent rheological models is not the primary topic of this thesis, only hysteretic (Maxwell) rheological models will be considered. Using the Fourier transforms in Eq. (2.34) along with the fundamental property F [dn f (z, t) /dtn ] (ω) = (iω)n f (z, ω) and Eq. (2.36), the variational statement in Eq. (2.32) can be reelaborated from the space-time to the space-frequency domain as δuT ρ (X) udv − ω2 V T s0 (X) dv V (δ (u))T C̃ (X, ω) (u) dv δu fc (X, ω)dv − T + δN L (u) V (2.39) V δuT tc (X, ω) da − + ∂V δuT p0 ∂V ∂n (u) uda = 0. ∂u The above equation is used as the basic equation for the semi-analytical ﬁnite element discretization procedure, which is exposed in the next section. 2.5 Domain discretization using semi-isoparametric ﬁnite elements The dimension of the problem represented by Eq. (2.39) can be reduced by one in the space domain by exploiting the translational invariance (or periodicity) of the geometric and mechanical properties of the waveguide. The volume and surface integrals are decomposed as follows +∞ f (X, t) dv = f (x, z, t) dxdydz (2.40) f (x, z, t) dsdz (2.41) −∞ Ωs +∞ V f (X, t) da = ∂V −∞ 21 ∂Ωs 2. WAVE PROPAGATION IN PRESTRESSED WAVEGUIDES: SAFE METHOD where Ωs and ∂Ωs denote the area and boundary of the waveguide cross-section, respectively, and x = [x, y]T is the generic in-plane conﬁguration vector for a material particle in the prestressed conﬁguration. Substituting the positions in Eqs. (2.40) and (2.41) into Eq. (2.39) yields ω 2 − +∞ −∞ Ωs +∞ −∞ Ωs +∞ + − (δu)T ρ (x, z) udxdydz −∞ Ωs +∞ −∞ Ωs +∞ δN L (u) (2.42) T (δ (u)) C̃ (x, z, ω) (u) dxdydz −∞ ∂Ωs +∞ −∞ s0 (x, z) dxdydz (δu)T fc (x, z, ω)dxdydz + − T ∂Ωs (δu)T tc (x, z, ω) dsdz (δu)T p0 ∂n (u) udsdz = 0. ∂u Since Eq. (2.42) holds for any virtual displacement δu, the integrals over the longitudinal coordinate z vanish, and Eq. (2.42) is therefore equivalent to ω T δN L (u) s0 (x, z) dxdy δu ρ (x, z) udxdy − 2 T Ωs Ωs δu fc (x, z, ω)dxdy − Ω s T δu tc (x, z, ω) ds − + (δ (u))T C̃ (x, z, ω) (u) dxdy T + ∂Ωs (2.43) Ωs ∂Ωs δuT p0 ∂n (u) uds = 0. ∂u The integral Eq. (2.43) can be solved by a Fourier transform of the longitudinal coordinate z to the axial wavenumber κz . Making use of Eq. (2.35), Eq. (2.43) is transformed to the wavenumber domain as T 2 T δu ρ (x, κz ) udxdy − δN L (u) s0 (x, κz ) dxdy ω Ωs Ωs δuT fc (x, κz , ω)dxdy − (δ (u))T C̃ (x, κz , ω) (u) dxdy + Ω Ω s s ∂n (u) uds = 0. δuT tc (x, κz , ω) ds − δuT p0 + ∂u ∂Ωs ∂Ωs (2.44) The cross-section of the waveguide is discretized in the prestressed conﬁguration C 22 2.5. DOMAIN DISCRETIZATION USING SEMI-ISOPARAMETRIC FINITE ELEMENTS Figure 2.1: Fundamental conﬁgurations for the wave propagation problem in prestressed waveguides. by means of a planar mesh of Nel bidimensional ﬁnite elements with area Ωes , boundary ∂Ωes and 3 degrees of freedom per node associated to the three displacements components ui . Assuming an in-plane linear mapping from the reference element identiﬁed ref e e by the area Ωref s and boundary ∂Ωs to the corresponding area Ωs and boundary ∂Ωs of the generic eth element of the mesh (see Fig. 2.1), the semi-isoparametric representation results in an uncoupled description of the out-of-plane and in-plane motion. The displacement vector u (x, z, t) within the eth element is approximated as u (ξ, z, t) = N (ξ) qe (z, t) on Ωes (2.45) u (η, z, t) = N (η) qe (z, t) on ∂Ωes (2.46) where N (ξ) and N are matrices containing the shape functions in the natural coordiref e nates ξ = (ξ1 , ξ2 ) and η on Ωref s and ∂Ωs , respectively, while q (z, t) is the vector of 23 2. WAVE PROPAGATION IN PRESTRESSED WAVEGUIDES: SAFE METHOD nodal displacements (see Fig. 2.1). Using the Fourier transforms in Eq. (2.34) and (2.35), the displacement vectors on Ωes and ∂Ωes in the wavenumber-frequency domain are rewritten as u (ξ, κz , ω) = N (ξ) qe (κz , ω) on Ωes (2.47) u (η, κz , ω) = N (η) qe (κz , ω) on ∂Ωes (2.48) while the Fourier-transformed vectors corresponding to the increments of volume and surface loads become fc (ξ, κz , ω) and tc (η, κz , ω). The transformed kinematic relation given in Eq. (2.7) is (ξ, κz , ω) = B xy (ξ) + iκz B z (ξ) qe (κz , ω) , (2.49) ∂N (ξ) ∂ξi ∂N (ξ) ∂ξi +Ly , B xy (ξ) = L x ∂ξi ∂x ∂ξi ∂y (2.50) where B z (ξ) = L z N (ξ) . (2.51) Finally, the vector of nonlinear strain components given in Eq. (2.8) transformed in the wavenumber-frequency domain takes the form ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ 1⎢ N L (ξ, κz , ω) = ⎢ 2⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ N (ξ) ∂ξi T N (ξ) ∂ξi (q ) qe ∂ξi ∂x ∂ξi ∂x T N (ξ) ∂ξi e T N (ξ) ∂ξi qe (q ) ∂ξi ∂y ∂ξi ∂y −κ2z (qe )T (N (ξ))T N (ξ) qe T e T N (ξ) ∂ξi 2iκz (q ) N (ξ) qe ∂ξi ∂y T e T N (ξ) ∂ξi 2iκz (q ) N (ξ) qe ∂ξi ∂x T N (ξ) ∂ξi e T N (ξ) ∂ξi qe 2 (q ) ∂ξi ∂x ∂ξi ∂y e T ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (2.52) The discretized equations of motion can be derived for the translational invariant waveguide by observing that the relationship between an inﬁnitesimal area dxdy and the corresponding volume in the reference system dξ1 dξ2 is given by dxdy = Jse (ξ) dξ1 dξ2 , with Jse (ξ) = det[∂x/∂ξ] denoting the Jacobian of the isoparametric mapping in the x − y plane. Using these relations, one can compute the area integrals over ﬁnite 24 2.5. DOMAIN DISCRETIZATION USING SEMI-ISOPARAMETRIC FINITE ELEMENTS elements as Ωes (·) dxdy = Ωref s (·) Jse (ξ) dξ1 dξ2 . Similarly, the relation between an inﬁnitesimal surface area of the waveguide in the prestressed and reference conﬁgurations can be written as ds = Jse (η) dη, where the Jacobian of the in-plane transformation is now given by Jse (η) = (∂x/∂η) × n3 , in which n3 denotes the unit vector along the z-direction. Therefore, each boundary integral can be written at the ﬁnite element level as ∂Ωe (·) ds = ∂Ωref (·) Jse (η) dη. s s The linearized change of orientation of the surface normal due to the displacement u (x, t) can be obtained as the vector product between the tangential displacement at x in the x − y plane and the unit vector in the z-direction. Introducing the linearization [Bonet and Wood, 2008; Wriggers, 2008] ∂u ∂n (u) ∂η × n3 , u = ∂x Du n (u) = ∂u ∂η × n3 (2.53) and using the relation Jse (η) = (∂x/∂η) × n3 , one obtains 1 ∂n (u) u= e ∂u Js (η) ∂u × n3 . ∂η (2.54) which allows to write the last boundary integral in Eq. (2.44) for the eth ﬁnite element as +∞ −∞ ∂Ωref s +∞ = −∞ ∂n (u) e uJs (η) dηdz ∂u T ∂u × n3 dz. −p0 (δu) ∂η −p0 (δu)T ∂Ωref s (2.55) As it can be noted, the ﬁnal form of the surface integral in Eq. (2.55) is irrespective of the actual geometry of the element in the prestressed conﬁguration. Substituting Eqs. (2.47), (2.48), (2.49), (2.52) and (2.55) into Eq. (2.44), after some algebra the following linear system of M equations in the (κz , ω) domain is obtained T T + K − K + K − K + iκz K2 − KT κ2z K3 + Kσzz 0 0 0 0 0 σyz σxz 2 σyz σxz T 0 + Kσ 0 + Kσ 0 + K 0 − Kp Q (κz , ω) + K1 + Kσxx σxy yy xy (2.56) = Fv (κz , ω) + Fb (κz , ω) where the dynamic stiﬀness matrices Ki , the mass matrix M and the global vectors of nodal displacements Q (κz , ω), volume forces Fv (κz , ω) and surface forces Fb (κz , ω) 25 2. WAVE PROPAGATION IN PRESTRESSED WAVEGUIDES: SAFE METHOD are expressed as K3 = K2 = K1 = M= Kσxx 0 = Kσyy 0 = Kσzz 0 = Kσyz 0 = Kσxz 0 = N el Ωref s e=1 N el ref e=1 Ωs N el Ωref s e=1 N el Ωref s e=1 N el ref e=1 Ωs N el Ωref s e=1 N el Ωref s e=1 N el ref e=1 Ωs N el Ωref s e=1 N el BzT (ξ) C̃e (ξ, ω) Bz (ξ) Jse (ξ) dξ1 dξ2 (2.57) (Bxy (ξ))T C̃e (ξ, ω) Bz (ξ) Jse (ξ) dξ1 dξ2 (2.58) (Bxy (ξ))T C̃e (ξ, ω) Bxy (ξ) Jse (ξ) dξ1 dξ2 (2.59) ρe (N (ξ))T N (ξ)Jse (ξ) dξ1 dξ2 (2.60) 0 σxx (ξ) 0 σyy (ξ) N (ξ) ∂ξi ∂ξi ∂x N (ξ) ∂ξi ∂ξi ∂y T T N (ξ) ∂ξi ∂ξi ∂x N (ξ) ∂ξi ∂ξi ∂y Jse (ξ) dξ1 dξ2 (2.61) Jse (ξ) dξ1 dξ2 (2.62) 0 σzz (ξ) (N (ξ))T N (ξ) Jse (ξ) dξ1 dξ2 0 σyz (ξ) 0 σxz (ξ) N (ξ) ∂ξi ∂ξi ∂y N (ξ) ∂ξi ∂ξi ∂x T T N (ξ) Jse (ξ) dξ1 dξ2 (2.64) N (ξ) Jse (ξ) dξ1 dξ2 (2.65) N (ξ) ∂ξi Jse (ξ) dξ1 dξ2 (ξ) Kσxy 0 = ∂ξ ∂y ref i Ω s e=1 ⎤ 0 1 0 N el ⎢ ⎥ ∂N dη −p0 (N (η))T ⎣ −1 0 0 ⎦ Kp = ∂η e ∂Ωs e=1 0 0 0 (e∈∂Ωs ) Q= Fv = Fb = N el 0 σxy N (ξ) ∂ξi ∂ξi ∂x ⎡ T qe (κz , ω) e=1 N el e e=1 (e∈∂Ωs ) ∂Ωes (2.66) (2.67) (2.68) (N (ξ))T fc (κz , ω) Jse (ξ) dξ1 dξ2 e=1 Ωs N el (2.63) (N (η))T tc (κz , ω) Jse (η) dη. 26 (2.69) (2.70) 2.5. DOMAIN DISCRETIZATION USING SEMI-ISOPARAMETRIC FINITE ELEMENTS in which Nel e=1 a ﬁnite element assembling procedure over the Nel elements of the mesh. The above integrals can be evaluated numerically using the Gauss-Legendre quadrature rule [Stroud and Secrest, 1996; Wriggers, 2008]. The algebraic system in Eq. (2.56) does not represents a complete general form of the possible load conditions since it has been derived making the assumption of invariant initial stresses and mechanical properties along the z direction, i.e. σ 0 (x, κz ) = σ0 (x) , (2.71) C̃ (x, κz , ω) = C̃ (x, ω) . (2.72) In some practical situations this statement may not be representative of the actual stress distribution in the waveguide. In these situations, the various operators deﬁned in Eqs. (2.61)-(2.66) still remain formally unchanged but their positions inside the ﬁnal system of Eq. (2.56) may vary. Moreover, it can be noted that the particular case of closed boundary ∂Ωs and constant hydrostatic pressure p0 preserves the symmetry of Kp , that is in general unsymmetric. In force of this property, one can assume an incremental pressure pseudo potential Du Vp = 12 QT Kp Q and Du δVp = δQT Kp Q, which is the particular case of nonconservative work considered in the rest of this chapter. Considering only the free vibrations, i.e. Fv (κz , ω) = 0 and Fb (κz , ω) = 0, the application of the FE-discretized waveguide Eq. (2.56) yields the following M -dimensional homogeneous wave equation T T + K − K + K − K + iκz K2 − KT κ2z K3 + Kσzz 0 0 0 0 0 σyz σxz 2 σyz σxz T + K1 + Kσxx 0 + Kσ 0 + Kσ 0 + K 0 − Kp Q (κz , ω) = 0. σxy yy xy (2.73) Eq. (2.73) represents a twin parameter generalized eigenproblem in κz and ω, where for dissipative materials the stiﬀness matrices Ki result to be complex. The eigenvectors Qm (κz , ω), describe the cross sectional deformation of the wave whereas the wavenumbers, κm z , describe the wave propagation and decay (κz = Re (κz ) + iRe (κz )). The frequency ω is assumed real and the frequency range is usually known. Consequently, complex valued wavenumbers and associated wavestructures are calculated as κz (ω) and Q (κz (ω)). For the sake of simplicity, Eq. (2.73) can be rewritten in a more compact form as κ2z K3 + iκz K2 + K1 − ω 2 M Q (κz , ω) = 0, 27 (2.74) 2. WAVE PROPAGATION IN PRESTRESSED WAVEGUIDES: SAFE METHOD where K3 = K3 + Kσzz 0 , K2 = K2 − (K2 )T + Kσyz 0 T T − Kσyz + Kσxz , 0 0 − Kσ 0 xz T K1 = K1 + Kσxx 0 + Kσ 0 + Kσ 0 + K 0 − Kp σxy yy xy (2.75) (2.76) (2.77) The second order eigenvalue problem in Eq. (2.74) can be recasted in the following eigensystem with ﬁrst-order wavenumber κz by doubling its algebraic size (state-space solution) A − κz B Q̄ (κz , ω) = 0 in which A= 0 K1 − ω 2 M B= (2.78) K1 − ω 2 M −iK2 K1 − ω 2 M 0 0 −K3 , (2.79) , (2.80) are complex matrices of dimension 2M × 2M and Q̄ = Q κz Q (2.81) is a complex vector of dimension 2M × 1. 2.6 2.6.1 Dispersion analysis General solutions for lossy and lossless materials The eigenvalue problem in Eq. (2.78) consists in ﬁnding the set of generally comm m plex valued scalars κm z = a + ib and the set of corresponding complex eigenvectors Q̄m = Φ̄m + iΨ̄m for a given real frequency ω > 0. The number of eigenvalues and corresponding eigenvectors is equal to 2M . For an attenuating system in vacuum, i.e. viscoelastic materials are considered and the stress-strain relation is described by the m m complex moduli matrix C̃ (ω), all the solutions are complex κm z = Re (κz ) + iIm (κz ), indicating that all the possible guided modes, propagative and evanescent, are attenuated. The complex eigenvalues appear in the following two diﬀerent types of pairs 28 2.6.1. GENERAL SOLUTIONS FOR LOSSY AND LOSSLESS MATERIALS m m κm z = ± (a + ib ) , m m κm z = ± (c − id ) , Q̄m = ± Φ̄m + iΨ̄m , Q̄m = ± Φ̄m − iΨ̄m , (2.82) where a, b, c and d are positive arbitrary values. Substituting Eq. (2.82) into Eq. (2.33) allows to rewrite the wavenumber-frequency dependence in the form exp [i (κz z − ωt)] = exp [i ((Re (κz ) + iIm (κz )) z − ωt)] = exp [i (Re (κz ) z − ωt)] exp [−Im (κz ) z] (2.83) (2.84) from which it possible to observe that the real part of any scalar or vectorial ﬁeld represents the propagative part while the imaginary part is an exponential envelope of the wave. The propagative harmonic part of the ﬁeld is described by a constant phase argument exp [iφ] = exp [i (Re (κz ) z − ωt)]. Thus for a given wavenumber and frequency, while t is increasing the value of z has to change in order to accommodate the property of constant phase φ. For example, for a wave with positive real part of the wavevector re (κz ) > 0, if the time increases the sign of z has to be positive and its value has to increase in order to maintain φ = cost. based on these considerations, a positive Re (κz ) indicate a wave traveling in the positive direction of the z-axis (right propagating wave), while a negative Re (κz ) denotes a wave traveling in the negative direction (left propagating wave). Since the solution is symmetric, for any wave propagating in the positive direction a corresponding wave is propagating in the negative direction. From the analysis of the solutions, one can observe that only the eigenvalues for which the exponential term exp[−iIm (κz )] < 1 are of interest. In fact, since for thermodynamic reasons no energy is added to the system during the free propagation phenomena, all the solutions with exp[−iIm (κz )] > 1 corresponding to waves increasing in magnitude while propagating are to be discarded because nonphysical. Thus, for waves propagating in the positive direction (Re (κz )) for which sign(z) = +, an acceptable physical solution requires Im (κz ) ≥ 0, so that exp[−iIm (κz )] ≤ 0 describes a decaying (Im (κz ) > 0) or a non-attenuated (Im (κz ) = 0) wave. In the opposite case, i.e. for a wave that propagates in the negative z-direction, a physical solution would require Im (κz ) ≤ 0 in order to have an attenuated or wave or a wave with constant amplitude. Thus, among all the possible solutions Eqs. (2.83) and 29 2. WAVE PROPAGATION IN PRESTRESSED WAVEGUIDES: SAFE METHOD (2.84) m m κm z = + (a + ib ) → exp [i (am z − ωt)] exp [−bm z] m m κm z = − (a + ib ) → exp [i (−am z − ωt)] exp [+bm z] (attenuated wave), (2.85) (attenuated wave), (2.86) m m κm z = + (c − id ) → exp [i (cm z − ωt)] exp [+dm z] m m κm z = − (c − id ) → exp [i (−cm z − ωt)] exp [+dm z] (nonphysical), (nonphysical), (2.87) (2.88) (2.89) only the solutions of type Eq. (2.83) are physical, while the type in Eq. (2.84) have no physical meaning, since correspond to waves with amplitude that grows while propagatm ing. Therefore, from the full set of eigensolutions [κm z (ω) , Q̄ (ω)] (m = 1, 2, ..., 2M ) m m obtained from Eq. (2.78), only those of type κm z (ω) = ±[Re (kz )+iIm (kz )] are selected along with their corresponding eigenvectors. For the situation in which elastic materials are considered, the stress-strain relation is governed only by the storage modulus C̃ = Re (C) since the loss moduli becomes null. In this case the wavenumbers can be real, imaginary or complex. The real eigenvalues correspond to real eigenvectors. In such a case, the real scalars are the wavenumbers of propagative elastic waves κm z (ω), while the upper part of the corresponding eigenvector Q̄m = [Qm , κz Qm ]T describes the propagative modes of the cross section of the waveguide. The real eigenvalues appear in pairs of opposite sign, which indicates two waves propagating in opposite directions, κm z = ±a, Q̄m = ±Φm . (2.90) The purely imaginary solutions for the wavenumber correspond to the exponentially decaying near ﬁelds, which generally do not transport any appreciable mechanical energy, unless the length of the waveguide is small. These evanescent modes are also known as end modes, referring to the fact that their presence is necessary to satisfy the condition of traction free in a boundary problem or in the study of wave reﬂection. They also appear in pairs of opposite sign. The corresponding eigenvectors are purely imaginary, κm z = ±ib, Q̄m = ±iΨ̄m . (2.91) The complex wavenumbers and corresponding complex modes appear in groups of four and are the evanescent modes. Each of the waves has two components. The real part 30 2.6.2. DISPERSIVE PARAMETERS of the solution represents the propagative part of the ﬁeld while the imaginary part is an exponential envelope of the wave, κm z = ± (a ∓ ib) , Q̄m = ±Φ̄m ∓ iΨ̄m . (2.92) Following what stated before in the general case, some of this solutions can be discarded because physically not acceptable. 2.6.2 Dispersive parameters The solutions obtained from the eigenvalue problem in Eq. (2.78) are post-processed to extract the dispersive characteristics of guided modes for the given geometry and materials of the waveguide. Since the eigenproblem is solved in the axial wavenumbers and associated wavestructures for any ﬁxed ω > 0, the dispersion spectra are graphically represented as continuous or discontinuous lines which are plotted as a function of the frequency. The dispersion spectra of more practical interest are those of the real wavenumber (Re [κz (ω)]), phase velocity (cp (ω)), attenuation (α (ω)), group velocity (cg (ω)) and energy velocity (ce (ω)). The ﬁrst three do not necessitate post-processing operations since can be directly obtained from the real and imaginary parts of the computed wavenumbers. On the other hand, the group and energy velocity require further elaborations. In the following sections, each dispersion parameter is described from the mathematical and physical point of view. The following sections give a description of each fundamental dispersion parameter from both the mathematical and physical point of view. In particular, in sec: 2.6.2.5 the derivation of a new modal formula for the energy velocity extraction is proposed. 2.6.2.1 Real wavenumber The real part of the axial wavenumber displays the relationship between the temporary and spatially varying wave characteristics of the guided mode along the direction of propagation, and is generally given in radians per meter [Pavlakovic, 1998]. The real wavenumber of the generic mth propagative mode is inversely related to its the wavelength by the equation wavelength m (ω) = 31 2π . Re [κm z (ω)] (2.93) 2. WAVE PROPAGATION IN PRESTRESSED WAVEGUIDES: SAFE METHOD As can be noted, the frequency-real wavenumbers spectra can be traced from the full set of eigensolutions without the need of post-processing operations. Moreover, since these spectra appear as straight lines, they are generally used along their corresponding eigenvectors in mode sorting operations. A routine that is able to track the various modes is particularly useful, especially at high frequencies where an high number of modes is generally found and it becomes diﬃcult to discriminate between various modes. To eﬀectively represent dispersion curves, a routine that is based on wavenumbers sorting and eigenvectors correlations has been used in this thesis to trace continuous dispersion curves. 2.6.2.2 Phase velocity The phase velocity of a guided wave describes the rate at which individual crests of the wave move [Pavlakovic, 1998] and is related to the real part of the axial wavenumber through the relation cm p (ω) = ω . Re [κm z (ω)] (2.94) Since the real part of the axial wavenumber appears in both Eqs. (2.93) and (2.94), the real wavenumber and phase velocity spectra show the same informations. However, the phase velocity view is more convenient to use for realistic ultrasonic testing, since it emphasizes the velocity changes due to the guided nature of the modes [Pavlakovic, 1998]. 2.6.2.3 Attenuation The attenuation parameter is expressed by the relationship αm (ω) = Im [κz (ω)] (2.95) and gives an information on the energy lost by the guided mode per unit distance traveled. This parameter is generally measured in Nepers per meter (Np/m) or, alternatively, in Decibels per meter, being the relation between the two: αm [dB/m] = 20log10 [exp(αm [Np/m] )]. Since the amplitude of the wave has an harmonic wavenumber-frequency dependence of type exp[i(κz z − ωt)], an attenuation of 1 Np/m denotes that the amplitude of the guided wave is reduced of exp(−α) after traveling one meter. The attenuation of guided mode can be related to various phenomena: if the waveguide is elastic and immersed in vacuum, then the attenuation is zero and the amplitude 32 2.6.2. DISPERSIVE PARAMETERS of the guided mode is constant along the direction of propagation. In case of dissipative materials, both the bulk and guided waves are attenuated along their corresponding propagation directions. 2.6.2.4 Group velocity As stated in [Rose, 2004], the group velocity corresponds to the propagation velocity of a group (or packet) of waves having similar frequency. Mathematically, it is deﬁned as derivative of the frequency-wavenumber dispersion relation cg (ω) = ∂ω ∂κz (ω) (2.96) and it may be numerically calculated by the values of wavenumbers and frequencies of two adjacent points in the spectra. The knowledge of the group velocity in practical application is of fundamental importance, since it provides an information on the dispersion of a wave packet generated in the structure in a certain frequency range. The dispersion of the wave packet is directly related to the slope of the group velocity in the given frequency range. If the diﬀerence between the group velocity of each frequency component is large, then the packet is formed by waves that travels at diﬀerent velocities, causing the signal to change shape while propagating in the structure. On the other hand, if the group velocities are similar, the shape of the wave packet is maintained, so that the signal recorded has the same shape of the signal generated at a certain distance. A closed formula for the computation of group velocity in lossless media and for 0 = 0) has been already proposed in the literature [Loveday, 2009]. axial loads only (σzz 0 and If the geometric stiﬀness terms related to the nonzero initial stress components σyz 0 in Eqs. (2.64) and (2.65) are introduced to take into account for a complete three σxz dimensional prestress ﬁeld, the formula proposed by Loveday [2009] becomes cm g (ω) = ∂ω = ∂κm z m (ω) K + K (Qm )T TH K2 + Kσyz Q T + 2κ 0 + Kσ 0 0 3 σzz z xz 2ω (Qm )T MQ , (2.97) whereas in the original formula only the geometric stiﬀness operator Kσzz 0 was taken into account. Eq. (2.97) is still valid for the case of general initial stress, with the exception that the operator Ki inglobes also the geometric stiﬀness terms related to 0 and σ 0 deﬁned in Eqs. (2.64) and (2.65). the nonzero initial stress components σyz xz In Eq. (2.97) T is an M × M identity matrix with the imaginary unit substituted in 33 2. WAVE PROPAGATION IN PRESTRESSED WAVEGUIDES: SAFE METHOD correspondence of each degree of freedom in the z direction and H denotes the complex conjugate transpose (Hermitian). As well stated in the literature [Achenbach, 1973; Brillouin, 1960; Whitam, 1974], the equivalence between the group velocity cgr and the velocity of energy transportation ce is guaranteed by the Lighthill theorem [Biot, 1957; Lighthill, 1965] only in the general case of dispersive uniform lossless media, for which the central wavenumber of the wave packets traveling at inﬁnitely close frequencies is conserved. On the contrary, the dissipation mechanism in nonconservative systems leads to complex wavenumbers and, as a consequence, the group velocity loses signiﬁcance and the meaningful parameter becomes the energy velocity [Davidovich, 2010; Gerasik and Stastna, 2010]. 2.6.2.5 Energy velocity The energy velocity is a real scalar that deﬁnes the velocity at which the energy carried by a wave packet at inﬁnitely close frequencies travels down the structure. This concept correspond to a generalization of the group velocity and therefore the energy velocity also represents a generalization of the group velocity, being ce (ω) = cg (ω) for nonattenuated modes [Achenbach, 1973; Davidovich, 2010; Gerasik and Stastna, 2010]. The rate of transfer of the energy is determined as the ratio between the energy ﬂux density per unit of time and the total energy density of the system, which follows from the application of the energy conservation law [Chang and Ho, 1995; Holzapfel, 2000] DK v s + Pint + PD = Pext + Pext Dt (2.98) where the stress power Pint , the viscous power loss PD , the power supplied on the v and the power supplied by the external system by the external volume forces Pext s , are expressed, in the order, as surface forces, Pext Pint + PD = v = Pext s = Pext Ξ (u) : Ḟ (u)dxdydz (2.99) V u̇T fc dxdydz (2.100) V u̇T (tc − pn)dsdz (2.101) ∂V in which Ξ (u) denotes the 1st Piola-Kirchhoﬀ stress tensor [Bonet and Wood, 2008; Wriggers, 2008]. Eq. (2.98) can be recasted in linear incremental form by introducing the positions in Eqs. (2.99)-(2.101) and applying the usual linearization concept. Using 34 2.6.2. DISPERSIVE PARAMETERS the power equivalence Ξ (u) : Ḟ (u)dxdydy = S (u) : Ė (u)dxdydy V (2.102) V and considering a constant pressure p = p0 applied to closed boundary conditions during the motion (i.e. no ﬂuid-structure interaction), yields to ∂ K + Du W − Du Vp = ∂t T u̇T tc (x, z, t)dsdz (2.103) u̇ fc (x, z, t)dxdydy + V ∂V which represents the incremental form of the balance of energy in material description. Looking at the second integral on the right hand side of Eq. (2.103) in terms of incremental equilibrium at the boundary surface of the solid, it can be recognized that Du Ξ (u) n = tc (x, z, t) (2.104) where, in force of the relation [Bonet and Wood, 2008; Wriggers, 2008] Ξ (u) = F (u) S (u) (2.105) the linearized incremental 1st Piola-Kirchhoﬀ stress tensor can be computed as Du Ξ (u) = Du [F (u) S (u)] = (Du F (u)) S (u)|u=0 + F (u)|u=0 Du S (u) , from which, substituting the following identities Du F (u) = ∇u, S (u)|u=0 = σ0 (x, z) , F (u)|u=0 = I, ∂e u (x, z, τ ) dτ, C (x, z, t − τ ) : Du S (u) = ∂τ −∞ t 35 (2.106) 2. WAVE PROPAGATION IN PRESTRESSED WAVEGUIDES: SAFE METHOD one obtains ∂e u (x, z, τ ) C (x, z, t − τ ) : dτ. Du Ξ (u) = ∇uσ0 (x, z) + ∂τ −∞ t (2.107) Multiplication of Eq. (2.107) by n and substitution inside the boundary integral on the right hand side of Eq. (2.103) leads ﬁnally to the incremental form of the Poynting theorem in material description ∂ K + Du W − Du Vp + ∂t ∂V T v − Du Ξ (u) u̇ · nda = Pext , (2.108) T where − Du P (u) u̇ is the linearized incremental Poynting vector in material description. This entity expresses the energy ﬂux in the current conﬁguration Ct per unit of area in the prestressed conﬁguration C when a small displacement u (x, z, t) is applied. Given the harmonic behaviour of the wave process, the time derivative can be replaced by the average over a time period [t, t + 2π/ω], leading to K + Du W − Du Vp + where = ω 2π ∂V v Du J (u) · n da = Pext , (2.109) t+ 2π ω dt denotes the time average operation and the incremental Poynt T ing vector Du J (u) = − Du Ξ (u) u̇ in the wavenumber-frequency domain takes the form t Du J (u) = −iω σ (x, κz ) ∇u 0 T + C̃ (x, κz , ω) : e (u) u. (2.110) Once the wave solution is known from the eigenvalue problem of Eq. (2.74) in terms m of κm z (ω) and Q (ω) for the mth propagating mode, the previous quantities are only function of the angular frequency ω. Based on the Umov’s deﬁnition [Davidovich, 2010], the energy velocity for the mth propagating mode is then obtained as the ratio between the average energy ﬂux component projected along the z-direction and the total energy density of the waveguide at the given angular frequency ω Du Jm (ω) · n3 dΩs Ωs cm e (ω) = K m (ω) + Du W m (ω) − Du Vpm (ω) Ωs . (2.111) ∂Ωs As shown in other works [Treyssède, 2008], Eq. (2.111) can be rewritten making use of the matrix operators previously deﬁned in Eqs. (2.57)-(2.66). For the incremental energy ﬂux in the z direction, using Eq. (2.110) and the compatibility operator Lz , one 36 2.7. NUMERICAL APPLICATIONS obtains m Du J (ω) · n3 m m m H ω 0 ∂u 0 ∂u m 0 m T m + σ23 + iκz (ω) σ33 u + Lz C̃ σ13 = Im u 2 ∂x ∂y (2.112) Substituting the expressions in Eq. (2.47) and recalling the operators in Eqs. (2.57), (2.58), (2.63), (2.64) and (2.65), the integral of the energy intensity ﬂux over the waveguide cross section reads Du Jm (ω) · n3 dΩs Ωs H ω m T T T m m = Im Q Q Kσxz 0 + Kσ 0 + K2 + iκz (ω) K3 + Kσ 0 zz yz 2 (2.113) while the time average incremental kinetic energy, stored and dissipated energy, as well as the average nonconservative work are deﬁned respectively as K m (ω)|Ωs Du W m (ω)|Ωs Du Vpm (ω) ∂Ω s H ω2 m m Re Q = MQ , 4 H 2 1 m = Re Q K3 + Kσzz + κm 0 z (ω) 4 T 0 + 2Kσ 0 + + iκm z (ω) K2 − K2 + 2Kσyz xz 0 + Kσ 0 + 2Kσ 0 Qm , + K1 + Kσxx yy xy H 1 m m = Re Q Kp Q . 4 (2.114) (2.115) (2.116) Substituting Eqs. (2.113), (2.114), (2.115) and (2.116) into Eq. (2.111) provides the energy velocity for the assumed mth guided mode at given frequency ω. This relation holds for a generic 3D prestress ﬁeld and linear elastic and viscoelastic materials. Moreover, it can be veriﬁed that Eq. (2.97) is exactly recovered by Eq. (2.111) for the case of lossless materials. 2.7 2.7.1 Numerical applications Viscoelastic rail under thermal-induced axial stress Residual stresses represent a fundamental issue in the railway production and maintenance since they aﬀect negatively the rail resistance, compromise integrity and reduce 37 2. WAVE PROPAGATION IN PRESTRESSED WAVEGUIDES: SAFE METHOD 0.15 [m] 0.1 0.05 0 −0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 [m] Figure 2.2: Finite element mesh used for the dispersion curves extraction in sections 2.7.1 and 2.7.2. durability. While the presence of high compressive stresses is generally related to buckling problems, especially under hot temperatures, tensile stresses represent a vehicle for crack initiation and propagation. Moreover, some geometrical characteristics of the rail such as straightness and ﬂatness of the running surface can be deteriorated with loss in comfort. Therefore, it is of great importance for railways companies to monitorize the state of stress of the generic cross section of the rail. 0 along Some numerical investigations on the eﬀect of a constant axial prestress σzz with some proposed techniques based on guided waves for the stress magnitude measurement can be found in [Bartoli et al., 2010; Chen and Wilcox, 2007; Loveday, 2009; Loveday and Wilcox, 2010]. In these works only perfectly elastic materials are considered. The purpose of this numerical example is to show the eﬀect of the material attenuation on the dispersive behaviour of guided waves propagating in the rail subjected to a positive axial elongation ε0zz = 0.1%. In the following examples a standard A113 rail is considered. The mesh used is represented in Fig. 2.2, which is composed of 125 nodes and 182 triangular elements with linear shape functions. The steel in the prestressed conﬁguration is considered as a hysteretic linear viscoelastic material with mass density ρ = 7800 kg/m3 , longitudinal and shear bulk waves equal to cL = 6005 m/s and cS = 3210 m/s respectively, longitudinal bulk wave attenuation κL = 0.003 Np/λ and shear bulk wave attenuation κS = 0.043 Np/λ. Following Lowe [1992], the complex bulk velocities, Young’s modulus 38 2.7.1. VISCOELASTIC RAIL UNDER THERMAL-INDUCED AXIAL STRESS and Poisson’s ratio can be expressed as c̃L,S = cL,S Ẽ = ρc̃S κL,S 1+i 2π −1 , 3c̃2L − 4c̃2S , c̃2L − c̃2S 1 c̃2L − 2c̃2S , ν̃ = 2 c̃2L − c̃2S (2.117) (2.118) (2.119) from which one obtains the complex Lamè constants and the tensor of complex moduli λ̃ = Ẽ ν̃ , 1 + ν̃ 1 − 2ν̃ (2.120) Ẽ , μ̃ = 2 1 + ν̃ (2.121) C̃ijkm = λ̃δij δkm + μ̃ (δik δjm + δim δjk ) , (2.122) to be used into the incremental stress-strain relations in Eq. (2.36). It should be noted that the tensor of complex moduli remains independent from the angular frequency ω, in agreement with the assumed hysteretic behaviour of the material. Therefore, there is no need to update it at each frequency step performed in the eigenvalue problem of Eq. (2.73) and the matrices in Eqs. (2.57)-(2.67) can be computed once at the beginning of the analysis. The dispersion results in the 0 ÷ 10 kHz frequency range are depicted in Fig. 2.3 for the ﬁrst ﬁve low order modes. The mode identiﬁcation assumed here is the same adopted in [Bartoli et al., 2006], where the ﬂexural-like modes m1 and m4 as well as the torsional-like mode m2 result to be antisymmetric with respect to the x − z plane while the ﬂexural-like mode m3 and the extensional-like mode m5 are symmetric. The three plots represented in the left hand side of Fig. 2.3 show the phase velocity, energy velocity and attenuation dispersion curves for the elastic and viscoelastic rail without applied loads. It can be noted that the phase velocity and the energy velocity of the ﬁrst ﬁve modes are almost unaﬀected by the presence of the material attenuation. The three graphs on the right hand side of Fig. 2.3 report the variations of the corresponding 0 = 0.001Re Ẽ . As it can be seen, the quantities due to the applied axial stress σzz presence of an axial load leads to an increase in the phase velocity for the two ﬂexural- 39 2. WAVE PROPAGATION IN PRESTRESSED WAVEGUIDES: SAFE METHOD like modes m1 and m3 at very low frequencies, which corresponds to a decrease of about 40 m/s in the energy velocity. It is interesting nothing that the maximum shift in the attenuation is located at about 4.5 kHz for the m1 mode and 6.2 kHz for the m3 mode. This trend is in contrast with the one observed for the shift in phase and energy velocity of the two modes, which present their maximum for a frequency value approaching zero. Due to the decrease in attenuation, it follows that mode m3 at 0 prestress in the around 6 kHz could be a good candidate for revealing the state of σzz waveguide. The torsional-like mode m2 shows a positive Δcp along the entire frequency range 0 . However, its energy velocity shows both positive considered due to the tensile σzz and negative variations. The frequency values in correspondence of the maximum and minimum shift in the attenuation for the m2 mode are approximatively those with minimum and maximum shift on the energy velocity. Similar behaviour can be observed also for the two ﬂexural-like modes m1 and m3. Similarly to the previous modes, the ﬂexural-like mode m4 presents an increase in the phase velocity for the entire frequency range, and an alternate trend for both energy velocity and attenuation. It can be noticed that while the ﬂexural-like modes present their maximum shift in the phase velocity at very low frequency values (about 0 kHz for the m1 and m3 modes and in correspondence of the cutoﬀ frequency for the m4 mode), the remaining two modes do not show this behaviour. This is particularly evident for the m5 extensional-like mode, which presents its maximum at about 6.7 kHz. Moreover, at the same frequency value of about 6.5 kHz, the mode shows the maximum increase in the energy velocity and the maximum decrease in the attenuation with respect to the unloaded case. The maximum negative shift in the attenuation is not shown in the frequency-Δatt spectra for representative reasons, and its value is −0.016 Np/m. 40 2.7.1. VISCOELASTIC RAIL UNDER THERMAL-INDUCED AXIAL STRESS 8000 35 elastic viscoelastic 0 σzz =0 7000 elastic viscoelastic 0 σzz = 0, 001E 30 6000 25 m5 4000 Δcp [m/s] cp [m/s] 5000 m4 3000 20 m4 15 m2 10 2000 m1 m3 m2 1000 m3 5 m1 m5 0 0 1 2 3 4 5 6 7 8 9 0 0 10 1 2 3 frequency [kHz] 4 5 6 7 8 9 10 frequency [kHz] 6000 40 elastic viscoelastic 0 σzz =0 m5 0 σzz = 0, 001E 5000 elastic viscoelastic m5 30 20 Δce [m/s] ce [m/s] 4000 3000 m3 2000 m2 10 m3 0 m4 m1 −10 m2 m1 −20 m4 1000 −30 0 0 1 2 3 4 5 6 7 8 9 10 −40 0 1 2 3 frequency [kHz] 4 5 6 7 8 9 10 frequency [kHz] −3 x 10 0 σzz elastic viscoelastic =0 1 elastic viscoelastic 0 σzz = 0, 001E m2 0.25 m4 Δatt [Np/m] att [Np/m] 0.2 0.15 m3 0.1 m5 0 m1 m2 −1 −2 m1 m3 m4 −3 0.05 −4 m5 0 0 1 2 3 4 5 6 7 8 9 10 −5 0 1 2 3 4 5 6 7 8 9 10 frequency [kHz] frequency [kHz] Figure 2.3: Phase velocity, energy velocity and attenuation for the loaded and unloaded cases. The ﬁrst ﬁve modes m1, m2, m3, m4 and m5 are identiﬁed as in [Bartoli et al., 2006]. 