MODELING OF ACOUSTIC-EMISSION WAVE PROPAGATION
IN LAYERED MEDIA
by
Aalhamid R. Alamin
c Copyright by Aalhamid R. Alamin, 2016
All Rights Reserved
A thesis submitted to the Faculty and the Board of Trustees of the Colorado School
of Mines in partial fulfillment of the requirements for the degree of Doctor of Philosophy
(Engineering - Systems).
Golden, Colorado
Date
Signed:
Aalhamid R. Alamin
Signed:
Dr. Ray R. Zhang
Thesis Advisor
Golden, Colorado
Date
Signed:
Dr. Greg Jackson
Professor and Head
Department of Mechanical Engineering
ii
ABSTRACT
This study proposes a model for acoustic-emission (AE) wave propagation in a layered
medium, which can be used for damage detection in structural health monitoring in general,
and crack identification in non-destructive testing and evaluation for materials in particular.
In the model, the materials are characterized by piecewise, layered media with one surface
as free and the other as free, fixed or infinite. The AE waves are generated by dislocation
over a fault area located in one of the layers or at a layer-to-layer interface.
In this study, three-dimensional (3D) wave motion in each homogeneous, isotropic and
linearly-elastic layer is solved with the use of an integral transformation approach. The 3D
wave motions are decomposed into 2D wave motion with coupled P-SV waves and 1D with SH
waves in a transformed, frequency-wavenumber domain, where P and S indicate respectively
compression and shear waves, SH denotes a shear wave component with particle motion
direction parallel to the free surface, and SV perpendicular to SH motion direction. Wave
reflection and transmission at layer-to-layer interfaces, boundaries and in a layered medium
are derived in terms of wave scattering matrices in a transformed domain. A closed-form
solution for wave responses to a point, unit-impulse dislocation is derived in the transformed
domain, which can then be used to construct the wave responses to a finite time-dependent
dislocation source and converted in a time-space domain.
With the model, this study examines AE wave propagation features in general, and wave
responses at a free surface in particular from the perspective of damage detection. Numerical
examples are provided for illustration.
iii
TABLE OF CONTENTS
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
LIST OF SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
LIST OF ABBREVIATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi
DEDICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
CHAPTER 2 BACKGROUND AND LITERATURE REVIEW . . . . . . . . . . . . . . 4
2.1
2.2
Background of AE Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.1
AE Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.2
Identification of Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.3
Parameters of AE Event . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.4
Application of AE
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7
Modeling of AE Wave Propagation . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.1
Integral Transform Approach . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.2
Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
CHAPTER 3 PROBLEM STATEMENT . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.1
Wave Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2
Constitutive Relationship
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
iv
3.3
Continuity Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.3.1
3.4
Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Dislocation Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
CHAPTER 4 DISPLACEMENT RESPONSE TO POINT, UNIT-IMPULSE
DISLOCATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.1
Integral Transformation Method . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.2
Wave Scattering Matrices in a Multi-layer Medium . . . . . . . . . . . . . . . 34
4.3
Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.4
Buried Point Source in Transformed Domain . . . . . . . . . . . . . . . . . . . 43
CHAPTER 5 NUMERICAL EXAMPLE . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.1
5.2
Features of Wave Responses to Point Impulsive Dislocation Source . . . . . . . 45
5.1.1
The Influence of Damping . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.1.2
Source at Different Depth . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.1.3
Influences of Boundary Conditions . . . . . . . . . . . . . . . . . . . . 66
Features of Wave Responses to General Dislocation Source . . . . . . . . . . . 70
CHAPTER 6 ON WAVE RESPONSE FEATURES WITH PROPAGATING
DISLOCATION SOURCE IN LAYERED MATERIALS . . . . . . . . . 74
6.1
A Model with Propagating Source Line . . . . . . . . . . . . . . . . . . . . . . 74
6.2
Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
CHAPTER 7 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
APPENDIX A - TRADITIONAL METHOD OF DISLOCATION SOURCE . . . . . . 85
A.1 Elastodynamic Representation Theorems . . . . . . . . . . . . . . . . . . . . . 85
A.1.1 Betti’s Theorem
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
A.1.2 Representation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 88
v
A.1.3 Representation Theorem for an Internal Surface . . . . . . . . . . . . . 88
A.2 Equivalent Body Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
A.3 Estimating The Slip on The Buried Fault . . . . . . . . . . . . . . . . . . . . . 91
A.3.1 A Mathematical Representation of Moment Tensor . . . . . . . . . . . 92
APPENDIX B - TRANSFORMATIONS COORDINATE SYSTEMS . . . . . . . . . . 95
APPENDIX C - NUMERICAL SIMULATION OF SURFACE MOTION FOR
LAYERED HALF SPACE . . . . . . . . . . . . . . . . . . . . . . . . 98
REFERENCES CITED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
vi
LIST OF FIGURES
Figure 2.1
Representation of the AE event. . . . . . . . . . . . . . . . . . . . . . . . . 6
Figure 2.2
Longitudinal wave: deformation parallel to the direction of
propagation, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Figure 2.3
Transverse wave: deformation perpendicular to the direction of
propagation, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Figure 2.4
Model for a dislocation source. The fault plane is in the (xy) plane.
