Mazzotti Matteo tesi

Mazzotti Matteo tesi
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.
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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
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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
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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].
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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
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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. Marzani, ”Nondestructive characterization of tie-rods by means of dynamic testing, added masses and Genetic Algorithms”, 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
A. Marzani, M. Mazzotti, E. Viola, L. De Marchi, N. Speciale, P. Rizzo, ”A Genetic Algorithm based
procedure for the constitutive characterization of composite plates using dispersive guided waves data”
in: , Advances in Structural Engineering, Mechanics and Computation, LEIDEN, CRC Press/Balkema,
2010, pp. 305 - 308 (4th International Conference on Structural Engineering, Mechanics and Computation - SEMC 2010, Cape Town, South Africa, 6-8 September 2010)
194
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