41 2. WAVE PROPAGATION IN PRESTRESSED WAVEGUIDES: SAFE METHOD 2 10 elastic viscoelastic 0 σzz = 0, 001E 1 10 m1 m2 10 m3 m1 m5 m3 m4 −1 m4 m2 10 Δc p | cUnloaded | [%] 0 p m5 −2 10 −3 10 −4 10 0 1 2 3 4 5 6 7 8 9 10 frequency [kHz] 1 10 0 σzz = 0, 001E m1 0 m2 m4 −1 10 m3 m5 m2 −2 10 m1 e Δce | cUnloaded | [%] 10 elastic viscoelastic m5 m3 −3 10 m4 −4 10 0 1 2 3 4 5 6 7 8 9 10 frequency [kHz] 1 0 σzz = 0, 001E 10 elastic viscoelastic m5 m1 m1 0 Δatt | attUnloaded | [%] 10 m2 m3 m4 m2 −1 10 m5 −2 10 m5 m4 −3 10 m3 m2 −4 10 0 1 2 3 4 5 6 7 8 9 10 frequency [kHz] Figure 2.4: Percent variations between the loaded and unloaded cases for the axially loaded rail. Thin lines denote positive variations while thick lines denote negative variations. 42 2.7.2. GUIDED WAVES PROPAGATION IN A NEW ROLL-STRAIGHTENED VISCOELASTIC RAIL 0 σyy (x, y) 0 σxx (x, y) −250 −200 −150 −100 −50 0 50 0 σzz (x, y) 100 150 200 250 300 MPa Figure 2.5: Reconstructed stress patterns for the roller straightened 113A standard proﬁle in [Keller et al., 2003]. 2.7.2 Guided waves propagation in a new roll-straightened viscoelastic rail Residual stresses in rails do not depend only on the loads occouring during the service life, but also with those arising from welding or manufacturing processes, which can be very large. A principal source of residual stresses is represented by the roller straightening, which is generally the last stage of the production cycle of the rail. The residual stress formation in rails due to roller straightening has been intensively investigated in the last years [Biempica et al., 2009; Keller et al., 2003; Ringsberg and Lindbäck, 2003; Schleinzer and Fischer, 2001] and non destructive techniques, such as guided waves, can be very useful to determine the state of stress. To show the eﬀect induced by the residual stress on the dispersive behaviour in new roll-straightened rails, the stress patterns obtained by Keller et al. [Keller et al., 2003] for the standard 113A proﬁle have been considered. In particular, transversal, vertical and longitudinal contours of the residual stress are shown in Fig. 2.5 along with the ﬁnite element mesh. 0 , σ 0 and σ 0 are assumed to vary linearly The nonzero initial stress components σxx yy zz over the generic ﬁnite element as a function of the stress value at each node, σii0 (ξ) = 0 0 !3 j=1 Nj (ξ) σii j , with the jth nodal stress value σii j depending on the position of the node itself inside a speciﬁc stress region. The remaining stress components are neglected since of low order of magnitude. The eﬀect of the stress patterns on the guided waves dispersive characteristics is presented in Fig. 2.6 in the frequency range 0 ÷ 10 kHz. As previously noticed for the axially loaded rail, the dispersive behaviour for the 43 2. WAVE PROPAGATION IN PRESTRESSED WAVEGUIDES: SAFE METHOD ﬁrst ﬁve low order modes is only slightly inﬂuenced except for the m2 mode, which phase and energy velocity tend asymptotically to plus inﬁnity and minus inﬁnity for a frequency value tending to zero, respectively. At the same time, the mode attenuation decreases. This particular behaviour is not observed in the axially loaded rail and is a conse0 and σ 0 . quence of the presence of the transverse and vertical stresses σxx yy 44 2.7.2. GUIDED WAVES PROPAGATION IN A NEW ROLL-STRAIGHTENED VISCOELASTIC RAIL 8000 20 unloaded roller straightened 7000 15 m4 m2 6000 10 m5 4000 5 Δcp [m/s] cp [m/s] 5000 m4 3000 0 m3 −5 m1 m5 m2 2000 m3 −10 1000 m1 −15 0 0 1 2 3 4 5 6 7 8 9 −20 0 10 1 2 3 frequency [kHz] 4 5 6 7 8 9 10 8 9 10 8 9 10 frequency [kHz] 6000 10 unloaded roller straightened m5 m5 0 5000 m3 −10 Δce [m/s] ce [m/s] 4000 3000 m3 2000 m1 −20 m4 m2 −30 m2 m1 −40 m4 1000 −50 0 0 1 2 3 4 5 6 7 8 9 0 10 1 2 3 4 5 6 7 frequency [kHz] frequency [kHz] −3 8 unloaded roller straightened x 10 6 0.25 m4 m1 4 m4 Δatt [Np/m] att [Np/m] 0.2 0.15 m3 0.1 m2 2 m1 m3 0 m5 −2 0.05 m2 m5 0 0 1 2 3 4 −4 5 6 7 8 9 10 frequency [kHz] 0 1 2 3 4 5 6 7 frequency [kHz] Figure 2.6: Phase velocity, energy velocity and attenuation for the unloaded rail and the roller straightened rail in [Keller et al., 2003]. 45 2. WAVE PROPAGATION IN PRESTRESSED WAVEGUIDES: SAFE METHOD 2 10 1 m4 0 10 m1 m2 m3 −1 10 m3 m5 p Δc p | cUnloaded | [%] 10 m4 −2 10 m2 m1 −3 10 m5 −4 10 0 1 2 3 4 5 6 7 8 9 10 frequency [kHz] 2 10 1 10 m2 m4 0 e Δce | cUnloaded | [%] 10 m1 m3 m2 m1 −1 10 m5 m3 −2 10 m3 m2 m2 −3 10 m5 m4 −4 10 0 1 2 3 4 5 6 7 8 9 10 frequency [kHz] 2 10 1 10 m2 m4 m1 0 m3 Δatt | attUnloaded | [%] 10 m2 m1 −1 m5 10 m4 −2 10 m3 m2 m5 −3 10 m4 −4 10 −5 10 0 1 2 3 4 5 6 7 8 9 10 frequency [kHz] Figure 2.7: Percent variations between the loaded and unloaded case for the roller straightened rail. Thin lines denote positive variations while thick lines denote negative variations. 46 2.7.3. PIPE UNDER INITIAL PRESSURE LOADING 2.7.3 Pipe under initial pressure loading In many practical situations the loads applied on the waveguide are dependent upon the deformation of the solid itself. This is the case, for instance, of a pressure acting at the inner and outer surfaces of a pipe when it undergoes to stress wave propagation, which is the case studied in this example. The pressure ﬂuctuations in the gas phase due to the solid-ﬂuid interaction are neglected and the pressure is assumed to be constant during the motion. The pipe is considered suﬃciently long to assume the cross-section in plain strain state in the prestressed conﬁguration C. For diﬀerent inner and outer pressures pi and pe , the generic point (xp , yp ) of the pipe cross section with center in (x = 0, y = 0) is subjected to the following nonzero components of initial stress (see Fig. 2.8) 0 0 , σyy σxx 0 σzz " # " # x2p , yp2 yp2 , x2p c1 c1 = − c2 − c2 + − 2 x2p + yp2 x2p + yp2 xp + yp2 x2p + yp2 pe Re2 − pi Ri2 = Re ν̃ σx0 + σy0 = −2Re ν̃ Re2 − Ri2 (2.123) where ν̃ is deﬁned as in Eq. (2.119) and the constants c1 and c2 take the form Ri2 Re2 pe − pi c1 = Re2 − Ri2 c2 = p2e Re2 − p2i Ri2 Re2 − Ri2 (2.124) 0 and Positive values for the two pressures pi and pe produce compressive stresses σxx 0 , which vary quadratically along the pipe wall thickness, while the axial stress σ 0 σyy zz is constant for each point of the waveguide. The geometric stiﬀness matrices keσ0 , keσ0 xx yy and keσ0 can be calculated by integrating via Gauss quadrature the stresses deﬁned in zz Eq. (2.123) over each ﬁnite element. The numerical application considers an ASME 11/2 Schedule 160 steel pipe (outside radius Re = 24.15 mm and inside radius Ri = 17.01 mm) subjected to a hydrostatic pressure gradient between the internal and the external surfaces. The steel in the prestressed conﬁguration is assumed as isotropic and hysteretic linear viscoelastic, having mass density ρ = 7800 kg/m3 , longitudinal and shear bulk waves equal to cL = 5963 m/s and cS = 3187 m/s respectively, longitudinal bulk wave attenuation κL = 0.003 Np/λ and shear bulk wave attenuation κS = 0.008 Np/λ. The complex bulk velocities as well as the tensor of complex moduli are computed as in Eq. (2.117) and Eq. (2.122). In Fig. 2.9 solutions relative to ﬁve cases are represented considering the mesh of 112 nodes and 150 linear triangular elements depicted 47 2. WAVE PROPAGATION IN PRESTRESSED WAVEGUIDES: SAFE METHOD y 0.03 y 0.03 0.02 0.02 0.01 0.01 8 6 4 0 x pi 2 0 0 x pi -0.01 -0.01 -0.02 -0.02 MPa -2 -4 -6 -0.02 -0.01 0 0.01 0.02 -8 pe pe -0.02 0.03 -0.01 0 0.01 0.02 0.03 -10 0 σyy 0 σxx Figure 2.8: Finite element mesh of 112 nodes and 150 linear triangular elements for the ASME 1-1/2 Schedule 160 pipe. The transversal stress contours are relative to an inner pressure pi = 10 MPa and an outer pressure pe = 5 MPa (case 3). Negative values denote compressive stresses. in Fig. 2.8. All the cases are studied by taking a reference pressure pref = 5 MPa. The continuous thick line denotes the stress free case (case 1), in which the pipe is not subjected to any pressure gradient. The solutions for the remaining four cases are obtained by varying the inner and outer pressures. In particular, the dashed line denotes an internal pressure pi = pref and pe = 0 (case 2); the dotted line denotes that pi = 2pref and pe = pref (case 3); the dash dotted line refers to pi = 0 and pe = pref (case 4), and ﬁnally, the continuous thin line indicates an internal pressure pi = pref and an external pressure pe = 2pref (case 5). As it can be seen in Fig. 2.9, the presence of a pressure gradient mostly aﬀects the low order modes, essentially the torsional mode T(0,1) and the two ﬂexural modes F(1,1). The most signiﬁcant eﬀect for this two modes is essentially related to changes in phase and energy velocities in the frequency range between 0 and 1000 Hz, which becomes larger if one assumes pref > 5 MPa. The presence of an internal pressure only (case 2) produces a decrease of the phase velocity in the frequency range 0÷1000 Hz for the torsional mode T(0,1), which become dispersive. At the same time, an increase of the phase velocity for the two ﬂexural modes F(1,1) is observed in the frequency range 0 ÷ 50 Hz, with a corresponding decrement in the energy velocity. This is principally due to the fact that an internal pressure produces a traction stress on the orthogonal direction z (see Eq. (2.124)), which translates into an additional geometric stiﬀness contribute and, as a consequence, into 48 2.7.3. PIPE UNDER INITIAL PRESSURE LOADING an increased ﬂexural waves velocity (see also [Chen and Wilcox, 2007; Loveday, 2009]). Moreover, an increase of the wave attenuation is observed for the torsional mode T(0,1) in the frequency range 0 ÷ 500 Hz, while a further drop in the wave attenuation for the longitudinal mode L(0,1) is observed in the range 0 ÷ 100 Hz (phase and energy velocities for this mode result to be substantially unchanged). Dispersion curves for the cases 3, 4 and 5 show a similar behaviour. In these cases the presence of an external pressure (lower than the internal pressure in the case 3 and higher in the cases 4 and 5) produces always a cutoﬀ frequency and an increment in the phase velocity for the torsional mode T(0,1), which is limited to the frequency range 25 ÷ 250 Hz for the case 3 and 190 ÷ 1500 Hz and 240 ÷ 1500 Hz for the cases 4 and 5, respectively. The related energy velocity is always increased for these cases. In the same frequency ranges the wave attenuation of the T(0,1) mode is highly reduced by the presence of the prestress ﬁeld and the phase velocity for the two ﬂexural modes F(1,1) results to be lower than the stress-free case for each of the three cases considered. An interesting observation can be made by noting the behaviour of the ﬂexural mode F(1,1) in case 2 and case 3. In fact, even if in both cases the axial stress 0 is positive, in case 2 the eﬀect of the internal pressure increases the mode phase σzz velocity as a consequence of an increased geometric stiﬀness while in case 3 the extra external pressure reduces the mode phase velocity. As previously noticed for the case 2, only very small changes can be observed for the F(1,1) wave attenuation at very low frequencies, with a decrease on the attenuation values for the cases 3 and 4 and an increase for the case 5. However, an increment in the wave attenuation (very small for the case 3 and much higher for the cases 4 and 5) is observed for the longitudinal mode L(0,1) while its speed is not substantially aﬀected by the presence of the pressure gradient. 49 2. WAVE PROPAGATION IN PRESTRESSED WAVEGUIDES: SAFE METHOD 8000 F(2,1) 7000 2 10 6000 T(0,1) L(0,1) |Δcp | [m/s] cp [m/s] 5000 4000 T(0,1) 3000 L(0,1) 0 10 F(1,1) -2 10 F(1,1) 2000 0 -2 10 -1 0 10 10 1 10 L(0,1) -4 F(1,1) 1000 T(0,1) 0 10 5 10 15 20 25 30 35 40 frequency [kHz] frequency [kHz] 6000 2 10 L(0,1) 5000 1 10 T(0,1) L(0,1) 0 |Δce | [m/s] ce [m/s] 4000 T(0,1) 3000 2000 F(1,1) 10 -1 10 L(0,1) F(1,1) -2 10 F(1,1) F(1,1) 1000 T(0,1) -3 10 F(2,1) L(0,1) 0 -4 -2 10 -1 0 10 10 1 10 10 0 5 10 15 20 25 30 35 40 frequency [kHz] frequency [kHz] 0 10 F(2,1) L(0,1) -1 10 -4 10 T(0,1) F(1,1) L(0,1) -3 F(1,1) T(0,1) 10 |Δatt| [m/s] att [Np/m] -2 10 -6 10 F(1,1) L(0,1) F(1,1) 30 35 T(0,1) -8 10 -4 10 L(0,1) -10 -5 10 10 -2 10 -1 10 0 10 1 0 10 5 10 15 20 25 40 frequency [kHz] frequency [kHz] unloaded pi = pref , pe = 0 pi = 2pref , pe = pref pi = 0, pe = pref pi = pref , pe = 2pref Figure 2.9: Phase velocity, energy velocity and attenuation for the ASME 1-1/2 Schedule 160 pipe under diﬀerent pressure gradients. 50 2.7.3. PIPE UNDER INITIAL PRESSURE LOADING 2 10 0 10 L(0,1) F(1,1) -2 10 p Δc p | cUnloaded | [%] T(0,1) -4 10 F(1,1) L(0,1) T(0,1) -6 10 0 5 10 15 20 25 30 35 40 frequency [kHz] 1 10 0 10 T(0,1) L(0,1) e Δce | cUnloaded | [%] -1 10 -2 10 -3 10 F(1,1) -4 10 F(1,1) L(0,1) F(1,1) T(0,1) -5 10 L(0,1) -6 10 0 5 10 15 20 25 30 35 40 frequency [kHz] T(0,1) 0 10 L(0,1) Δatt | attUnloaded | [%] F(1,1) -2 10 F(1,1) L(0,1) F(1,1) 30 35 -4 10 T(0,1) L(0,1) -6 10 0 5 10 15 20 25 40 frequency [kHz] Figure 2.10: Percent variations between the loaded and unloaded cases for the pressurized pipe. Thin lines denote positive variations while thick lines denote negative variations. 51 2. WAVE PROPAGATION IN PRESTRESSED WAVEGUIDES: SAFE METHOD 2.8 Conclusions An extension of the Semi Analytical Finite Element (SAFE) formulation has been proposed to include the eﬀect of a three dimensional prestress ﬁeld in viscoelastic waveguides. Based on a semi-isoparametric discretization, the formulation of the problem has been extended by taking into account high order terms in the strain-displacement relations and complex elastic constants in the incremental stress-strain relations. The energy velocity formula has been also revisited to include initial stress terms starting from the balance law of the mechanical energy in material description. Some numerical investigation have been conducted on a 113A standard rail, considering hysteretic materials. The rail has been analyzed in the 0 ÷ 10 kHz frequency range, but knowledge of high-frequency dispersion data (up to 100 kHz) can be very helpful for axial load measurement. For the case of an axial load only, the ﬁrst ﬂexural modes in the low frequency range are the most inﬂuenced, showing generally an increase in the phase velocity and a corresponding decrease in the energy velocity when a tensile load is applied. The phase and energy velocities of the ﬁrst modes are mostly sensitive in the very low frequency range, although this does not happen for their corresponding wave attenuation, which show the highest changes in magnitude for higher frequency values. In the case of a roller-straightened rail, the simultaneous presence of both longitudinal and transversal stresses modiﬁes signiﬁcantly the behaviour of the fundamental torsional mode, while the sensitivity of the ﬁrst ﬂexural modes to the residual stress results to be highly mitigated with respect to the constant axial stress case. Although the analysis have been conducted in a low frequency range, the knowledge of high-frequency dispersion data (up to 100 kHz) can be very helpful for axial load measurement since some higher order modes remain considerable sensitive, providing useful informations in load detection schemes based on the measurement of the shift in phase produced by the load itself. The dispersive characteristic of guided waves propagating in a hysteretic ASME 1-1/2 Schedule 160 pipe have been also analyzed by considering the eﬀect of a pressure gradient between the inner and outer surfaces. Similarly to the roller-straightened rail, the presence of the transversal (radial and circumferential) initial stresses aﬀects principally the ﬁrst torsional modes, which becomes dispersive, while the principal ﬂexural mode are slightly inﬂuenced by the axial load which arise by considering the pipe in plane stress state. 52 2.8. CONCLUSIONS Finally, it appears that the inﬂuence of the initial stress on the dispersive characteristics of compact sections is large for low order modes at low frequencies while higher order modes are generally less inﬂuenced. The reason is that at high frequencies the geometric stiﬀness contribution becomes very small if compared with the elastic stiﬀness contribution and therefore the wave propagation behaviour mainly depends on the waveguide properties and it is slightly aﬀected by the prestress state Chen and Wilcox [2007]. Based on the proposed numerical examples, the frequency values corresponding to the highest shift in the attenuation for the principal modes seem generally far to those at which the highest shift in the phase and energy velocity are observed. This particular behaviour could be deepened by assuming a diﬀerent viscoelastic model as, for example, the Kelvin-Voigt model or the Linear Standard Solid. The proposed formulation is suﬃciently general to cover also prestressed waveguides of viscoelastic anisotropic materials and can be relevant in the design of several long range non-destructive techniques based on guided waves. In particular, it can be extremely helpful in the prediction of testing results for ultrasonic guided wave based screening of roller straightened rails, where the stress state has to be limited to prevent crack propagations and rail failures, as well as in pressurized pipelines carrying gases, where the distance of propagation of guided waves is of primary importance. 53 2. WAVE PROPAGATION IN PRESTRESSED WAVEGUIDES: SAFE METHOD 54 Chapter 3 Wave propagation in bounded and unbounded waveguides: 2.5D Boundary Element Method 3.1 Sommario Nel presente capitolo viene descritta una formulazione basata sul metodo degli elementi di contorno (Boundary Element Method) per il calcolo delle caratteristiche di dispersione di onde guidate in guide d’onda con sezione trasversale di estensione ﬁnita ed inﬁnita. La geometria dell’elemento longitudinale o della cavità si considera arbitraria, mentre il materiale della guida è assunto isotropo e viscoelastico lineare. L’attenuazione dell’onda guidata viene descritta in maniera spaziale attraverso la componente immaginaria del vettore d’onda e si considera dovuta unicamente a meccanismi di dissipazione interna del materiale. La formulazione agli elementi di contorno viene ricavata dal caso elastodinamico tridimensionale mediante trasformate di Fourier nel tempo e nello spazio. La trasformazione dal dominio spazio-tempo a quello numero d’onda-frequenza consente di rappresentare il problema tridimensionale mediante una mesh di elementi al contorno monodimensionali, utilizzata per modellare il contorno della guida. Come ben noto, gli integrali di contorno presentano delle singolarità legate alla natura delle funzioni di Green. Utilizzando una tecnica basata su moti di corpo rigido e sulla corrispondenza fra le singolarità delle funzioni di Green dinamiche e statiche, gli integrali non convergenti in senso classico sono regolarizzati, rendendo possibile l’utilizzo di tecniche convenzionali di quadratura numerica. 55 3. WAVE PROPAGATION IN BOUNDED AND UNBOUNDED WAVEGUIDES: 2.5D BOUNDARY ELEMENT METHOD L’equazione d’onda si conﬁgura come un problema non lineare agli autovalori, il quale viene risolto nelle incognite numeri d’onda complessi per ogni ﬁssata frequenza reale e positiva. Mediante l’utilizzo di un algoritmo basato su integrali di contorno (Contour Integral Method), il problema in parola viene trasformato in un problema olomorﬁco e lineare all’interno di una ﬁssata regione nel piano complesso del numero d’onda assiale. Basandosi sul principio di radiazione di Sommerfeld e sulla natura del fenomeno di propagazione, viene dimostrato che le radici corrispondenti ai modi guidati reali giacciono su tre dei quattro possibili fogli di Riemann. I risultati ottenuti con il metodo proposto per due diﬀerenti guide d’onda di sezione ﬁnita risultano in ottimo accordo con quelli ottenuti con il metodo SAFE. Mediante studi numerici condotti su cavità di due diﬀerenti geometrie, viene inﬁne dimostrato che l’attenuazione dei modi guidati tende asintoticamente al valore di attenuazione dell’onda di Rayleigh per interfacce piane. 56 3.2. INTRODUCTION AND LITERATURE REVIEW 3.2 Introduction and literature review In Guided Waves based nondestructive testing and structural health monitoring, the computation of the waves dispersion properties is indispensable for the design of both actuation and sensing systems, as well as to tune experimental set-up. To date, for the prediction of the dispersion properties several formulations are available. Widely adopted numerical techniques are represented by analytical methods [Lowe, 1995] and semi-analytical methods [Bartoli et al., 2006; Gavric, 1995; Hayashi et al., 2003]. The analytical methods generally provide accurate solutions for problems involving energy losses due to both internal damping and leakage, but their application is generally restricted to waveguides of regular geometry. In addition, while looking for roots of the dispersive equation they suﬀer from numerical instabilities, missing roots and can require large computational time for multilayered waveguides (e.g. composite laminates). On the other hand, semi-analytical methods hardly handle problems involving unbounded domains. Moreover, at high frequency they become time consuming due to the large number of degrees of freedom involved to support accurate solutions. The Boundary Element Method can enter in this context as a possible alternative for the dispersive data computation in both bounded and unbounded waveguides. Unlike FE-based formulations, which operate by discretizing the entire domain, the BEM can achieve a better accuracy by only discretizing the boundary of the waveguide, reducing the dimension of the problem. At the same time, some features of the analytical methods are conserved, in particular the potential to deal with problems involving unbounded mediums. In the last years a large number of works have been published, in which diﬀerent boundary element formulations have been proposed for the wave propagation problem in waveguide-like structures. This problem is sometimes referred in literature as the twoand-a-half (2.5D) problem [Costa et al., 2012; François et al., 2010; Godinho et al., 2003; Lu et al., 2008b; Rieckh et al., 2012; Sheng et al., 2005, 2006; Tadeu and Kausel, 2000], since the geometric and mechanical translational invariance allows a two dimensional description of the geometry, while the body motion still completely retains its three dimensional characteristic. While most of these studies are focused on wave scattering problems [Cho and Rose, 2000, 1996; Galán and Abascal, 2005, 2004; Godinho et al., 2003; Pedersen et al., 1994; Wang et al., 2011; Zhao and Rose, 2003], vibrations induced problems [Costa et al., 2012; François et al., 2010; Rieckh et al., 2012; Sheng et al., 2005, 2006] or dynamic response problems [Lu et al., 2008b], minor attention has been dedicated to the study of 57 3. WAVE PROPAGATION IN BOUNDED AND UNBOUNDED WAVEGUIDES: 2.5D BOUNDARY ELEMENT METHOD dispersive characteristics of guided waves, especially when attenuation is involved. For such task, FE-based formulations are preferred to BEM due to (i) numerical diﬃculties with the treatment of the characteristic singularities of the fundamental solutions and (ii) complexities related to the solution of a nonlinear eigenvalue problem resulting from the boundary element modal analysis. However, these two problems have been successfully tackled in diﬀerent manners in recent years. In their work, Tadeu and Santos [2001] used a 2.5D boundary element formulation to extract the phase and group velocity dispersion curves for both slow and fast elastic formations by solving an eigenvalue problem in absence of an incident waveﬁeld. The leaky modes poles have been found using complex frequencies, i.e. by describing decay in time, and no attenuation dispersion curves were provided. Godinho et al. [2003] used a similar 2.5D boundary element formulation to extract the phase velocity dispersion curves in cylindrical shell structures immersed in ﬂuids. The dispersion curves were obtained by computing the response of the system for diﬀerent values of the axial wavenumber at a given frequency and by considering a source and receiver line inside the cavity. It is well known that when the wavenumber approaches a modal wavenumber of the system, a peak in the system response is obtained [Wu, 2000], providing a tool to extract the dispersion curves. Unfortunately, attenuation information is generally diﬃcult to obtain using this method. Gunawan and Hirose [2005] proposed a boundary element formulation for waveguides of arbitrary cross-section, using discontinuous quadratic elements and subdividing the singular integrals into regular and singular parts, which were treated separately. To extract dispersion curves, they used a Newton’s scheme where the eigensolutions, the real wavenumbers, were searched at diﬀerent frequency steps by starting from the highest frequency of interest and proceeding backwards, exploiting the relative straightness of the axial wavenumbers dispersion curves. Moreover, their scheme was made more robust by using the group velocity extrapolation formula during the iterative search. The method has demonstrated to work properly for real wavenumbers. However, the extension of this approach for complex wavenumbers would imply a substantial increment of operations. In addition, the convergence of the method strongly depends on the initial guesses, which are diﬃcult to estimate when attenuation is involved. To account for attenuation, the modiﬁed bisection method proposed by Lowe [1995] represents an excellent variation into the curve tracking algorithms family; this approach is very robust, although its convergence rate is lower than that of a Newton-like method. The main limitations in using Newton-like methods and iterative methods are represented by the risk to follow the wrong curve when the spectrum is densely 58 3.2. INTRODUCTION AND LITERATURE REVIEW populated and by the fact that the solution at the previous step is needed as starting point to ﬁnd the solution at the subsequent step. Therefore, for densely populated dispersion spectra, very small incremental steps are required. In addition, extraction of eigenvalues with multiplicity higher than one is generally complicated. A substantially diﬀerent method is represented by the Modiﬁed Matrix Pencil Algorithm proposed by Ekstrom [1995], who estimated dispersion data from borehole acoustic arrays. The method estimates dispersion properties by ﬁrst performing a time Fourier transform of a space-time array resulting from multiple receivers. Then, at each temporal frequency, the complex wavenumbers are extracted using a forward/backward averaging matrix pencil method [Hua and Sarkar, 1990]. The method has been applied by Zengxi et al. [2007] for the dispersion parameters extraction in ﬂuid-ﬁlled boreholes with irregular shapes using a 2.5D Boundary Element Formulation. More recently, Badsar et al. [2010] used a half-power bandwidth method for the determination of the material damping ratio in shallow soil layers. This method uses computed or measured waveﬁelds to extract the frequency-wavenumber and frequencyattenuation spectra. In particular, the wavenumber dispersion curves are derived from the peaks positions of the FFT-transformed waveﬁeld, whereas the attenuation curves are derived from their width using the half-power bandwidth method. In this chapter, a 2.5D regularized boundary element formulation [François et al., 2010; Lu et al., 2008b] is used to extract dispersion curves for homogeneous damped waveguides. The attenuation is spatially described through the imaginary part of the axial wavenumber [Bartoli et al., 2006; Lowe, 1995] and the dispersive parameters, i.e. complex wavenumbers, phase velocity and energy velocity, are computed by solving a nonlinear eigenvalue problem using a contour integral algorithm [Amako et al., 2008; Asakura et al., 2009; Beyn, 2012]. This algorithm does not require an initial guess of eigenvalues and eigenvectors. Moreover, this method is particularly suitable when the number of roots in the complex region of interest is much smaller than the eigenvalue problem dimension, as it appears in the dispersion curves extraction of fundamental modes. At low frequencies, in fact, few fundamental modes generally exist, while eigenvalue problems designed also for waves computation at high frequencies may be characterized by a large number of equations. A recent application of a contour integral algorithm in 2D-BEM acoustic problems can be found in the work of Gao et al. [2011]. The extraction of the energy velocity in post-processing is also discussed, and some numerical examples are presented, comparing the obtained results with those provided by the SAFE method [Bartoli et al., 2006]. 59 3. WAVE PROPAGATION IN BOUNDED AND UNBOUNDED WAVEGUIDES: 2.5D BOUNDARY ELEMENT METHOD The chapter is concluded with a study of the dispersion properties of cavities with arbitrary cross-section in unbounded linear isotropic viscoelastic mediums. The knowledge of surface waves dispersion properties in the vicinity of cavities can be useful in some practical applications such as, for example, geophysical and seismic prospecting techniques or in the study of vibrations in underground tunnels. 3.3 Problem statement The problem under consideration consists in an isotropic viscoelastic waveguide of general shape which is interested by a wavefront propagating along its longitudinal axis (Fig. 3.1). The boundary of the waveguide is considered to be in contact with vacuum and, as a consequence, no energy losses due to radiation of bulk waves occur, since the generic bulk wave incident at the solid-vacuum interface is totally reﬂected Im and mode-converted. The wavenumber vector kz = kRe z + ikz associated to the guided Im wave results from the sum of the projections of the bulk wavenumbers kL = kRe L + ikL Im and kS = kRe S + ikS onto the z-direction, so that the guided wave represents the wavefront that propagates along the z-axis as a result of the superimposition of the bulk waves traveling obliquely and continuously reﬂected and mode-converted at the boundary of the waveguide. Since the bulk waves travel at some incidence angle with respect to the z-axis, it can be recognized the presence of a wavefront in the x − y plane, which wavenumber vector is intended as the sum of the projections of kL and kS to the generic z=cost plane. The associated radial components of the longitudinal and shear bulk wavenumber vectors Im Re Im are denoted with kα = kRe α + ikα and kβ = kβ + ikβ respectively. Given a time-harmonic excitation and the translational invariance of geometric and mechanical characteristics along the z-axis, the wave propagation process is assumed with dependence exp [i (ωt − κz z)], (3.1) where the angular frequency ω is real and positive while, for a generic dissipative Im system, κz = |kRe z | + i|kz |. The real component of the axial wavenumber represents the modulus of the harmonic propagation vector kRe z , while the imaginary component is the modulus of the spatial attenuation vector kIm z , which describes the exponential amplitude decay per unit of distance traveled. Focusing only on guided waves propagating in the positive z-direction, it can be noted from Eq. (3.1) that, in order to have an amplitude decay for z > 0, the conditions 60 3.4. 2.5D INTEGRAL REPRESENTATION THEOREM Re (κz ) > 0 and Im (κz ) < 0 must be satisﬁed for any ﬁxed real positive frequency. In such case, and according to the Correspondence Principle [Christensen, 2010], the complex velocities for the longitudinal and shear bulk waves, c̃L (ω) and c̃S (ω), can be written in the following form [Luo and Rose, 2007] c̃L,S (ω) = cL,S (ω) ω = , Re(κL,S ) + iIm(κL,S ) 1 − iαL,S (ω) /2π (3.2) Im where κL,S = |kRe L,S | + i|kL,S | are the complex moduli of the longitudinal and shear bulk wavevectors, while the bulk attenuation coeﬃcients αL,S (ω) = −Im(κL,S )/Re(κL,S ) represent the exponential amplitude decay of the wave after traveling one wavelength. If a linear viscoelastic Maxwell rheological model is adopted, c̃L,S result to be frequency independent [Christensen, 2010]. The corresponding complex material constants are evaluated as [Luo and Rose, 2007] 2μ̃ − Ẽ , λ̃ = Ẽ/μ̃ − 3 μ̃ = c̃2S ρ, 3 − 4 (c̃S /c̃L )2 μ̃, Ẽ = 1 − (c̃S /c̃L )2 ν̃ = λ̃ , (3.3) (3.4) 2 λ̃ + μ̃ where λ̃ and μ̃ denote the ﬁrst and second complex Lamé constants, ρ the material density, Ẽ is the complex Young’s modulus and ν̃ the complex Poisson’s ratio. To be consistent with Eq. (3.2) and the analysis of Lowe [1995], in the following it is assumed that the attenuation component of the wavenumber vectors is always parallel to the propagation component, i.e. the maximum amplitude decay due to material damping occurs along the propagation direction. 