Motion of the upper surface is in the x direction as indicated by the
arrows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Figure 2.5
Nine possible couples for the components of the moment tensor. . . . . . 16
Figure 3.1
Profile of layered materials with a dislocation source that is buried in
layer k. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Figure 3.2
Overall representation of the model set up for treatment of a
dislocation source. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Figure 3.3
Description of dislocation in an inclined-to-surface fault. . . . . . . . . . . 25
Figure 3.4
Profile of layered materials with a dislocation source that is buried at
an interface between layer (k) and (k+1). . . . . . . . . . . . . . . . . . . 26
Figure 4.1
P-wave, longitudinal, corresponds to a displacement in the direction of
propagation. The S wave, cross, corresponds to a displacement
perpendicular to the direction of propagation, SH and SV
decomposable components. . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Figure 4.2
P-SV waves with displacement in the vertical plane. . . . . . . . . . . . . 30
Figure 4.3
SH waves with displacement in the horizontal plane. . . . . . . . . . . . . 31
Figure 4.4
Schematic of relation between out-going wave vectors µ
~ u (i) and µ
~ d (k)
and input wave vectors µ
~ d (i) and µ
~ u (k) through reflection and
~
~
transmission vectors R and T . . . . . . . . . . . . . . . . . . . . . . . . . 37
Figure 4.5
The reflection and transmission matrices in layer i. . . . . . . . . . . . . . 38
vii
Figure 4.6
Transmission matrices at the interface. . . . . . . . . . . . . . . . . . . . 41
Figure 4.7
Composition Rule in locations (k,j,i)
Figure 5.1
Representation of wave propagation and the reflection. A source emits
a signal through the medium and the boundary retransmits the signal,
returned in the reverse direction. . . . . . . . . . . . . . . . . . . . . . . . 48
Figure 5.2
One-layer half space for numerical example. Dashed line shows the
direction of the observation positions 1, 2 and 3 on the surface. . . . . . . 49
Figure 5.3
P-SV and SH wave responses at surface with 5 M rad/s for a uniform
media. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Figure 5.4
P-SV and SH wave responses at surface with 5 M rad/s for a layered
media. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Figure 5.5
P-SV and SH wave responses at surface with 20 M rad/s for a uniform
media. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Figure 5.6
P-SV and SH wave responses at surface with 20 M rad/s for a layered
media. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Figure 5.7
The x- and z-direction displacement amplitudes with 5 M rad/s in the
frequency-wave number domain in a uniform medium. . . . . . . . . . . . 52
Figure 5.8
The x- and z-direction displacement amplitudes with 20 M rad/s in the
frequency-wave number domain in a uniform medium. . . . . . . . . . . . 53
Figure 5.9
The x- and z-direction displacement amplitudes with 5 M rad/s in the
frequency-wave number domain in a layered medium. . . . . . . . . . . . 54
Figure 5.10
The x- and z-direction displacement amplitudes with 20 M rad/s in the
frequency-wave number domain in a layered medium. . . . . . . . . . . . 55
Figure 5.11
The x-direction displacement amplitude spectra in a uniform and a
three-layer media. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Figure 5.12
Displacement wave responses at observation locations 1,2 and 3, in time
domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Figure 5.13
Displacement wave responses at observation locations 1,2 and 3, in
frequency domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Figure 5.14
Displacements vs time for a three-layer medium in x, y and z directions. . 59
viii
. . . . . . . . . . . . . . . . . . . . 42
Figure 5.15
Displacements vs frequency for a three-layer medium in x, y and z
directions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Figure 5.16
Affecting different damping ratios on the response in layered medium in
frequency domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Figure 5.17
Affecting different damping ratios on the response in layered medium in
time domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Figure 5.18
Comparison of displacement signals at observation point located at
(x = 15 mm, y = 20 mm) from the source centered at a depth of
(1.723 mm) below the surface in frequency domain. . . . . . . . . . . . . 64
Figure 5.19
Comparison of displacement signals at observation point located at
(x = 15 mm, y = 20 mm) from the source centered at a depth of
(1.723 mm) below the surface in time domain. . . . . . . . . . . . . . . . 65
Figure 5.20
Displacement due to point source in layered medium with free-fixed
boundary condition in frequency domain. . . . . . . . . . . . . . . . . . . 67
Figure 5.21
Displacement due to point source in layered medium with free-fixed
boundary condition in time domain. . . . . . . . . . . . . . . . . . . . . . 68
Figure 5.22
Comparison of displacement in the x-direction for layered medium at
fixed boundary condition (solid line) to infinity boundary condition
(dashed line) in frequency domain. . . . . . . . . . . . . . . . . . . . . . . 69
Figure 5.23
A ramp source function with triggering time τ , rise time Tr , and slip
amplitude S. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Figure 5.24
Fourier transform of the time-variation of the moment in which
described by R(t, τ ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Figure 5.25
Displacement response to general dislocation source at location 1 in x, y
and z directions in frequency domain. . . . . . . . . . . . . . . . . . . . . 72
Figure 5.26
Displacement response to general dislocation source at location 1 in x, y
and z directions in time domain. . . . . . . . . . . . . . . . . . . . . . . . 73
Figure 6.1
Geometry and discretization of dislocation in an inclined-to-surface
fault. The dislocation at the j − th sub-area denoted with a star is
characterized with a ramp source function with triggering time tsj , rise
time Trj , and slip amplitude Sj . . . . . . . . . . . . . . . . . . . . . . . . 75
Figure 6.2
Profile of layered materials with buried line source. . . . . . . . . . . . . . 77
ix
Figure 6.3
Profile of layered materials with buried line source in x-direction. . . . . . 78
Figure 6.4
The x- direction displacement amplitudes in the frequency-wave
number domain in a layered medium. . . . . . . . . . . . . . . . . . . . . 79
Figure 6.5
The z-direction displacement amplitudes in the frequency-wave number
domain in a layered medium. . . . . . . . . . . . . . . . . . . . . . . . . . 80
Figure 6.6
Displacement responses in frequency domain to line source at location
(x = 15, y = 20) in the x-, y- and z-directions. . . . . . . . . . . . . . . . 81
Figure 6.7
Displacement responses in time domain to line source at location
(x = 15mm, y = 20mm) in the x-, y- and z-directions. . . . . . . . . . . . 82
Figure A.1
An elastic medium of volume V surrounded by a surface S . . . . . . . . . 89
Figure A.2
Afault surface Σ is lying in the ξ3 = 0 plane . . . . . . . . . . . . . . . . . 91
Figure A.3
Double couple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
Figure B.1
Transformation from Local coordinate system to Global coordinate
system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
x
LIST OF TABLES
Table 3.1
The relationship between local and global coordinate system . . . . . . . . 24
Table 4.1
Relations between the different elastic constants of an isotropic solid . . . . 33
Table 5.1
Observation positions at the surface . . . . . . . . . . . . . . . . . . . . . . 46
Table 5.2
Physical properties of layers . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Table 5.3
Comparison with different damping . . . . . . . . . . . . . . . . . . . . . . 61
Table B.1
The relationship between local and global coordinate system . . . . . . . . 97
xi
LIST OF SYMBOLS
Fault area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A
Coefficient matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [A]
Eigenvector matrix of [A] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [D]
Orthogonal unit vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ex , ey , ez
Orthogonal unit vector in transformed domain . . . . . . . . . . . . . . . . . . . eR , eS , eT
Body force per unit mass