3.4 2.5D integral representation theorem The integral representation theorem is ﬁrst recalled in the 3D case for an isotropic linear viscoelastic body of volume V and external surface ∂V , with mechanical properties deﬁned by the material density ρ and the complex Lamé constants λ̃ and μ̃ (or, equivalently, the complex bulk velocities c̃L and c̃S ). The body is assumed to be either bounded or unbounded and allows for the presence of edges, corners and internal cavities. The body is considered to be subjected to a unitary harmonic point load p (X , t) = δ (X − X ) exp(iωt) applied at X = [x , y , z ]T ∈ V , where δ (·) denotes the Dirac 61 3. WAVE PROPAGATION IN BOUNDED AND UNBOUNDED WAVEGUIDES: 2.5D BOUNDARY ELEMENT METHOD Figure 3.1: Analytical model of the bounded waveguide. delta function. The spatial coordinate X = [x, y, z]T ∈ ∂V describes a receiver point located on the boundary. In this step, no distinctions are made on whether the body is bounded or unbounded. Denoting with R = |X − X | the spatial distance between the receiver point and the source point, the 3D integral representation theorem in the frequency domain is given by the well known relation [Andersen, 2006; Bonnet, 1999; Dominguez, 1993] cij X uj X , ω) = ∂V − UijD (R, ω) tj (X, ω)da (X) TijD (R, ω) uj (X, ω)da (X) , (3.5) X ∈ V, X ∈ ∂V ∂V which expresses the relation between the displacements uj (X, ω) and tractions tj (X, ω) at any X located on the boundary ∂V (state of unknown boundary variables) and the Green’s functions for the displacements UijD (R, ω) and tractions TijD (R, ω) [Andersen, 2006; Bonnet, 1999; Dominguez, 1993] (known state of fundamental solutions). The fundamental displacements UijD (R, ω) correspond to a second order tensor that satisﬁes the diﬀerential equation ∂ 2 U D (R, ω) ∂ 2 UijD (R, ω) ij D + μ̃ + δkj δ X − X = −ρω 2 Ukj (R, ω) λ̃ + μ̃ ∂xk ∂xi ∂xi ∂xi (3.6) in the isotropic viscoelastic full space. The corresponding dynamic fundamental tractions TijD (R, ω) at point X ∈ ∂V with outward normal n (X) = [nx , ny , nz ]T ∈ ∂V are obtained from the fundamental displacements via constitutive relations. The general subscript notation (·)ij , with i, j = 1, 2, 3, stands for the eﬀect in the jth direction at the receiver point X when the unitary harmonic point load is acting at the source point X in the ith direction. As usual, the subscripts 1, 2, 3 are freely 62 3.4. 2.5D INTEGRAL REPRESENTATION THEOREM interchanged with x, y, z for convenience of representation. The coeﬃcients cij (X ) in / V , while Eq. (3.5) take the values Cij (X ) = δij if X ∈ V and Cij (X ) = 0 if X ∈ Eq. (3.5) is not deﬁned for X ∈ ∂V . The derivation of the 2.5D integral representation theorem follows from the work of François et al. [2010]. To this end, the geometry and mechanical properties of the body are now assumed to be invariant in the z-direction and the intersections of V and ∂V with the z = 0 plane are denoted with Ωb and ∂Ωb , respectively. Accordingly, the projection of the receiver point X and the source point X on the z = 0 plane are denoted respectively with x = [x, y]T and x = [x , y ]T . The point x ∈ ∂Ωb is understood as the intersection of an observer line inﬁnitely extended along the zdirection with the z = 0 plane, while x ∈ Ωb is intended as the intersection of a unitary harmonic line load with the same plane. Since the distance between the observer and source lines is constant throughout z, the spatial vector R is replaced by r = x−x , with r = |x − x | denoting the in-plane source-receiver distance. The line load is assumed to be harmonic in time and space and assumes the following representation p x , y , z , t = δ x − x δ y − y exp i ωt − κz z . (3.7) where z denotes the out-of-plane coordinate along the line of projection x on the x − y plane. Given the longitudinal invariance, the surface integrals in Eq. (3.5) can be decomposed as +∞ (·) da (X) = (·) ds (x) dz −∞ ∂V (3.8) ∂Ωb leading to the following integral representation cij x uj x , z , ω = +∞ −∞ − UijD r, z , z, ω tj (x, z, ω)ds (x) dz ∂Ωb +∞ −∞ ∂Ωb TijD (3.9) r, z , z, ω uj (x, z, ω)ds (x) dz, / Ωb , respectively. Using where cij (x ) are equal to δij and 0 for x ∈ Ωb and x ∈ the translational invariance property of the Green’s functions [Andersen, 2006; Bonnet, 1999; Kobayashy, 1987], the source point of in-plane coordinates x and out-of-plane 63 3. WAVE PROPAGATION IN BOUNDED AND UNBOUNDED WAVEGUIDES: 2.5D BOUNDARY ELEMENT METHOD coordinate z can be shifted to the plane z = 0. Eq. (3.9) is then rewritten as cij x uj x , z , ω = +∞ −∞ − UijD r, 0, z − z , ω tj (x, z, ω)ds (x) dz ∂Ωb +∞ −∞ TijD (3.10) r, 0, z − z , ω uj (x, z, ω)ds (x) dz. ∂Ωb The spatial harmonic dependence assumed in Eq. (3.7) allows any scalar or vectorial ﬁeld to be contracted in the wavenumber domain using the Fourier transform f (κz ) = +∞ −∞ f (z) exp iκz z dz, (3.11) which, applied to Eq. (3.10), leads to cij x uj x , z , ω = +∞ +∞ −∞ −∞ ∂Ωb × exp iκz z dzds (x) dz +∞ +∞ − TijD r, 0, z − z , ω uj (x, z, ω) −∞ UijD r, 0, z − z , ω tj (x, z, ω)dz −∞ (3.12) ∂Ωb × exp iκz z dzds (x) dz , Substituting the identity exp iκz z = exp −iκz z − z exp (iκz z) (3.13) inside Eq. (3.12), the following expression is obtained cij x uj x , z , ω = +∞ +∞ ∂Ωb −∞ −∞ exp −iκz z − z UijD r, 0, z − z , ω dz × tj (x, z, ω) exp (iκz z) dzds (x) +∞ +∞ D exp −iκz z − z Tij r, 0, z − z , ω dz − ∂Ωb −∞ −∞ × uj (x, z, ω) exp (iκz z) dzds (x) , (3.14) where the terms inside the square brackets are recognized as the space Fourier transforms of the 3D Green’s functions. These functions represent the fundamental solutions for the time and spatial harmonic line load problem of Eq. (3.7) in the isotropic linear 64 3.5. GREEN’S FUNCTIONS viscoelastic full space. Denoting the 2.5D fundamental displacements with UijD (r, κz , ω) and the corresponding fundamental tractions with TijD (r, κz , ω), Eq. (3.53) is reelaborated as follows cij x uj x , z , ω = UijD (r, −κz , ω) ∂Ωb × +∞ −∞ − tj (x, z, ω) exp (iκz z) dz ds (x) (3.15) TijD (r, −κz , ω) ∂Ω b+∞ × −∞ uj (x, z, ω) exp (iκz z) dz ds (x) , where the terms in the square brackets represent the space Fourier transforms of the displacements and tractions on the boundary, denoted with uj (x, κz , ω) and tj (x, κz , ω), respectively. Eq. (3.15) is ﬁnally recasted in the following form cij x uj x , κz , ω = UijD (r, −κz , ω) tj (x, κz , ω) ds (x) ∂Ωb TijD (r, −κz , ω) uj (x, κz , ω) ds (x) , − (3.16) ∂Ωb x ∈ Ωb , x ∈ ∂Ωb which corresponds to the 2.5D integral domain representation theorem. This result has also been found by Sheng et al. [2005] and Lu et al. [2008b] using the 2.5D reciprocal theorem for the cases of isotropic elastic and poroelastic materials, respectively. 3.5 Green’s functions The dynamic fundamental solutions for the 2.5D elastodynamic problem have been presented in recent years by diﬀerent authors. Pedersen et al. [1994] and Sheng et al. [2005] derived the 2.5D Green’s functions for an unbounded medium considering an harmonic load moving along the propagation direction. These functions also recover the stationary case when the velocity of the moving load is set to zero and can be applied to both isotropic linear elastic and viscoelastic media. The 2.5D Green’s functions for a stationary line load in the full space have been derived by Li et al. [1992] and Tadeu and Kausel [2000] for linear viscoelastic isotropic media and by Tadeu et al. [2001] for an isotropic linear viscoelastic half space. In their work, Lu et al. [2008a] derived 65 3. WAVE PROPAGATION IN BOUNDED AND UNBOUNDED WAVEGUIDES: 2.5D BOUNDARY ELEMENT METHOD the Green’s function for the 2.5D problem involving stationary line loads applied on poroelastic media. The Green’s functions adopted here are those proposed by Tadeu and Kausel [2000] for a inﬁnite homogeneous medium by using the method of potentials. Considering the line load in Eq. (3.7) passing through the point x on the x − y plane and extending along the z-direction, the displacement components at a receiver point x on the x − y plane are given by [Kausel, 2006; Tadeu and Kausel, 2000] 1 2 =A − B1 + γ1 B2 r 1 D 2 2 U22 (r, κz , ω) = A κS H0β − B1 + γ2 B2 r D (r, κz , ω) = A κ2S H0β − κ2z B0 U33 D U11 (r, κz , ω) κ2S H0β (3.17) D D (r, κz , ω) = U21 = γ1 γ2 AB2 U12 D D U13 (r, κz , ω) = U31 = iκz γ1 AB1 D D U23 (r, κz , ω) = U32 = iκz γ2 AB1 where the various terms take the following expressions $ r= (x1 − x1 )2 + (x2 − x2 )2 1 A= 4iρω 2 ∂r xi − xi = γi = ∂xi r source-receiver distance in the x − y plane, (3.18) Bn = κnβ Hnβ − κnα Hnα composition of Hankel functions, (3.21) Hnα = Hn(2) (κα r) nth order Hankel function of the 2nd kind, (3.22) Hnβ = Hn(2) (κβ r) $ κα = ± κ2L − κ2z $ κβ = ± κ2S − κ2z ω κL = c̃L ω κS = c̃S nth order Hankel function of the 2nd kind, (3.23) amplitude, (3.19) direction cosines in the x − y plane, (3.20) radial longitudinal wavenumber, (3.24) radial shear wavenumber, (3.25) longitudinal bulk wavenumber, (3.26) shear bulk wavenumber. (3.27) The solution along the observer line at a coordinate z = 0 in the (κz , ω) domain is obtained by multiplying Eq. (3.17) for Eq. (3.1), while the corresponding solution in the (z, t) domain can be recovered by means of the inverse Fourier transforms in space and time. The presence of the double sign ± in Eqs. (3.24) and (3.25) has a precise 66 3.5. GREEN’S FUNCTIONS meaning: since the interest is focused on waves with amplitude decay in space due to both material attenuation and geometric spreading, it is required, coherently with the assumption in Eq. (3.1), that Im(κα ) 0 and Im(κβ ) 0. This requirement follows directly from the condition of zero amplitude at inﬁnite distance (r → ∞) from the origin, which must reﬂect the fact that no sources are located at inﬁnity. This condition is also known as the Sommerfeld radiation condition [Bonnet, 1999]. Given the harmonic behaviour in time (exp (iωt)) and space (exp (−iκz z)) the Han(2) (2) kel functions Hn (κα r) and Hn (κα r) physically represent wavefronts that propagate away from the origin. Thus, the negativeness of the imaginary part of the ra(2) dial wavenumbers ensures that the Hankel function of the second kind Hn (·) behave asymptotically as complex exponential (exp (−iκz z)), approaching zero as its argument approaches inﬁnity. However, this condition is not automatically ensured in a numerical implementation, but it must be guaranteed by an appropriate choice of the phase of the complex arguments κα r and κβ r, which depends on the nature of the wave. This aspect is discussed in Sec. 3.8.2. The second set of fundamental solutions, the tractions Green’s functions TijD (r, κz , ω), are obtained as D (r, κz , ω) nk (x) , TijD (r, κz , ω) = σijk i, j, k = 1, 2, 3 (3.28) being nk (x) the kth component of the external normal n (x) = [n1 , n2 ]T at x ∈ ∂Ωb , while D D (r, κz , ω) = λ̃ (ω) εD σijk ivol (r, κz , ω) δjk + 2μ̃ (ω) εijk (r, κz , ω) , i, j, k = 1, 2, 3 (3.29) is the third order tensor of fundamental stresses, i.e. the jkth stress component at x when the line load is acting at x in the ith direction. It is noted that the stressstrain relation in Eq. (3.29) is equivalent to Eq. (2.36) with the substitutions given in Eq. (2.122) and the replacement of the linearized second order strain tensor εij (x, κz , ω) with εD ijk (r, κz , ω). Applying the deﬁnition in Eq. (2.4), the third order tensor of fundamental linear D strains εD ijk (r, κz , ω) can be expressed in terms of fundamental displacements Uij (r, κz , ω) 67 3. WAVE PROPAGATION IN BOUNDED AND UNBOUNDED WAVEGUIDES: 2.5D BOUNDARY ELEMENT METHOD as follows [Kausel, 2006; Tadeu and Kausel, 2000] D (r, κ , ω) D (r, κ , ω) ∂U ∂U 1 z z ij ik + εD ijk (r, κz , ω) = 2 ∂xk ∂xj H0β H0β ∂ 3 B0 1 + δkl . i, j, k = 1, 2, 3 +A = kS2 A δjl 2 ∂xk ∂xj ∂xl ∂xj ∂xk (3.30) Substituting Eqs. (3.18)-(3.27) into Eq. (3.30) leads to the following fundamental strains for the harmonic line load acting in the x − y plane [Kausel, 2006; Tadeu and Kausel, 2000] 4 = γi A + + B2 − B3 r B2 2 D 2 2 B2 − κS κβ H1β δ1i + − γ1 B3 εi11 (r, κz , ω) = γi A r r 2 B2 D 2 2 B2 − κS κβ H1β δ2i + − γ2 B3 εi22 (r, κz , ω) = γi A r r εD ivol (r, κz , ω) −κ2S κβ H1β κ2z B1 2 εD i33 (r, κz , ω) = γi κz AB1 B2 1 2 D − κS κβ H1β (δ1i γ2 − δ2i γ1 ) − γ1 γ2 γi B3 εi12 (r, κz , ω) = A r 2 B1 1 2 D − κS H0β δ1i − γ1 γi B2 εi13 (r, κz , ω) = iκz A r 2 B1 1 2 D − κS H0β δ2i − γ2 γi B2 εi23 (r, κz , ω) = iκz A r 2 i = 1, 2 68 (3.31) 3.5. GREEN’S FUNCTIONS while for the line load acting in the z-direction (i = 3) one has εD 3vol (r, κz , ω) = iκz A −κ2S H0β + κ2z B0 B1 2 − γ1 B2 (r, κz , ω) = iκz A r B1 D 2 − γ2 B2 ε322 (r, κz , ω) = iκz A r 2 2 εD 333 (r, κz , ω) = iκz A −κS H0β + κz B0 2 + B1 − B2 r εD 311 (3.32) εD 312 (r, κz , ω) = −iκz γ1 γ2 AB2 1 2 D 2 ε313 (r, κz , ω) = γ1 A − κS κβ H1β + κz B1 2 1 2 D 2 ε323 (r, κz , ω) = γ2 A − κS κβ H1β + κz B1 . 2 Eqs. (3.31) and (3.32) can be ﬁnally substituted into Eq. (3.29) and then into Eq. (3.28) to evaluate the fundamental tractions. Alternatively, the following compact form has been presented by Castro and Tadeu [2012] D D D D D D (r, κz , ω) = 2μ̃ χUi1,1 + (χ − 1) Ui2,2 + Ui3,3 + Ui1,2 n1 + μ̃ Ui2,1 n2 Ti1 D D D D D D Ti2 (r, κz , ω) = 2μ̃ (χ − 1) Ui1,1 + Ui3,3 + χUi2,2 n2 + μ̃ Ui2,1 + Ui1,2 n1 D D D D D Ti3 (r, κz , ω) = μ̃ Ui1,3 + Ui3,1 + Ui3,2 n1 + μ̃ Ui2,3 n2 , i = 1, 2, 3 (3.33) where χ = c̃2L /2c̃2S . When the source point approaches the receiver point, Eqs. (3.17) and (3.33) become singular. In particular, the asymptotic expressions of the Green’s displacements and tractions for ω = 0, κz = 0 and r → 0 are 1 r→0 UijD (κz , ω) −−−−−−−−→ ln , r (ω=0, κz =0) 1 r→0 TijD (κz , ω) −−−−−−−−→ . (ω=0, κz =0) r (3.34) (3.35) Based on Eqs. (3.34) and (3.35), the ﬁrst integral on the right hand side of Eq. (3.16) becomes weakly singular when x → x, while the second integral has a strong singularity. This singularity needs a special treatment when the source points are taken on the boundary ∂Ωb , which is the problem addressed in Sec. 3.6. For ω = 0, r = 0 and κz = 0, the Green’s functions in Eqs. (3.17) and (3.33) recover those of the line load problem in plane-strain. Therefore, these expressions satisfy the 69 3. WAVE PROPAGATION IN BOUNDED AND UNBOUNDED WAVEGUIDES: 2.5D BOUNDARY ELEMENT METHOD plane-strain model as special case [Tadeu and Kausel, 2000]. 3.6 Regularized 2.5D boundary integral equation 3.6.1 Limiting process The integral representation theorem in Eq. (3.16) allows to calculate the displacements ui (x , κz , ω) at any x ∈ Ωb once the displacements uj (x, κz , ω) and tractions tj (x, κz , ω) are known at any x ∈ ∂Ωb . However, due to the possible unique assignment of either uj (x, κz , ω) (Dirichlet boundary conditions) or tj (x, κz , ω) (Neumann boundary conditions) at x, the boundary conditions are half-determined. The remaining boundary variables are computed by extending Eq. (3.16) to ∂Ωb , which is usually accomplished by performing a limiting process x ∈ Ωb → x ∈ ∂Ωb . As stated in Sec. 3.5, this operation necessarily introduces singularities into the boundary integrals when the source point x approaches the receiver point x, since UijD (r, κz , ω) ∼ ln (1/r) and TijD (r, κz , ω) ∼ 1/r for r → 0. The ﬁrst singularity is a weak singularity, and therefore the integrals involving the fundamental displacements converge in the ordinary sense. The second singularity can be studied by introducing a circular neighborhood Ωε (x ) of x as shown in Fig. 3.2, and subdividing the integral involving the fundamental tractions as follows ∂Ωb TijD (r, −κz , ω)uj (x, κz , ω) ds (x) TijD (r, −κz , ω) uj (x, κz , ω) ds (x) = ∂Ω −e (x ) b ε TijD (r, −κz , ω) uj (x, κz , ω) ds (x) , + (3.36) ∂Ωε (x ) where eε (x ) = ∂Ωb ∩ Ωε (x ) and ∂Ωε (x ) = Ωb ∩ ∂Ωε (x ). Taking the limit ε → 0 in the representation formula Eq. (3.16), leads to [Bonnet, 1999] cij x uj x , κz , ω = UijD (r, −κz , ω) tj (x, κz , ω) ds (x) TijD (r, −κz , ω) uj (x, κz , ω) ds (x) , − C.P.V. ∂Ωb (3.37) ∂Ωb where (·) ds (x) = − C.P.V. ∂Ωb (·) ds (x) = lim →0 ∂Ω −eε (x ) b ∂Ωb 70 (·) ds (x) (3.38) 3.6.2. REGULARIZATION PROCEDURE Figure 3.2: Exclusion neighborhood used for the limiting process ε → 0. is the Cauchy Principal Value of the integral over ∂Ωb . From the limiting process, the free term cij (x ) is also determined, which assumes the following expression cij x = lim →0 ∂Ωε (x ) TijD (r, −κz , ω) ds (x) ∈ ∂Ωb . (3.39) For x ∈ ∂Ωb and i = j, the free term represents the ratio between the angle subtended by ∂Ωε and the angle of a complete circle when ε → 0 [Bonnet, 1999; Brebbia and Dominguez, 1989; Zimmerman and Stern, 1993]. In the special case for which the boundary ∂Ωb is smooth at x , Eq. (3.39) simpliﬁes to cij (x ) = 1/2δij . 3.6.2 Regularization procedure The analytical treatment of Cauchy Principal Value integrals in Eqs. (3.37) and (3.39) may result diﬃcult due to (i) the analytical treatment of the strong singularity in the fundamental tractions, (ii) the presence of boundary corners, where the external normal is not uniquely deﬁned and (iii) the shape functions used to interpolate the unknown boundary displacements. Lately, researchers have proposed special integration methods to account for singularities in the integral kernels [Sheng et al., 2005] or to simplify the treatment of the corners at the discretization level by using discontinuous boundary elements [Gunawan and Hirose, 2005]. To overcome analytical and implementation diﬃculties, in this study the singular integrals and free terms are evaluated making use of the so-called rigid body motion technique [Banerjee, 1981; Brebbia and Dominguez, 1989; Dominguez, 1993]. This technique has been extended to the 2.5D case by [Lu et al., 2008b] for wave propagation problems involving poroelastic materials, and by François et al. [2010] for sound 71 3. WAVE PROPAGATION IN BOUNDED AND UNBOUNDED WAVEGUIDES: 2.5D BOUNDARY ELEMENT METHOD radiation problems involving translation invariant structures embedded in elastic and viscoelastic layered media. The basic idea is to identify a second boundary value problem which uses simple fundamental solutions with the same asymptotic behaviour of the original boundary value problem when r → 0. Then, if the solution of the second problem is adjusted so that the singular coeﬃcients cij (x ) have the same value at x = x , the equations of the two problems can be subtracted and the singularity removed. For the dynamic problem under consideration, this can be accomplished by (i) choosing the domain as complementary to the original domain to R2 , i.e. of the auxiliary problem Ωaux b = R2 − Ωb and ∂Ωaux = ∂Ωb , and (ii) applying a rigid body displacement u0 on Ωaux b b ∂Ωaux b . Since the auxiliary volume V aux = R3 − V is translational invariant, a constant can be understood as a time and spatial harmonic disdisplacement applied on Ωaux b placement with inﬁnite wavelength in z-direction (κz = 0), and represents therefore a constant solution for the entire three dimensional domain. In this case, the original dynamic problem reduces to a combination of a static plane-strain line load problem p (x ) = δ (x − x ) with fundamental solutions 1 1 (3 − 4Re (μ̃)) ln δij + γi γj (r) = 8πRe (μ̃) (1 − Re (ν̃)) r 1 {γk nk [(2Re (μ̃) δij + 2γi γj ) TijS (r) = − 4π (1 − Re (μ̃)) r UijS − (1 − 2Re (μ̃)) (γi nj − γj ni )]} , (3.40) i, j, k = 1, 2 and an anti-plane line load problem p (z ) = δ (z − z ) with fundamental solutions 1 1 ln 2πRe (μ̃) r 1 S γk nk . k = 1, 2 (r) = − T33 2πr S (r) = U33 (3.41) The asymptotic expressions of the fundamental displacements and tractions in Eqs. (3.40) and (3.41) when the source point approaches the receiver point are 1 S r→0 −−−→ ln , i, j = 1, 2 UijS , U33 r S S r→0 1 Tij , T33 −−−→ , i, j = 1, 2 r (3.42) (3.43) which correspond to those in Eq. (3.34) and Eq. (3.35) for the 2.5D elastodynamic 72 3.6.2. REGULARIZATION PROCEDURE Figure 3.3: Auxiliary domain for a bounded waveguide. problem. The rigid body displacement u0 (x, κz = 0, ω = 0) = u0 (x) is now considered of an auxiliary domain Ωaux corresponding to an to be applied at the boundary ∂Ωaux b b (see Fig. 3.3). Observing inﬁnite space bounded by ∂Ω∞ at inﬁnite distance from ∂Ωaux b that for a rigid body motion tj (x, κz = 0, ω = 0) = 0 and taking into account the equivalences UijD (r, κz = 0, ω = 0) = U S (r) and TijD (r, κz = 0, ω = 0) = T S (r), the application of Eq. (3.37) for the auxiliary domain results in the following expression caux ij 0 0 x u x = −u (x) − TijS TijS (r) ds (x) + ∂Ωb (r) ds (x) , (3.44) ∂Ω∞ where the boundary integral involving the fundamental displacements is vanished while the boundary integral involving the fundamental tractions has been extended in order to include ∂Ω∞ . Since the rigid displacement is arbitrary, Eq. (3.44) still holds for the particular choice u0 (x) = uj (x , κz , ω), leading to caux ij x uj x , κz , ω = − uj x , κz , ω − TijS (r) ds (x) ∂Ωb S Tij (r) ds (x) . + (3.45) ∂Ω∞ Considering now that the unitary static force p (x , z ) = δ (x − x ) δ (z − z ) at the must form an equilibrated system with the reaction forces source point x ∈ ∂Ωaux b distributed along ∂Ω∞ , it follows from equilibrium considerations that the integral of the fundamental tractions over ∂Ω∞ results in an opposite force of the same unitary magnitude [Andersen, 2006; Brebbia and Dominguez, 1989]. The second integral in the 73 3. WAVE PROPAGATION IN BOUNDED AND UNBOUNDED WAVEGUIDES: 2.5D BOUNDARY ELEMENT METHOD right hand side of Eq. (3.45) becomes TijS (r) ds (x) = −δij (3.46) ∂Ω∞ Substituting Eq. (3.46) into Eq. (3.45) gives caux ij x uj x , κz , ω = −uj x , κz , ω − TijS (r) ds (x) − δij . (3.47) ∂Ωb Taking into account the opposite sign of the outward normals between the original and the auxiliary problems at x, i.e. n (x)|∂Ωb = −n (x)|∂Ωaux , the signs inside the square b brackets in Eq. (3.47) can be reversed, leading to caux ij x − δij uj x , κz , ω = uj x , κz , ω − TijS (r) ds (x) . (3.48) ∂Ωb If Ωb correspond to an unbounded domain, the second integral in the right-hand side = R2 − Ωb is bounded. In this case Eq. (3.49) reduces of Eq. (3.44) vanishes since Ωaux b , κ , ω = u , κ , ω − x u x x caux j z j z ij to TijS (r) ds (x) . (3.49) ∂Ωb Generalizing the results in Eqs. (3.48) and (3.49) as proposed by Lu et al. [2008b] results in the following expression aux cij x − c∞ δij uj x , κz , ω = uj x , κz , ω − TijS (r) ds (x) , (3.50) ∂Ωb where c∞ is a coeﬃcient equal to 1 if Ωb is bounded and 0 if Ωb is unbounded. Eq. (3.50) can be added to Eq. (3.37) without altering the original problem, since the rigid body motion does not involve physical tractions on the auxiliary domain.. As result, the following relation is obtained x − c∞ δij uj x , κz , ω = cij x + caux ij UijD (r, −κz , ω) tj (x, κz , ω) ds (x) + = ∂Ωb D S Tij (r, −κz , ω) uj (x, κz , ω) − Tij (r) uj x , κz , ω ds (x) . − (3.51) ∂Ωb Recalling Eq. (3.39), the sum of the free terms cij (x ) + caux ij (x ) can be expressed as 74 3.6.2. REGULARIZATION PROCEDURE follows [Brebbia and Dominguez, 1989; Zimmerman and Stern, 1993] x = lim cij x + caux ij →0 + TijD (r, −κz , ω) ds (x) ∈ ∂Ωb D Tij (r, −κz , ω) ds (x) ∈ ∂Ωb ∂Ωε (x ) ∂Ωaux ε (x ) (3.52) = δij , where ∂Ωaux ε (x ) is deﬁned as in Fig. 3.3. Substitution of the fundamental property in Eq. (3.52) into Eq. (3.51) cancels out the free term, and the regularized 2.5D boundary integral equation becomes UijD (r, −κz , ω) tj (x, κz , ω) ds (x) + (1 − c∞ ) ui x , κz , ω = ∂Ωb TijD (r, −κz , ω) uj (x, κz , ω) − TijS (r) uj x , κz , ω ds (x) , − (3.53) ∂Ωb x, x ∈ ∂Ωb in which the ﬁrst integral on the right hand side contains a weak, integrable, singularity of order ln (1/r), while the strong singularity of order 1/r in the second boundary integral has been removed. The ﬁrst and second integrals in Eq. (3.53) can therefore be evaluated numerically in a boundary element discretization scheme by using the Gauss-Laguerre and the Gauss-Legendre quadrature formulae, respectively [Stroud and Secrest, 1996]. However, as pointed out by Lu et al. [2008b], due to the presence of the Hankel function the Gauss-Laguerre quadrature formula for the integral involving UijD (r, −κz , ω) may result inaccurate. Since the order of the singularities in both fundamental dynamic and static displacements is the same, Eq. (3.53) can be further modiﬁed using the addition-subtraction technique proposed by Lu et al. [2008b], leading to D S Uij (r, −κz , ω) − Uij (r) tj (x, κz , ω) ds (x) (1 − c∞ ) ui x , κz , ω = ∂Ωb UijS (r) tj (x, κz , ω) ds (x) + ∂Ω b D S Tij (r, −κz , ω) uj (x, κz , ω) − Tij (r) uj x , κz , ω ds (x) , − (3.54) ∂Ωb x, x ∈ ∂Ωb where the ﬁrst and third integral can be evaluated using the Gauss-Legendre quadrature 75 3. WAVE PROPAGATION IN BOUNDED AND UNBOUNDED WAVEGUIDES: 2.5D BOUNDARY ELEMENT METHOD formula while the second integral can now be evaluated analytically or via the GaussLaguerre quadrature formula. It is remarked that Eq. (3.54) holds only for source points belonging to the boundary ∂Ωb . Once the solution is found in terms of displacements and tractions at any x ∈ ∂Ωb , the complete displacement waveﬁeld inside the domain Ωb is readily obtained from Eq. (3.16) by posing cij (x ) = δij . The remaining quantities, i.e. strains and stresses, can be recovered from Eq. (3.16) via compatibility and constitutive relations. 3.7 Boundary discretization using semi-isoparametric boundary elements The boundary ∂Ωb is subdivided into Nb mono-dimensional boundary elements. The generic qth element of domain ∂Ωqb is assumed to be isoparametric in the plane of the domain Ωb , i.e. the in-plane components of both uj and tj are interpolated with the same polynomial functions used to interpolate the cross-section geometry of the boundary. Assuming an in-plane linear mapping from the two-nodes reference element identiq ﬁed by ∂Ωref b to the corresponding two-nodes generic element ∂Ωb as shown in Fig. 3.4, the semi-isoparametric representation of both displacements and tractions at the generic boundary point x ∈ ∂Ωqb results in an uncoupled description of the out-of-plane from the in-plane motion of the form u (η, κz , ω) = N (η) qq (κz , ω) t (η, κz , ω) = N (η) hq (κz , ω) % at x (η) = N (η) xq ∈ ∂Ωqb (3.55) where N (η) is the 3 × 6 matrix containing the linear shape functions in the natural q q q coordinate η ∈ ∂Ωref b , while x , q (κz , ω) and h (κz , ω) are the 6 × 1 vectors of nodal coordinates, displacements and tractions, respectively. The discretized global system of algebraic equations is constructed from Eq. (3.54) by applying a point collocation scheme where the collocation points x are assumed to be coincident with the nodes of the boundary element mesh [Brebbia and Dominguez, 1989]. Denoting by xc the cth collocation node of the boundary element mesh and using Eqs. (3.55), the recursive collocation procedure over the total number of nodes Nn = Nb 76 3.7. BOUNDARY DISCRETIZATION USING SEMI-ISOPARAMETRIC BOUNDARY ELEMENTS Figure 3.4: Semi-isoparametric discretization using mono-dimensional elements with linear shape functions. allows to rewrite Eq. (3.54) in the following discrete form Nn (1 − c∞ ) uc (κz , ω) c=1 = Nn & c=1 − Nb Uq1 (rc (η) , κz , ω) − Uq2 (rc (η)) q=1 Nb Tq1 (rc (η) , κz , ω) qq (κz , ω) q=1 q / (xc ∈∂Ω b) − Nb hq (κz , ω) (3.56) Tq2 (rc (η) , κz , ω) qq (κz , ω) q=1 (xc ∈∂Ωqb ) + Nb % Tq3 (rc (η)) uc (κz , ω) q=1 q / (xc ∈∂Ω b) where rc (η) = |x (η) − xc | denotes the in-plane distance between the collocation point and the integration point, uc (κz , ω) is the 3 × 1 vector of boundary displacements at stands for the assembling operation over the nodes (subscript c) and point xc and 77 3. WAVE PROPAGATION IN BOUNDED AND UNBOUNDED WAVEGUIDES: 2.5D BOUNDARY ELEMENT METHOD the elements (subscript q) of the mesh. The operators in Eq. (3.56) are deﬁned as Ue1 (rc (η) , κz , ω) = Uq2 (rc (η)) = Tq1 (rc (η) , κz , ω) = Tq2 (rc (η) , κz , ω) = Tq3 (rc (η)) = ∂Ωqb ∂Ωqb ∂Ωqb ∂Ωqb ∂Ωqb UD (rc (η) , −κz , ω) − US (rc (η)) N (η) Jbq (η) dη (3.57) US (rc (η)) N (η) Jbq (η) dη (3.58) TD (rc (η) , −κz , ω) N (η) Jbq (η) dη (3.59) TD (rc (η) , −κz , ω) N (η) − TS (rc (η)) L Jbq (η) dη TS (rc (η)) Jbq (η) dη (3.60) (3.61) where Jbq (η) = |∂x (η) /∂η| is the Jacobian of the semi-isoparametric transformation. The 3 × 3 displacement and traction Green’s tensors in Eq. (3.57)-(3.61) are deﬁned as ⎡ D UD UD U11 12 13 ⎡ ⎤ ⎢ D D UD ⎥ UD = ⎣ U21 U22 23 ⎦ D D D U31 U32 U33 ⎡ S ⎤ S U11 U12 0 ⎢ S ⎥ S US = ⎣ U21 U22 0 ⎦ 0 0 D TD TD T11 12 13 ⎤ ⎢ D D TD ⎥ TD = ⎣ T21 T22 23 ⎦ D D D T31 T32 T33 ⎡ S ⎤ S T11 T12 0 ⎢ S ⎥ S TS = ⎣ T21 T22 0 ⎦. S U33 0 0 (3.62) (3.63) S T33 where the entries of the 2 × 2 blocks in the static Green’s tensors (identiﬁed by the ﬁrst and second rows and columns) correspond to the fundamental solutions of the two-dimensional plane-strain line load problem given in Eq. (3.40) while the remaining nonzero terms correspond to the fundamental solutions for the case of the elastic antiplane line load given in Eq. (3.41). The 3 × 6 operator L is introduced to collocate the static Green’s tractions tensor TS (rc (η)) on the 3 × 3 diagonal blocks of the global system, which contain singular terms. Such operator varies according to the position of the collocation node xc inside the element and is denoted by & L = [ β1 I3×3 , β2 I3×3 ] , with β1 = 1, β2 = 0 if xc ≡ xq1 β1 = 0, β2 = 1 if xc ≡ xq2 , (3.64) where xq1 and xq2 are, respectively, the coordinate vectors for the ﬁrst and second node 78 3.7. BOUNDARY DISCRETIZATION USING SEMI-ISOPARAMETRIC BOUNDARY ELEMENTS of the element including the collocation node xc (x = [xq1 , xq2 ]T ). From Eq. (3.56), by grouping the local displacement and traction operators into the global inﬂuence operators according to the mesh topology Ub (κz , ω) = Nb ' Nn 2 Uqi (rc , κz , ω) , (3.65) Tqi (rc , κz , ω) , (3.66) c=1 q=1 i=1 Tb (κz , ω) = Nb ' Nn 2 c=1 q=1 i=3 and by assembling the local displacement and tractions vectors into the global vectors Qb (κz , ω) = Nb qq (κz , ω) , (3.67) hq (κz , ω) , (3.68) q=1 Hb (κz , ω) = Nb q=1 the following set of linear algebraic equations is obtained [Tb (κz , ω) + (1 − c∞ ) I] Qb (κz , ω) = Ub (κz , ω) Hb (κz , ω) (3.69) where I denotes the identity matrix. If no tractions discontinuities exist at a generic node, the operators in Eq. (3.69) have dimension N × N , with N = Nn × 3 denoting the total number of displacement variables, while the global vectors Qb (κz , ω) and Hb (κz , ω) have dimension N × 1. Once Eq. (3.69) is formed, the boundary conditions must be imposed in terms of displacements and tractions on the discretized boundary nodes. Since only one between the Dirichlet uj (κz , ω) and Neumann tj (κz , ω) boundary conditions can be imposed at any xc , the eﬀective system takes generally the form of a mixed linear system in which the rows of the matrix operators corresponding to the unknown variables are rearranged to form the matrix of coeﬃcients, while the rows of the matrix operators corresponding to the assigned boundary conditions are selected to form the vector of constant terms. After the above system has been solved for all the unknown boundary variables, the solution over ∂Ωb is fully determined and the waveﬁeld at a generic point x ∈ Ωb can be computed by using Eq. (3.16), with cij (x ) = δij . Since the integrals in Eq. (3.16) are not singular, the standard Gauss-Legendre quadrature formula can be used for their 79 3. WAVE PROPAGATION IN BOUNDED AND UNBOUNDED WAVEGUIDES: 2.5D BOUNDARY ELEMENT METHOD evaluation. The discretized representation of Eq. (3.16) reads ud x , κz , ω = Ud (κz , ω) Hb (κz , ω) − Td (κz , ω) Qb (κz , ω) (3.70) where Ud (κz , ω) = Nb q=1 Td (κz , ω) = ∂Ωqb Nb q=1 ∂Ωqb UD (r (η) , −κz , ω) N (η) Jbq (η) dη (3.71) TD (r (η) , −κz , ω) N (η) Jbq (η) dη, (3.72) in which r = |x (η)−x |, while Qb (κz , ω) and Hb (κz , ω) are the vectors of displacements and tractions for the boundary nodes obtained as solution of Eq. (3.69). It is noted that the linear system in Eq. (3.70) is valid only for x ∈ Ωb and the operators Ud (κz , ω) and Td (κz , ω) have dimension 3 × N . 3.8 Nonlinear eigenvalue problem The dispersion characteristics for each normal mode are determined from the wave equation of the external traction-free problem. Thus, by imposing homogeneous Neumann boundary conditions on the system Eq. (3.69), i.e. Hb (κz , ω) = 0, the following eigenvalue problem is obtained. Z (κz , ω) Qb (κz , ω) = 0, (3.73) Z (κz , ω) = U−1 b (κz , ω) [Tb (κz , ω) + (1 − c∞ ) I] . (3.74) where corresponds to the dynamic stiﬀness matrix of the bounded (c∞ = 1) or unbounded (c∞ = 0) waveguide. For any ﬁxed positive real frequency ω, the nonlinear eigenvalue problem Eq. (3.73) can be solved in the complex wavenumbers κz (ω) by using algorithms of the contour integral family [Amako et al., 2008; Asakura et al., 2009; Beyn, 2012]. These algorithms can extract the roots of the nonlinear problem Eq. (3.73) without the need of an initial guess for the eigensolutions, which is a limiting property of more classical algorithms such those of the Newton-Raphson family. To solve the eigenvalue problem Eq. (3.73), the contour integral method proposed by Beyn [2012] is adopted. The method is able to compute all the eigenvalues and 80 3.8.1. CONTOUR INTEGRAL METHOD associated eigenvectors for an holomorphic eigenvalue problem that lies within a given close contour in the complex plane, including eigenvalues with multiplicity higher than one. The algorithm proposed by Beyn [2012], in the form where the sum of the algebraic multiplicities of the eigenvalues does not exceed the system dimension, is recalled in Sec. 3.8.1. The singular and multivalued character of the operator Z (κz , ω) are also discussed. Such properties are of fundamental importance in the deﬁnition of the region in the complex κz -plane where the roots of the fundamental modes must be sought. 