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . f
Stress components of f associate with P-SV waves . . . . . . . . . . . . . . . . . . FR , FS
Stress components of f associate with SH waves . . . . . . . . . . . . . . . . . . . . . . FT
Body force component associate with SH waves in the space-time domain . . . . . . . . fH
Body force component associate with P-SV waves in the space-time domain . . . . . . fV
Green’s function in the i-th direction due to an impulsive point force in the n-th
direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gin
Identity matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [I]
Wave numbers in the x, y and z directions . . . . . . . . . . . . . . . . . . . . . kx , ky , kz
Magnitude of seismic moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M0
Submatices of eigenvector matrix [D] . . . . . . . . . . . . . . . . . . Mu , Md , Nu and Nd
Slowness parameter for P and S waves in the Z direction . . . . . . . . . . . . . qα and qβ
Wave propagator matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
[Q]
Submatrices of matrix [Q] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [Qij ]
Reflection matrix between two levels i and j . . . . . . . . . . . . . . . . . . . . . . . Rij
xii
Orthogonal vector harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . R, S and T
Time variation of moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S(t, τ )
Stress discontinuity components for the P-SV wave . . . . . . . . . . . . . . . TR and TS
Stress discontinuity component for the SH wave . . . . . . . . . . . . . . . . . . . . . . TT
Transmission matrix between levels i and j . . . . . . . . . . . . . . . . . . . . . . . T (ij)
Transformation matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [Tij ]
Rise time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tr
Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . t
Stress vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . t
Displacement vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
u
Displacement components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ux , uy , uz
Displacement component for SH wave in space-time domain . . . . . . . . . . . . . . . uH
Displacement component for P-SV wave in space-time domain . . . . . . . . . . . . . . uV
Displacement components for the P-SV wave . . . . . . . . . . . . . . . . . . . . . WR , WS
Displacement components for the SH wave . . . . . . . . . . . . . . . . . . . . . . . . WT
Depth of dislocation source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zs
P wave speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . α
S wave speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . β
Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ω
Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . τ
Small quantity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ε
Lame elastic constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . λ, µ
Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ρ
xiii
Poisson’s ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ν
Young’s modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E
Kronecker delta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . δij
Wave vectors in the upward and downward directions . . . . . . . . . . . . . . . µu and µd
Distance between two levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∆zi
Stress components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . τxx , τxy etc.
Stress variable for the SH wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . τHz
Stress variable for the P-SV wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . τV z
Truncated distance in the x and y directions . . . . . . . . . . . . . . . . . . . . xt and yt
Truncated time in t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . tt
Shear slip direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Θs
Laplace operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∆
Diagonal matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . diag
Imaginary part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Im
Mean or average wave field
. . . . . . . . . . . . . . . . . . . . . . . . . . . “ Overbar ”
SH wave motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Subscript“SH”
P-SV wave motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . Subscript“P − SV ”
Complex conjugate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Superscript“ ∗ ”
Transpose of a matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . “ T ”
Counterpart in the transformed domain . . . . . . . . . . . . . . . . . . . . . . “ T ilde ”
A discontinuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [ ]+
−
xiv
LIST OF ABBREVIATIONS
The Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . GF
Generalized Impulse Response Function . . . . . . . . . . . . . . . . . . . . . . . . GIRF
xv
ACKNOWLEDGMENTS
I would like to express my deepest appreciation and my sincere thanks to my advisor,
Dr. Ray R. Zhang for his support and encouragement. He has been a friend, teacher,
advisor, and source of guidance for my study and research toward the Ph.D. degree. His
leadership, knowledge, attention to detail, ability to share and hard work have set an example
I hope to match some day. His recommendations and suggestions have been invaluable and
tremendous.
In addition, I would also like to thank my committee members, Dr. John Berger, Dr. Tim
Ohno and Dr. Paul Martin, Dr. John Steele, for their help and comments. I would also like
to express my gratitude for the financial support from my sponsor, the Libyan Government,
the Engineering College at the Colorado School of Mines and the CSM-Petroleum Institute
joint research project under the auspices of Abu Dhabi National Oil Company.
Finally, I would like to thank my family especially my wife, for her support, love and
understanding throughout this study time. Without the encourage and support of these
people, this work would have never been accomplished and achieved in this way.
xvi
To my family and many friends.A special feeling of gratitude to my parents.
xvii
CHAPTER 1
INTRODUCTION
Acoustic emission (AE) is a naturally occurring phenomenon that occurs when there is
a crack, or dislocation source, in a material [1]. A significant amount of energy is released
due to a loss of cohesion and changes in the internal-structure of the crack. Part of the
energy released can be transformed into an acoustic wave, also known as an AE signal.
This signal can be used in investigations of whether materials have been damaged, because
the signals emit different amplitudes depending on the type and/or amount of damage to
the material. The detection of the changes in the internal structure of the materials is
nondestructive, recognized as AE technique. The AE technique is typically based on the
detection and conversion of high-frequency acoustic waves to electrical signals using a sensor
in real time. Following the conversion, the signals are conditioned (preamplifier, amplifier
and filter) prior to being analyzed. Analyzing AE signals can help locate and identify the
cracking or dislocation source inside the material. The main advantage of AE technique is
to identify the crack location, but the results are usually difficult to interpret in terms of
microscopic failure mechanisms, and often they are unrepeatable.
AE signals are not repeatable because in real world conditions, once a load is applied to
an experimental sample, the conditions of the experimental environment change. Successive
applications of a load will result in different AE responses. This makes it difficult to test
an environmental condition for various loads. Therefore, appropriate modeling for AE wave
motion in materials becomes crucially important for synthesizing realistic AE wave motion
propagation and source location, which has broad-based applications in structural health
monitoring and damage detection.
In this dissertation, my study addresses modeling of AE wave propagation in layered
media due to dislocation source. The traditional method, in place for many years, was
1
established for wave propagation in a half-space subjected to a buried dislocation source.
The model consists of a half space with a free surface, subject to a dislocation source buried in
a layer where the waves are originally generated, and then propagated through the medium.