3.8.1 Contour integral method The algorithm is initialized by computing the two moment matrices ( 1 Z−1 (κz , ω) Vdκz ∈ CN,L A0 = 2πi Γ(κz ) ( 1 κz Z−1 (κz , ω) Vdκz ∈ CN,L A1 = 2πi Γ(κz ) (3.75) (3.76) over the simple closed curve Γ (κz ) (Jordan curve) deﬁned in the complex κz -plane. In Eqs. (3.75) and (3.76), V ∈ CN,L is chosen randomly. The positive integer L is chosen to satisfy the requirement K L N , denoting K the supposed number of eigenvalues inside the contour. The integrals Eqs. (3.75) and (3.76) are evaluated numerically by discretizing the complex contour into Np integration points and applying the trapezoidal rule. Once the two moment matrices in Eqs. (3.75) and (3.76) are formed, a Singular Value Decomposition (SVD) A0 = VΣWH (3.77) is computed, where V is a N × N complex unitary matrix, Σ is N × L is a diagonal matrix with non-negative entries along the diagonal and WH is a L×L complex unitary matrix. Since small singular values σl (l = 1, ..., L) of the diagonal matrix Σ determine a bad conditioning of the eigenvalues computation, a rank test is then performed and only the ﬁrst M singular values higher than a ﬁxed tolerance tolrank are retained. The remaining singular values are eliminated from Σ along with their corresponding columns in V and W. After the rank test has been performed, the following operator is constructed ∈ CM,M B = V0H A1 W0 Σ−1 0 81 (3.78) 3. WAVE PROPAGATION IN BOUNDED AND UNBOUNDED WAVEGUIDES: 2.5D BOUNDARY ELEMENT METHOD where Σ0 = diag (σ1 , ..., σM ) , (3.79) V0 = V (1 : N, 1 : M ) , (3.80) W0 = W (1 : L, 1 : M ) . (3.81) The operator B in Eq. (3.78) is diagonalizable and has as eigenvalues the eigenvalues of Z (κz , ω) inside Γ (κz ) [Beyn, 2012]. Solving a standard linear eigenvalue problem m for B leads to a set of eigenvalues κm z (ω) and corresponding eigenvectors y (ω) (m = 1, 2, .., M ) where, due to the choice of tolrank and Np , generally results K M L. Of the remaining M − K spurious eigensolutions, those lying outside the contour are directly discarded, while the remaining are ﬁltered out by establish ﬁrst a suitable threshold value tolres that is used next to perform the following relative residual test m Z (κm z (ω) , ω) Qb (ω)∞ ) tolres ) ) ) m Z (κm z (ω) , ω)∞ Qb (ω) ∞ (3.82) m m where Qm b (ω) = V0 y (ω) is the approximative eigenvector associated to κz (ω) and ·∞ denotes the inﬁnity norm. Since the matrix B also retains the complete multiplicity structure of eigenvalues inside the contour Γ (κz ), some eigenvalues may result ill-conditioned with the corresponding eigenvectors. In this case, a Schur decomposition BQ = QR (3.83) is performed, with R block-diagonalized such that the diagonal blocks belong to different eigenvalues. Then, the eigenvectors ym (ω) are selected from the ﬁrst column of each mth diagonal-block in R to compute the associated true eigenvector Qm b (ω). m The eigenpairs [κm z (ω) , Qb (ω)] that satisfy the inequality Eq. (3.82) are then ac- cepted as ﬁnal solution. 3.8.2 Deﬁnition of the integral path and permissible Riemann sheets The procedure proposed by Beyn [2012] and reported in Sec. 3.8.1 allows to extract all the eigenvalues for the holomorphic problem Z (κz , ω) ∈ H Ω∗ , CK,K , where Ω∗ denotes the region of the complex κz -plane enclosed by Γ (κz ). From the inspection of 82 3.8.2. DEFINITION OF THE INTEGRAL PATH AND PERMISSIBLE RIEMANN SHEETS Eqs. (3.22)-(3.27) it appears that, due to the presence of the radial wavenumbers κα = ± $ κ2L − κβ = ± κ2z , $ κ2S − κ2z . (3.84) in the arguments of the Hankel functions, the operator Z (κz , ω): results to be singular at points κz = ±κL and κz = ±κS , since the Hankel function is not deﬁned as its argument κα,β r becomes zero; is multivalued due to the signs ± of κα and κβ for any ﬁxed couple (κz , ω). To fulﬁll the holomorphicity requirement inside Ω∗ , the operator Z (κz , ω) must be made single-valued and the points of the complex plane corresponding to singularities and discontinuities must be excluded. Firstly, it is recalled that, in order to have a wave that is attenuated in the direction of propagation, the imaginary component of its wavenumber must be negative in accordance with the position in Eq. (3.1). Then, the single valued deﬁnition of Z (κz , ω) follows directly from the the imposition of the Sommerfeld radiation condition Im(κα ) < 0, Im(κβ ) < 0, ∀κz ∈ C (Re (κz ) > 0, Im (κz ) ≤ 0) (3.85) (2) which ensures that Hn (κα,β r) → 0 for r → ∞, i.e. the amplitude of the outgoing radial waves becomes zero at inﬁnite radial distance from the origin. Imposition of the Sommerfeld radiation condition determines the correct choice of the permissible Riemann sheets, i.e. the portions of the Riemann surface on which the physical solutions are located [He and Hu, 2009, 2010; van Dalen et al., 2010; Zhang et al., 2009]. On the Riemann surface, the operator Z (κz , ω) is analytic and single-valued everywhere, except in correspondence of the poles of guided modes. Since the Riemann sheets are deﬁned on the κz -plane by the possible combinations of signs (±, ±) for (κα , κβ ), the whole Riemann surface for the isotropic case is composed of four sheets [Ewing et al., 1957]. However, the poles of the function Z (κz , ω) corresponding to the physical solutions must be searched only on the permissible sheets, which are selected according to the requirements in Eq. (3.85). The choice of the signs for each region of the κz -plane enclosed by the contour is shown in Fig. 3.5. It can be noted that these assumptions result into a discontinuity of the operator Z (κz , ω) along two hyperbolic trajectories that depart from κL and κS and extend to inﬁnity along the negative imaginary axis. These hyperbolas can be determined by letting vanish the imaginary component of the radial wavenumbers. 83 3. WAVE PROPAGATION IN BOUNDED AND UNBOUNDED WAVEGUIDES: 2.5D BOUNDARY ELEMENT METHOD Figure 3.5: Complex κz -plane with branch points, branch cuts and integration path Γ (κz ). The notation (·, ·) stands for the choice of the signs in Eq. (3.84) for κα and κβ respectively. From Eq. (3.84), by posing Im (κα ) = 0 and Im (κβ ) = 0, one obtains Im (κz ) = Re (κL ) Im (κL ) , Re (κz ) Im (κz ) = Re (κS ) Im (κS ) , Re (κz ) (3.86) which delimitate zone of the complex plane in which the phase of the radial wavenumbers satisfying at the Sommerfeld radiation condition Eq. (3.85) has a shift in phase of π radians. In fact, for any ﬁxed Re (κz ) ∈ [0, Re (κL,S )] and Im (κz ) ∈ [−∞, +∞], the shift in phase is not a continuous function if the conditions in Eq. (3.85) are imposed, but shows a jump when it crosses the hyperbolic trajectories in Eq. (3.86). Therefore, along these line the stiﬀness operator Z (κz , ω) does not satisfy the requirement of holomorphicity. The strategy to remove these discontinuities is to perform two cuts and closing the contour around the branches as shown in Fig. 3.5 (note that the total number of cuts is four since there are two other symmetric branches in the second quadrant of the complex plane, with branch points −κL and −κS ). These cuts are generally indicated as Sommerfeld (or fundamental) branch cuts [Ewing et al., 1957; van Dalen et al., 2010; Zhang et al., 2009] and vary in the κz -plane with κα (ω) and κβ (ω), which move along 84 3.9. DISPERSION CHARACTERISTICS EXTRACTION the lines a and b, respectively, while the frequency ω increases or decreases. Along the generic qth branch cut (q = α, β), Re(κq ) > 0 on the left side and Re(κq ) < 0 on the right side, while Im(κq ) < 0 on both sides. The two signs in each area in Fig. 3.5 represent the chosen Riemann sheet and correspond to the sign of the real part of the two radial wavenumbers (κα , κβ ) necessary to satisfy the condition in Eq. (3.85). It can be noted that the second Riemann sheet, denoted by (+, −), is excluded from the search space, since it does not satisfy the Sommerfeld radiation condition. If an elastic medium is considered, the bulk wavenumbers become real quantities, so that a and b rotate around the origin to overlap the real axis. In this case, the two branch cuts collapse on the negative imaginary axis and the portion of the real axis between the origin and the two corresponding bulk wavenumbers, becoming in fact a single branch cut. Since for the elastic case the poles of the normal modes lie on the real axis, it should be noted that the roots included in the range 0 κz κS are excluded from the contour region. However, due to the conditions in Eq. (3.85), the elastic case can only be treated by adding a small value of material attenuation (numerical attenuation) for both the bulk waves, so that the same considerations for the viscoelastic case can be applied. As ﬁnal remark, it is noted that the only other branch cut in the complex plane is represented by the negative real axis, which is a branch cut of the Hankel function as it presents a discontinuity along this axis. However, if only wavenumbers with strictly positive real part (right-propagating waves) are considered, this branch cut is unnecessary and can be directly avoided by assuming the integration path as in Fig. 3.5. 3.9 Dispersion characteristics extraction m Once the complete set of eigensolutions [κm z (ω) , Qb (ω)] has been determined from Eq. (3.73) for the frequency of interest, the dispersion characteristics cm p (ω) = ω Re(κm z (ω)) αm (ω) = −Im(κm z (ω)) m Ωb J (ω) · n3 dxdy (ω) = cm e m (ω) + W m (ω)dxdy Ωb K phase velocity, (3.87) attenuation, (3.88) energy velocity, (3.89) for the mth propagating or evanescent normal mode can be extracted. It can be noted that the phase velocity and attenuation in Eqs. (3.87) and (3.88) are directly 85 3. WAVE PROPAGATION IN BOUNDED AND UNBOUNDED WAVEGUIDES: 2.5D BOUNDARY ELEMENT METHOD Figure 3.6: Subdivision of the domain Ωb by means of integration cells. derived from the obtained set of eigensolutions, while the energy velocity Eq. (3.89) is computed in post-processing. The various terms that appear in Eq. (3.89) are the acoustic Poynting vector Jm (ω), the kinetic energy K energy W m (ω), m (ω) and the pseudo-potential which includes both the energy stored and dissipated via internal damping mechanisms. Their deﬁnitions have been given in Sec. 2.6.2.5, while their expressions are m (ω) um Jim (ω) = −iωσji j (ω) (3.90) ω2 m ρu (ω) conj (um i (ω)) 2 i 1 m (ω) conj εm W m (ω) = σij ij (ω) . 2 K m (ω) = (3.91) (3.92) Since it is generally diﬃcult to obtain a boundary integral representation of these quantities, the integrals in Eq. (3.89) can be evaluated by partitioning the domain Ωb into an arbitrary number Ncells of integration cells as shown in Fig. 3.6. The shape of a single cell is deﬁned as the mapping of a parent cell which is geometrically suitable to support a Gaussian quadrature scheme. Assuming an internal quadrature rule as shown in Fig. 3.1, the displacement ﬁeld can be obtained at any quadrature point xp ∈ Ωb using Eqs. (3.70) and (3.72), which yield the following 3 × N linear system ⎡ ⎣ um p (ω) = − ⎤ Nb q=1 ∂Ωqb q ⎦ m TD (rp (η) , −κm z (ω) , ω) N (η) Jb (η) dη Qb (ω) , (3.93) where rp (η) = |x (η) − xp |. It can be noted that in Eq. (3.93) the term involving the fundamental displacements has dropped since Hm b (κz , ω) = 0. 86 3.9. DISPERSION CHARACTERISTICS EXTRACTION The derived ﬁeld variables at point xp are denoted with the 6 × 1 vectors collecting the independent components of the Cauchy stress tensor expressed in Voigt notation, m m m m m m T sm p (ω) = [σ11 , σ22 σ33 , σ23 , σ13 , σ12 ] , and the corresponding symmetric linear strain m m m m m m T components, m p (ω) = [ε11 , ε22 ε33 , ε23 , ε13 , ε12 ] (cf. Sec. 2.3). The compatibility and constitutive relations can be rearranged in the general compact form m m m p (ω) = B (ω) up (ω) , m sm p (ω) = C̃p (ω) , (3.94) where the 6 × 3 compatibility operator Bm (ω) and the 6 × 6 fourth order tensor of viscoelastic moduli C̃ijkm are expressed by ∂ ∂ m + Ly − iκz (ω) Lz , B (ω) = Lx ∂x ∂y m (3.95) C̃ijkm = λ̃δij δkm + μ̃ (δik δjm + δim δjk ) , in which the Li operators are deﬁned as in Eq. (2.9). Substituting Eq. (3.93) into Eq. (3.94), the following 6 × N linear systems are obtained for the strain and stress vectors at xp ⎡ ⎣ m p (ω) = − ⎡ ⎣ sm p (ω) = − Nb q=1 ∂Ωqb Nb q q=1 ∂Ωb ⎤ q ⎦ m B m (ω) TD (rp (η) , −κm z (ω) , ω) N (η) Jb (η) dη Qb (ω) (3.96) ⎤ q ⎦ m C̃B m (ω) TD (rp (η) , −κm z (ω) , ω) N (η) Jb (η) dη Qb (ω) (3.97) where the compatibility operator B m (ω) applies only on the fundamental solutions TD (rp (η) , −κm z (ω) , ω), since the derivative is intended as a variation around the point xp [Dominguez, 1993]. The diﬀerent operators in Eqs. (3.90)-(3.92) are integrated over the cross-section by considering the contribute of each cell. The time-averaged Poynting vector polarized in the z-direction, the pseudo-potential energy and the kinetic energy for the waveguide cross-section are obtained as follows ⎧ ⎫ NGp N' ⎨ ⎬ cells ' H T m ω Jm (ω) · n3 dxdy = Im Jcs (ξp ) wp um (ω) L s (ω) p z p ⎩ ⎭ 2 Ωb s=1 p=1 87 (3.98) 3. WAVE PROPAGATION IN BOUNDED AND UNBOUNDED WAVEGUIDES: 2.5D BOUNDARY ELEMENT METHOD K m (ω)dxdy = ρ Ωb ω2 4 Re ⎧ NGp cells ' ⎨N' ⎩ s=1 p=1 ⎫ ⎬ H m Jcs (ξp ) wp um up (ω) p (ω) ⎭ ⎧ ⎫ NGp cells ' ⎨N' ⎬ 1 H m W m (ω)dxdy = Re Jcs (ξp ) wp m (ω) s (ω) p p ⎭ 4 ⎩ s=1 p=1 Ωb (3.99) (3.100) where NGp is the total number of Gauss points for the sth cell, ξp and Jcs (ξp ) are the natural coordinates and the Jacobian of the in-plane mapping for the sth cell at point p, respectively, wp is the corresponding integration weight and the expressions for um p (ω), m m p (ω) and sp (ω) are given by Eqs. (3.93), (3.96) and (3.97), respectively. Substitution of Eqs. (3.98)-(3.100) into Eq. (3.89) allows to compute the energy velocity for the mth normal mode. 3.10 Numerical analyses of bounded waveguides To show the capability of the 2.5D BEM formulation, the dispersion curves obtained for a standard 113A rail section and a square section are compared with those extracted using the Semi-Analytical Finite Element (SAFE) method. In both the examples, the waveguides are considered to be made of steel with mass density ρ = 7800 Kg/m3 , longitudinal and shear bulk wave velocities equal to cL = 5744.7 m/s and cS = 3224.6 m/s, respectively, longitudinal wave attenuation κL = 0.003 Np/wavelength and shear bulk wave attenuation κS = 0.008 Np/wavelength. Since the accuracy of the eigensolutions is strongly dependent on the number of integration points and the extension of the region enclosed by the curve Γ (κz ), an adaptive scheme has been implemented for the contour algorithm. The extension of the contour region, as well as the number of integration points, can be chosen by observing that at low frequencies only the ﬁrst low order modes with small wavenumbers are expected, which allows to reduce the dimension of the complex contour and the number of integration points. When frequency increases, the extension of the spectrum including propagative modes increases and the contour has to be adjusted in order to capture the complete set of roots. As the region enlarges, an increased number of integration points is needed, thus making the algorithm computationally more expensive at high frequencies. The rank and residual tolerances have been chosen on the bases of convergence tests in which the number of integration points has been increased until a stable trend was 88 3.10.1. STANDARD BS11-113A RAIL observable on the separation of singular values as well as the relative residuals given in Eq. (3.82). 3.10.1 Standard BS11-113A rail In the ﬁrst numerical application, the dispersion curves for a standard 113A rail are compared with those obtained using the SAFE method. The boundary element mesh used for the numerical test is illustrated in Fig. 3.7(a). It is composed of 146 semiisoparametric linear elements and 146 nodes, corresponding to a total of 438 degrees of freedom. The numerical integrations have been carried out at the element level using 10 quadrature points, while the internal waveﬁeld and its derivatives have been computed using a subdivision of the internal area into 1158 cells. Each cell is represented by a 3-nodes triangular element with linear shape functions. The SAFE solution has been obtained by using a mesh of 1496 semi-isoparametric linear triangular elements and 835 nodes (2505 dof) of Fig. 3.7(b). Both the boundary element mesh and the ﬁnite element mesh have been chosen from a convergence test performed over a frequency range of 0 ÷ 10 kHz, decreasing the mesh size until a stable dispersion solution was reached. The contour Γ (κz ) has been deﬁned as in Fig. 3.5, with L = 40, a ﬁxed vertical range of −3.0 ≤ Im (κz ) ≤ +1.0 and a horizontal range varying linearly with frequency, with constant minimum value of Re (κz ) = 0.001 and maximum varying between 20.0 (f = 0 kHz) and 50.0 (f = 10.0 kHz). The singular values separation after 50 frequency steps can be observed in Fig. 3.8, where σi denotes the ith singular value and σmax represents the maximum singular value at the current frequency step. For the ﬁrst frequency, a total number of 500 integration points was used for the trapezoidal rule, then linearly increased to 1400 for the maximum frequency. As it can be noted, the gap is generally included in the range −2.0 ≤ (σi /σmax ) ≤ −4.0, although it becomes smaller at around f = 2.0 kHz and in the range 5.0 ÷ 6.0 kHz, where cutoﬀs of the modes m4, m6, m7 and m8 occur. The term cutoﬀ is used to indicate the frequency value corresponding to a noticeable increase of real part of the axial wavenumber, bearing in mind that the concept of cutoﬀ does not have meaning in the viscoelastic case. In order to minimize the negative eﬀect of spurious solutions in the computation of the linear eigenvalue problem Eq. (3.78), the rank tolerance has been chosen equal to tolrank = −2.0, while the tolerance for the residual test Eq. (3.82) has been assumed 89 3. WAVE PROPAGATION IN BOUNDED AND UNBOUNDED WAVEGUIDES: 2.5D BOUNDARY ELEMENT METHOD (a) (b) Figure 3.7: (a) Boundary element mesh with internal cells subdivision and (b) SAFE mesh of the BS11-113A rail. equal to tolres = 1.0 × 10−6 . As observed in Figs. 3.9-3.12, the BEM solution (continuous lines) is in very good agreement with the SAFE solution (dots). Considerations on the dispersion characteristics of the various m modes can be found in a number of works (see, for example, [Bartoli et al., 2006; Gavric, 1995; Hayashi et al., 2003]) and are not repeated here. The attenuation for all the guided modes are in very good agreement with those provided by the SAFE method (Fig. 3.11), although the m5 and m7 modes exhibit a slightly unstable behaviour in the 5.0 ÷ 5.6 kHz and the 6.0 ÷ 7.0 kHz frequency range, respectively. It is noted that, that attenuation curves for the viscoelastic rail have been found in literature only by means of Finite Element-based analyses. The comparison between the BEM and the SAFE method for the energy velocity ((Fig. 3.12)) also indicates a good correspondence. The only noticeable diﬀerences are represented by the m3 mode in the 1.0 ÷ 4.0 kHz frequency range and the m8 mode for its entire frequency range. 3.10.2 Square bar The second numerical test is performed on a square bar with 20 mm side length. The dispersion curves have been extracted in the 0 ÷ 200.0 kHz frequency range. The goal of this numerical test is to verify the performances of the method when the spectra are densely populated and in the presence of eigenvalues with multiplicity higher than one. After a convergence test, a boundary mesh of 148 semi-isoparametric linear elements and 148 nodes has been chosen, along with an internal subdivision into 4418 triangular 90 3.10.2. SQUARE BAR log 10 (σi/σmax ) í í í í í í ith singular values frequency [kHz] Figure 3.8: Singular values distribution after 50 frequency steps for the standard 113A rail in Sec. 3.10.1. cells with associated linear shape functions (see Fig. 3.13(a)). As comparison, the square section has been analyzed using the SAFE method with a mesh of 4096 semiisoparametric linear triangular elements and 2113 nodes (see Fig. 3.13(b)), which also gave a convergent solution in the considered frequency range. The separation of singular values after 50 frequency steps can be observed in Fig. 3.14. For the eigenvalues computation, a dynamically adaptive contour window Γ (κz ) has been used as previously illustrated for the rail example. The number of integration point has been linearly increased from a minimum of 500 at the ﬁrst frequency step to a maximum of 1500 at the last frequency step. As already noted, the jump in the singular values is strongly reduced at cutoﬀ frequencies, where the solution appears more prone to numerical instabilities. The comparison between the dispersion curves obtained via BEM and those extracted using the SAFE method is shown in Figs. 3.15-3.18. As can be noted, the solutions in terms of real part of the axial wavenumbers and energy velocity are in very good agreement. The solutions in terms of attenuation show some discrepancies for the 91 3. WAVE PROPAGATION IN BOUNDED AND UNBOUNDED WAVEGUIDES: 2.5D BOUNDARY ELEMENT METHOD BEM 45 SAFE 40 (κ z ) [rad/m] 35 m2 m3 30 m1 25 20 m6 15 m7 10 m4 5 0 0 m5 m8 1 2 3 4 5 6 7 8 9 10 frequency [kHz] Figure 3.9: Real wavenumber dispersion curves for the viscoelastic steel BS11-113A rail. ﬂexural mode F2 in the frequency range 85.0 ÷ 105.0 kHz, for the ﬂexural mode F3 in the 115.0 ÷ 150.0 kHz range and the longitudinal mode L2 in the 150.0 ÷ 165.0 kHz range. Larger discrepancies are observed for the attenuation of the screw S31 mode for its entire frequency range. Small diﬀerences can be observed also in the attenuation curve for the L3 mode. It is ﬁnally emphasized that the contour algorithm correctly identiﬁes the eigenvalues with multiplicity 2, corresponding to the ﬂexural Fi modes. 3.11 Surface waves along cavities of arbitrary cross-section A ﬁrst investigation of surface dispersion characteristics for axially symmetric modes in cylindrical cavities of circular cross-section can be found in the work of Biot [1952], that demonstrated the existence of a cutoﬀ for all the pseudo-Rayleigh modes with wavelength corresponding to the bulk shear wavelength. The existence of the ﬁrst ﬂexural mode at all frequencies has been proved analytically by Boström and Burden 92 3.11.1. CIRCULAR CAVITY IN VISCOELASTIC FULL-SPACE 9000 BEM 8000 SAFE m8 7000 cp [m/s] 6000 m5 m7 5000 4000 m4 m6 3000 2000 m2 m3 1000 0 0 m1 1 2 3 4 5 6 7 8 9 10 frequency [kHz] Figure 3.10: Phase velocity dispersion curves for the viscoelastic steel BS11-113A rail. [1982] in cylindrical cavities of circular cross-section and by Burden [1985a,b] for cavities with circular indented, hyperelliptical and elliptical cross-section. However, in these works only elastic mediums are considered. More recently, Tadeu et al. [2002b] used a 2.5D boundary element formulation to study borehole cavities of diﬀerent cross-sections in elastic mediums, extracting the phase velocity spectra from the response of the system to a blast load in the frequencywavenumber domain. Degrande et al. [2006] used a coupled boundary element-ﬁnite element formulation for the prediction of vibrations in the free ﬁeld from excitations due to metro trains in tunnels, extracting the slowness dispersion curves for a layered soil medium with a cylindrical cavity. In this section it is shown that for κz > κS (non radiating region) the attenuation curves of surface normal modes approach asymptotically the attenuation of the Rayleigh wave. 93 3. WAVE PROPAGATION IN BOUNDED AND UNBOUNDED WAVEGUIDES: 2.5D BOUNDARY ELEMENT METHOD 0.1 BEM 0.09 SAFE m6 0.08 α [Np/m] 0.07 m4 0.06 m3 0.05 0.04 0.03 m1 0.02 m7 0.01 0 0 m8 m2 m5 1 2 3 4 5 6 7 8 9 10 frequency [kHz] Figure 3.11: Attenuation dispersion curves for the viscoelastic steel BS11-113A rail. 3.11.1 Circular cavity in viscoelastic full-space In the following numerical example, a cylindrical cavity with circular cross-section of radius a = 1 m and immersed in a viscoelastic medium is studied. As demonstrated by Biot [1952] and Boström and Burden [1982], surface waves in cylindrical cavities propagate with phase velocity varying between the shear wave speed cS and the Rayleigh wave speed cR . Following Biot [1952], the normal modes ranging between these velocities are classiﬁed as pseudo-Rayleigh waves, that do not exhibit attenuation if the medium is elastic. On the other hand, any disturbance propagating at the surface with wavelength longer than the shear wavelength causes shear waves to be radiated, and the energy carried by the surface waves is therefore geometrically attenuated [Botter and van Arkel, 1982]. As a consequence, a cutoﬀ occurs when the axial wavenumbers of the propagating modes become equal to the shear wavenumber. Since surface waves are characterized by a displacement amplitude decreasing with increasing depth, the corresponding axial wavenumbers κz must be larger than the shear wavenumber κS , so that the second Hankel functions give the typical exponential decay in the radial direction. As the inferior speed limit at which surface 94 3.11.1. CIRCULAR CAVITY IN VISCOELASTIC FULL-SPACE 6000 BEM SAFE m5 5000 m7 4000 ce [m/s] m8 3000 m3 2000 m2 m1 1000 0 0 m4 1 2 m6 3 4 5 6 7 8 9 10 frequency [kHz] Figure 3.12: Energy velocity dispersion curves for the viscoelastic steel BS11-113A rail. waves propagate without geometric attenuation is given by the Rayleigh wave speed cR [Biot, 1952], the associated dispersion curves can be obtained by choosing the contour Γ (κz ) such that the real part of the axial wavenumber is always included in the range Re (κS ) ≤ Re (κz ) ≤ Re (κR ), where κR ≈ κS (0.87 + 1.12ν̃) / (1 + ν̃) denotes the Rayleigh wavenumber [Rose, 2004]. Note that, for this particular choice, the only singular point in the κz -plane is given by the shear wavenumber κS , which can be easily excluded from the complex region with an appropriate deformation of the contour Γ (κz ), while the operator Z (κz , ω) does not present discontinuities in the included complex region. Moreover, only a small number of integration points is required, since for Re (κz ) > Re (κS ) both the real and imaginary part of the displacement Green’s functions are strongly attenuated with the amplitude approaching zero, so that in this subregion the matrix Z (κz , ω) is almost constant, except near the bulk shear wavenumber. The normalized dispersion curves in Fig. 3.19(a) and 3.19(b) have been extracted using a mesh of 150 boundary elements and a rectangular contour with a total of 120 integration points and limited by 1.0 ≤ κz /κS ≤ 1.1. The rank and residual tolerances 95 3. WAVE PROPAGATION IN BOUNDED AND UNBOUNDED WAVEGUIDES: 2.5D BOUNDARY ELEMENT METHOD (a) (b) Figure 3.13: (a) Boundary element mesh with internal cells subdivision and (b) SAFE mesh of the square bar. have been set to tolrank = −4.0 and tolres = 5.0 × 10−5 . The viscoelastic medium is deﬁned through its complex Poisson’s ratio ν̃ = 0.3 − i4.5 × 10−4 . As discussed by Boström and Burden [1982], the ﬂexural mode m = 1 is the only one existing in the whole frequency range, while the longitudinal mode m = 0 and the ﬂexural modes m = 2, 3, 4 have a cutoﬀ at κz = κS . From this value, each mode approaches asymptotically the Rayleigh wavenumber Re (κR ) which, for the case Re (ν̃) = 0.3 is Re (κR ) /Re (κz ) = 1.0779. The dispersion curves for the real part of the axial wavenumber agree with those presented by Boström and Burden [1982], while the corresponding attenuation curves due to material damping are reported in Fig. 3.19(b). At the best of the authors knowledge, these curves were not previously reported in the literature. As expected, the attenuations of the various modes approach asymptotically the attenuation of the Rayleigh wave, which is approximately αR /αS = 1.0178. It is worth noting that, despite the exclusion of the shear wavenumber from the contour region, the algorithm is able to extract a root which is very close to the shear wavenumber itself when Re (κS ) a < 1, although it is not shown in Fig. 3.19(a) and 3.19(b). In fact, for these values of the dimensionless frequency, the solution for the m = 1 is very close to the shear wavenumber and it is correctly detected by the algorithm. However, since the root lies in proximity of a singular point, its residual given by Eq. (3.82) is poor, and the root itself is consequently discarded by the residual test. 96 3.11.2. SQUARE CAVITY IN A VISCOELASTIC FULL-SPACE í σi /σmax í í í í í ith singular value frequency [kHz] Figure 3.14: Singular values distribution after 50 frequency steps for the square bar in Sec. 3.10.2. 3.11.2 Square cavity in a viscoelastic full-space In this numerical example, a square cavity with side length equal to 2a is considered. The cavity is immersed in a viscoelastic medium with complex Poisson’s ratio ν̃ = 0.3 − i4.5 × 10−4 . The dispersion curves for the real part of the axial wavenumber and the attenuation are depicted in Fig. 3.20(a) and 3.20(b), respectively. The curves have been obtained using a mesh of 146 semi-isoparametric linear elements, while the scheme used for the contour algorithm is the same of Sec. 3.11.1. As expected, the dispersive behaviour of the guided modes is very similar to that observed for the circular cavity for both the real part of the axial wavenumber and the attenuation. Furthermore, all the normal modes approach asymptotically with their real and imaginary parts the Rayleigh wavenumber. The normal modes appear as grouped in separate families. The ﬁrst family is formed by the ﬁrst ﬂexural mode F1 , which does not have a cutoﬀ, the ﬁrst longitudinal mode L1 and the ﬁrst screw mode S11 . The second family is formed by the screw mode S12 , the ﬂexural mode F2 97 3. WAVE PROPAGATION IN BOUNDED AND UNBOUNDED WAVEGUIDES: 2.5D BOUNDARY ELEMENT METHOD 450 BEM 400 SAFE 350 (κ z ) [rad/m] 300 250 F4 S11 S21 200 150 F1 100 T2 F3 T1 L2 S12 L1 50 L3 S22 F2 F5 S31 0 0 20 40 60 80 100 120 140 160 180 200 frequency [kHz] Figure 3.15: Real wavenumber dispersion curves for the viscoelastic steel square bar of 2.0 mm side length. and the ﬁrst torsional mode T1 . Finally, the third family includes the screw mode S22 , the ﬂexural mode F3 and the screw mode S12 . As previously observed for the rail and the square bar, the separation reduces at the cutoﬀ frequencies. In these cases, some eigensolutions may result slightly less accurate (see mode F3 ). Finally, a moderate numerical instability in the attenuations can be observed for the ﬁrst family of normal modes in the dimensionless frequency range 8.0 ÷ 9.0. Note that this frequency range corresponds in fact to the cutoﬀs for the modes S22 and F3 . 3.12 Conclusions in this chapter, a 2.5D regularized Boundary Element formulation has been proposed to compute the dispersion curves for isotropic linear viscoelastic waveguides of arbitrary cross-section. The attenuation has been taken into account by adding an imaginary part to the axial wavenumber vector, which has been considered parallel to the real (propagative) component. The dispersive parameters have been extracted by solving a 98 3.12. CONCLUSIONS 15000 S31 BEM SAFE S22 F3 F2 F5 L3 10000 cp [m/s] T2 F4 S21 L2 S12 S11 L1 5000 T1 F1 0 0 20 40 60 80 100 120 140 160 180 200 frequency [kHz] Figure 3.16: Phase velocity dispersion curves for the viscoelastic steel square bar of 2.0 mm side length. nonlinear eigenvalue problem in absence of external applied tractions using the contour integral algorithm proposed by Beyn [2012]. The energy velocity has been obtained in post-processing using the method of the cells. Due to the singular characteristics and the multivalued nature of the Green’s functions, the Sommerfeld branch cuts have been introduced and the signs for the real and imaginary parts of the axial wavenumber have been selected in order to satisfy both the Sommerfeld radiation condition and the holomorphicity requirement for the resolvent operator inside the complex region enclosed by the contour. Numerical tests performed on a rail cross-section and a square cross-section have shown that the real part of the eigensolution always matches the corresponding solution obtained via the SAFE method, while some larger discrepancies have been observed for both the attenuation and the energy velocity. The dispersion data extracted for surface normal modes propagating along cylindrical cavities of circular cross-section are in very good agreement with those available in the literature. As expected with the introduction of the material damping, the attenuation dispersion curves of the surface normal modes approach the value of the attenuation of the non-dispersive Rayleigh waves. A similar behaviour has been ob- 99 3. WAVE PROPAGATION IN BOUNDED AND UNBOUNDED WAVEGUIDES: 2.5D BOUNDARY ELEMENT METHOD 1 S12 BEM 0.9 S11 S21 L3 F4 SAFE 0.8 α [Np/m] 0.7 0.6 S22 0.5 0.4 F2 S31 0.3 0.2 T1 F1 F3 L2 0.1 0 0 L1 20 40 60 80 100 120 140 160 180 200 frequency [kHz] Figure 3.17: Attenuation dispersion curves for the viscoelastic steel square bar of 2.0 mm side length. tained for a square cavity, where the normal modes appear to be grouped into families. In line with other works [François et al., 2010; Rieckh et al., 2012], the proposed method could be equally used in the dispersion analysis of cavities embedded in layered media, providing that the fundamental solutions for the isotropic elastic full space are replaced by a numerically computed solution for a layered halfspace. 100 3.12. CONCLUSIONS 6000 BEM SAFE L1 5000 L2 4000 ce [m/s] F3 3000 2000 T1 S22 F1 F2 L3 1000 F5 S21 S12 0 0 S31 F4 T2 S11 20 40 60 80 100 120 140 160 180 200 frequency [kHz] Figure 3.18: Energy velocity dispersion curves for the viscoelastic steel square bar of 2.0 mm side length. 101 3. WAVE PROPAGATION IN BOUNDED AND UNBOUNDED WAVEGUIDES: 2.5D BOUNDARY ELEMENT METHOD 1.08 (κ R ) / (κ S ) 1.07 (κ z ) / (κ S ) 1.06 1.05 m=0 1.04 1 2 1.03 3 1.02 1.01 1 0 4 1 2 3 4 5 (κ S ) a 6 7 8 9 10 (a) 1.08 1.07 1.06 m=0 4 α/αS 1.05 3 1 1.04 2 1.03 αR /αS 1.02 1.01 1 0 1 2 3 4 5 (κ S ) a 6 7 8 9 10 (b) Figure 3.19: Dimensionless (a) real axial wavenumbers and (b) attenuations versus dimensionless frequency for ν̃ = 0.3 − i4.5 × 10− 4. The normal modes are identiﬁed as 102 in Boström and Burden [1982]. 3.12. CONCLUSIONS 1.08 (κR ) / (κS ) 1.07 (κz ) / (κS ) 1.06 1.05 1.04 1.03 1.02 F1 S11 L1 1.01 1 0 S12 1 2 3 4 5 (κS ) a S22 F3 F2 T1 6 S21 7 8 9 10 (a) 1.08 1.07 1.06 T1 L1 1.05 α/αS S21 F2 1.04 F3 1.03 S11 1.02 1.01 1 0 αR /αS S12 F1 1 2 3 4 S22 5 (κS ) a 6 7 8 9 10 (b) Figure 3.20: Dimensionless (a) real axial wavenumbers and (b) attenuations versus dimensionless frequency for ν̃ = 0.3 − i4.5 × 10−4 . 103 3. WAVE PROPAGATION IN BOUNDED AND UNBOUNDED WAVEGUIDES: 2.5D BOUNDARY ELEMENT METHOD 104 Chapter 4 Leaky Guided Waves in waveguides embedded in solid media: coupled SAFE-2.5D BEM formulation 4.1 Sommario In questo capitolo viene presentata una formulazione accoppiata SAFE-2.5D BEM per il calcolo delle caratteristiche di dispersione di onde guidate che si propagano in guide d’onda viscoelastiche immerse in mezzi isotropi, viscoelastici ed inﬁnitamente estesi (leaky guided waves). La natura dispersiva delle leaky guided waves è stato studiato in letteratura utilizzando metodi analitici [Lowe, 1992; Pavlakovic, 1998; Simmons et al., 1992; Viens et al., 1994] e metodi semi-analitici agli elementi ﬁniti accoppiati con metodi delle regioni di assorbimento [Castaings and Lowe, 2008], elementi semi-analitici inﬁniti [Jia et al., 2011] e Perfectly Matched Layers (PML) [Treyssède et al., 2012]. Tuttavia, mentre i metodi analitici sono applicabili solamente a guide d’onda di geometrie semplici, i metodi semi-analitici presentano delle diﬃcoltà nella modellazione del campo irradiato a causa di riﬂessioni spurie (metodo delle regioni di assorbimento), arbitrarietà nella scelta delle funzioni di forma (elementi inﬁniti semi-analitici) o scelta delle funzioni di smorzamento (PML). La formulazione accoppiata SAFE-2.5D BEM descritta in questo capitolo consente 105 4. LEAKY GUIDED WAVES IN WAVEGUIDES EMBEDDED IN SOLID MEDIA: COUPLED SAFE-2.5D BEM FORMULATION di superare i problemi dei metodi analitici e di quelli numerici sopra citati. Infatti, mentre il metodo SAFE consente di rappresentare guide d’onda immerse di geometrie e caratteristiche meccaniche complesse, mediante il BEM è possibile descrivere accuratamente il campo irradiato nel mezzo solido circostante. In particolare, la formulazione SAFE diﬀerisce da quella descritta nel Capitolo 2 nell’utilizzo di elementi ﬁniti quadratici in luogo di quelli lineari, mentre l’eﬀetto di uno stato di stress iniziale non viene considerato. Poichè l’accoppiamento delle regioni SAFE e BEM prevede la compatibilit degli spostamenti e la continuit delle trazioni lungo all’interfaccia, anche nella formulazione BEM viene fatto uso di elementi quadratici. Inoltre, le funzioni di Green 2.5D utilizzate nel Capitolo 3 vengono sostituite da un set di funzioni simili ma che consentono un risparmio in termini di tempi computazionali. Le singolarità delle funzioni nucleo negli integrali di contorno vengono trattate utilizzando la procedura di regolarizzazione descritta nel Capitolo 3. Poichè il numero di gradi di libertà del dominio discretizzato BEM generalmente inferiore a quello del dominio discretizzato SAFE, l’accoppiamento delle due formulazioni viene eseguito trasformando il dominio BEM in un singolo elemento ﬁnito avente lo stesso numero di gradi di libertà del dominio originale. La matrice di impedenza del mezzo circostante viene pertanto trasformata in una matrice di rigidezza dinamica equivalente, la quale è successivamente assemblata nel sistema SAFE. L’equazione d’onda ottenuta si conﬁgura come un problema non lineare agli autovalori. Tale problema viene risolto utilizzando il metodo degli integrali di contorno proposto da Beyn [2012] e descritto nel Capitolo 3. Date le profonde diﬀerenze tra la natura delle onde guidate di tipo leaky e quelle che si propagano in guide d’onda immerse nel vuoto, una nuova analisi delle superﬁci di Riemann viene presentata, nella quale si tiene conto delle condizioni aggiuntive di interfaccia (legge di Snell generalizzata). I risultati ottenuti con il metodo proposto vengono dapprima validati con due risultati noti in letteratura, nei quali solamente geometrie cilindriche vengono considerate. Inﬁne, le potenzialità del metodo proposto vengono dimostrate attraverso tre esempi numerici di interesse pratico, presentati in letteratura per la prima volta. 106 4.2. INTRODUCTION AND LITERATURE REVIEW 4.2 Introduction and literature review As discussed in previous chapters, the dispersion properties of guided waves in tractionfree waveguides can be eﬃciently computed by means of analytical methods [Chaki and Bourse, 2009; Knopoﬀ, 1964; Lowe, 1995; Pavlakovic et al., 1997; Pavlakovic, 1998; Shin and Rose, 1999] and ﬁnite element-based methods [Bartoli et al., 2006; Chen and Wilcox, 2007; Gavric, 1995; Hayashi et al., 2003, 2006; Loveday, 2009; Sorohan et al., 2011; Treyssède, 2008]. The 2.5D BEM formulation described in Chapter 3 also assumes that the boundary of the waveguide is in contact with vacuum, so that only reﬂection and mode conversion of bulk waves occur at the solid-vacuum interface. However, in several circumstances waveguides are embedded in solid media. These are, for examples, the cases of tendons, foundation piles, buried pipes, railways or embedded ﬁbers. In these cases, guided modes traveling with phase speed greater than the bulk speed of the surrounding media radiates energy into it. As a consequence, inspection ranges are generally reduced since the energy radiated in the surrounding media causes high attenuation rates of the guided modes (leaky modes). The knowledge of dispersion properties of leaky modes is therefore fundamental in NDE testing of civil, mechanical and aerospace structures and mathematical tools able to describe waveguides with diﬀerent geometric and mechanical characteristics are needed. In this context, several studies can be found in literature involving simple geometries, i.e. plate and cylindrical structures, in which analytical methods have been extensively applied. A comprehensive study of matrix techniques for the computation of dispersion curves in free, embedded and immersed plates can be found in the work of Lowe [1995]. The propagation of leaky Lamb waves in plates embedded in solids have been studied by Dayal and Kinra [1989, 1991] and Lowe [1992]. The propagation of non-leaky guided waves in elastic circular waveguides embedded in elastic media have been investigated by Parnes [1981, 1982] and Kleczewski and Parnes [1987]. Dispersion relations for leaky modes in elastic circular rods embedded in isotropic elastic solids have been extracted by Thurston [1978], Simmons et al. [1992] and Viens et al. [1994] using analytic dispersive equations. In their work, Nayfeh and Nagy [1996] have applied the Transfer Matrix Method to investigate the propagation of axisymmetric waves in coaxial layered anisotropic ﬁbers embedded in solids and immersed in ﬂuids. General studies on wave propagation in transversely isotropic and 107 4. LEAKY GUIDED WAVES IN WAVEGUIDES EMBEDDED IN SOLID MEDIA: COUPLED SAFE-2.5D BEM FORMULATION homogeneous anisotropic circular rods immersed in ﬂuids have been conducted by Dayal [1993], Nagy [1995], Berliner and Solecki [1996a,b] and Ahmad [2001]. The dispersion properties of leaky guided waves in both embedded and immersed cylindrical structures have been in depth analyzed by Pavlakovic [1998] using the Global Matrix Method (GMM). This method has been used next to perform numerical analyses and support experimental investigations involving free pipes with defects [Lowe et al., 1998], buried pipes [Long et al., 2003b], embedded circular bars [Pavlakovic et al., 2001] and embedded tendons and bolts Beard et al. [2003]; Beard and Lowe [2003]. An analytical method has been proposed by Laguerre et al. [2007] to predict dispersion curves and to interpret the ultrasonic transient bounded-beam propagation in a solid cylindrical waveguide embedded in a solid medium. Although very attractive for simple geometries, analytical approaches are generally unsuitable to extract dispersion properties for waveguides with irregular cross-section and, in these cases, one must resort to numerical methods. Due to the capability to represent domains with diﬀerent materials and arbitrary geometries while forming well posed polynomial eigenvalue problems, the SAFE method has been also extended in recent years to wave propagation problems involving unbounded domains. In their work, Castaings and Lowe [2008] have used a SAFE mesh to discretize both the waveguide and the embedding medium. The material surrounding the waveguide was simulated by introducing a ﬁnite absorbing region of length proportional to the largest radial wavelength of the existing leaky waves. The method eliminates the well known problem of non-physical reﬂections which would arise using a ﬁnite mesh to model the unbounded surrounding domain. However, this method may require very large meshes to properly model waves radiating in the surrounding media and guided modes with high rates of energy conﬁned in the embedded cross-section need to be selected from a large set of eigensolutions. A hybrid SAFE formulation has been proposed by Jia et al. [2011] to study double layer hollow cylinders embedded in inﬁnite media. In this study, the unbounded medium has been discretized by means of inﬁnite elements, which overcomes the problem of energy reﬂection. However, the capability of inﬁnite elements to correctly represent the physics of leaky waves is strongly related to the choice of the elements shape functions. Moreover, complicated geometries, such as H shaped beams, may result diﬃcult to treat. In their work, Lin et al. [2011] have considered the presence of two isotropic elastic half spaces at the top and bottom interface of a SAFE-modeled layer by introducing appropriate analytical boundary conditions. The analytical boundary conditions have 108 4.2. INTRODUCTION AND LITERATURE REVIEW been adopted in order to satisfy the Snell’s law for radiated longitudinal and shear waves. Only solutions relative to evanescent waveﬁelds in the surrounding medium have been considered in this study. A further numerical technique that allows to model radiated waves without reﬂections has been proposed by Treyssède et al. [2012] by coupling the SAFE method with the Perfectly Matched Layer (PML) method. Using the PML, leaky modes are deﬁned through analytic extensions in terms of complex spatial coordinates. Although this method allows to preserve the original dimension of the problem as well as the nature of the dispersive wave equation, the radiation eﬃciency strictly depends on the choice of the complex-valued function used to represent geometric decay inside the PML domain. A possible alternative is represented by the Boundary Element Method (BEM). Unlike FE-based formulations, the BEM allows to describe the unbounded surrounding domain by means of a boundary mesh only. Moreover, since the weight functions are represented by the fundamental solutions of the dynamic problem (Green’s function), no approximations are introduced in the deﬁnition of the radiated waveﬁeld. In recent years, diﬀerent coupled FEM-BEM formulations have been proposed to investigate the wave propagation in waveguide-like structures. Such formulations are sometimes referred in literature as the wavenumber ﬁnite-boundary element method [Sheng et al., 2005, 2006], the waveguide ﬁnite-boundary element method [Nilsson et al., 2009] or the 2.5D ﬁnite-boundary element method [Costa et al., 2012; François et al., 2010]. While most of these studies are focused on forced or induced vibrations problems, minor attention has been dedicated to the study of dispersive characteristics of guided waves, especially when attenuation is involved. Some exceptions are represented by the work of Tadeu and Santos [2001] and Zengxi et al. [2007], which have adopted a 2.5D BEM for the computation of dispersion relations in ﬂuid ﬁlled boreholes. However, attenuation information is not provided in these works. More recently, Nilsson et al. [2009] have proposed a waveguide FEM-BEM formulation to study the radiation eﬃciency of open and embedded rails. In such work, dispersion relations for radiating modes have been obtained by considering complex wavenumbers, thus taking into account the amplitude decay due to attenuation. However, since the acoustic impedance mismatch between the rail and the air was very high, the authors have considered in their model only the inﬂuence of the rail on the ﬂuid vibrations and not the one of the air on the rail (the model is not fully coupled). In this chapter, the SAFE method is coupled with the regularized 2.5D BEM to extract dispersion curves for viscoelastic waveguides of arbitrary cross-section embedded in viscoelastic isotropic materials. With respect to the SAFE formulations that 109 4. LEAKY GUIDED WAVES IN WAVEGUIDES EMBEDDED IN SOLID MEDIA: COUPLED SAFE-2.5D BEM FORMULATION use absorbing regions, inﬁnite elements and PMLs, the proposed SAFE-2.5D BEM formulation represents exactly the radiated waveﬁeld from waveguides of arbitrary crosssection while preserving the dimension of the SAFE problem and without the need of special complex functions. The complex axial wavenumbers and the corresponding wavestructures are computed from a nonlinear eigenvalue problem solved via a contour the Contour Integral Method proposed by [Beyn, 2012] and described in Sec. 3.8.1. The complex poles associated to leaky and evanescent modes are obtained by choosing the arguments of the wavenumbers in the embedding medium consistently with the nature of the radiated waves and removing points of singularities and discontinuities from the complex plane of the axial wavenumber. The method is ﬁrst validated against available results obtained, for embedded circular bars, by means of alternative approaches [Castaings and Lowe, 2008; Pavlakovic et al., 2001]. Next, dispersion curves are extracted for a viscoelastic square steel bar embedded in viscoelastic grout and for a viscoelastic HP200 steel pile embedded in a viscoelastic soil. To the best of author’s knowledge, these cases are never been studied in literature. The proposed method can be useful to understand the physical behaviour of leaky guided waves as well as to design testing conditions in GUW-based inspections and experiments involving embedded beams or foundation piles. 4.3 Wave equation In this section, the guided wave equation is derived for the system with translational invariant geometric and mechanical properties of Fig. 4.1. The wavenumber-frequency dependence is assumed in the form exp [i (κz z − ωt)] , (4.1) from which, the following conditions > 0, Re (κz ) = kRe z > 0, Im (κz ) = kIm z (4.2) must be satisﬁed in order to ensure the amplitude decay of guided modes propagating in the positive direction of the z-axis (cf. Sec. 2.6.1). As seen in Chapter 2, the longitudinal invariance allows to describe the threedimensional wave propagation problem in the x − y plane, while the third dimension is accounted by contraction of any z-dependent scalar or vectorial ﬁeld in the axial wavenumber domain through the spatial Fourier transform in Eq.2.35. In particular, 110 4.3. WAVE EQUATION the waveguide cross-section of area Ωs is discretized using the SAFE method while the external medium of inﬁnite extent Ωb is modeled via a 2.5D regularized boundary integral formulation. The in-plane position vector x = [x, y]T (x ∈ Ωs ∪ Ωb ) is used to denote a generic point located at the cross-section of axial coordinate z = 0. The SAFE and BEM meshes are deﬁned with coincident nodes and matching shape functions at the coupling interface ∂Ω = ∂Ωs = ∂Ωb , where compatibility of displacements and equilibrium of tractions are enforced through the relationships u (x, z, t)|∂Ωs = u (x, z, t)|∂Ωb , (4.3) t (x, z, t)|∂Ωs = −t (x, z, t)|∂Ωb , (4.4) denoting with u (x, z, t) = [u1 , u2 , u3 ]T the displacements vector and t (x, z, t) = [t1 , t2 , t3 ]T the tractions vector. The minus on the right hand side of Eq. (4.4) accounts for the opposite sign of the outward normals of the SAFE and BEM regions at the boundary point x, i.e. n (x)|∂Ωs = −n (x)|∂Ωb (see Fig. 4.1). The equilibrium equation for a waveguide embedded in an inﬁnite medium can be obtained in the wavenumber-frequency domain by following the same procedure described in Chapter 2. Under the hypotheses of (i) translational invariant mechanical characteristics, (ii) initial stress-free state and (iii), absence of body forces, the equilibrium equation (2.44) reduces to ω (δ (u))T C̃ (x, ω) (u) dxdy δu ρ (x) udxdy − 2 T Ωs Ωs (4.5) δuT [ts (x, κz , ω) − tb (x, κz , ω)] ds = 0, + ∂Ωs where ρ (x) is the material density at point x ∈ Ωs , (u) is the vector of the independent linear strain components, deﬁned in Eq. (2.7), and C̃ (x, ω) is the fourth order tensor of complex moduli deﬁned as in Eqs. (2.36) and (2.72). In the derivation of Eq. (4.5), the vector of surface loads tc (x, κz , ω) appearing in Eq. (2.44) has been replaced by tc (x, z, ω) = ts (x, ω) − tb (x, z, ω) , (4.6) where ts (x, κz , ω) is the vector of the external surface loads applied at the interface 111 4. LEAKY GUIDED WAVES IN WAVEGUIDES EMBEDDED IN SOLID MEDIA: COUPLED SAFE-2.5D BEM FORMULATION while tb (x, κz , ω) is the vector of the interface tractions resulting from the mutual interaction of the waveguide with the surrounding medium. The strategy adopted to solve Eq. (4.5) is based on a SAFE discretization of the embedded waveguide and the computation of the vector of interface tractions tb (x, κz , ω) by means of a 2.5D BEM formulation. Compared to well stated analytical methods such as the Transfer Matrix Method (TMM) or the Global matrix Method (GMM), this approach allows to model embedded waveguides of any geometry and material through the SAFE. At the same time, the BEM allows to exactly compute the radiated waveﬁeld, which is the main drawback of FE-based techniques. 112 4.3. WAVE EQUATION Figure 4.1: Analytical model of the embedded waveguide. 113 4. LEAKY GUIDED WAVES IN WAVEGUIDES EMBEDDED IN SOLID MEDIA: COUPLED SAFE-2.5D BEM FORMULATION 4.4 SAFE model of the embedded waveguide The domain Ωs is discretized into a number Nel of quadratic semi-isoparametric ﬁnite elements of area Ωes , with 3 degrees of freedom per node associated to the three displacement components ui . The displacement vector at point x ∈ (Ωes ∪ ∂Ωes ) is approximated as u (ξ, z, t) = N (ξ) qe (z, t) , (4.7) where N (ξ) is the matrix collecting the quadratic shape functions for the parent element T of area Ωref s in the natural reference system, ξ = [ ξ1 , ξ2 ] is the vector of the natural e coordinates deﬁned on Ωref s , and q (z, t) is the vector of nodal displacements. A list of various quadratic elements used in this study is given in Table4.1. From Eq. (4.7), using the space-time Fourier transform in Eq. (2.35) the vector of linear strain components (u) in the (κz , ω) domain is obtained as (cf. Eq.2.49) (ξ, κz , ω) = [Bxy (ξ) + iκz Bz (ξ)] qe (κz , ω) , (4.8) in which the compatibility operators Bxy (ξ) and Bz (ξ) are expressed as in Eqs. (2.50) and (2.51). Following the analysis in Chapter 3, the complex tensor of viscoelastic moduli C̃e (ξ, ω) is deﬁned so that the material attenuation vectors for both longitudinal and shear bulk waves in the embedded waveguide and surrounding medium are assumed perpendicular to their wavefronts. Physically, this means that the directions of propagation and maximum decay due to material damping are mutually parallel. The complex Lamé constants can be expressed as λ̃ (ω) = ρ c̃2L (ω) − 2c̃2S (ω) , μ̃ (ω) = ρc̃2S (ω) , cL cS , c̃S (ω) = , c̃L (ω) = 1 + iβL (ω) /2π 1 + iβS (ω) /2π (4.9) e (ω) = λ̃ (ω) δij δkm + μ̃ (ω) (δik δjm + δim δjk ). from which one derives C̃ijkm Substituting Eqs. (4.7), (4.8) into Eq. (4.5) and using Eq. (4.9), algebraic manipulations lead to the following N -dimensional linear system of equations in the (κz , ω) domain 2 − ω M Q (κz , ω) + Fb (κz , ω) = Fs (κz , ω) , + K κ2z K3 + iκz K2 − KT 1 2 (4.10) where the diﬀerent matrix operators, which result from the application of a ﬁnite element assembling procedure for all the Nel elements of the mesh, take the following 114 4.4. SAFE MODEL OF THE EMBEDDED WAVEGUIDE N1 = ζ (2ζ − 1) N1 = 1 4 (1 − ξ1 ) (1 − ξ2 ) (−ξ1 − ξ2 − 1) N1 = 1 4 2 ξ1 − ξ1 ξ22 − ξ2 N2 = ξ1 (2ξ1 − 1) N2 = 1 4 (1 + ξ1 ) (1 − ξ2 ) (ξ1 − ξ2 − 1) N2 = 1 4 2 ξ1 + ξ1 ξ22 − ξ2 N3 = ξ2 (2ξ2 − 1) N3 = 1 4 (1 + ξ1 ) (1 + ξ2 ) (ξ1 + ξ2 − 1) N3 = 1 4 2 ξ1 + ξ1 ξ22 + ξ2 N4 = 4ξ1 ζ N4 = 1 4 (1 − ξ1 ) (1 + ξ2 ) (−ξ1 + ξ2 − 1) N4 = 1 4 2 ξ1 − ξ1 ξ22 + ξ2 N5 = 4ξ1 ξ2 N5 = 1 2 (1 − ξ2 ) (1 + ξ1 ) (1 − ξ1 ) N5 = 1 2 2 ξ2 − ξ2 1 − ξ12 N6 = 4ξ2 ζ N6 = 1 2 (1 + ξ1 ) (1 + ξ2 ) (1 − ξ2 ) N6 = 1 2 2 ξ1 + ξ1 1 − ξ22 ζ = 1 − ξ − 1 − ξ2 N7 = 1 2 (1 + ξ2 ) (1 + ξ1 ) (1 − ξ1 ) N7 = 1 2 2 ξ2 + ξ2 1 − ξ12 N8 = 1 2 (1 − ξ1 ) (1 + ξ2 ) (1 − ξ1 ) N8 = 1 2 2 ξ1 + ξ1 1 − ξ22 N9 = 1 − ξ12 1 − ξ22 Table 4.1: Shape functions for diﬀerent quadratic isoparametric ﬁnite elements. representations (cf. Eqs. (2.57)-(2.60)) K3 = K2 = K1 = M= N el e=1 N el e=1 N el Ωref s Ωref s ref e=1 Ωs N el e=1 Ωref s BzT (ξ) C̃e (ξ, ω) Bz (ξ) Jse (ξ) dξ1 dξ2 (4.11) (Bxy (ξ))T C̃e (ξ, ω) Bz (ξ) Jse (ξ) dξ1 dξ2 (4.12) (Bxy (ξ))T C̃e (ξ, ω) Bxy (ξ) Jse (ξ) dξ1 dξ2 (4.13) ρe (ξ) (N (ξ))T N (ξ)Jse (ξ) dξ1 dξ2 (4.14) in which Jse (ξ) = det [∂x (ξ) /∂ξ] represents the Jacobian of the isoparametric mapping in the x − y plane for the eth semi-isoparametric ﬁnite element. The vectors of nodal displacements on Ωs ∪∂Ωs , Q (κz , ω), and nodal forces on ∂Ωs , Fs (κz , ω) and Fb (κz , ω), 115 4. LEAKY GUIDED WAVES IN WAVEGUIDES EMBEDDED IN SOLID MEDIA: COUPLED SAFE-2.5D BEM FORMULATION are expressed as Q (κz , ω) = Fs (κz , ω) = Fb (κz , ω) = N el qe (κz , ω) , e=1 Nb ref q=1 ∂Ωs Nb ref q=1 ∂Ωs (4.15) (N (ξ (η)))T tqs (κz , ω) Jsq (ξ (η)) dη, (4.16) (N (ξ (η)))T tqb (κz , ω) Jsq (ξ (η)) dη, (4.17) where ξ (η) is a coordinate transformation for the in-plane mapping of the edge of an element which nodes belong to ∂Ωs , Jsq (ξ (η)) = |(∂x/∂ξ) (∂ξ/∂η)| the corresponding Jacobian and Nb the total number of edges that discretize ∂Ωs . It is worth noting that, while the matrix operators K1 , K2 and K3 can be either dependent or independent on the frequency with varying rheological models (see the description given in Sec. 2.4), the vector Fb (κz , ω) always depends on wavenumber and frequency since it accounts for the acoustic mechanical and geometric properties of the external medium. This vector is determined via a 2.5D BEM formulation, which is described in the next section. 4.5 4.5.1 BEM model of the surrounding medium Regularized 2.5D boundary integral equation The surrounding medium of unbounded domain Ωb is assumed to be isotropic and linear viscoelastic, with mechanical properties deﬁned by mass density ρ and complex bulk velocities c̃L and c̃S . As shown in Chapter 3, the 2.5D boundary integral formulation is obtained from the corresponding integral representation theorem in which two diﬀerent dynamic states are considered. The ﬁrst state is represented by the unknown displacements u (x, κz , ω) and tractions t (x, κz , ω) at a receiver point x ∈ ∂Ωb (see Fig. 4.1). The second state is assumed as the state of fundamental solutions in the full space for the spatial and time harmonic problem, i.e. the dynamic Green’s functions in terms of displacements and tractions at x due to a harmonic line load p (x , z , t) = δ (x − x ) exp [i (κz z − ωt)] with plane coordinates x ∈ Ωb (see Fig. 4.1). The procedure adopted to extend the boundary integral formulation to source points x belonging to the boundary involves the limiting process x ∈ Ωb → x ∈ ∂Ωb and is described in Sec. 3.6.1. As a result, the boundary integrals are convergent in the 116 4.5.1. REGULARIZED 2.5D BOUNDARY INTEGRAL EQUATION Cauchy Principal Value sense. In Chapter 3 it has been illustrated how numerical diﬃculties in treating Cauchy principal value integrals and boundary corners can be overcome by using the so called rigid body motion technique [François et al., 2010; Lu et al., 2008b]. From Eq. (3.54), by posing c∞ = 0, the regularized 2.5D boundary integral equation in the (κz , ω) domain for a source point x ∈ ∂Ωb and in absence of body forces is expressed as D U (r, κz , ω) − US (r) t (x, κz , ω) ds (x) u x , κz , ω = ∂Ωb US (r) t (x, κz , ω) ds (x) + ∂Ωb − D T (r, κz , ω) u (x, κz , ω) − TS (r) u x , κz , ω ds (x) , (4.18) ∂Ωb x, x ∈ ∂Ωb , where r = |x − x | is the source-receiver distance in the z = 0 plane (see Fig. 4.1). The diﬀerent sign for the axial wavenumber κz in the arguments of the dynamic Green’s functions between Eq. (3.54) and Eq. (4.18) follows directly from the assumptions in Eqs. (3.1) and (4.1), respectively. The fundamental dynamic solutions UijD (r, κz , ω) in Eq. (4.18) express the jth displacement component at x when the harmonic line load of plane coordinates x is acting in the ith direction. In this chapter, the dynamic Green’s functions proposed by Li et al. [1992] for a homogeneous isotropic linear viscoelastic full space are adopted instead of those derived by Tadeu and Kausel [2000] and used in Chapter 3, since they present an advantage in terms of computational times. In fact, while the Green’s functions proposed by Tadeu and Kausel [2000] require the evaluation of a set of four Hankel functions (in this case from order 0 to order 3), those proposed by Li et al. [1992] need only the evaluation of the zero and one order Hankel functions. In boundary element codes that operate with dynamic analyses, the evaluation of the Green’s functions (and therefore of the Hankel functions) represent a time-consuming operation, especially for large meshes and large numbers of integration points, since the evaluation must be performed for several combinations of κα,β r. The fundamental solution derived by Li et al. [1992] for the harmonic wave motion in time and space reads UijD (r, κz , ω) = 1 i 0 (1) (1) (1) H0 (κβ r)δij + Lij H0 (κβ r) − H0 (κα r) , 4μ̃ i, j = 1, 2, 3 117 (4.19) 4. LEAKY GUIDED WAVES IN WAVEGUIDES EMBEDDED IN SOLID MEDIA: COUPLED SAFE-2.5D BEM FORMULATION where $ κα = κβ = $ κ2L − κ2z , (4.20) κ2S − κ2z , (4.21) are the wavenumbers normal to the interface ∂Ω and ω , c̃L ω κS = , c̃L κL = (4.22) (4.23) (1) denote the complex longitudinal and shear bulk wavenumbers. In Eq. (4.19), H0 (·) is the zero order Hankel function of ﬁrst kind and 1 ∂ ∂2 2 − iκz (δ3j δki + δ3i δkj ) − κz δ3j δ3i . Lij = 2 δkj δqi ∂xk ∂xq ∂xk κS (4.24) i, j, k, q = 1, 2, 3 Following Gunawan and Hirose [2005], the fundamental displacements in Eq. (4.19) can be further elaborated to give 2P2 ∂r ∂r δij − P3 − κS r ∂xi ∂xj −κz ∂r D D (r, κz , ω) = U3i (r, κz , ω) = P2 i = 1, 2 Ui3 4μ̃κS ∂xi i κ2z D Q1 − 2 P1 . U33 (r, κz , ω) = 4μ̃ κS UijD 1 (r, κz , ω) = 4μ̃ P2 Q1 − κS r i, j = 1, 2 (4.25) The second set of fundamental solutions, the tractions Green’s functions TijD (r, κz , ω), are obtained as D (r, κz , ω) nk (x) , TijD (r, κz , ω) = σijk i, jk = 1, 2, 3 (4.26) being nk (x) the kth component of the outward normal at x ∈ ∂Ωb and D D (r, κz , ω) = λ̃εD σijk ill (r, κz , ω) δjk + 2μ̃εijk (r, κz , ω) , i, j, k, = 1, 2, 3 (4.27) the jkth component of the Cauchy stress tensor at x when the line load of projection 118 4.5.1. REGULARIZED 2.5D BOUNDARY INTEGRAL EQUATION x is acting in direction i, while D (r, κ , ω) D (r, κ , ω) ∂U ∂U 1 z z ij ik + , εD ijk (r, κz , ω) = 2 ∂xk ∂xj i, j, k, = 1, 2, 3 (4.28) is the associated Green’s tensor of linear strains. Substituting Eq. (4.25) into Eqs. (4.28), (4.27) and (4.26) leads to the following expressions for the fundamental tractions [Gunawan and Hirose, 2005] κβ λ̃ iκS κ2z 2P3 ∂r 4P2 P4 − (r, κz , ω) = Q2 + 2 P2 + − ni 4 μ̃ κS κS (κS r)2 κS r ∂xj κβ ∂r 4P2 2P3 ∂r Q2 + − n + n + − j i κS ∂xi ∂xj (κS r)2 κS r 8P3 ∂r ∂r ∂r 16P2 i, j = 1, 2 + − 2 + κ r + 2P4 ∂x ∂x ∂n , (κS r) S i j λ̃ κz κ2z 2P2 D P3 + 2 P1 − Q1 + ni Ti3 (r, κz , ω) = − 4 μ̃ κS r κS 2P2 ∂r ∂r − P3 , i = 1, 2 −2 κS r xi ∂n 2P2 κz 2P2 ∂r ∂r D − Q1 n i − 2 − P3 , i = 1, 2 T3i (r, κz , ω) = − 4 κS r κS r ∂xi ∂n κβ iκS 2κ2z ∂r D − , Q2 + 2 P2 T33 (r, κz , ω) = 4 κS ∂n κS TijD (4.29) in which (1) Q1 = H0 (κβ r) , (1) (1) Q2 = H1 (κβ r) , (1) P1 = H0 (κβ r) − H0 (κα r) , κβ κα (1) (1) H1 (κβ r) − H1 (κα r) , P2 = κS κL 2 2 κβ κα (1) (1) H0 (κβ r) − H0 (κα r) , P3 = κS κL 3 3 κβ κα (1) (1) H1 (κβ r) − H1 (κα r) . P4 = κS κL (4.30) The static fundamental displacements and tractions in Eq. (4.18), UijS (r) and TijS (r), respectively, correspond to the fundamental solutions for the in-plane line load problem 119 4. LEAKY GUIDED WAVES IN WAVEGUIDES EMBEDDED IN SOLID MEDIA: COUPLED SAFE-2.5D BEM FORMULATION in plane strain (cf. Eq. (3.40)) 1 1 ∂r ∂r (3 − 4Re (μ̃)) ln δij + (r) = , 8πRe (μ̃) (1 − Re (ν̃)) r ∂xi ∂xj ∂r 1 ∂r ∂r S nk 2Re (μ̃) δij + 2 Tij (r) = − 4π (1 − Re (μ̃)) r ∂xk ∂xi ∂xj ∂r ∂r nj − ni , i, j, k = 1, 2 − (1 − 2Re (μ̃)) ∂xi ∂xj UijS (4.31) (4.32) and those for the anti-plane line load problem in plane strain (cf. Eq. (3.41)) 1 1 ln , = 2πRe (μ̃) r 1 ∂r S (r) = − nk . k = 1, 2 T33 2πr ∂xk S (r) U33 (4.33) (4.34) Since the asymptotic behaviour of the dynamic and static fundamental solutions correspond when r → 0, the dominant singularities of the kernel functions in the ﬁrst and last integral of Eq. (4.18) cancel each other out when the source point x approaches the receiver point x. Consequently, these integrals can be evaluated numerically using the standard Gauss-Legendre quadrature formula [Stroud and Secrest, 1996]. The second integral in Eq. (4.18) behaves asymptotically as ln (1/r) for r → 0 and can be evaluated using the Gauss-Laguerre and Gauss-Legendre quadrature formulae (see Sec. 4.5.3). 4.5.2 Boundary element discretization The boundary ∂Ωb is subdivided into a number Nb of quadratic semi-isoparametric monodimensional elements with shape functions as indicated in Table4.2. In order to satisfy the compatibility conditions Eq. (4.3), the nodes of the generic boundary element ∂Ωqb are chosen to coincide with those belonging to one edge of an adjacent semi-analytical ﬁnite element Ωes . The boundary geometry, displacements and tractions are interpolated as follows x (η) = N (η) xq , (4.35) u (η, κz , ω) = N (η) q (κz , ω) , (4.36) t (η, κz , ω) = N (η) hq (κz , ω) , (4.37) q where N (η) is the matrix containing the quadratic shape functions in the natural q q q coordinate η ∈ ∂Ωref b (see Table4.2), while x , q (κz , ω) and h (κz , ω) are the vectors 120 4.5.2. BOUNDARY ELEMENT DISCRETIZATION of nodal coordinates, displacements and tractions, respectively. The regularized boundary integral formulation Eq. (4.18) is rewritten in discretized form by applying a point collocation scheme [Brebbia and Dominguez, 1989], where collocation points x are assumed to be coincident with the nodes of the boundary element mesh. Denoting by xc the cth collocation node and introducing Eqs. (4.36) and (4.37) into Eq. (4.18), the recursive collocation procedure over the total number of nodes Nn = Nb × 2 of the boundary element leads to ⎧ Nb Nn ⎨ c=1 ⎩ − [Uq1 (rc (η) , κz , ω) + Uq2 (rc (η))] hq (κz , ω) q=1 Nb q=1 q) (xc ∈∂Ω / + Nb q=1 q) (xc ∈∂Ω / [Tq1 (rc (η) , κz , ω) qq Nb (κz , ω)] − [Tq2 (rc (η) , κz , ω) qq (κz , ω)] q=1 (xc ∈∂Ωq ) ⎫ ⎪ Nn ⎬ q [T3 (rc (η)) uc (κz , ω)] = uc (κz , ω) , ⎪ ⎭ c=1 (x, xc ) ∈ ∂Ωb (4.38) where uc (κz , ω) is the displacement vector at xc and Uq1 (rc (η) , κz , ω) = ∂Ωref b UD (rc (η) , κz , ω) − US (rc (η)) N (η) Jbq (η) dη, (4.39) Uq2 (rc (η)) = ∂Ωref b US (rc (η)) N (η) Jbq (η) dη, (4.40) TD (rc (η) , κz , ω) N (η) Jbq (η) dη, (4.41) Tq1 (rc (η) , κz , ω) = ∂Ωref b Tq2 (rc (η) , κz , ω) = ∂Ωref b TD (rc (η) , κz , ω) N (η) − TS (rc (η)) N (ηc ) Jbq (η) dη, (4.42) Tq3 (rc (η)) = ∂Ωref b TS (rc (η)) Jbq (η) dη, (4.43) are inﬂuence operators, in which rc (η) = |x (η) − xc | denotes the in-plane distance between the integration point x (η) and the collocation point xc , ηc is the adimensional coordinate evaluated at the element’s node coincident with xc and Jbq (η) = |∂x (η) /∂η| is the Jacobian of the semi-isoparametric transformation. 121 4. LEAKY GUIDED WAVES IN WAVEGUIDES EMBEDDED IN SOLID MEDIA: COUPLED SAFE-2.5D BEM FORMULATION From Eq. (4.38), by grouping the displacements and tractions operators into the ! global inﬂuence operators Ub (κz , ω) = c q 2j=1 Uqj (rc , κz , ω) and Tb (κz , ω) = !3 q j=1 Tj (rc , κz , ω), and by assembling the displacements and tractions vectors c q into the global vectors Qb (κz , ω) = q qq (κz , ω) and Hb (κz , ω) = q hq (κz , ω) according to the mesh topology, the following set of linear algebraic equations is obtained [Tb (κz , ω) + I] Qb (κz , ω) = Ub (κz , ω) Hb (κz , ω) , (x, xc ) ∈ ∂Ωb (4.44) which is deﬁned only for source points xc belonging to the boundary. Once the vectors of boundary displacements Qb (κz , ω) and tractions Hb (κz , ω) have been determined from Eq. (4.44), the radiated waveﬁeld ud (x , κz , ω) at any x ∈ Ωb can be computed using the 2.5D integral representation theorem in Eq. (3.16). The discretized form of the 2.5D integral representation theorem is given in Eq. (3.71) and is repeated here for convenience ud x , κz , ω = Ud (κz , ω) Hb (κz , ω) − Td (κz , ω) Qb (κz , ω) , x ∈ Ωb , (4.45) where the inﬂuence operators Ud (κz , ω) and Td (κz , ω) result from the following element assembling procedure Ud (κz , ω) = Td (κz , ω) = Nb ref q=1 ∂Ωq Nb ref q=1 ∂Ωq UD r (η) , κz , ω N (η) Jbq (η) dη, (4.46) TD r (η) , κz , ω N (η) Jbq (η) dη, (4.47) in which r (η) = |x (η) − x |. Since the dynamic Green’s functions are nonsingular for x ∈ Ωb , the integrals in Eqs. (4.46) and (4.47) can be evaluated numerically using the standard Gauss-Legendre quadrature formula. 4.5.3 Evaluation of weakly singular integrals As can be noted from Eqs. (4.31) and (4.33), the displacement kernels in Eq. (4.40) are weakly singular of order ln (1/r). Following [Gao and Davies, 2001], the strategy adopted in this case is to isolate the logarithmic singularity and integrate it using the Gauss-Laguerre quadrature rule, while the nonsingular residual can be integrated using the Gauss-Legendre quadrature rule. For the quadratic element of Table4.2 three cases need to be considered because the source point xc = [xc , yc ]T may be located at 122 4.5.3. EVALUATION OF WEAKLY SINGULAR INTEGRALS N1 = 12 η (η − 1) xc = xq1 N2 = 12 η (η + 1) xc = xq2 xc = xq3 N3 = 1 − η 2 Table 4.2: Shape functions for the quadratic monodimensional boundary element and logarithmic singularities for various conﬁgurations of the source point. the initial node (node 1) of coordinates xq1 = [x1 , y1 ]T , at the end node (node 2) of coordinates xq2 = [x2 , y2 ]T or the mid-side node (node 3), of coordinates xq3 = [x3 , y3 ]T . The distance rc (η) between an arbitrary point of coordinates x (η) (see Eq. (4.35)) and the source xc is obtained from the equation rc (η) = [x (η) − xc ]2 − [y (η) − yc ]2 . (4.48) If xc is located at node 1, the substitution of Eq. (4.35) into Eq. (4.48) along with the quadratic shape functions given in Table4.2 leads to the following expression 2 0 1 2 (1 + η) = [− (2 − η) xq1 + ηxq2 + 2 (1 − η) xq3 ] 2 1 2 + [− (2 − η) y1q + ηy2q + 2 (1 − η) y3q ] rc2 (4.49) Similarly, if xc is located at node 2, one obtains 2 0 1 2 (1 + η) = [− (2 + η) xq1 − ηxq2 + 2 (1 + η) xq3 ] 2 1 2 + [− (2 + η) y1q − ηy2q + 2 (1 + η) y3q ] rc2 (4.50) Eqs. (4.49) and (4.50) can be expressed in the uniﬁed form rc2 = ϕ2 f12 + f22 123 (4.51) 4. LEAKY GUIDED WAVES IN WAVEGUIDES EMBEDDED IN SOLID MEDIA: COUPLED SAFE-2.5D BEM FORMULATION where ϕ is an adimensional coordinate with origin at the same point in which the source point is located, obtained via the change of variables ϕ= 1 (1 − ηc η) 2 (4.52) in which ηc = −1 if the source point xc is located at node 1 of the element ∂Ωq and ηc = +1 if it is located at node 2. The functions f1 and f2 in Eq. (4.51) are then f1 = − (2 + ηc η) xa − ηc ηxb + 2 (1 + ηc η) x3 , f2 = − (2 + ηc η) ya − ηc ηyb + 2 (1 + ηc η) y3 , (4.53) where a = 1, b = 2 when the collocation point is located at node 1, and a = 2, b = 1 when it is located at node 2. For the case when the collocation point is located at the mid-side node xq3 , one obtains rc2 = η 2 g12 + g22 (4.54) where the functions g1 and g2 take the following representation 1 [(η − 1) x1 + (η + 1) x2 ] − ηx3 , 2 1 g2 = [(η − 1) y1 + (η + 1) y2 ] − ηy3 , 2 g1 = (4.55) Taking the logarithm of Eq. (4.51), the following expression can be obtained ln 1 rc = 1 ln ϕ − Gauss−Laguerre 2 1 2 2 ln f1 (η) + f2 (η) 2 (4.56) Gauss−Legendre The expression for the logarithm given in Eq. (4.56) can be substituted into Eqs. (4.31) and (4.33), and then into Eq. (4.40). The resulting integral can be subdivided into a singular part, containing the ﬁrst term on the right hand side of Eq. (4.56), and the regular part, containing the second term. The singular and regular integral are evaluated numerically using the Gauss-Laguerre and Gauss-Legendre quadrature formulae, respectively. An expression equivalent to Eq. (4.56) can be obtained from Eq. (4.54), valid for the case in which the source point is located in the mid-side node of the element. Finally, the boundary integrals in Eqs. (4.39), (4.41), (4.42) and (4.43) are evaluated by means of the Gauss-Legendre quadrature formula, since they are nonsingular. 