The dislocation source is represented as nine double-couple body forces equivalent to the
dislocation source, with amplitude mathematically represented as the product of the rigidity
modulus, the fault area, and the average slip. There are many comprehensive methods
involved in equivalent body forces, well developed and efficiently implemented in the area of
seismology (e.g., Lay and Wallace, [2], Sheriff and Geldart, [3], Koyama, [4], and Zhang [5]).
Recently these techniques attempted to apply to fracture mechanics (e.g., Grosse, [6]). Our
intention is to use these techniques in acoustic emission fields.
My improved model builds upon the traditional model by finding an AE response at a free
surface to the propagating waves originating at a dislocation source. The dislocation source
description is directly given with a dislocation form, and subsequently used for finding wave
motion instead of using equivalent body forces. In addition, the equivalent body forces are
valid only for describing a dislocation within a medium, which is not applicable to generalized
dislocations at the interface of two materials such as debonding damage.
In addition, the AE source is modelled as an arbitrary orientation dislocation over an
inclined-to-surface fault within one layer or at the layer-to-layer interface. Each layer of the
medium is assumed to be homogeneous, linear elastic, and isotropic. The boundary conditions are continuity of stress and displacement at the layers and across the layer interfaces
and the vanishing of the traction at the free surface. An integral transformation method
is used to determine the wave response in frequency-wave number domain, which is then
converted to time-space domain. The wave response in the half-space with a dislocation
source in a layer can be found by solving the wave equation with the use of the constitutive
relationship between displacements and stresses. The propagation of wave motion in the
medium is governed by an equation typically referred to as the wave equation. Using the
integral transform, the wave equation can be solved in the frequency wavenumber domain.
2
Then, the response at the surface in the time space domain is obtained through the inverse
integral transform.
The remainder of this dissertation is organized as follows: Chapter 2 is a literature review
on AE wave propagation models and its applications. Chapter 3 presents general methods
for finding a solution to the governing equations of the three-dimensional wave motion in an
elastic medium. In chapter 4, the construction of the wave scattering matrices in a layered
medium for individual layers and for the layer-to-layer interfaces is discussed. In chapter 5,
numerical examples are provided for further illustration. Chapter 6 presents a continuummechanics model for AE wave propagation in layered materials, which is generated by timedependent dislocation-source in the form of a line buried within one of the layers or at the
layer-to-layer interface. Finally, general conclusions are presented in chapter 7.
3
CHAPTER 2
BACKGROUND AND LITERATURE REVIEW
This chapter presents a review and background of AE wave propagation models. Several
fundamental models are discussed and a brief introduction is given about the AE method and
its applications. Specific features pertaining to AE signals are presented including sources,
signals and applications.
2.1
Background of AE Technology
AE is a phenomenon of energy release in the form of transient elastic waves generated
within a solid as a consequence of deformation and fracture processes (cracks, inclusions,
corrosion, delamination, etc.). This phenomenon occurs in many materials, when subjected
to mechanical loading or stress. AE technique is a comprehensive and simple non-destructive
test method commonly used to detect and locate faults in mechanically loaded structures.
The AE technique has been studied for decades, a few of which are reviewed in the following
sections.
2.1.1
AE Sources
Every phenomenon causing internal displacement or localized irreversible deformation
generates elastic waves that propagate in a medium; this phenomenon is then called an AE
source. Generally, different kinds of AE sources can be attributed to different mechanisms
of damage in the materials. However, many physical phenomena can be the cause of AE
waves, such as plastic deformation, dislocation movement, twinning, initiation and crack
propagation (fatigue, corrosion under stress, etc.), corrosion, micro and macroscopic fractures
in composites, friction, mechanical impacts, leaks (liquid and gas), cavitations and boiling
[7, 8].
4
The location of naturally occurring AE sources is generally unknown. If one seeks information to identify the temporal nature of a source and its orientation, in addition to its
magnitude, the first step is locating the source in three dimensional space. Whether for the
location of the epicenter of a crack or for applications in materials science and the mechanics
of rupture, the use of multiple sensors allows the location of the AE source to be found.
More complex geometries may require a larger number of sensors. The position of the source
in relation to a received signal is calculated based on the time differences of the arrival of
signals to the sensors and the speed of wave propagation in the material considered [9].
2.1.2
Identification of Sources
Many studies, including those that have researched the needs of industrial control applications, require the use of simple procedures to distinguish the AE signals collected during
testing of material loaded by the use of a single parameter calculated on the forms of waves.
Most of them concern the amplitude of AE signals. Chen et al. [10] studied the evolution of
the amplitude of AE signals received during bending tests and tensile monotonic on samples
of short-fiber composite carbon and glass matrix. Kotsikos et al. [11, 12] studied the received acoustic emission during fatigue tests on samples of laminate glass fiber and polyester
matrix. Ceysson et al. [13] performed different types of bending tests on laminates of carbon
fiber/epoxy matrix. Benzeggagh et al. and Barré et al. [14, 15] focused on the study of
signal amplitudes and different types of composites.
2.1.3
Parameters of AE Event
Detection of AE events is typically performed with respect to a threshold below which
no signal is recorded. The threshold is a constant level that is used to derive several of
the characteristics of AE event. Parameters characterizing AE events are many. The main
parameters [16], shown in Figure 2.1, are:
- Duration: The time between the initial rise of AE energy above the threshold and
the time at which the AE energy decays below the threshold. The duration is expressed in
5
microseconds.
- Amplitude: The maximum amplitude attained in the duration of an AE event.
- Rise time: The time passed between when the first threshold is exceeded and the peak
amplitude of the signal. The rise time is expressed in microseconds.
- Number of counts: The number of times, the signal crosses a threshold for the duration
of the AE event.
- Energy: The integral of the squared signal over the duration of the AE event.
Figure 2.1: Representation of the AE event.
The study of the relationship between sources and AE signals can be oriented by analyzing
the frequency content of received waves. Calabrò et al. and Giordano et al. [17, 18] studied
the diagrams of the signals received in acoustic emission tests on unidirectional composite
carbon and epoxy in order to reconstruct the entire fracture process. The study by Hamstad
et al. [19] focused particularly on the effect of the distance between the source and the sensor.