124 4.6. SAFE-BE COUPLING 4.6 SAFE-BE coupling The coupling between the SAFE and the BEM regions is established via the compatibility conditions in Eqs. (4.3) and (4.4), and is carried out in a ﬁnite element sense [Andersen, 2006]. On these bases, the inﬁnite boundary element domain is converted into a single, wavenumber and frequency dependent, ﬁnite element-like domain with Nn nodes. The dynamic stiﬀness matrix of this pseudo ﬁnite element, relating nodal tractions to nodal displacements, is obtained by recasting Eq. (4.44) in the following form Hb (κz , ω) = Kb (κz , ω) Qb (κz , ω) , (4.57) where the dynamic stiﬀness matrix relating the nodal displacements and tractions Kb (κz , ω) = U−1 b (κz , ω) [Tb (κz , ω) + I] (4.58) is complex and non symmetric. The nodal tractions are then converted into nodal forces by following the procedure indicated by Andersen [2006], which uses the equivalence between the virtual work done by the integral of the surface tractions over the boundary for the virtual displacements and the virtual work resulting from the application of the equivalent nodal forces for the same virtual displacements. The work done by the surface tractions t (x, κz , ω) over the boundary in applying a virtual displacement δu (x, κz , ω) is given as [δu (x, κz , ω)]T t (x, κz , ω) ds (x) . δWb = (4.59) ∂Ωb Using the element shape functions to interpolate the displacements and the tractions, the ﬁeld quantities at any point x ∈ ∂Ωb remains determined from Eqs. (4.36) and (4.37), respectively. Substituting these equations into Eq. (4.59) and applying the discretization procedure lead to δWb = Nb q=1 q [δq (κz , ω)] T ∂Ωref b (N (η)) T N (η) Jbq (η) dη h (κz , ω) . q (4.60) Since the work done by the surface tractions for the qth element is equal to the work done by the equivalent nodal forces fbq (κz , ω) for the same virtual displacement, the 125 4. LEAKY GUIDED WAVES IN WAVEGUIDES EMBEDDED IN SOLID MEDIA: COUPLED SAFE-2.5D BEM FORMULATION following relation holds δWb = Nb [δqq (κz , ω)]T fbq (κz , ω) . (4.61) q=1 Combining Eqs. (4.60) and (4.61), the following relationship is derived Fb (κz , ω) = Tb Hb (κz , ω) , where Tb (κz , ω) = Nb ref q=1 ∂Ωb (4.62) (N (η))T N (η) Jbq (η) dη. (4.63) is a distribution matrix that relates the nodal tractions to nodal forces on the boundary. Substituting Eq. (4.57) in Eq. (4.63) leads to the following relation between nodal displacements and nodal forces Fb (κz , ω) = Tb Kb (κz , ω) Qb (κz , ω) . (4.64) Introduction of Eq. (4.64) in Eq. (4.10) gives the following complex and nonsymmetric N -dimensional linear system 0 κ2z K3 + iκz K2 − KT 2 + K1 1 + LbT [Tb Kb (κz , ω)] Lb − ω 2 M Q (κz , ω) = Fs (κz , ω) , (4.65) in which Qb (κz , ω) = Lb Q (κz , ω) (4.66) is a matrix that collocates the global vector of nodal displacements on the boundary into the global vector of nodal displacements of the SAFE mesh. The displacement ﬁeld at any x ∈ (Ωs ∪ ∂Ωs ) can be obtained by solving the N × N linear system Eq. (4.65) in the unknown nodal displacements Q (κz , ω) and using the interpolation in Eq. (4.7). In addition, substituting Eq. (4.57) into Eq. (4.45) leads to the following 3 × N linear system of equations ud x , κz , ω = [Ud (κz , ω) Kb (κz , ω) − Td (κz , ω)] Lb Q (κz , ω) , x ∈ Ωb (4.67) which allows to compute the radiated displacement waveﬁeld at any source point belonging to the surrounding domain. 126 4.7. DISPERSION ANALYSIS 4.7 Dispersion analysis The dispersion properties of guided modes are determined in terms of complex wavenumbers κz (ω) for any ﬁxed ω > 0 in absence of external forces applied at the interface ∂Ωs . Substituting Fs (κz , ω) = 0 in Eq. (4.65), the dispersive equation for the unbounded waveguide of domain Ωs ∪ Ωb results in the following nonlinear eigenvalue problem in κz (ω) Z (κz , ω) Q (κz , ω) = 0, (4.68) where 0 Z (κz , ω) = κ2z K3 + iκz K2 − KT 2 + K1 1 + LbT [Tb Kb (κz , ω)] Lb − ω 2 M , ∈ CN,N (4.69) is the dynamic stiﬀness matrix of the coupled SAFE-2.5D BEM model. As shown in Chapter 3 the Contour Integral Method proposed by Beyn [2012] can be applied to transform the nonlinear eigenvalue problem Eq. (4.68) into a linear one inside a simple closed curve Γ (κz ) ∈ C where poles of the guided modes must be sought. Since the algorithm described in Sec. 3.8.1 remains unchanged and requires only the substitution of Z (κz , ) with the expression in Eq. (4.69), the procedure will not be repeated here. On the other hand, the analysis of Sec. 3.8.2 is no longer valid for the case of embedded waveguides, and the single-valued deﬁnition of the dynamic stiﬀness matrix Eq. (4.69) must be revised. 4.7.1 Single-valued deﬁnition of the dynamic stiﬀness matrix The procedure reported in Sec. 3.8.2 allows to extract all the eigenvalues for a holomorphic problem Z (κz , ω) ∈ H(Ω∗ , CK,K ), where Ω∗ denotes the region of the complex κz plane enclosed by Γ (κz ). However, this condition is not generally satisﬁed as Z (κz , ω) (1) is singular and multivalued due to the properties of the Hankel functions Hn (·) as well as the two wavenumbers κα = ±(κ2L − κ2z )1/2 and κβ = ±(κ2S − κ2z )1/2 . Before performing the contour integration in Eqs. (3.75) and (3.76), the operator Z (κz , ω) must be made single valued and analytic everywhere inside Ω∗ . This task is accomplished by choosing the phase of κα and κβ consistently with the nature of the existing partial bulk waves in the surrounding medium, and by removing points of singularity and discontinuity in the κz -plane. The signs of the wavenumbers normal to the interface, with reference to the more 127 4. LEAKY GUIDED WAVES IN WAVEGUIDES EMBEDDED IN SOLID MEDIA: COUPLED SAFE-2.5D BEM FORMULATION Figure 4.2: Complex κz -plane with bulk wavenumbers, vertical branch cuts and integration path for an external isotropic viscoelastic medium. The signs of κα and κβ on Ω∗ and along Γ (κz ) are determined by imposing the conditions on their imaginary parts as indicated in the diﬀerent regions. general viscoelastic case of Fig. 4.2, are established as in the following: for Re(κz ) > Re(κS ), the Snell-Descartes law [Borcherdt, 2009; Rose, 2004] enforces a total reﬂection, with possible mode conversion at the interface, of the longitudinal and shear bulk waves traveling inside the waveguide (non-leaky region). In this case, the particles motion in the surrounding medium remains conﬁned in proximity of the interface, with amplitude decaying exponentially in the direction normal to the interface [Auld, 1973; Pavlakovic, 1998]. Since (1) the propagation process is represented by the Hankel functions Hn (κα r) and (1) Hn (κβ r) and assumes a dependence exp[i(κz z − ωt)], in order to have outgoing waves satisfying the radiation condition at inﬁnity, the signs of κα and κβ must be chosen so that Im(κα ) > 0 and Im(κβ ) > 0. In the range Re(κL ) < Re(κz ) < Re(κS ), the longitudinal bulk waves are still totally reﬂected at the interface, and the sign of κα is then selected in order to preserve the positiveness of its imaginary component, which satisﬁes the radiation condition at inﬁnity. On the other hand, shear bulk waves are also refracted at −1 Re(κz ) Leak = sin some leakage angle ϑS Re(κS ) with respect to the normal at the 128 4.7.1. SINGLE-VALUED DEFINITION OF THE DYNAMIC STIFFNESS MATRIX Figure 4.3: Wave vectors conﬁgurations for the point P 3 of Fig. 4.2. The propagation vector kRe S is oriented along the radiation direction (dashed gray lines), while the attenuation vector kIm S is perpendicular to equi-amplitude lines (solid gray lines) and oriented in the direction of maximum decay. Magnitude of displacements is proportional to the thickness of equi-amplitude lines. interface [Auld, 1973; Castaings and Lowe, 2008] and therefore, for the properties of the nth order Hankel function of the ﬁrst kind, sgn(κβ ) must be chosen in order to satisfy the condition Re(κβ ) > 0. Regarding the imaginary component of κβ , it must be observed that for any ﬁxed positive Re(κz ) ∈ [0, Re(κS )], Im(κβ ) changes monotonically as a function of Im(κz ) and vanishes for values of Im(κz ) = Re(κS )Im(κS )/Re(κz ), which deﬁne a branch of hyperbola passing through the point κS . This branch of hyperbola, indicated with qβ in Fig. 4.2, determines the transition between an outgoing growing (Im(κβ ) < 0) and an outgoing decaying (Im(κβ ) > 0) shear waves waveﬁeld along the orthogonal direction to the interface. These physical states are represented by points P 1 and P 3 in Fig. 4.2, respectively, while the transition state (Im(κβ ) = 0) is represented by point P 2 on qβ . The wavevector conﬁgurations for points P 1, P 2 and P 3 for a planar interface, are shown in Figs. 4.5-4.3 in terms of propagation and attenuation vectors, kRe S Re Im Re Im and kIm S , respectively, with |kS ||kS |cos(γS ) = kS · kS and 0 < γS < π/2 [Carcione et al., 1988; Caviglia et al., 1990]. The attenuation vector is given Im Im Im by kIm S = kSd + kSi , where kSd is the component due to material damping (homogeneous component), parallel to kRe according to the material damping 129 4. LEAKY GUIDED WAVES IN WAVEGUIDES EMBEDDED IN SOLID MEDIA: COUPLED SAFE-2.5D BEM FORMULATION Figure 4.4: Wave vectors conﬁgurations for the point P 2 of Fig. 4.2. The propagation vector kRe S is oriented along the radiation direction (dashed gray lines), while the attenuation vector kIm S is perpendicular to equi-amplitude lines (solid gray lines) and oriented in the direction of maximum decay. Magnitude of displacements is proportional to the thickness of equi-amplitude lines. model of Eq. (4.9), and kIm Si is the component associated to energy radiation (inhomogeneous component), which is normal to kRe S [Cervený and Pšencı́k, 2011]. All the wavenumber vectors lie on the plane containing the z-axis and the outward normal n at the interface. For any κz above qβ (point P 1), imposition of Re(κβ ) > 0 implies that Im(κβ ) < 0 Im and the propagation and attenuation vectors normal to the interface, kRe β and kβ , respectively, result in opposite directions (Fig. 4.5). Since kIm is perpendicular to the lines of constant amplitudes in the z − n plane and is oriented in the direction of the maximum decay of amplitude [Cervený and Pšencı́k, 2011], a well known characteristic of leaky waves can be observed: while material damping (homogeneous component) causes the amplitude of the partial shear wave to decrease along the radiation direction (dashed lines), due to the inhomogeneous component the wave amplitude increases in direction n. This behaviour can be observed from the intersections of the equi-amplitude lines (solid lines) with the normal to the interface and has been already discussed by diﬀerent authors [Simmons et al., 1992; Viens et al., 1994; Vogt et al., 2003] in the special case of isotropic elastic open waveguides, for which kIm S = 0 and γS = π/2 [Carcione et al., 1988; Caviglia et al., 1990]. 130 4.7.1. SINGLE-VALUED DEFINITION OF THE DYNAMIC STIFFNESS MATRIX Figure 4.5: Wave vectors conﬁgurations for the point P 1 of Fig. 4.2. The propagation vector kRe S is oriented along the radiation direction (dashed gray lines), while the attenuation vector kIm S is perpendicular to equi-amplitude lines (solid gray lines) and oriented in the direction of maximum decay. Magnitude of displacements is proportional to the thickness of equi-amplitude lines. If κz lies below qβ (point P 3), then Re(κβ ) > 0 implies that Im(κβ ) > 0, and (1) Hn (κβ r) is therefore convergent. In this case, the shear wave amplitude decreases along both the radiation direction and the direction normal to the interface (Fig. 4.3). Analogous considerations apply in the range Re(κz ) < Re(κL ), where both longitudinal and shear bulk waves are radiated in the surrounding medium. Once the signs for κα and κβ have been determined, the operator Z (κz , ω) results singlevalued everywhere on Ω∗ . To make it also analytical, it is necessary to remove points corresponding to singularities and discontinuities. In this study, an approach similar to the Vertical Branch Cut Integration (VBCI) method [He and Hu, 2010; Kurkjian, 1985; Liu and Chang, 1996; Zhang et al., 2009] has been adopted. For the given choices of sgn(κα ) and sgn(κβ ), points of discontinuities are represented by the two vertical lines departing from κL = ω/c̃L and κS = ω/c̃S and extending along the positive direction of the imaginary axis (see Fig. 4.2). These lines delimit zones of Ω∗ where κα and κβ change sign in both their real and imaginary components. The two vertical cuts are therefore introduced to remove these discontinuities. (1) These cuts also include the bulk wavenumbers, since Hn (κα,β r) are not deﬁned for κz → κα,β . The last branch cut is represented by the whole negative real axis, which 131 4. LEAKY GUIDED WAVES IN WAVEGUIDES EMBEDDED IN SOLID MEDIA: COUPLED SAFE-2.5D BEM FORMULATION Figure 4.6: Complex κz -plane with bulk wavenumbers, vertical branch cuts and integration path for an external isotropic elastic medium. The signs of κα and κβ on Ω∗ and along Γ (κz ) are determined by imposing the conditions on their imaginary parts as indicated in the diﬀerent regions. is a branch cut of the Hankel function, and is easily avoided by restricting the contour Γ (κz ) only to positive real values of the axial wavenumbers (right propagating waves). The integral path in Fig. 4.6 represents a special case of that in Fig. 4.2 when an isotropic elastic surrounding medium is considered. In this case, κL and κS are purely real and the hyperbolic lines qα and qβ collapse on the positive imaginary axis and part of the real axis. Since in this case kIm L,S = 0, lines of constant amplitude become parallel to the radiation direction (γL,S = π/2), causing the displacement ﬁeld to grow with distance in the direction normal to the interface [Vogt et al., 2003]. 4.7.2 Dispersion characteristics extraction m Once the complete set of eigensolutions [κm z (ω) , Q (ω)] has been determined from Eq. (4.68) for the frequency of interest, the dispersion characteristics for the mth guided mode are computed as 132 4.8. NUMERICAL APPLICATIONS cm p (ω) = ω , Re[κm z (ω)] αm (ω) = Im[κm z (ω)], cm e (ω) = 1 4 Re 0 ω 2 Im (4.70) 0 Ωs [um (ω)]H LzT C̃ (ω) m (ω) dxdy (4.71) 1 H m H 2 m m m Ωs ω ρ [u (ω)] u (ω) + [ (ω)] C̃ (ω) (ω) dxdy 1, (4.72) where the displacements um (ω) and strains m (ω) on Ωs are recovered from Qm (ω) using the interpolations Eq. (4.7) and the strain-displacement relations Eq. (4.8), respectively. It should be noted that Eq. (4.72) does not represent the exact expression of the energy velocity for leaky guided modes. In this case, energy ﬂow curves bend away from the waveguide into the surrounding medium, determining an axial component of the energy ﬂow on Ωb [Auld, 1973; Castaings and Lowe, 2008; Molz and Beamish, 1996; Simmons et al., 1992]. In such circumstances, the domain integrals in Eq. (4.72) should be rigorously evaluated on Ωs ∪ Ωb , which has inﬁnite extension. However, Eq. (4.72) is commonly accepted as suﬃciently accurate in GUW applications [Pavlakovic, 1998; Pavlakovic and Lowe, 2003] and becomes exact for non-leaky modes (Re(κz ) > Re(κS )), being the waveﬁeld on Ωb constituted by evanescent waves [Auld, 1973]. In this case there is no energy ﬂux through Ωb , with the total energy remaining conﬁned within Ωs and ﬂowing parallel to the interface. 4.8 Numerical applications In this section, ﬁve numerical applications are presented. The ﬁrst two, which have been studied in literature using the Global Matrix Method and the SAFE method with absorbing regions, are used as validation cases, while the remaining three applications are proposed to show the unique capabilities of the coupled SAFE-2.5D BEM formulation to compute dispersive properties of leaky waves in embedded waveguides of arbitrary cross-section. The material properties used in the analyses are listed in Tab. 4.3. Since only the Maxwell rheological model is considered in this study, the material constants are independent from frequency [Bartoli et al., 2006]. The settings of the contour algorithm have been deﬁned on the basis of single analysis performed at few frequencies, by changing the parameters (Np , tolrank and tolres ) until a stable trend was observable in the separation of the singular values as well as the relative residuals of eigensolutions. 133 4. LEAKY GUIDED WAVES IN WAVEGUIDES EMBEDDED IN SOLID MEDIA: COUPLED SAFE-2.5D BEM FORMULATION material steel 1 concrete grout soil 1 2 3 ρi ciL ciS βLi βSi (Kg/m3 ) (m/s) (m/s) (Np/wavelength) (Np/wavelength) st 7932 5960 3260 0.003 0.008 co 2300 4222.1 2637.5 - - gr 1600 2810 1700 0.043 0.100 so 1750 1000 577 0.126 0.349 i 2 1 3 Pavlakovic et al. [2001] Castaings and Lowe [2008] Ketcham et al. [2001] Table 4.3: Materials constants used for the numerical analyses in Sec. 4.8 4.8.1 Elastic steel bar of circular cross section embedded in elastic concrete In the ﬁrst example, the coupled SAFE-BEM formulation is validated with respect to the FEM solution proposed by Castaings and Lowe [2008] for a 20 mm diameter elastic steel (st) bar embedded in elastic concrete (co). The SAFE mesh used in the analysis is composed of 48 six-node triangular elements and 32 nine-node quadrilateral elements, as shown in Fig. 4.7. The BEM mesh matches the SAFE mesh at the interface and is composed of 32 three-node monodimensional elements. The steel longitudinal and shear bulk wave attenuations listed in Tab. 4.3, βLst and βSst , respectively, are neglected. The dispersion curves, represented in Figs. 4.8-4.10 in terms of phase velocity, attenuation and energy velocity, have been obtained by considering the upper limit of the integration path in Fig. 4.6 equal to 200 Np/m (1737.18 dB/m), while the horizontal extension has been limited to Re(κco S ) at each frequency step. The attenuation value has been added to the phase and energy velocity curves ﬁlling the circular markers with diﬀerent blue levels. Light and dark levels denote higher and lower values of the attenuations, respectively. The results for the L(0, 1), F (1, 1) and F (1, 2) modes are in very good agreement with those in Ref. [Castaings and Lowe, 2008]. Of the remaining modes, it is interesting to observe the global behaviour of the F (2, 1), which experiences three discontinuities in the range 0 − 200 kHz. The ﬁrst discontinuity is located at about 40 kHz, where F (2,1) the mode becomes leaky (cp > cco S ). Moreover, the energy velocity in the frequency 134 4.8.1. ELASTIC STEEL BAR OF CIRCULAR CROSS SECTION EMBEDDED IN ELASTIC CONCRETE Figure 4.7: SAFE-BEM mesh of the elastic steel bar of circular cross section embedded in elastic concrete. F (2,1) range corresponding to cco S < cp < cco L is negative. The second discontinuity occurs when the mode crosses the longitudinal bulk velocity of the concrete. In the frequency range 82−130 kHz, where the mode is indicated as F (2, 1) , both longitudinal and shear bulk waves are leaked in the concrete. The third discontinuity occurs in the frequency range 130 − 136 kHz, where the phase velocity becomes lower than the longitudinal bulk velocity of the concrete, thus corresponding to radiation of shear bulk waves only. 135 4. LEAKY GUIDED WAVES IN WAVEGUIDES EMBEDDED IN SOLID MEDIA: COUPLED SAFE-2.5D BEM FORMULATION 10000 α [dB/m] 9000 0 F (2, 2) 1750 F (1, 3) 8000 F (1, 2) cp [m/s] 7000 6000 F (3, 1) T (0, 1) L(0, 2) L(0, 1) 5000 F (2, 1) cco L F (2, 1) 4000 F (2, 1) 3000 2000 0 L(0, 1) F (1, 1) 50 100 150 Frequency [kHz] cco S 200 Figure 4.8: Phase velocity dispersion curves for the elastic steel bar of circular cross section embedded in elastic concrete of Fig. 4.7. Modes are indicated as in Ref. [Castaings and Lowe, 2008]. 1600 F (3, 1) F (2, 1) 1400 F (1, 3) α [dB/m] 1200 L(0, 2) F (2, 1) 1000 800 F (1, 2) T (0, 1) F (2, 2) 600 F (2, 1) 400 200 0 0 L(0, 1) F (1, 1) 50 L(0, 1) 100 150 200 Frequency [kHz] Figure 4.9: Attenuation dispersion curves for the elastic steel bar of circular cross section embedded in elastic concrete of Fig. 4.7. Modes are indicated as in Ref. [Castaings and Lowe, 2008]. 136 4.8.1. ELASTIC STEEL BAR OF CIRCULAR CROSS SECTION EMBEDDED IN ELASTIC CONCRETE 5000 α [dB/m] L(0, 1) cco L 1750 0 4000 ce [m/s] F (1, 1) 3000 cco S L(0, 1) F (2, 1) 2000 F (1, 2) T (0, 1) F (2, 2) L(0, 2) F (1, 3) 1000 F (2, 1) 0 0 F (3, 1) F (2, 1) 50 100 150 200 Frequency [kHz] Figure 4.10: Energy velocity dispersion curves for the elastic steel bar of circular cross section embedded in elastic concrete of Fig. 4.7. Modes are indicated as in Ref. [Castaings and Lowe, 2008]. 137 4. LEAKY GUIDED WAVES IN WAVEGUIDES EMBEDDED IN SOLID MEDIA: COUPLED SAFE-2.5D BEM FORMULATION 4.8.2 Viscoelastic steel bar of circular cross section embedded in viscoelastic grout In the second example, a 20 mm diameter viscoelastic steel bar (st) embedded in viscoelastic grout (gr) is considered. This example is used to validate the proposed SAFE-BEM formulation for a case in which all the materials are viscoelastic. The cross section is discretized with the same type and number of elements used in Sec. 4.8.1. The obtained dispersion curves, shown in Figs. 4.12-4.13, are very similar to those in Ref. [Pavlakovic et al., 2001], in which the same problem has been solved by using the software DISPERSE [Pavlakovic et al., 1997]. In the analysis, the imaginary part of the integral path in Fig. 4.2 has been limited to 200 Np/m (1737.18 dB/m). The dispersion curves for the fundamental longitudinal mode, L(0, 1), and fundamental ﬂexural mode, F (1, 1), are in very good agreement with those in the Ref. [Pavlakovic et al., 2001]. It is also worth noting that the contour integral method is able to detect the portion of the F (1, 1) mode in the frequency range 0 − 15 kHz, although the non-leaky poles lie in this case very close to κgr S . As indicated by Beyn [2012], the contour integral method is indeed able to detect the roots if they lie outside but close to the contour, although the accuracy becomes strongly dependent on the number of integration points used in proximity of the same roots. Since Z (κz , ω) is not deﬁned for κz → κgr S , the solutions provided by the contour integral method can be inaccurate. In fact, some of these solutions have been found to lie in the non-leaky region, while Pavlakovic et al. [2001] have excluded the existence of the F (1, 1) mode in this region. Therefore, to get precise and reliable solutions for κz → κgr S , the roots obtained by the contour integral method were improved by using them as initial guesses in the Muller’s root ﬁnding algorithm [Press et al., 1992]. As in the elastic case of Sec. 4.8.1, discontinuities occur when the modes cross the bulk velocities of the external medium. The discontinuities for the F (1, 1) mode in the phase velocity spectra are mild compared with those of the F (2, 1) mode. The corresponding jumps in attenuation are clearly observable. As for the F (2, 1) mode in F (2,1) Sec. 4.8.1, the branch of the mode that satisﬁes the condition cgr S < cp a negative energy velocity. 138 < cgr L shows 4.8.2. VISCOELASTIC STEEL BAR OF CIRCULAR CROSS SECTION EMBEDDED IN VISCOELASTIC GROUT Figure 4.11: SAFE-BEM mesh of the viscoelastic steel bar of circular cross section embedded in viscoelastic grout. 11000 10000 9000 α [dB/m] F (2, 2) 0 1750 F (1, 3) F (1, 2) 8000 T (0, 1) cp [m/s] 7000 6000 F (2, 1) L(0, 1) L(0, 2) 5000 F (3, 1) F (2, 1) 4000 3000 cgr L 1000 0 0 cgr S F (1, 1) 2000 discontinuity in attenuation due to leakage of S-waves 50 discontinuity in attenuation due to leakage of L-waves 100 150 200 Frequency [kHz] Figure 4.12: Phase velocity dispersion curves for the viscoelastic steel circular bar embedded in viscoelastic grout of Fig. 4.11. Modes are indicated as in Ref. [Pavlakovic et al., 2001]. 139 4. LEAKY GUIDED WAVES IN WAVEGUIDES EMBEDDED IN SOLID MEDIA: COUPLED SAFE-2.5D BEM FORMULATION 1000 900 F (1, 3) 800 L(0, 2) α [dB/m] 700 F (1, 2) F (2, 1) 600 500 T (0, 1) F (3, 1) 400 F (2, 2) 300 200 F (1, 1) F (1, 1) 100 F (1, 1) L(0, 1) 0 0 50 100 150 200 Frequency [kHz] Figure 4.13: Attenuation dispersion curves for the viscoelastic steel circular bar embedded in viscoelastic grout of Fig. 4.11. Modes are indicated as in Ref. [Pavlakovic et al., 2001]. 6000 α [dB/m] L(0, 1) 5000 1750 0 ce [m/s] 4000 F (1, 1) 3000 cgr L F (2, 2) cgr S 2000 T (0, 1) 1000 F (1, 3) F (1, 2) L(0, 2) F (2, 1) 0 0 F (2, 1) F (3, 1) 50 100 150 200 Frequency [kHz] Figure 4.14: Energy velocity dispersion curves for the viscoelastic steel circular bar embedded in viscoelastic grout of Fig. 4.11. Modes are indicated as in Ref. [Pavlakovic et al., 2001]. 140 4.8.3. VISCOELASTIC STEEL BAR OF SQUARE CROSS SECTION EMBEDDED IN VISCOELASTIC GROUT 4.8.3 Viscoelastic steel bar of square cross section embedded in viscoelastic grout The third example considers a square viscoelastic steel bar (st), of 20 mm side length, embedded in viscoelastic grout (gr) (see Fig. 4.15). Actually, in the considered frequency range the steel does not generally exhibit material damping, thus, to test the presented method a small damping was artiﬁcially added by considering the material as viscoelastic. The bar is discretized with 100 eight-node quadrilateral elements, while a boundary mesh of 40 three-node monodimensional elements is adopted to model the surrounding space. A maximum attenuation of 200 Np/m (1737.18 dB/m) has been considered in the analysis. The modes in the dispersion spectra of Fig. 4.16-4.18 have been labeled as in Ref. [Gunawan and Hirose, 2005], where a square waveguide in vacuum was considered. It can be observed that, in the frequency range 0 − 13 kHz, the ﬁrst ﬂexural mode, F1 , behaves similarly to the F (1, 1) mode for the circular bar in Sec. 4.8.2. Also in this case the Muller’s method has been applied to improve the accuracy of these solutions. It is also interesting noting the existence of a non-leaky section for the two skew modes S11 and S12 . Similarly to the F (2, 1) mode in both the examples of Secs. 4.8.1 and 4.8.2, the energy velocity of these modes becomes positive S1 S2 gr 1 for values of cp 1 > cgr L and cp > cL . 141 4. LEAKY GUIDED WAVES IN WAVEGUIDES EMBEDDED IN SOLID MEDIA: COUPLED SAFE-2.5D BEM FORMULATION Figure 4.15: SAFE-BEM mesh of the viscoelastic steel bar of square cross-section embedded in viscoelastic grout. 11000 10000 9000 α [dB/m] S22 F2 1750 0 S31 8000 cp [m/s] F3 T1 7000 S12 6000 F4 5000 4000 3000 L1 L2 cgr L F1 2000 1000 0 0 S21 S11 S11 S12 S12 50 cgr S S11 100 150 200 Frequency [kHz] Figure 4.16: Phase velocity dispersion curves for the viscoelastic steel square bar embedded in viscoelastic grout of Fig. 4.15. Modes are indicated as in Ref. [Gunawan and Hirose, 2005], where a square bar in vacuum was considered. 142 4.8.3. VISCOELASTIC STEEL BAR OF SQUARE CROSS SECTION EMBEDDED IN VISCOELASTIC GROUT 1000 900 800 α [dB/m] 700 S12 F4 F2 600 500 F3 S11 S21 T1 L2 400 300 200 S22 F1 F1 100 L1 0 0 50 100 150 200 Frequency [kHz] Figure 4.17: Attenuation dispersion curves for the viscoelastic steel square bar embedded in viscoelastic grout of Fig. 4.15. Modes are indicated as in Ref. [Gunawan and Hirose, 2005], where a square bar in vacuum was considered. α [dB/m] 5000 L1 1750 0 S22 4000 L2 ce [m/s] F1 3000 cgr L 1000 0 F4 T1 F2 S12 0 S11 cgr S 2000 S12 S11 S21 S31 S12 F3 S11 50 100 150 200 Frequency [kHz] Figure 4.18: Energy velocity dispersion curves for the viscoelastic steel square bar embedded in viscoelastic grout of Fig. 4.15. Modes are indicated as in Ref. [Gunawan and Hirose, 2005], where a square bar in vacuum was considered. 143 4. LEAKY GUIDED WAVES IN WAVEGUIDES EMBEDDED IN SOLID MEDIA: COUPLED SAFE-2.5D BEM FORMULATION 4.8.4 Viscoelastic steel HP200 beam embedded in viscoelastic soil Despite the fact that non-destructive evaluation of pile-integrity is an important topic in geotechnical engineering [Ding et al., 2011; Liu, 2012; Ni et al., 2008], dispersion analyses of guided waves propagating in foundation piles seem to be limited in literature to simple geometries [Finno and Chao, 2005; Finno et al., 2001]. In this example, the proposed formulation is exploited to predict the dispersion curves for an HP200 steel (st) pile embedded in soil (so). Both steel and surrounding soil are treated as linear viscoelastic materials. The attenuations of L and S waves in soils have been investigated by Ketcham et al. [2001] and are reported in Tab. 4.3 for a surface soil layer. The pile is discretized with 52 eight-node quadrilateral elements and 2 six-node triangular elements, as shown in Fig. 4.19. The BEM mesh is composed of 108 threenode monodimensional elements. Since only the ﬁrst low order modes are of interest in practical applications, the analysis has been carried out by considering a maximum attenuation of 9.2 Np/m (80.86 dB/m), where these modes have been found to exist in the frequency range 0 − 1000 Hz. The low order modes are indicated with m1, m2, m3 and m4. It can be noted that all these modes are discontinuous in correspondence of the soil bulk velocities. The ﬂexural-like mode m4, which is indicated with m4 for m4 > cso , exists in both the two leaky zones of the > cso cm4 p S and with m4 for cp L spectra in the frequency range 600 − 870 Hz. The longitudinal-like mode m1 becomes almost non-dispersive in the frequency range 700 − 1000 Hz, while its attenuation so remains almost constant in the frequency range corresponding to cm1 p > cL . Since this mode also shows the highest energy velocity combined with the minimum attenuation if compared to the remaining low order modes, it can be particularly suitable in practical inspection applications. The behaviour of the radiated waveﬁeld for the ﬂexo-torsional m2 mode is examined with reference to the analysis of Sec. 4.7.1. From Fig. 4.21, it can be noted that the attenuation of this mode is always greater than the attenuations of both longitudinal so so so and shear bulk waves, αso L (ω) = Im[κL (ω)] and αS (ω) = Im[κS (ω)], respectively. It is therefore expected that the amplitudes of both longitudinal and shear waves in their corresponding leaky zones must increase with distance along the direction normal to so the boundary. On the other hand, for cm2 p < cS , the radiated waveﬁeld must decay with distance from the pile-soil interface. These behaviours can be observed in Fig. 4.23, where the wavestructures Qm2 (ω) of the m2 mode at various frequencies have been substituted in Eq. (4.67) to compute the radiated waveﬁeld at the nodes x of an external mesh. As can be noted, the waveﬁeld in soil for the m2 mode at 88.38 Hz 144 4.8.4. VISCOELASTIC STEEL HP200 BEAM EMBEDDED IN VISCOELASTIC SOIL Figure 4.19: SAFE-BEM mesh of the HP200 viscoelastic steel beam embedded in viscoelastic soil. decays rapidly away from the interface, while the waveﬁeld amplitudes for the m2 mode at 616.16 Hz (radiated S waves) and the m2 mode at 952.02 Hz (radiated S and L waves) increase with distance from the interface. It should be kept in mind that the modes attenuations resulting from the present analysis are probably overestimated due to the assumptions in Eq. (4.3). Lower and more realistic attenuations could be predicted by inserting an appropriate thin layer between the two media as previously done by Nayfeh and Nagy [Nayfeh and Nagy, 1996] and by Pavlakovic [Pavlakovic, 1998]. 145 4. LEAKY GUIDED WAVES IN WAVEGUIDES EMBEDDED IN SOLID MEDIA: COUPLED SAFE-2.5D BEM FORMULATION 4000 α [dB/m] 3500 0 80 3000 m1 cp [m/s] 2500 2000 m4 m2 1500 1000 m3 cso L m4 500 m2 m2 0 0 0.2 0.4 m4 0.6 cso S m3 0.8 1 Frequency [kHz] Figure 4.20: Phase velocity velocity dispersion curves for the viscoelastic steel HP200 beam embedded in viscoelastic soil of Fig. 4.19. 80 70 α [dB/m] 60 m2 50 m2 40 m2 m4 30 m4 20 m3 m3 m4 m1 10 αSso 0 0 0.2 0.4 αso L 0.6 0.8 1 Frequency [kHz] Figure 4.21: Attenuation dispersion curves for the viscoelastic steel HP200 beam embedded in viscoelastic soil of Fig. 4.19. 146 4.8.4. VISCOELASTIC STEEL HP200 BEAM EMBEDDED IN VISCOELASTIC SOIL 4000 α [dB/m] 3500 0 80 m1 3000 ce [m/s] 2500 m3 2000 m3 m4 1500 1000 500 m4 cso L m4 cso S 0 0 m2 m2 m2 0.2 0.4 0.6 0.8 1 Frequency [kHz] Figure 4.22: Energy velocity dispersion curves for the viscoelastic steel HP200 beam embedded in viscoelastic soil of Fig. 4.19. 