They were able to show that the characteristic parameters of waveforms such as amplitude,
rise time, and the spectral content of signals for similar sources are largely influenced by the
distance sensor source. The work of Kinjo et al. and Suzuki et al. [20, 21] was performed
6
on the frequency content of acoustic emission associated with different damages. Their
technique requires experimental conditions but the results are very promising. Different
interfacial qualities were tested by Mäder et al. [22] on microcomposite glass fiber and
composites of the same nature. Analysis of the frequency content is associated with a more
traditional amplitude analysis. The frequency content of signals of low amplitude evolves
with different fiber sizes. Another way is that the frequency of the signals can be used as a
parameter, in which the average frequency is calculated directly from the waveform studied
[23, 24].
2.1.4
Application of AE
The different techniques of characterization and identification play an essential role in
the study of AE phenomena that are involved in the development of materials and systems
in general. The choice of technique that is used to study a system is mainly related to the
type of phenomenon to characterize. Concerning the materials science, acoustic techniques
have been widely implemented as non-destructive methods of investigation; this is either to
evaluate the mechanical proprieties of studied systems or to detect the internal defaults of
material. In the past thirty years, many studies have been undertaken to develop the AE
technique in terms of reliability and characterization of damage [25–32].
Numerous studies using AE techniques have been conducted to understand the microstructural mechanisms, such as rupture during mechanical loading of functional materials, [33, 34], dislocation motion [35], the gas bubbles [36], the organization of magnetic
domains, phase transformations (in particular martensitic) [37, 38], the debonding of fibers
in composites [39, 40] or in situ monitoring of a cement [41, 42].
Andreykiv et al. [43] studied the initiation and propagation of a crack, by proposing a
mathematical model to locate the source of AE. For that reason, they studied the relationship
between the geometric characteristics of a crack and the parameters of the recorded signals.
Other researchers have studied the shape parameters associated with the signals to identify
the damage mechanisms (microcracking, defects, etc). This is usually achieved by analyses
7
of amplitude distribution, rise time, or energy parameters [17, 44–46].
Johnson and Gudmundson [47] have used the technique of AE to identify different types
of damage occurring in a laminate composed of glass and epoxy. Different stratifications
were adopted to favor the formation of certain types of defects during tensile tests. They
observed that for certain types of damage, AE signals with different parameters (amplitude,
duration and frequency content) could be identified. Most signals could be grouped into
several main classes. Particular attention was paid to the extraction of signal information
in the initiation and propagation of cracks in the composite.
In order to study microstructural phenomena, most work focuses on temporal parameters. It was shown specifically by Ni and Iwamoto [48] that these parameters are strongly
dependent on signal propagation distances, which is not the case with frequency parameters.
Åberg and Gudmundson [49], using frequency analysis, found that the acoustic signals
associated with cracks corresponded to low frequency signals, while those associated with
the rupture of fibers had higher frequencies. Another study on damage of layered materials,
[50], showed that the inter-layer delamination could be characterized and the signals relating
to the damage identified using the AE technique.
The movement of dislocations is one of the mechanisms potentially emissive in terms
of AE on the microscopic scale [51, 52]. The recorded signals can give an image of the
relaxation of the involved energy during the movement of dislocations. Weiss et al [53]
presented a statistical analysis of signals induced by the movement of dislocation. The
performance in compression and torsion in their experiments indicates that the deformation
in single crystals of ice is complex and inhomogeneous. This deformation is characterized
by sudden and rapid movement of dislocation. they suggest that the amplitude of these
movements may be associated with the energy and amplitude of the recorded signals. Other
studies have shown that, by using AE techniques, it is possible to distinguish the strain
produced either from a screw dislocation, or from an edge dislocation, [54].
8
A relationship between the rate of crack propagation and the count rate of emission sound
was established by using AE monitoring [55, 56]. This relationship is very interesting in the
context of predicting the lifetime of a structure. Bashir et al [57] have developed a simple
theoretical model to describe how the acoustic energy associated with the propagation of
a microcrack develops in the complex mechanical system (steel bearing in the gearbox of a
helicopter). Lin et al [58] studied the acoustic signals emitted by alloys based on titanium and
aluminum. Some differences observed in the microstructure were explained by differences in
the acoustic response, and this was mainly in the energy and signal amplitude.
In recent years, the use of AE has significantly increased and more operators are interested in many applications. Mehan and Mullin [28] indicated the successful identification
of AE sources and show the use of AE signals as the means to identify the type of source
that generated the signals. Hamstad et al., [59, 60] examined the application of a wavelet
transform to identify AE sources by using a database of wideband AE modeled signals.
Martin and Dimopoulos [61] have shown how AE may be implemented in risk-based
assessment. They also present the basics of AE and its application that could be used
for evaluating damage and assessing the structural integrity of a range of materials and
structures including fiberglass booms, pressure vessels and above ground storage tanks. As
AE technique is capable of detecting various types of damage mechanisms in composite
materials, and the use of composite materials in many different systems has increased, a
quantitative approach has been adopted to study damage mechanisms in composite materials
(e.g., [62]). The availability of AE signals for early detection could be effectively applied in
early fault detection.
Yuan et al.[63] proposed an experimental method to collect AE signals and pre-process
them by Empirical Mode Decomposition (EMD). They found that, applying AE techniques
on the fault diagnosis of the Rolling Element Bearing could improve the detection accuracy.
In particular, this method could be effectively applied in early fault detection and real-time
diagnosis of bearing fault.
9
While AE techniques have been used in extensive applications in materials, the results
are usually difficult to interpret in terms of microscopic failure mechanisms, and often they
are not reproducible. These difficulties are perhaps not too surprising due to the nature
of the signal source. Since AE testing is not reproducible due to the nature of the source,
exemplified as sudden and sometimes random formation of a crack, modeling of the AE
source and wave propagation is essential.
2.2
Modeling of AE Wave Propagation
Theoretical studies on wave propagation in a layered medium have long been a subject
of fundamental interest in fields such as earthquake engineering, soil structure interaction,
dynamic loads on pavements, non-destructive testing, and identification of energy resources.
An immense body of literature examining elastic wave propagation problems from theoretical, computational, and observational viewpoints have followed the first research dates back
to the pioneering work by Lamb in 1904 [64]. Since then a difference of elastodynamic problems related to isotropic materials have been studied by many researchers. Early analytical
treatment of the subject can be found in Achenbach [65], Aki and Richards [66], Sheriff and
Geldart [3], Thomson [67], Avesth and Mavko [68], Haskell [69, 70], Dunkin [71], Kausel and
Roesset [72], Kausel and Peek [73], Kennett [74], Bouchon and Aki [75], Bouchon [76, 77],
Luco and Aspel [78], Hisada [79], Wang and Herrmann [80], Yao and Harkrider [81], Wang
[82], Chapman [83], and Wang and Kümpel [84].
Extensive studies have been made recently on methods for seismic inversion and wave
propagation in stratified and porous media. For example, Pak [85] studied an isotropic halfspace subjected to an arbitrary, time-harmonic, finite, buried source analytically by using a
method of displacement potentials. This work is extended to determine the Green’s function
for an isotropic multilayered half-space by Pak and Guzina [86]. Stokoe et al. [87] discussed
the case of one layer above a solid homogenous half-space and analyzed the dispersion curve
of the ground roll to produce near surface S-wave velocity profiles. Roesset et al. [88]
discussed the determination of the depth to bedrock, computation of the natural period of
10
vibration of the pavement system, and the estimation of the wave propagation velocity and
modulus. Sun et al. [89] developed a numerical model for predicting transient response of
beam under a non-destructive tests impact load. Sun and Luo [90] extended the classical
problem of transient wave propagation in multilayered solids to transient wave propagation
in multilayered viscoelastic solids. Luo et al. [91] developed an inverse analysis computer
program that integrates nonlinear optimization algorithm and forward dynamic analysis of
multilayered medium for inferring structural and material properties of the medium. Ditri
[92] deduced a formula of medium that was applied to measure surface stress of material, and
this formula based on small phase change of Rayleigh wave that propagated along the surface
of an initially stressed isotropic medium. Sun and Greenberg [93] developed a general theory
to solve for the dynamic response of pavement structures under moving stochastic loads.
Liang and Zeng [94] presented an efficient analytical solution technique for the transient
wave propagation in a multilayered soil medium based on axisymmetric dynamic loads. De
Barros et al [95] investigated the feasibility of a full waveform inversion of the seismic response
of the stratified medium.
The analysis and modeling of wave propagation in layered media have been extensively
investigated in the fields of engineering [65, 96–98]. Modeling of wave propagation requires
knowledge of the wave characteristics, starting with the properties of the medium through
which the wave travels. The structure of the medium for each layer is assumed to be homogenous, linear elastic and isotropic. Such an assumption of a layered structure makes it
possible to obtain a wave-response solution, which is usually not the case for more complicated structures. This study will deal with AE waves produced by a known source, when
the medium comprises of unlimited layers. AE signals reflect and transmit from one layer to
another. That is, waves from an early disturbance certainly meet and interact with boundaries. The boundaries are free surfaces at the upper layer and infinity at the lower layer.
The continuity conditions in terms of displacement and traction at each layer and interface
are considered. The propagation of AE waves is governed by an equation commonly referred
11
to as the wave equation. The solution process of the model is to solve the wave equation in
each layer. The source is within one layer or at layer-to-layer interface. The wave equation
in a uniform medium is governed by three coupled partial differential equations. With the
use of the constitutive relationship, the wave equation can be solved.
2.2.1
Integral Transform Approach
In general, partial differential equations are much more difficult to solve analytically than
ordinary differential equations. Fortunately, partial differential equations of second-order are
often amenable to analytical solution. Solving the governing equations of wave motion was
suggested by many procedures (eg. Chin et al. [99]). The integral transform procedure
was used as the most useful one to change partial differential equations of wave motion into
sets of ordinary differential equations (eg. Haskell [100]). In these applications, the multiple
transform has been applied to have the transformed equations in the frequency-wave number
domain.
Transformed equations are ordinary differential equations with the depth variable z as
the only variable. Two decoupled sets of ordinary differential equations can be found; one set
corresponds to the coupled P and SV waves, the other to the SH waves (eg. Hudson [101]).
There exist only two kinds of waves propagating in a homogeneous, isotropic and linear elastic
medium [65], that is, the P and S waves. The P wave is also known as the compressive wave,
longitudinal wave, primary wave, irrotational wave, or dilatational wave as seen in Figure 2.2.
The S wave is also called the shear wave, transverse wave, rotational wave, or equivoluminal
wave as seen in Figure 2.3. The particle motion of a P wave coincides with the direction of
propagation, whereas the particle motion of an S wave is perpendicular to that direction. In
the half-space medium, an S wave may be decomposed into two components: SV and SH
waves. The particle motion of an SH wave is parallel to the horizontal ground surface and
the particle motion of an SV wave is perpendicular to that of an SH wave.
12
Figure 2.2: Longitudinal wave: deformation parallel to the direction of propagation, [2].
Figure 2.3: Transverse wave: deformation perpendicular to the direction of propagation, [2].
As a result, all problems of elastodynamic wave propagation in layered media can be
decomposed into a problem governed by P-SV-wave propagation and a problem governed by
SH-wave propagation, which can be solved independently. These equations are represented
13
in term of displacements, strains, and stresses in an elastodynamic medium, which can
be solved through integral transformations from the time-space domain to the frequency
wavenumber domain. Integral transformation is a common procedure used to solve partial
differential equations that describe elastodynamic wave propagation.
Many numerical methods have been proposed to compute elastic wave propagation in
homogeneous media, [6, 65, 96, 97]. The solution of the elastodynamic equation, consisting
of the equations of motion and the constitutive (stress - strain) relationship under prescribed
boundary and radiation conditions, is required to simulate elastic wave propagation. The
simulation of elastic wave propagation is an effective way to further our understanding of
the physical fundamentals of AE.
2.2.2
Source
Previous research [1, 102] suggests that AE is typically considered as a form of microseismicity generated during the failure process as the material is loaded. Therefore, modeling of
earthquake ground motion with a dislocation source in seismology can be, and has already
been, used with AE waves [6]. Maruyama [103] and Burridge et al. [104] have used a method
that is based on replacing the dislocation acting within the source region by an equivalent
body force acting in an elastic medium. The advantage of representing the dislocation source
in terms of equivalent body forces is to simplify the problem by avoiding complex boundary conditions at the fault, to give the equivalent force system, and computing the surface
motion as a linear problem.