147 4. LEAKY GUIDED WAVES IN WAVEGUIDES EMBEDDED IN SOLID MEDIA: COUPLED SAFE-2.5D BEM FORMULATION (a) (b) (c) Figure 4.23: Mode shapes and waveﬁeld in soil for (a) mode m2 at 88.38 Hz, (b) mode m2 at 616.16 Hz and (c) mode m2 at 952.02 Hz. 148 4.8.5. RECTANGULAR HSS40 20 2 VISCOELASTIC STEEL TUBE EMBEDDED IN VISCOELASTIC GROUT 4.8.5 Rectangular HSS40 20 2 viscoelastic steel tube embedded in viscoelastic grout The last numerical example considers the rectangular HSS40 20 2 viscoelastic steel tube embedded in viscoelastic grout of Fig. 4.24. The cross-section is discretized using a mesh of 36 eight-nodes quadratic elements for the steel section and a boundary element mesh of 36 three-nodes monodimensional elements for the surrounding grout. The dispersion curves, represented in Fig. 4.25 in terms of real wavenumber, phase velocity, attenuation and energy velocity, are extracted in the frequency range 0 ÷ 35 kHz by considering modes with attenuation lower than 903.33 dB/m, i.e. by limiting the contour Γ (κz ) to values of κz with imaginary component Im (κz ) ≤ 104 Np/m at each frequency step. The ﬁrst four fundamental modes are identiﬁed with m1 (longitudinal-like mode), m2 (torsional-like mode), m3 (ﬁrst ﬂexural-like mode) and m4 (second ﬂexural-like mode). From the inspection of the dispersive spectra of real wavenumbers (Fig. 4.25(a)) it can be noted that when the real part of a guided mode crosses the lines Re[κgr L,S (ω)] = ω/Re(c̃gr L,S ) a jump in both the real and imaginary part of the mode is observed, which reﬂects into a jump in the phase velocity, attenuation and energy velocity. The existence of such jumps has been explained by Pavlakovic [1998] and it’s essentially due to the diﬀerent amount of energy radiated while passing from the non-leaky to the leaky regime or between two leaky regimes. From a physical point of view, their existence may indicate the presence of some modes with diﬀerent wavenumbers at the same frequency (backward jumps) or that the phase velocity of the mode correspond to the bulk phase velocity of the surrounding material. However, as noted by Pavlakovic [1998], sections of curves corresponding to jumps are unsuitable in non-destructive testing because of the unstable characteristics of the corresponding guided modes. Moreover, a mathematical link between diﬀerent branches of the same mode is generally diﬃcult to obtain, since the dynamic stiﬀness matrix Kb (κz , ω) is numerically unstable for values of κz close to the two bulk wavenumbers of the surrounding medium. Besides the fact that the bulk wavenumbers of the surrounding medium are excluded by the integration path of Fig. 4.2, the contour integral method has however the additional capability to extract roots located outside the contour but close to it [Beyn, 2012]. The accuracy depends in this case on the number of integration points used to discretize the contour near the bulk wavenumbers. An example is given by the second ﬂexural mode m4 in the frequency range 0 ÷ 8 kHz. In 149 4. LEAKY GUIDED WAVES IN WAVEGUIDES EMBEDDED IN SOLID MEDIA: COUPLED SAFE-2.5D BEM FORMULATION Figure 4.24: SAFE-BEM mesh of the embedded HSS40 20 2 rectangular steel tube . this range, the phase velocity of the mode is almost equal to the shear bulk velocity (1) in the cement grout and therefore the Hankel function Hn (κβ r) is divergent, leading to the numerical instability. A root searching analysis has been performed in this frequency range by comparing the performances of the contour integral method with those of the Muller’s algorithm [Press et al., 1992], which uses a quadratic interpolation to ﬁnd the minimum of det [Z (κz , ω)]. Both the algorithms are able to extract complex roots, but the Muller’s algorithm leads to more inaccurate results due to the numerical instabilities in the computation of the determinant. On the other hand, the contour integral method can extract the roots more accurately by performing the integration along the non-singular path around the bulk wavenumbers. However, from the energy velocity spectra of Fig. 4.25(d) it can be noted that the computation of eigenvectors for the mode m4 in the frequency range 0 ÷ 8 kHz should be further improved in order to obtain acceptable results. This can be achieved by increasing the number of integration points along the integration path in proximity of the bulk wavenumbers. The frequency range 8 ÷ 24 kHz corresponds to a gap in the mode m4, while in the frequency range 24 ÷ 35 kHz the phase velocity of the mode decreases and the mode becomes non-leaky. The ﬁrst fundamental pseudo-ﬂexural mode m3 shows a phase velocity always bounded between the two bulk velocities of the grout in the frequency range 11 ÷ 35 kHz. Although not shown in the dispersion spectra, the behaviour of this mode in the frequency range 0 ÷ 11 kHz is expected to be similar to that of the m4 mode, with the phase velocity almost equal to the shear bulk velocity of the grout. The fundamental pseudo-torsional mode m2 behaves similarly to the mode m3, al- 150 4.8.5. RECTANGULAR HSS40 20 2 VISCOELASTIC STEEL TUBE EMBEDDED IN VISCOELASTIC GROUT though it can be noted that in the frequency range 20 ÷ 35 kHz both its phase velocity and attenuation are almost constant, thus showing a nearly-nondispersive characteristic. This mode, represents therefore a particularly suitable guided mode for nondestructive testing, along with the pseudo-longitudinal mode m1. The latter shows in fact the highest phase velocity and is weakly dispersive in the frequency range 30 ÷ 35 kHz, where it also shows the highest energy velocity. However, from the attenuation spectra of Fig. 4.25(c) it can be noted that in the above frequency range the mode m2 has a lower attenuation. The remaining modes of the spectra correspond to higher order modes, some of which are not shown in Figs. 4.25(b) and 4.25(c) for representative reasons. As can be noted, higher order modes generally present several jumps and are characterized by higher attenuation values compared with the fundamental modes. 151 4. LEAKY GUIDED WAVES IN WAVEGUIDES EMBEDDED IN SOLID MEDIA: COUPLED SAFE-2.5D BEM FORMULATION 150 5000 m4 4500 m1 m2 4000 m3 cp [m/s] Re(κz ) [rad/m] 100 Re[κgr S (ω)] 50 3500 3000 cgr L Re[κgr L (ω)] 2500 m3 m4 m1 m2 2000 cgr S 0 0 5 10 15 20 25 Frequency [kHz] 30 m4 1500 0 35 5 10 15 20 Frequency [kHz] (a) 25 30 35 (b) 5000 500 4500 450 4000 m1 400 3500 350 ce [m/s] α [dB/m] 3000 300 250 m3 cgr L 2500 m3 2000 m2 200 150 m2 m4 1000 100 m1 m1 500 50 0 0 m4 1500 m1 cgr S m4 5 10 m4 15 20 Frequency [kHz] 25 30 35 (c) 0 0 5 10 15 20 Frequency [kHz] 25 30 35 (d) Figure 4.25: Dispersion curves for the viscoelastic HSS40 20 2 steel section embedded in viscoelastic grout of Fig. 4.24. 152 4.9. CONCLUSIONS 4.9 Conclusions In this chapter, a Semi-Analytical Finite Element (SAFE) method coupled with a regularized 2.5D Boundary Element Method (BEM) has been applied to derive the dispersive equation for a viscoelastic waveguide of arbitrary cross-section embedded in a viscoelastic isotropic unbounded medium. The coupling between the SAFE and BEM domains has been established in a ﬁnite element sense, by converting the inﬁnite BEM domain into a single, wavenumber and frequency dependent, SAFE-like element. The discretized wave equation, which is conﬁgured as a nonlinear eigenvalue problem in the complex axial wavenumber, has been solved using a Contour Integral Method. In order to fulﬁll the requirement of holomorphicity of the dynamic stiﬀness matrix Z (κz , ω) inside the complex contour, the phase of the wavenumbers normal to the interface have been chosen consistently with the nature of the waves existing in the surrounding medium. The method has been ﬁrst validated against literature results for an elastic circular steel bar embedded in elastic concrete and a viscoelastic circular steel bar embedded in viscoelastic grout, for which results relative to some modes were available. In both cases, a very good agreement between the solutions has been observed. Next, four new cases have been investigated. The dispersion curves obtained for a viscoelastic steel bar of square cross section embedded in viscoelastic grout show some analogies with those of the viscoelastic steel bar of circular cross section embedded in viscoelastic grout, especially for the longitudinal, torsional and ﬂexural modes. The dispersion analysis performed for a HP200 viscoelastic steel beam embedded in viscoelastic soil show that the ﬁrst longitudinal mode is the most suitable in practical guided waves-based inspections. The dispersion curves for a rectangular HSS40 20 2 viscoelastic steel tube embedded in viscoelastic grout have been extracted in the 0 ÷ 35 kHz frequency range. The results show that the fundamental torsional mode is almost nondispersive in the frequency range 25 ÷ 35 kHz and can be therefore suitable for nondestructive testings. The fundamental longitudinal mode presents similar characteristics in the same frequency range, with similar values of attenuation and higher values of energy velocity, but with a more pronounced dispersive behavior compared to the torsional guided mode. The coupled SAFE-2.5D BEM formulation can be further extended to problems with embedded thin walled sections [Shah et al., 2001], immersed waveguides [Godinho et al., 2003], poroelastic surrounding media Lu et al. [2008a,b] and waveguides embedded in both isotropic and layered half spaces [François et al., 2010; Rieckh et al., 2012]. 153 4. LEAKY GUIDED WAVES IN WAVEGUIDES EMBEDDED IN SOLID MEDIA: COUPLED SAFE-2.5D BEM FORMULATION 154 Chapter 5 Leaky Guided Waves in waveguides immersed in perfect ﬂuids: coupled SAFE-2.5D BEM formulation 5.1 Sommario Una formulazione accoppiata SAFE-2.5D BEM viene proposta per lo studio delle caratteristiche di dispersione di guide d’onda viscoelastiche immerse in ﬂuidi ideali. Il metodo semi-analitico agli elementi ﬁniti è utilizzato per modellare la guida d’onda immersa, mentre la formulazione spettrale agli elementi di contorno, impiegata per modellare il ﬂuido circostante di estensione inﬁnita, consente di superare il problema delle riﬂessioni spurie tipico di altre tecniche basate sulla discretizzazione del dominio [Fan et al., 2008]. Contrariamente al caso elastodinamico descritto nei Capitoli 3 e 4, le singolarità asintotiche delle soluzioni fondamentali per il problema di Helmholtz non corrispondono quando i punti sorgente e ricevente risultano inﬁnitamente vicini sul contorno. Tuttavia, una formulazione regolarizzata risulta comunque possibile e gli integrali di contorno possono pertanto essere valutati con tecniche di quadratura numerica convenzionali. In maniera simile a quanto fatto nel Capitolo 4, l’accoppiamento delle regioni SAFE e BEM viene eseguito nel senso degli elementi ﬁniti trasformando le matrici di impedenza della regione BEM in una matrice di rigidezza dinamica equivalente, la quale risulta dipendente dal numero d’onda e dalla frequenza. 155 5. LEAKY GUIDED WAVES IN WAVEGUIDES IMMERSED IN PERFECT FLUIDS: COUPLED SAFE-2.5D BEM FORMULATION Pur ammettendo soluzione unica in linea teorica, risulta ben noto dalla letteratura [Bonnet, 1999; Schenck, 1968] che il problema esterno di Helmholtz modellato con gli elementi di contorno soﬀre di non unicità della soluzione per speciﬁci autovalori di un corrispondente problema interno. Questo problema, di tipo puramente numerico, si manifesta in maniera maggiore alle alte frequenze, dove lo spettro di dispersione risulta più popolato. L’eﬀetto delle frequenze ﬁttizie viene mitigato usando il metodo CHIEF proposto da [Schenck, 1968], il quale consiste nella scrittura di un sistema di equazioni sovradeterminato che viene in seguito risolto nel senso dei minimi quadrati. L’equazione d’onda ottenuta dal sistema accoppiato SAFE-2.5D BEM si conﬁgura come un problema non lineare agli autovalori, dal quale i numeri d’onda complessi sono per diverse ﬁssate frequenze utilizzando il metodo degli integrali di contorno. Un’analisi delle superﬁci di Riemann viene descritta in maniera analoga a quella del Capitolo 4, nella quale si tiene conto della legge di Snell-Descartes all’interfaccia solido-ﬂuido. La formulazione proposta viene dapprima validata confrontando i risultati ottenuti con quelli generati dal software DISPERSE [Pavlakovic and Lowe, 2011] per una barra circolare di titanio immersa nell’olio. Inﬁne, vengono mostrati i risultati relativi ad una barra quadrata di ed una barra ad L di acciaio immerse nell’acqua. 156 5.2. INTRODUCTION AND LITERATURE REVIEW 5.2 Introduction and literature review Guided Ultrasonic Waves (GUW) are widely used as an eﬃcient tool for the nondestructive diagnostic and ultrasonic characterization of ﬂuid-loaded waveguides [Aristégui et al., 2001; Fan et al., 2008; Fan, 2010; Kažys et al., 2010; Long et al., 2003a,c; Lowe, 1992; Ma, 2007; Pavlakovic, 1998; Siqueira et al., 2004; Zernov et al., 2011]. As in the embedded case discussed in Chapter 4, guided waves propagating in immersed waveguides are referred as trapped or leaky. If the phase velocity of the guided wave is lower than the bulk velocity of the surrounding ﬂuid, its energy is totally reﬂected and mode-converted at the interface and the wave remains trapped within the waveguide. Therefore, the attenuation of such a wave is due only to material damping mechanisms. On the other hand, if the phase velocity of the guided wave exceeds the bulk velocity of the surrounding ﬂuid, its mechanical energy is only partially reﬂected and mode-converted at the interface. The remaining part is refracted and travels away in the ﬂuid medium in form of bulk waves. Such mechanism, also known as energy leakage, causes the leaky guided waves to be generally highly attenuated, with signiﬁcant reduction of inspection ranges. The knowledge of the attenuation of guided waves, as well as their phase and energy velocity, is of paramount importance in guided ultrasonic applications. To this end, dispersion characteristics of immersed waveguides of regular cross-section (plates, rods, cylinders) have been investigated in depth in recent years by means of analytical methods. The behaviour of immersed plate-like structures has been analyzed using analytical expressions by Nayfeh and Chimenti [1988] for the case of ﬁber-reinforced composite immersed plates and by Ahmad et al. [2002] in the case of ﬂuid-loaded transversely isotropic plates. In their studies, Belloncle et al. [2003, 2004] have extracted the dispersion properties of poroelastic plates immersed in ﬂuids. Guided waves in circular rods immersed in ﬂuids have been studied by several researchers [Ahmad, 2001; Berliner and Solecki, 1996a,b; Dayal, 1993; Honarvar et al., 2011; Nagy, 1995; Nagy and Nayfeh, 1996]. For multilayered cylindrical and plate-like systems, the matrix family methods have been widely applied in the literature for the dispersion analysis of both embedded and immersed waveguides. A comparison between the Transfer Matrix Method (TMM) [Haskell, 1953; Thomson, 1950] and the Global Matrix Method (GMM) [Knopoﬀ, 1964; Randall, 1967] applied to free, embedded and immersed plates can be found in the work of Lowe [1995]. Using the Transfer Matrix Method, Nayfeh and Nagy [1996] have investigated the propagation of axisymmetric waves in coaxial layered anisotropic ﬁbers 157 5. LEAKY GUIDED WAVES IN WAVEGUIDES IMMERSED IN PERFECT FLUIDS: COUPLED SAFE-2.5D BEM FORMULATION embedded in solids and immersed in ﬂuids. The Global Matrix Method for plate-like structures has been used by Lowe [1992] for the detection of a brittle layer in diﬀusion bonded titanium and by Bernard et al. [2001] to study the energy velocity in nonabsorbing plates immersed ﬂuids. In his work, Pavlakovic [1998] has applied the GMM to study free, embedded and immersed multilayered cylindrical systems. This method has also been used for the numerical analyses and experimental investigations of ﬂuidﬁlled pipes surrounded by ﬂuids [Aristégui et al., 2001] and buried water pipes [Long et al., 2003a,c]. In the context of numerical methods, and in particular spectral Finite Element Methods, Hladky-Hennion et al. [1997, 1998] have proposed a ﬁnite element formulation enriched with a non-reﬂective boundary condition applied on the perimeter of the ﬂuid domain. More recently, Fan et al. [2008] have proposed a SAFE formulation in which absorbing regions are used to simulate the unbounded surrounding ﬂuid. The absorbing region has been modeled considering the same mass density of the ﬂuid, but increasing damping properties with increasing distance from the central axis of the waveguide. The two methods above present the advantages that only small changes are required in existing FEM/SAFE codes, while the discrete dispersive equation remains formulated as a polynomial eigenvalue problem, which can be solved by standard routines. However, due to the large number of elements required, the dimension of the problem increases considerably, thus leading to a large set of eigensolutions from which only meaningful guided modes with energy concentrated in the waveguide must be ﬁltered. Other hybrid SAFE formulations have been proposed in literature for waveguides embedded in solid media, which use inﬁnite elements [Jia et al., 2011] and Perfectly Matched Layers (PML) [Treyssède et al., 2012]. Although these formulations could be equally extended to the case of inﬁnite surrounding ﬂuids, they suﬀer of some drawbacks. For instance the wave attenuation is described by user-deﬁned shape functions in inﬁnite elements, while in the PML is deﬁned by analytical continuation of the equilibrium equations into the complex spatial coordinates, which is introduced by an arbitrary complex function. On the contrary, a Boundary Element Method (BEM) approach allows for a more natural description of the radiation problem. The 2.5D BEM has been used by diﬀerent authors for the extraction of dispersion curves of ﬂuid-ﬁlled boreholes in solid formations [Tadeu et al., 2002a; Tadeu and Santos, 2001; Zengxi et al., 2007] and submerged cylindrical solids with irregular crosssection geometry [Godinho et al., 2003; Pereira et al., 2002]. However, attenuation spectra are not obtained in these works. 158 5.3. DISCRETIZED WAVE EQUATION In this chapter, a coupled SAFE-2.5D BEM formulation to model guided waves in immersed waveguides is proposed, in which the SAFE method is used to model the viscoelastic waveguide while the BEM is used to account for surrounding inﬁnite inviscid ﬂuids. The formulation described in this chapter is the natural extension of that proposed in Chapter 4. In the case of sourronding ﬂuids, the well known problem of spurious solutions due to the non-uniqueness of the external Helmholtz boundary integrals is addressed by means of the CHIEF method [Schenck, 1968]. Numerical results obtained for a titanium bar immersed in oil are in perfect agreement with those obtained using the GMM [Pavlakovic and Lowe, 2011], while new results are presented for viscoelastic steel bars with square and L-shaped cross-section immersed in water. 5.3 5.3.1 Discretized wave equation Problem statement The equation governing the wave propagation problem under consideration is obtained for the translational invariant system of Fig. 5.1, in which Ωs denotes the cross-section of the immersed waveguide in the x−y plane, while Ωf is used to indicate the cross-section of the inﬁnite surrounding ﬂuid. The in-plane ﬂuid-structure interface is denoted by ∂Ω = ∂Ωs = ∂Ωf . The geometric an and mechanical parameters used to describe the waveguide are assumed as in Sec. 4.3, and will not be repeated here. Since only non-viscous ideal ﬂuids are considered, the acoustic properties of the surrounding liquid are identiﬁed by the mass density ρf and the phase speed of the longitudinal bulk wave, cfL . Finally, the wave propagation process is assumed with a wavenumber-frequency dependence of any scalar and vectorial ﬁeld of the form exp [i (κz z − ωt)]. 5.3.2 SAFE model of the waveguide The equilibrium equation of the ﬂuid-loaded waveguide can be obtained from Eq. (4.5) in absence of externally applied surface loads. By letting ts (x, κz , ω, ) = 0, the following integral equation is obtained ω (δ (u))T C̃ (x, ω) (u) dxdy δu ρ (x) udxdy − 2 T Ωs Ωs − (5.1) δuT tb (x, κz , ω) ds = 0, ∂Ωs 159 5. LEAKY GUIDED WAVES IN WAVEGUIDES IMMERSED IN PERFECT FLUIDS: COUPLED SAFE-2.5D BEM FORMULATION where the various terms are deﬁned as in Sec. 4.3. Following the procedure described in Sec. 4.4, the discretization of the immersed waveguide is carried out at the cross-section level by using semi-isoparametric quadratic ﬁnite elements. The displacement within the eth SAFE element is interpolated as ue (x, z, t) = N (ξ) qe (z, t) (5.2) which can be substituted in the compatibility relations Eq. (4.8) and then in Eq. (5.1). After applying the usual standard ﬁnite element assembling procedure, the wave equation results in the following expression 0 1 2 κ2z K3 + iκz K2 − KT 2 + K1 − ω M Q (κz , ω) + Fb (κz , ω) = 0, (5.3) where the operators inside the braces are deﬁned as in Eqs. (4.11)-(4.14). In Eq. (5.3), Q (κz , ω) represents the global vector of nodal displacements while Fb (κz , ω) is the vector of nodal forces at the ﬂuid-structure interface, which is derived in the following sections via a boundary element formulation. 160 5.3.2. SAFE MODEL OF THE WAVEGUIDE Figure 5.1: Analytical model of the immersed waveguide. 161 5. LEAKY GUIDED WAVES IN WAVEGUIDES IMMERSED IN PERFECT FLUIDS: COUPLED SAFE-2.5D BEM FORMULATION 5.3.3 2.5D BEM model of the ﬂuid domain The linear acoustic problem in the translational invariant ﬂuid domain is governed by the 2.5D Helmholtz equation. As shown in other works [Nilsson et al., 2009; Tadeu et al., 2012], this equation can be readily obtained from the three-dimensional case by applying the space Fourier transform along the longitudinal coordinate (z-coordinate). The resulting espression is formally indentical to the Helmholtz equation in three dimensions, providing that the 3D Laplace operator is substituded by the corresponding 2D one and the bulk wavenumber of the ﬂuid is substituted by the wavenumber normal to the interface. The 2.5D Helmholtz equation then becomes ∂2 ∂2 + ∂x2 ∂y 2 p (x, κz , ω) + κ2α p (x, κz , ω) = 0, x ∈ Ωf (5.4) where p (x, κz , ω) is the acoustic pressure in the ﬂuid and κα = ±[(κfL )2 − κ2z ]1/2 denotes the wavenumber in the x − y plane, being κfL = ω/cfL the real wavenumber of the longitudinal wave in the ﬂuid medium. The equilibrium of normal tractions and continuity of normal displacements are prescribed via the boundary conditions [Fan et al., 2008] −p (x, κz , ω) n (x) = tb (x, κz , ω) , (5.5) ∂p (x, κz , ω) = ρf ω 2 un (x, κz , ω) , ∂n x ∈ ∂Ω (5.6) where ∂(·)/∂n denotes the directional derivative along the outward pointing normal to the ﬂuid domain n (x) and un (x, κz , ω) = u (x, κz , ω) · n (x) (5.7) is the normal displacement component at the same point. The boundary integral equation can be derived from the Green’s second identity applied to an unknown physical state ∂p (x, κz , ω) p (x, κz , ω) , ∂n and the state of the fundamental solutions ∂P D (r, κz , ω) D P (r, κz , ω) , ∂n that satisfy the 2.5D Helmholtz equation in the full space when a unitary line load 162 5.3.3. 2.5D BEM MODEL OF THE FLUID DOMAIN p (x , y , z , t) = δ (x − x ) exp [i (κz z − ωt)] is acting at a ﬁxed point x [Bonnet, 1999]. Using the Green’s second identity and Eq. (5.6), the integral representation theorem for source points x ∈ Ωf results [Zengxi et al., 2007] c x p x , κz , ω =ρf ω 2 − ∂Ω ∂Ω P D (r, κz , ω) un (x, κz , ω) ds (x) ∂P D (r, κz , ω) p (x, κz , ω) ds (x) , ∂n (5.8) x ∈ Ωf , x ∈ ∂Ω where r = |x − x | is the usual in-plane source-receiver distance while c (x ) is equal to 1 when x ∈ Ωf and 0 otherwise. The dynamic fundamental solutions P D (r, κz , ω) and ∂P D (r, κz , ω) /∂n represent, respectively, the pressure and ﬂux at x when the harmonic line source passes through x . These solutions can be recovered from the corresponding 3D solutions (see, for example, [Bonnet, 1999; Brebbia and Dominguez, 1989]) by applying the space Fourier transform in the z-coordinate. The resulting expressions are formally identical to those of the 3D case, but require the substitution of the bulk wavenumber κfL with the radial wavenumber κα , leading to i (1) P D (r, κz , ω) = H0 (κα r) , 4 i ∂r ∂P D (r, κz , ω) (1) = − κα H1 (κα r) nk . k = 1, 2 ∂n 4 ∂xk (5.9) (5.10) In order to extend Eq. (5.8) to source points x located on the boundary ∂Ω, the limiting process x ∈ Ωf → x ∈ ∂Ω can be followed, in which boundary integrals involving the fundamental ﬂuxes are convergent in the Cauchy Principal Value sense [Bonnet, 1999; Chen et al., 2005]. A regularized boundary integral equation can be obtained by applying the so called equi-potential condition for external domains, which reads [Brebbia and Dominguez, 1989; Tomioka and Nishiyama, 2010] c x = − P S (r) ds (x) + 1 ∂n (5.11) P S (r) = P D (r, κz = 0, ω = 0) , (5.12) ∂Ω where is the fundamental solution of the 2D Laplace equation. Substituting Eq. (5.11) into Eq. (5.8) and making use of the subtraction-addition 163 5. LEAKY GUIDED WAVES IN WAVEGUIDES IMMERSED IN PERFECT FLUIDS: COUPLED SAFE-2.5D BEM FORMULATION technique proposed by Lu et al. [2008b] for weakly singular integrals, the following boundary integral equation is obtained D f 2 P (r, κz , ω) − P S (r) un (x, κz , ω) ds (x) p x , κz , ω =ρ ω ∂Ω f 2 P S (r) un (x, κz , ω) ds (x) +ρ ω ∂Ω (5.13) D ∂P (r, κz , ω) ∂P S (r) p (x, κz , ω) − p x , κz , ω ds (x) , − ∂n ∂n ∂Ω x , x ∈ ∂Ω where the fundamental solutions of the 2D potential problem are deﬁned as [Brebbia and Dominguez, 1989] 1 1 ln , 2π r 1 ∂r ∂P S (r) =− nk . k = 1, 2 ∂n 2πr ∂xk P S (r) = (5.14) (5.15) Since P D (r, κz , ω) and P S (r) behave asymptotically as ln 1/r for r → 0, the kernel in the ﬁrst boundary integral of Eq. (5.13) is nonsingular with the highest order term of O (1), while the kernel in the second boundary integral has a weak singularity of order ln 1/r. Therefore, the ﬁrst and second boundary integrals in Eq. (5.13) can be evaluated numerically using the Gauss-Legendre and Gauss-Laguerre quadrature formulas, respectively [Gao and Davies, 2001; Stroud and Secrest, 1996]. From the inspection of Eqs. (5.10) and (5.15) it can be noted that the asymptotic singularities of the dynamic and static fundamental solutions do not correspond when the source point approaches the receiver point. This is in contrast with the 2.5D elastodynamic case of Chapters 3 and 4. However, Eq. (5.13) still represents a regular boundary integral equation [Tomioka and Nishiyama, 2010]. In fact, the kernels in the last boundary integral can be rewritten as (superscripts D and S omitted) ∂P ∂P = er · n, ∂n ∂r in which er = x − x . |x − x | (5.16) (5.17) is the unit vector between the receiver and source points. Since er ⊥ n for r → 0, 164 5.3.4. BOUNDARY ELEMENT DISCRETIZATION the inner vector product becomes zero and the strong singularities in Eqs. (5.10) and (5.15) vanish. Therefore, the last boundary integral in Eq. (5.13) is nonsingular and can be evaluated by means of the Gauss-Legendre quadrature formula. 5.3.4 Boundary element discretization The ﬂuid-structure interface ∂Ω is discretized with Nb quadratic monodimensional elements with nodes coincident to the boundary nodes of the SAFE mesh. Pressures and normal displacements are interpolated within the generic qth boundary element of domain ∂Ωqb as follows pq (η, κz , ω) = N (η) pq (κz , ω) (5.18) N (η) qqn (κz , ω) (5.19) uqn (η, κz , ω) = where N (η) is the matrix of shape functions in the natural coordinate η, while pq (κz , ω) and qqn (κz , ω) are vectors of nodal pressures and normal displacements, respectively. Applying a point collocation scheme in which collocation points are assumed coincident with the nodes of the boundary element mesh and making use of Eqs. (5.18) and (5.19), the boundary integral Eq. (5.13) is recasted in the following discretized form Wb (κz , ω) Pb (κz , ω) = ρf ω 2 Gb (κz , ω) Q⊥ b (κz , ω) , (5.20) where Gb (κz , ω) and Wb (κz , ω) are inﬂuence operators involving the fundamental pressures and ﬂuxes, respectively, Pb (κz , ω) denotes the global vector of nodal pressures and Q⊥ b (κz , ω) represents the global vector of normal displacements at the boundary nodes. The acoustic pressure at any x ∈ Ωf can be computed from the discrete representation of Eq. (5.8) c x pd x , κz , ω = ρf ω 2 Gd (κz , ω) Q⊥ b (κz , ω) − Wd (κz , ω) Pb (κz , ω) , (5.21) where Gd (κz , ω) and Wd (κz , ω) are inﬂuence operators that are built on a elementby-element basis for the given source point x ∈ Ωf . 5.3.5 Non-uniqueness problem As well known [Schenck, 1968], the operator Wb (κz , ω) in Eq. (5.20) may become illconditioned for wavenumbers close to the resonance wavenumbers of the corresponding internal problem, thus leading to an inaccurate representation of the impedance of the 165 5. LEAKY GUIDED WAVES IN WAVEGUIDES IMMERSED IN PERFECT FLUIDS: COUPLED SAFE-2.5D BEM FORMULATION surrounding ﬂuid. As shown by Nilsson et al. [2009], this numerical problem can be overcome by means of the so-called CHIEF method [Schenck, 1968]. The method uses Eq. (5.8) with x ∈ Ωs as additional constraint that must be satisﬁed along with the surface Helmholtz integral in Eq. (5.13). Since c (x ) = 0 for source points located outside Ωf , an additional zero-pressure condition is obtained from Eq. (5.21) and added to Eq. (5.20) to form an over-determined system. A unique solution can then be obtained in a least square sense, providing that the CHIEF points are not distributed over modal lines of the internal eigenstructures. 5.4 Fluid-structure coupling The coupling between the SAFE and BEM regions is established in a ﬁnite element sense [Andersen, 2006], i.e. by transforming the discretized BEM domain into an equivalent single ﬁnite element that relates nodal forces to nodal displacements at the ﬂuidstructure interface. The pressure values at the boundary nodes can be obtained in terms of normal displacements by inverting Eq. (5.20) as follows Pb (κz , ω) = ρf ω 2 Wb−1 (κz , ω) Gb (κz , ω) Q⊥ b (κz , ω) . (5.22) in which a least square procedure can be used when CHIEF points are adopted. The relation between nodal pressures Pb (κz , ω) in the ﬂuid and nodal forces Fb (κz , ω) on the waveguide is obtained by using Eq. (5.5) and the Principle of Virtual Displacements over the ﬂuid-structure interface [Schneider, 2008] δuT (x, κz , ω) tb (x, κz , ω) + p (x, κz , ω) n (x) ds (x) = 0. (5.23) ∂Ω Eq. (5.23) is rewritten in discretized form using the interpolations in Eqs. (5.2) and (5.18), leading to Fb (κz , ω) = −RP Pb (κz , ω) , (5.24) where RP is a distribution matrix deriving from the following ﬁnite element assembling procedure RP = Nb q=1 ∂Ωqb NT (ξ (η)) n (η) N (η) Jbq (η) dη, (5.25) in which Jbq (η) is the Jacobian of the in-plane mapping for the qth semi-isoparametric 166 5.4. FLUID-STRUCTURE COUPLING boundary element. The relation between nodal displacements Qb (κz , ω) and their normal components Q⊥ b (κz , ω) can be obtained from the energy balance of the ﬂuxes at the ﬂuid-structure interface f ρ ω 2 ∂Ω δp (x, κz , ω) un (x, κz , ω) − u (x, κz , ω) · n (x) ds (x) = 0. (5.26) Using Eqs. (5.2), (5.18) and (5.19), the discretized form of Eq. (5.26) is derived as −1 T Q⊥ b (κz , ω) = Tb RP Qb (κz , ω) (5.27) where the matrix Tb is obtained from the following ﬁnite element assembling procedure (cf. Eq. (4.63)) Tb = Nb q=1 ∂Ωqb NT (η) N (η) Jbq (η) dη. (5.28) Substituting Eq. (5.24) into Eq. (5.22) and introducing the resulting expression into Eq. (5.27) lead to the following equivalent dynamic stiﬀness matrix for the ﬂuid domain Fb (κz , ω) = −ρf ω 2 Kb (κz , ω) Qb (κz , ω) (5.29) T Kb (κz , ω) = RP Wb−1 (κz , ω) Gb (κz , ω) T−1 b RP (5.30) where Eq. (5.29) is ﬁnally substituted into Eq. (5.3) to form the following homogeneous system 0 1 2 f T Q (κz , ω) = 0 (5.31) κ2z K3 + iκz K2 − KT 2 + K1 − ω M + ρ Lb Kb (κz , ω) Lb where Lb is a collocation matrix so that Qb (κz , ω) = Lb Q (κz , ω). The operator inside the braces in Eq. (5.31) represents the wavenumber and frequency dependent dynamic stiﬀness matrix for a waveguide that is immersed in an inﬁnite inviscid ﬂuid. The nodal displacements Q (κz , ω) that represent the nontrivial solution of Eq. (5.31) for a ﬁxed couple (κz , ω) can be used to extract the pressure at any point x ∈ Ωf . Recalling Eqs. (5.22) and (5.30), Eq. (5.21) can be rewritten as follows pd x , κz , ω = ρf ω 2 Gd (κz , ω) − Wd (κz , ω) R−1 p Kb (κz , ω) Lb Q (κz , ω) , which is valid only for x ∈ Ωf . 167 (5.32) 5. LEAKY GUIDED WAVES IN WAVEGUIDES IMMERSED IN PERFECT FLUIDS: COUPLED SAFE-2.5D BEM FORMULATION 5.5 Single valued deﬁnition of the dynamic stiﬀness matrix The dynamic stiﬀness matrix in Eq. (5.31) is multivalued due to the possible signs of the radial wavenumber κα = ±[(κfL )2 − κ2z ]1/2 . Poles corresponding to leaky and trapped modes can be determined by imposing the Snell-Descartes law at the ﬂuid-structure interface, i.e. by imposing the continuity of Re (κz ) and Im (κz ), and by choosing the sign of κα according to the characteristics of the partial wave in the ﬂuid medium. For Re (κz ) > κfL (non-leaky region) the angle of incidence formed by the longitudinal bulk wave in the immersed waveguide with the normal at the interface is larger than the critical angle, resulting into a total internal energy reﬂection [Rose, 2004]. The partial wave in the ﬂuid medium is thus evanescent and its amplitude decays exponentially in the x − y plane. Being the wave propagation process in the ﬂuid medium represented by the Hankel function of the ﬁrst kind, the sign of κα is chosen to satisfy Im (κα ) > 0. Since all the energy remains conﬁned within the waveguide, the attenuation of guided modes in the non-leaky region is due only to the material damping of the waveguide. In the wavenumber range 0 ≤ Re (κz ) < κfL (leaky region) the angle of incidence of the longitudinal bulk wave is lower than the critical angle and the wave is partially reﬂected and mode-converted at the interface, while a longitudinal bulk wave is also refracted in the ﬂuid medium. Therefore, part of the wave energy leaks from the waveguide into the ﬂuid, which causes the guided wave that exhibit an in-plane displacement component to be attenuated even in the case of non-dissipative materials. In absence of external sources, the propagation process in the ﬂuid ﬁeld corresponds = sin−1 [Re(κz )/κfL ] to a longitudinal wave that propagates away at a leakage angle ϑLeak L (1) with respect to the x − y plane. In order for Hn (κα r) to represent such a process, the sign of κα must be chosen so that Re (κα ) > 0. This implies in turn Im(κα ) ≤ 0 for any Re (κz ) ∈ [0, κfL ] and Im (κz ) > 0, i.e. the far ﬁeld amplitude of the radial wave increases with distance from the interface. The corresponding conﬁguration of the wavenumber vectors in the z − r plane Im is shown in Fig. 5.1, where kRe L and kLi represent the propagation vector and the inhomogeneous attenuation vector, respectively, while l denotes the intersection line of Im the x − y plane with the z − r plane. Since the ﬂuid is non dissipative, kRe L and kLi are orthogonal and the partial longitudinal wave propagates along leaky rays (dashed lines in Fig. 5.1) without attenuation [Hladky-Hennion et al., 2000]. As it can be observed, in order to have a guided mode that is attenuated in the propagation direction and that satisﬁes the Snell-Descartes law at the ﬂuid-structure interface, the attenuation 168 5.6. EIGENVALUE ANALYSIS component of the radial wavenumber in the ﬂuid medium (kIm α ) must be necessarily oriented in the opposite direction of the propagation component (kRe α ). Due to the continuous energy loss along the positive z-direction, the wave amplitude along a generic internal leaky ray, e.g. the line B-B , is always lower than the amplitude along an external leaky ray, for example A-A ; consequently, an increasing amplitude along the line l can be observed. Similar considerations can also be found in diﬀerent journal articles Hladky-Hennion et al. [1998, 2000]; Mozhaev and Weihnacht [2002]; Simmons et al. [1992]; Vogt et al. [2003] and textbooks [Caviglia and Morro, 1992]. It can be noted that the radial waveﬁeld obtained from the modal analysis keeps growing to inﬁnity while moving along the line l, which would imply that leaky bulk waves with inﬁnite amplitude are radiated at z = −∞. This unphysical behaviour derives from the assumption of translational invariance in the mathematical model. In reality, the phenomenon starts at a precise location, e.g. section A, where a leaky wave with the largest but ﬁnite amplitude φA is radiated in the surrounding ﬂuid. Therefore, for any considered distance r > A along the line l, no leaky bulk waves with amplitude greater than φA can be found. From the energetic point of view, the presence of a radial waveﬁeld that grows while propagating away from the interface does not violate thermodynamics. In fact, the total energy carried through the ﬂuid domain Ωf at a generic distance z from the origin is given by the sum of the energy previously radiated through the lateral surface ∂Ω × [−∞, z], so that the energy balance is preserved. 