In other words, material crack, seismic rupture source, and the like, share the same
source mechanism, that is, time-dependent dislocation over a finite fault area, or simply the
finite dislocation source. The finite dislocation source can be modeled as the summation (or
integration in the limit) of point sources, each of which accounts for the evolutionary dislocation over a discretized sub-area triggered at a different time instant. Therefore, truthfully
characterizing the point source is a key in understanding the nature of the dislocation source.
14
The mechanism of the above point source is typically modeled as the product of a source
time function characterizing the dislocation growth (e.g., a ramp function), a factor combining nine couples of impulse forces that are equivalent to the unit dislocation, and a scaling
factor or magnitude (=final dislocation x material rigidity x fault area). Each couple (or
dipole) can be represented mathematically by two impulses acting in opposite directions,
with an infinitesimal separation distance either along or perpendicular to the impulse direction or, in the limit, by the derivative of the impulse with respect to the separation-distance
parameter, as seen in Figure 2.4.
Figure 2.4: Model for a dislocation source. The fault plane is in the (xy) plane. Motion of
the upper surface is in the x direction as indicated by the arrows.
The combination of the nine couples in Figure 2.5 can be theoretically proven to be
equivalent to any kind of a dislocation (normal and shear) in an arbitrary orientation from
an elastodynamic approach, as first described in Burridge et al. [104]. In this approach, using
the double couple model, Kennett [74] solved the governing equations of wave motion and
obtained an equivalent set of stress-discontinuities in the transformed domain. These can
be then used to obtain the response at any depth in the transformed domain. The existence
of both displacement-and stress-discontinuities is due to the fact that, mathematically, a
15
double couple is derivative of a delta function with respect to separation distance [105].
Figure 2.5: Nine possible couples for the components of the moment tensor.
To simplify the analysis, Aki and Richards [66] disregard the inelastic behavior of the
medium around the source region and apply the elastodynamics theory to obtain the response
displacement field in terms of Green’s functions. If the medium is linear and the system
is conservative, then the reciprocal theorem (Betti’s theory) holds, and is used together
with a Green’s function, to obtain a representation of motion at a general point in the
medium in terms of body forces and information on boundaries. Using reciprocal properties
16
of Green’s functions satisfies homogeneous boundary conditions. The Green’s function (i.e.,
displacement response to a unit force at one source location) is assumed to be continuous
everywhere, even on both sides of a fault as if the dislocation never happens. Alternatively
and more generally, Backus and Mulcahy [106, 107] use the concept of stress glut to derive
the same results. That derivation uses the assumption that the earthquake and indigenous
sources are considered to be the result of a localized, transient failure of the linearized elastic
constitutive relation, which leads to the stress glut as a function of the dislocation quantity
(see Dahlen and Tromp [108]).
The above point-source characterization based on equivalent forces is useful in solving
wave motion equations, which have been well developed and widely used in studies of seismology and recently in fracture mechanics [6, 102]. This dissertation builds upon above-point
source characterization by using the developed model but replacing the equivalent body
force with a dislocation source. The next chapter describes how the source description can
be directly given with a dislocation form and subsequently used for finding wave motion.
17
CHAPTER 3
PROBLEM STATEMENT
This chapter discusses a model that is developed for this research. The model is threedimensional and consists of (n) horizontal layers separated by (n − 1) interfaces. In the
model, the materials are characterized by a layer of homogeneous, isotropic and linearlyelastic media with one surface free and the other as free, fixed or infinite boundary. Each
layer is characterized by the mass density (ρ), the P -wave velocity (α) and the S-wave
velocity (β), as schematically shown in Figure 3.1. This model assumes the AE source as a
dislocation rupturing over a certain area of a fault. A dislocation source is buried in layer k
where the waves are originally generated and then propagated through the layers of medium.
The dislocation source is introduced in the medium as a discontinuity in the displacement and assumed to be planar fault. In order to treat the system mathematically, the fault
is considered as a point source. In case of far field response under consideration, a single
point source can sufficiently describe the dislocation event. However, when the near field
responses are under consideration, the planar fault requires to be discretized into N number
of sub-faults, with each sub-fault being represented as a point source (Figure 3.2 center).
The response (e.g., displacement) to the dislocation event is then considered to be the summation of all responses from N sub-faults, with appropriate time-delays incorporated in to
the response excited by each sub-fault. The source function, describing the evolutionary
discontinuity of the displacement across the fault, is approximated with a ramp function.
This ramp function that describes the local slip is characterized through three parameters:
the start time of the rupture, the rise time, and the slip amplitude (Figure 3.2 bottom).
18
Figure 3.1: Profile of layered materials with a dislocation source that is buried in layer k.
19
Figure 3.2: Overall representation of the model set up for treatment of a dislocation source.
AE wave responses can then be obtained by solving force-free wave equations of motion
with a dislocation over a fault area. In doing so, it is also required that there are continuity
conditions in term of displacement and traction at each layer-to-layer interface. For ease
in describing the solution procedure, the following displacement and traction vectors are
defined:
20
~u(x, y, z, t) = ux~ex + uy~ey + uz~ez ;
(3.1)
~t(x, y, z, t) = τxz~ex + τyz~ey + τzz~ez .
(3.2)
where ~ex , ~ey and ~ez are, respectively, orthogonal unit vectors in the x, y and z directions,
u the displacement and τ the stress.
Since the dislocation source can be represented as a summation, or integral in its limiting
case, of unit-impulse dislocation at a point over discretized fault area, this study first examines the displacement wave response to a point, unit-impulse dislocation. Analog to the
traditional terminology such as Green’s function in wave propagation and impulse response
function in vibration which is the response to an impulsive force, the displacement response
to the point, unit-impulse dislocation here is referred to as wave-based or generalized impulse
response function (GIRF).
3.1
Wave Equations
The 3D wave motion in each layer is governed by
∂ 2 ux
∂τxx ∂τxy ∂τxz
+
+
+ ρfx = ρ 2 ,
∂x
∂y
∂z
∂t
(3.3)
∂τxy ∂τyy ∂τzy
∂ 2 uy
+
+
+ ρfy = ρ 2 ,
∂x
∂y
∂z
∂t
(3.4)
∂ 2 uz
∂τxz ∂τyz ∂τzz
+
+
+ ρfz = ρ 2 .