5.6 Eigenvalue analysis Once the admissible signs of the radial wavenumber κα have been determined for the non-leaky and leaky regions, the N -dimensional nonlinear eigenvalue problem in Eq. (5.31) can be solved in terms of complex wavenumbers and associated wavestructures for any ﬁxed real positive frequency. As in Chapters 3 and 4, also in this case the Contour Integral Method proposed by Beyn [2012] is adopted. The requirement of holomorphicity for the dynamic stiﬀness matrix in Eq. (5.31) is fulﬁlled by deforming the contour in the complex plane in order to avoid points of singularity or discontinuity that do not correspond to poles of guided modes. If the contour is limited only to the ﬁrst and fourth quadrants of the complex κz -plane, these points are represented by the longitudinal wavenumber (1) κfL , in correspondence of which Hn (κα r) is not deﬁned, and the vertical branch cut κfL + iIm(κz ) (Im(κz ) > 0), along which the dynamic Green’s functions show a jump 169 5. LEAKY GUIDED WAVES IN WAVEGUIDES IMMERSED IN PERFECT FLUIDS: COUPLED SAFE-2.5D BEM FORMULATION material i ρi ciL ciS βLi βSi (Kg/m3 ) (m/s) (m/s) (Np/wavelength) (Np/wavelength) titanium ti 4460 6060 3230 - - steel st 7932 5960 3260 0.003 0.008 oil oi 870 1740 - - - water wt 998.2 1478 - - - Table 5.1: Materials constants used in Sec. 5.7 (from Pavlakovic and Lowe [2003]). related to the abrupt change of phase in κα . After the contour integration has been carried out numerically, the SVD decomposition followed by the rank test is performed to separate physical from spurious eigensolutions. The accuracy of the eigenvalues computation is ﬁnally checked by means of the residual test. m Once the full set of eigensolutions [κm z (ω), Qb (ω)] (m = 1, .., M ) has been obtained for the frequency of interest, the phase velocity (cp ), attenuation (α) and energy velocity (ce ) for the mth guided mode can be obtained as described in Sec. 4.7.2. 5.7 Numerical examples In this section, some numerical applications are performed on waveguides with geometries typically encountered in NDE tests. The diﬀerent materials considered are reported in Table 5.1, where only the Maxwell rheological model is considered. The optimal set of parameters of the contour integral algorithm, i.e. the number of integration points, the rank test tolerance and the residual test tolerance Beyn [2012], have been determined on the basis of single analyses performed at few frequencies. At the same frequencies, the complex contour Γ (κz ) has been designed to include only guided modes with moderate attenuations. Both the contour integral method parameters and the extension of the complex contour have been assumed to vary with frequency. Due to the low frequency ranges and number of modes considered in the diﬀerent examples, the eigenvalues computation has always proved to be numerically stable even without the use of CHIEF points. However, if dispersion data are needed at higher frequencies, the use of CHIEF points can substantially improve the condition number of the boundary element matrices, which in turn results into a better separa- 170 5.7.1. VALIDATION CASE: ELASTIC TITANIUM BAR OF CIRCULAR CROSS-SECTION IMMERSED IN OIL tion of the singular values and facilitate the distinction between physical and spurious eigensolutions. 5.7.1 Validation case: elastic titanium bar of circular cross-section immersed in oil The SAFE-2.5D BEM formulation is ﬁrst validated for the case of a 10 mm diameter titanium (ti) bar immersed in oil (oi), for which the obtained solution is compared with that given by the software DISPERSE (evaluation copy) [Pavlakovic and Lowe, 2011]. The SAFE mesh used in the analysis, shown in Fig. 5.2, is composed of 48 six-nodes triangular elements plus 32 nine-node quadrilateral elements for the embedded (SAFE) section and 32 three-nodes monodimensional elements for the external ﬂuid domain (BEM). The real wavenumber, phase velocity, attenuation and energy velocity dispersion curves in the frequency range 0−500 kHz are shown in Figs. 5.2-5.5. For the comparison with DISPERSE, only the modes L(0, 1), L(0, 2), F (1, 1), F (1, 2) and F (1, 3) have been considered (dashed lines). As can be noted, the SAFE-2.5D BEM solutions (continuous lines) are in good agreement with the DISPERSE predictions. The behaviour of immersed circular bars has been analyzed by diﬀerent authors [Ahmad, 2001; Fan et al., 2008; Pavlakovic et al., 1997; Pavlakovic, 1998] and it is not reexamined here. However, it is worth mentioning that some high order modes with phase velocity greater than the oil bulk velocity experience zero attenuation values at certain frequencies. These are the cases of the F (1, 3) and L(0, 2) modes at about 330 kHz and 465 kHz, respectively, for which the radial displacements at the interface vanish. An analogous behaviour of the same modes in a elastic steel bar embedded in water has been previously observed by Pavlakovic et al. [1997]. 171 5. LEAKY GUIDED WAVES IN WAVEGUIDES IMMERSED IN PERFECT FLUIDS: COUPLED SAFE-2.5D BEM FORMULATION 1200 κoi L (ω) Re (κz ) [rad/m] 1000 F (1, 1) 800 L(0, 1) 600 400 L(0, 2) 200 F (1, 3) F (1, 2) 0 0 100 200 300 400 500 Frequency [kHz] Figure 5.2: Real wavenumber dispersion curves for the elastic steel bar of circular cross section immersed in oil. 12000 10000 DISPERSE SAFE-2.5D BEM cp [m/s] 8000 F (1, 2) F (1, 3) L(0, 2) 6000 L(0, 1) 4000 F (1, 1) 2000 0 0 100 200 coi L 300 400 500 Frequency [kHz] Figure 5.3: Phase velocity dispersion curves for the elastic steel bar of circular cross section immersed in oil. 172 5.7.1. VALIDATION CASE: ELASTIC TITANIUM BAR OF CIRCULAR CROSS-SECTION IMMERSED IN OIL 350 300 DISPERSE SAFE-2.5D BEM L(0, 2) α [dB/m] 250 200 F (1, 3) 150 F (1, 2) 100 F (1, 1) 50 0 0 L(0, 1) 100 200 300 400 500 Frequency [kHz] Figure 5.4: Attenuation dispersion curves for the elastic steel bar of circular cross section immersed in oil. 6000 DISPERSE L(0, 1) SAFE-2.5D BEM 5000 L(0, 2) ce [m/s] 4000 3000 F (1, 1) 2000 F (1, 2) 1000 F (1, 3) 0 0 100 200 300 400 500 Frequency [kHz] Figure 5.5: Energy velocity dispersion curves for the elastic steel bar of circular cross section immersed in oil. 173 5. LEAKY GUIDED WAVES IN WAVEGUIDES IMMERSED IN PERFECT FLUIDS: COUPLED SAFE-2.5D BEM FORMULATION 5.7.2 Viscoelastic steel bar of square cross-section immersed in water The following example considers a 10mm wide square bar immersed in water (wt). The bar is assumed to be made of steel (st) and small material damping component is considered (see Table 5.1). The mesh used for the solid region is composed of 100 eight-nodes quadrilateral elements, while the ﬂuid region has been modeled by means of 40 three-nodes monodimensional boundary elements (see Fig. 5.6). The dispersion spectra of the bar in the 0 − 400 kHz frequency range are reported in Figs. 5.6-5.9, where the comparison between the immersed case (continuous blue lines) and the in-vacuum case (continuous gray lines) is also shown. The low order modes have been named as in Ref. [Gunawan and Hirose, 2005]. The second main diﬀerence with respect to the immersed circular bar is that the ﬁrst torsional mode T1 becomes slightly dispersive w.r.t. the in-vacuum case, whereas the T (0, 1) mode of the circular bar in both the immersed and in-vacuum cases remains non-dispersive. In particular, it can be observed that the phase velocity of the T1 mode in the 0 − 200 kHz frequency range is in the order of 50 − 150 m/s lower with respect to the in-vacuum case (non-dispersive), while in the 200 − 400 kHz the two phase velocities correspond. Another distinction between the diﬀerent behaviour of the fundamental torsional mode in the square and circular bars is that, while in the circular bar the torsional displacements are always orthogonal to the normal at the interface, in the square bar this condition is no longer veriﬁed. Since the displacement component along the normal causes the displacement of the ﬂuid, the square (and, more generally, the non-circular sections) experiences attenuation due to leakage of bulk waves in the ﬂuid. Concerning the remaining low order modes, the ﬁrst ﬂexural (F1 ) mode presents, similarly to the F (1, 1) in the circular bar, a non-leaky section in the frequency range 0−30 kHz, in which the mode is non-attenuated. The phase velocity of the longitudinal mode L1 is similar to the phase velocity of the corresponding in-vacuum mode, while in the 0 − 100 kHz frequency range it also shows the lower attenuation and the highest energy velocity. Therefore, in this frequency range the L1 mode is particularly suitable for guided ultrasonic applications. 174 5.7.2. VISCOELASTIC STEEL BAR OF SQUARE CROSS-SECTION IMMERSED IN WATER 1200 κwt L (ω) Re (κz ) [rad/m] 1000 800 S11 T1 600 S12 400 F1 200 L1 0 0 50 100 150 200 250 300 350 400 Frequency [kHz] Figure 5.6: Real wavenumber dispersion curves for the viscoelastic steel square bar immersed in water. Guided modes are named as in Ref. [Gunawan and Hirose, 2005]. 9000 bar in vacuum bar in water 8000 7000 cp [m/s] 6000 S12 L1 5000 4000 3000 S11 T1 F1 2000 cwt L 1000 0 0 50 100 150 200 250 300 350 400 Frequency [kHz] Figure 5.7: Phase velocity dispersion curves for the viscoelastic steel square bar immersed in water. Guided modes are named as in Ref. [Gunawan and Hirose, 2005]. 175 5. LEAKY GUIDED WAVES IN WAVEGUIDES IMMERSED IN PERFECT FLUIDS: COUPLED SAFE-2.5D BEM FORMULATION 220 bar in vacuum bar in water 200 180 S11 S12 α [dB/m] 160 140 120 T1 L1 100 80 F1 60 40 20 0 0 50 100 150 200 250 300 350 400 Frequency [kHz] Figure 5.8: Attenuation dispersion curves for the viscoelastic steel square bar immersed in water. Guided modes are named as in Ref. [Gunawan and Hirose, 2005]. 6000 bar in vacuum bar in water L1 5000 ce [m/s] 4000 3000 2000 T1 F1 S12 1000 0 0 S11 50 100 150 200 250 300 350 400 Frequency [kHz] Figure 5.9: Energy velocity dispersion curves for the viscoelastic steel square bar immersed in water. Guided modes are named as in Ref. [Gunawan and Hirose, 2005]. 176 5.7.3. L-SHAPED VISCOELASTIC STEEL BAR IMMERSED IN WATER 5.7.3 L-shaped viscoelastic steel bar immersed in water In the third example, a L-shaped bar with dimensions 30 20 4 mm is considered. The bar is assumed to be made of viscoelastic steel and immersed in water. The bar crosssection is discretized by means of 32 eight-nodes quadrilateral elements and 2 six-nodes triangular elements, while the surrounding water is represented through a boundary element mesh of 36 three-nodes monodimensional elements (see Fig. 5.10). The dispersion curves for the in-vacuum case (continuous gray lines) and the immersed case (continuous blue lines) are shown in Figs. 5.10-5.13. The ﬁrst four fundamental modes are indicated in the spectra with m1 (longitudinal mode), m2 (ﬁrst pseudo-ﬂexural mode), m3 (second pseudo-ﬂexural mode) and m4 (pseudo-torsional mode). In this case, a substantial modiﬁcation of the dispersion curves for the immersed conﬁguration with respect to the in-vacuum case is observed. The m1 no longer shows the jump at about 29.5 kHz, but couples with the high order modes h2 and h3 at 26.7 kHz and 51.2 kHz, respectively. A similar phenomenon has been observed in plate waves due to the addition of material damping [Bartoli et al., 2006; Bernard et al., 2001; Ma, 2007; Simonetti and Cawley, 2004]. As it can be noted from Fig. 5.13, at such frequencies the m1 mode shows local minima of energy velocity as well as local maxima of attenuation. The normalized in-plane displacement and pressure ﬁelds for the m1 and m3 modes at 51.2 kHz are depicted in Figs. 5.14(a) and 5.14(b), respectively, showing that the mode shapes and pressure ﬁelds of the two modes are very similar at this frequency. In the frequency range 51.2 − 57 kHz, the attenuation of the m1 mode increases rapidly whereas its phase and energy velocities decreases. The maximum peak of attenuation occurs at about 77.0 kHz, which corresponds to the minimum value of the energy velocity and therefore to the frequency of maximum radiation for this mode. The normalized in-plane displacement and pressure at this frequency are shown in Fig. 5.15. Since the entity of the in-plane displacements decreases in the frequency range 77 − 120 kHz, the attenuation of the mode drops while its energy velocity increases. The m1 mode appears as the most suitable for NDT applications, in particular in the 0 − 48 kHz frequency range, where it is almost non-dispersive and characterized by high values of the energy velocity and low values of attenuation. 177 5. LEAKY GUIDED WAVES IN WAVEGUIDES IMMERSED IN PERFECT FLUIDS: COUPLED SAFE-2.5D BEM FORMULATION 500 m2 κwt L (ω) Re (κz ) [rad/m] 450 400 m1 350 m3 m4 300 250 200 150 h1 100 50 h3 60 h2 0 0 20 40 80 100 120 Frequency [kHz] Figure 5.10: Real wavenumber dispersion curves for the L-shaped viscoelastic steel bar immersed in water. 9000 8000 h3 h2 7000 h1 cp [m/s] 6000 m1 5000 4000 3000 cwt L 2000 1000 m2 0 0 20 m3 bar in vacuum bar in water m4 40 60 80 100 120 Frequency [kHz] Figure 5.11: Phase velocity dispersion curves for the L-shaped viscoelastic steel bar immersed in water. 178 5.7.3. L-SHAPED VISCOELASTIC STEEL BAR IMMERSED IN WATER 180 bar in vacuum bar in water 160 h3 140 α [dB/m] 120 h2 100 80 h1 m2 60 m3 40 m4 20 m1 0 0 20 40 60 80 100 120 Frequency [kHz] Figure 5.12: Attenuation dispersion curves for the L-shaped viscoelastic steel bar immersed in water. 6000 bar in vacuum bar in water m1 5000 ce [m/s] 4000 m4 3000 h2 m2 2000 1000 h1 0 0 h3 m3 20 40 60 80 100 120 Frequency [kHz] Figure 5.13: Energy velocity dispersion curves for the L-shaped viscoelastic steel bar immersed in water. 179 5. LEAKY GUIDED WAVES IN WAVEGUIDES IMMERSED IN PERFECT FLUIDS: COUPLED SAFE-2.5D BEM FORMULATION Due to their complexity, the m2, m3 and m4 modes may be of less interest for ultrasonic applications. However, they show some noteworthy features. The m2 mode shows alternate frequency intervals with radiating and non-radiating properties. The mode is ﬁrst non-leaky (0 − 8.5 kHz), with attenuation only due to the small material damping of the steel. Then it becomes leaky in the frequency range 8.5 − 11 kHz. In Fig. 5.16, the normalized in-plane displacements and pressure waveﬁelds for the m2 mode at 9.7 kHz are represented on the x − y plane at diﬀerent scales (the normalization is the same in both ﬁgures). From Fig. 5.16(a) it can be noted that, near the interface, the partial wave behaves similarly to an evanescent wave, i.e. the amplitude decreases almost exponentially away from the interface. However, in Fig. 5.16(b) it can be observed that at about 10.0 m the radial wave reaches an amplitude comparable to the maximum amplitude in Fig. 5.16(a) (near ﬁeld). An explanation of this behaviour can be given by observing ﬁrst that cm2 p (9.7 kHz) = 1520 m/s, from which a leakage angle of ϑm2 L (9.7 kHz) 80 deg is deﬁned with respect to the x − y plane. Therefore, the propagation vector of the partial wave in the ﬂuid medium lies close to the interface, which correspond to a conﬁguration similar to that of an evanescent wave. Moreover, from the computed axial wavenumber κm2 z (9.7 kHz) = 40.09 rad/m + i0.047 Np/m, the radial bulk wavenumber in the water results κwt α (9.7 kHz) = 6.62 rad/m − i0.28 Np/m. The small propagation component in the radial direction and the large angle of radiation determine a radial wavelength approximately equal to 0.95 m, which can be directly observed in Fig. 5.16(b), while the small negative attenuation component determines a slow increasing amplitude with the distance from the waveguide. Other than the leaky section discussed above, the m2 mode becomes non-leaky in the frequency range 11 − 20 kHz and then again leaky in the frequency range 20 − 120, in which is weakly dispersive with attenuation increasing monotonically to reach the value of about 168 dB/m at 120 kHz. The second pseudo-ﬂexural mode m3 and the ﬂexo-torsional mode m4 show similar characteristics. Both modes have large non-leaky branches characterized by a strongly evanescent waveﬁeld in the ﬂuid. For example, in Figs. 5.17(a) and 5.17(b) the normalized in-plane displacement and pressure ﬁelds are reported for the two modes at 40 kHz. The mode m3 crosses the phase bulk velocity of the water at about 84 kHz, from which its attenuation increases drastically (see Fig. 5.12) since leakage of longitudinal wt bulk waves occurs. In the 84 − 120 kHz range (cm3 p cL ), the imaginary component of κm4 z shows a slight numerical instability. Physically, such mode branch corresponds to a transition zone, which would not be suitable for experimental or application purposes due to its unstable behaviour. 180 5.7.3. L-SHAPED VISCOELASTIC STEEL BAR IMMERSED IN WATER (a) (b) Figure 5.14: Normalized in-plane displacement and pressure ﬁelds for (a) the m1 mode and (b) the h3 mode at 51.2 kHz. 181 5. LEAKY GUIDED WAVES IN WAVEGUIDES IMMERSED IN PERFECT FLUIDS: COUPLED SAFE-2.5D BEM FORMULATION Figure 5.15: Normalized in-plane displacement and pressure ﬁelds for the m1 mode at 77.0 kHz. 182 5.7.3. L-SHAPED VISCOELASTIC STEEL BAR IMMERSED IN WATER (a) (b) Figure 5.16: Normalized in-plane displacement and pressure ﬁelds for the m2 mode at 9.7 kHz in (a) the near ﬁeld and (b) the far ﬁeld (the normalized scale is the same in (a) and (b)). 183 5. LEAKY GUIDED WAVES IN WAVEGUIDES IMMERSED IN PERFECT FLUIDS: COUPLED SAFE-2.5D BEM FORMULATION (a) (b) Figure 5.17: Normalized in-plane displacement and evanescent pressure ﬁelds for (a) the m3 and (b) the m4 mode at 40 kHz. 184 5.8. CONCLUSIONS 5.8 Conclusions In this Chapter, a fully coupled SAFE-2.5D BEM formulation has been proposed for the computation of dispersion curves for viscoelastic waveguides immersed in inviscid ﬂuids. The solid region has been modeled via a standard SAFE procedure, while a regularized 2.5D BEM formulation has been used for the ﬂuid region, which allows to treat implicitly boundary corners as well as singular integrals. The dispersive equation is conﬁgured as a nonlinear eigenvalue problem, which has been solved by means of a contour integral algorithm. Complex poles for leaky and trapped modes can be obtained by imposing the correct phase of the radial wavenumber in the ﬂuid medium, which must represent a radially decaying ﬁeld in the non-leaky region (κz > κfL ) and a radially growing ﬁeld in the leaky region (κz < κfL ). Numerical experiments have been performed for waveguides of diﬀerent geometries, for which the BEM matrices involving fundamental ﬂuxes have proven to be numerically stable even without the use of CHIEF points. However, for computations in frequency ranges higher than those used in the numerical examples, the use of CHIEF points could improve the numerical stability and the separation of the singular values for the reduced linear problem inside the complex contour. The comparison between the dispersion curves obtained using software DISPERSE (evaluation copy) [Pavlakovic and Lowe, 2011] and the SAFE-2.5D BEM formulation for a circular titanium bar immersed in oil are in excellent agreement. The dispersion curves for a square steel bar immersed in water show that the ﬁrst torsional mode becomes slightly dispersive, while for the in-vacuum case the same mode is non-dispersive. This behaviour is not observed in the immersed circular bar and is mainly due to the square geometry, which allows for the coupling of the in-plane displacements with the ﬂuid medium. Therefore, the ﬁrst torsional mode in the square bar is also attenuated, whereas it is not in the immersed circular bar. The dispersion curves of a L-shaped viscoelastic steel bar immersed in water show a more noticeable shift in the phase velocity of the diﬀerent modes. In particular, the longitudinal mode does not present a clear jump as in the in-vacuum case and also shows a nearly non-dispersive behaviour in a wider frequency range compared to the in-vacuum case. As shown in other works Fan et al. [2008]; Fan [2010], the proposed method can be extended to the case of surrounding viscous ﬂuids. 185 5. LEAKY GUIDED WAVES IN WAVEGUIDES IMMERSED IN PERFECT FLUIDS: COUPLED SAFE-2.5D BEM FORMULATION 186 Chapter 6 Conclusions 6.1 Sommario In questa tesi sono stati presentati tre diﬀerenti metodi numerici per lo studio delle caratteristiche di dispersione di onde guidate ultrasoniche che si propagano in guide d’onda di geometria complessa. Lo scopo principale della ricerca svolta è stato quello di sviluppare delle formulazioni in grado di superare i problemi classici dei metodi utilizzati in letteratura per lo studio dei fenomeni di propagazione e il calcolo delle curve di dispersione. Come descritto nel Capitolo 1, i metodi analitici [Haskell, 1953; Knopoﬀ, 1964; Lowe, 1992; Pavlakovic, 1998; Simmons et al., 1992; Thomson, 1950], a fronte di risultati tipicamente molto accurati, non consento lo studio di guide d’onda aventi geometrie complesse. Quest’ultime sono generalmente studiate in letteratura utilizzando tecniche basate su elementi ﬁniti di tipo spettrale [Bartoli et al., 2006; Gavric, 1995; Hayashi et al., 2006; Sorohan et al., 2011], le quali tuttavia non sono in grado di rappresentare correttamente fenomeni di propagazione che coinvolgono domini illimitati [Castaings and Lowe, 2008; Fan et al., 2008; Jia et al., 2011; Treyssède et al., 2012]. Pertanto, si rende necessaria la disponibilità di modelli in grado di rappresentare accuratamente alcune situazioni d’interesse pratico e scientiﬁco, come ad esempio i fenomeni di propagazione in guide d’onda complesse soggette a stati di stress iniziale non nulli ed i fenomeni di radiazione dell’energia in guide d’onda immerse in mezzi solidi o ﬂuidi. Nel Capitolo 2, l’eﬀetto di uno stato di stress iniziale è stato studiato mediante un’estensione del metodo Semi-Analitico agli Elementi Finiti (SAFE) in descrizione Lagrangiana aggiornata, È stata inoltre derivata una nuova formula modale per il cal- 187 6. CONCLUSIONS colo della velocità dell’energia e nuovi risultati di interesse pratico sono stati proposti. Il Capitolo 3 è stato dedicato all’applicazione di un metodo spettrale (2.5D) agli elementi di contorno (BEM) come alternativa al metodo SAFE e per lo studio delle caratteristiche di dispersione in domini illimitati. Nei Capitoli 4 e 5 due formulazioni accoppiate SAFE-2.5D BEM sono state proposte per lo studio delle leaky guided waves in guide d’onda generiche immerse in mezzi solidi e ﬂuidi. I risultati ottenuti in questa tesi possono risultare utili nella comprensione della natura dei fenomeni propagativi ed in molte applicazioni reali basate sull’utilizzo di onde guidate, per le quali la conoscenza dei parametri di dispersione risulta di fondamentale importanza. 6.2 Conclusions and future works This thesis focuses on the development of numerical tools for the dispersion analysis of guided waves in complex translational invariant systems. As discussed in Chapter 1, the knowledge of the dispersion curves in terms of phase velocity, attenuation and group/energy velocity, is of crucial importance in nondestructive evaluation testing and structural health monitoring strategies based on guided waves. In particular, the phase velocity spectra shows the changes in velocity of existing modes at various frequencies, thus revealing the dispersive nature of the wave propagation process. The attenuation spectra gives an information on the amplitude decay per unit distance traveled. In Chapters 4 and 5 it has been shown that the attenuation can increase of one or more orders of magnitude if the waveguide is surrounded by solid or ﬂuid media, depending on the interface conditions and the speciﬁc mode considered. Since the energy losses cause the signal to be attenuated while propagating, it is easily argued the importance of the attenuation information in practical applications. The third dispersive parameter, the group velocity, describes the rate at which packets of waves at inﬁnitely close frequency propagates along the waveguide. Although this parameter has been discussed in Chapter 2 for elastic prestressed waveguides, it as not been considered in the rest of the thesis, being replaced by the more general energy velocity concept. The energy velocity expresses the rate at which the energy carried by a wave moves along the propagation direction. A generalized formula for the energy velocity computation has been given in Chapter 2. However, as shown in Chapters 4 and 5, such formula is approximated for leaky modes in embedded or 188 6.2. CONCLUSIONS AND FUTURE WORKS immersed waveguides, while still retains its general validity for trapped modes. At the beginning of each chapter, the advantages and drawbacks of available analytical and numerical tools have been discussed in relation to speciﬁc problems. The basic characteristics of these methods can be summarized as follows: analytical methods, although very accurate, are limited to simple geometries, whereas numerical ﬁnite element-based techniques, in particular the SAFE method, fail in the description of unbounded domains. In order to overcome such problems, three diﬀerent numerical tools have been described throughout Chapters 2-5. The major novelties and results of each chapter are summarized in the following, along with some suggestions and purposes for future works. In Chapter 2 a classical SAFE formulation for viscoelastic waveguide has been extended to include the eﬀect of a three-dimensional initial stress with translational invariant properties along the propagation direction. SAFE formulations proposed in literature [Loveday, 2009] account for initial axial stresses only and consider elastic materials. The eﬀect of pressure-type nonconservative loads have also been taken into account, which was never been treated in literature. Since stresses due to ultrasonic pulses are orders of magnitude lower than those commonly produced by service loads, the wave equation has been derived in linearized incremental form within an Updated Lagrangian framework. Given the dissipative properties of the materials considered, an energy velocity formula based on the Umov’s deﬁnition and the balance of energy in material description has been proposed. Changes in the dispersion curves have been shown for residual stresses in rails due to roller-straightening processes and initial stresses in pipelines due to the presence of gradients of pressure along the pipe walls. The study proposed in this chapter can be further extended to plasticized waveguides, for which a fully-nonlinear Lagrangian framework is needed. In Chapter 3 the drawback of ﬁnite element-based techniques in representing inﬁnite domains has been addressed by using a 2.5D BEM formulation. The well known analytical and numerical diﬃculties presented by the BEM in treating singular integrals and non-smooth boundary geometries have been overcome using a regularization procedure. The main novelty introduced with respect to other works proposed in literature [Gunawan and Hirose, 2005] is the introduction of material damping and the use of the Contour Integral Method [Beyn, 2012] to solve the resulting nonlinear eigenvalue problem. It has been shown how the presence of the material damping inﬂuences the choice of appropriate Riemann sheets as well as the contour integration path. The method, validated against a reliable SAFE formulation, has been used to compute for 189 6. CONCLUSIONS the ﬁrst time the dispersion curves for surface guided waves along cavity of arbitrary cross-section. The method proposed in this chapter can be further extended to the cases of cavities embedded in layered media, in which a numerically computed solution for a half space can be used [François et al., 2010], and poroelastic materials [Lu et al., 2008a,b]. Chapter 4 has been dedicated to the study of leaky guided waves in viscoelastic waveguides of arbitrary cross-section embedded in viscoelastic media. To this end, a coupled SAFE-2.5D BEM formulation has been proposed, exploits the capability of the SAFE to model geometrically and mechanically complex waveguides and the unique capability of the BEM to correctly model the radiated waveﬁeld. The major novelty introduced is in the deﬁnition of the interface conditions for the complex wavevectors, which must satisfy the Snell-Descartes law. This in turn leads to the deﬁnition of the proper phases of the wavenumbers normal to the interface, which are fundamental in the extraction of leaky poles. The integration path used in the Contour Integral Method has been deﬁned as generally assumed in the Vertical Branch Cut Integration (VBCI) method [Kurkjian, 1985]. The obtained results for waveguides of simple geometry have been validated against those obtained in literature using the Global Matrix Method (GMM) Pavlakovic [1998]; Pavlakovic and Lowe [2003] and the SAFE method with absorbing regions Castaings and Lowe [2008], while new results have been proposed for a square, an H-shaped and a hollow rectangular beams. As suggested for the method in Chapter 3, also the SAFE-2.5D BEM formulation can be extended to the case of surrounding layered media. Moreover, the eﬀect of an initial stress can be included by exploiting the method proposed in Chapter 2 and deﬁning appropriate interface conditions. Such a method could be particularly useful for the dispersion analysis of pre-tensioned or post-tensioned embedded cables and strands. Finally, in Chapter 5 attention has been focused on the dispersion properties of guided waves in immersed viscoelastic waveguides, for which a SAFE formulation coupled with a regularized 2.5D BEM formulation, used to represent the exterior 2.5D Helmholtz problem, has been proposed. In addition to the analysis of Chapter 4, the CHIEF method has been implemented to avoid numerical instabilities due to the nonuniqueness problem [Schenck, 1968]. The complex leaky poles have been found by means of the Contour Integral Method after the imposition of the correct phase of the wavenumbers normal to the interface. The numerical results for immersed waveguides of circular cross-section have been compared with those obtained using well-stated softwares [Pavlakovic and Lowe, 2011], while new results have been presented for a square and a L-shaped bar. The method proposed in this chapter can be further extended to 190 6.2. CONCLUSIONS AND FUTURE WORKS the case of viscous ﬂuids. Moreover, the eﬀect of the initial pressure could be added following the analysis of Chapter 2, which can be useful for the design of guided wavesbased inspections of subsea transportation pipelines. 191 6. CONCLUSIONS 192 Appendix A List of publications Journal papers M. Mazzotti, A. Marzani, I. Bartoli, ”Dispersion analysis of leaky guided waves in ﬂuid-loaded waveguides of generic shape”, submitted to Ultrasonics, 2013. M. Mazzotti, I. Bartoli, A. Marzani, E. Viola, ”A 2.5D Boundary Element formulation for modeling damped waves in arbitrary cross-section waveguides and cavities”, Journal of Computational Physics, 2013. M. Mazzotti, I. Bartoli, A. Marzani, E. Viola, ”A coupled SAFE-2.5D BEM approach for the dispersion analysis of damped leaky guided waves in embedded waveguides of arbitrary cross-section”, Ultrasonics, accepted for publication, 2013. C. Gentilini, A. Marzani, M. Mazzotti, ”Nondestructive characterization of tie-rods by means of dynamic testing, added masses and genetic algorithms”, Journal of Sound and Vibration, 332(1), 2012, pp. 76-101 M. Mazzotti, A. Marzani, I. Bartoli, E. Viola, ”Guided waves dispersion analysis for prestressed viscoelastic waveguides by means of the SAFE method”, International Journal of Solids and Structures, 49(18), 2012, pp. 2359-2372 A. Marzani, M. Mazzotti, E. Viola, P. Vittori, I. Elishakoﬀ, ”FEM formulation for dynamic instability of ﬂuid-conveying pipe on non-uniform elastic foundation”, Mechanics Based Design of Structures and Machines, 40(1), 2012, pp. 83-95 Conference proceedings M. Mazzotti, A. Marzani, I. Bartoli, E. Viola, ”A coupled SAFE-BEM formulation for modeling leaky waves in waveguides of arbitrary cross-section surrounded by isotropic media”, Proceedings of 193 A. LIST OF PUBLICATIONS SPIE, Vol. 8695 . M. Mazzotti, I. Bartoli, A. Marzani, E. Viola, ”A Boundary Element formulation for the computation of damped Guided Waves”, Review of progress In Quantitative Nondestructive Evaluation- AIP Conference Proceedings, 1511(1), 2013, pp. 113-120 M. Mazzotti, A. Marzani, I. Bartoli, E. Viola, ”A SAFE formulation for modeling stress waves in elastic waveguides subjected to an initial 3D prestress”, Proceedings of the 20th Conference of the Italian Association for Theoretical and Applied Mechanics - AIMETA 2011, Bologna, Italy, 12-15 September 2011, pp. 1-10 M. Miniaci, M. Mazzotti, A. Marzani, E. Viola, ”Mechanical waves in simply and multiply connected thin-walled beams”, Proceedings of the 20th Conference of the Italian Association for Theoretical and Applied Mechanics - AIMETA 2011, Bologna, Italy, 12-15 September 2011, pp. 1-10 C. Gentilini, M. Mazzotti, A. 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