∂x
∂y
∂z
∂t
(3.5)
where ρ is the density of the medium, the u’s and τ ’s denote the displacements and stresses,
respectively. and the f ’s denotes the body forces per unit mass.
21
3.2
Constitutive Relationship
The constitutive relationship between displacements and stresses is expressed as:
τxx = λ(
∂ux ∂uy ∂uz
∂ux
+
+
) + 2µ
,
∂x
∂y
∂z
∂x
(3.6)
τyy = λ(
∂uy
∂ux ∂uy ∂uz
+
+
) + 2µ
,
∂x
∂y
∂z
∂y
(3.7)
τzz = λ(
∂ux ∂uy ∂uz
∂uz
+
+
) + 2µ
,
∂x
∂y
∂z
∂z
(3.8)
τxy = µ(
∂ux ∂uy
+
),
∂y
∂x
(3.9)
τyz = µ(
∂uy ∂uz
+
),
∂z
∂y
(3.10)
τxz = µ(
∂ux ∂uz
+
).
∂z
∂x
(3.11)
in which λ and µ are the Lame elastic constants.
3.3
Continuity Conditions
The continuity condition at each layer-to-layer interface requires that displacement and
traction vectors be continuous across z = zi :
~u(x, y, zi− , t) = ~u(x, y, zi+ , t);
(3.12)
~t(x, y, zi− , t) = ~t(x, y, zi+ , t)
(3.13)
where zi− and zi+ represent respectively the negative and positive sides of interface z = zi .
22
3.3.1
Boundary Conditions
The boundary condition at the free surface is
~t(x, y, z = 0, t) = 0
(3.14)
+
~t(x, y, zn−1
, t) = 0
(3.15)
+
~u(x, y, zn−1
, t) = 0
(3.16)
and the other at free boundary
or fixed boundary as
or radiation condition with no propagating waves coming from the place where z is infinity.
3.4
Dislocation Source
The point, unit-impulse dislocation source can be described as normal and shear dis-
location components (∆un , ∆us ) with shear slip direction (θs ) over a fault area A at
(xs = 0, ys = 0, zs ) with normal direction (nz′ x , nz′ y , nz′ z ) or (cos θz′ x , cos θz′ y , cos θz′ z )
as shown in Figure 3.3. It can be expressed as:
~u ′ (xs , ys , zs+ , t) − ~u ′ (xs , ys , zs− , t) = ~us′ δ(x)δ(y)δ(z − zs )δ(t)
(3.17)
~us′ = A[∆us (cos θs~ex′ + sin θs~ey′ ) + ∆un~ez′ ])
(3.18)
where quantities with prime indicate that they are described with local x′ − y ′ − z ′
coordinate system in the fault and can be converted in the global x − y − z coordinates using
coordinate transformation system (Appendix B). The relationship between local coordinate
system and global coordinate system is given by direction cosine table, which is summarized
as follows:
23
Table 3.1: The relationship between local and global coordinate system
x
= cos θxx′
y
= cos θxy′
z
= cos θxz′
′
nxx′
′
nyx′ = cos θyx′
nyy′ = cos θyy′
nyz′ = cos θyz′
′
nzx′ = cos θzx′
nzy′ = cos θzy′
nzz′ = cos θzz′
x =
y =
z =
nxy′
nxz′
After applying the coordinate transformation system, the dislocation source equation can
be expressed as:
~u (xs , ys , zs+ , t) − ~u (xs , ys , zs− , t) = ~us δ(x)δ(y)δ(z − zs )δ(t)
(3.19)
~us = (∆ux~ex + ∆uy~ey + ∆uz~ez )
(3.20)
∆~ux = A[∆us (cos θs nx′ x + sin θs ny′ x ) + ∆un nz′ x ]
(3.21)
∆~uy = A[∆us (cos θs nx′ y + sin θs ny′ y ) + ∆un nz′ y ]
(3.22)
∆~uz = A[∆us (cos θs nx′ z + sin θs ny′ z ) + ∆un nz′ z ]
(3.23)
As an example, for ∆un = 0, θs = 0 and local x′ − y ′ − z ′ coordinate system consistent
with global x − y − z one, the equivalent force vector at source location can be found based
on Eqs. (3.6)-(3.11) as:
∂
f~e = [Aµk ∆us ][δ(x)δ(y) δ(z − zs )~ex ]
∂z
(3.24)
which are the double couples of impulsive forces in x− and z− directions with seismic moment
as M0 = Aµk ∆us . It can be verified that the equivalent forces to the general dislocation in
24
Eq. (3.19) are consistent with the traditional seismic-moment based nine couples.
Figure 3.3: Description of dislocation in an inclined-to-surface fault.
The result obtained with this method is the same as that obtained with the traditional
approach (see Appendix A). Similarly, we can use the displacement and traction representation to model a dislocation source at an interface between two materials of different elastic
moduli; layer (k) and (k+1), Figure 3.4. The configuration of the model consists of positive
and negative sides of different elastic properties connected along a planar interface. Each side
is characterized as an isotropic elastic material while also taking account the various different
25
elastic moduli.The treatment of the other layers are identical to the procedure adopted for
the isotropic homogenous medium Figure 3.1. Following the same calculation procedure, the
result can be generalized to solve for GIRF, which is described in more detail in the following
chapter.
Figure 3.4: Profile of layered materials with a dislocation source that is buried at an interface
between layer (k) and (k+1).
26
CHAPTER 4
DISPLACEMENT RESPONSE TO POINT, UNIT-IMPULSE DISLOCATION
This chapter focuses on the solution of a wave motion equation in a homogeneous,
isotropic and linear elastic medium. Once we model the discontinuity conditions in a dislocation source, the response to the source can be determined via the 3D equation of motion
via the integral transform approach, Alamin and Zhang [109].
4.1
Integral Transformation Method
In this section we illustrate how the Fourier transform method can be used to solve a
set of partial differential equations in (x, y, z) and t in order to reduce them to ordinary
differential equations in z.
To solve for GIRF, integral transform procedure is applied. Specifically, Fourier transforms are used between time-space domain (x, y, z, t) and frequency-wavenumber domain
(kx , ky , z, ω), in other words,
~u(x, y, z, t) =
~t(x, y, z, t) =
Z
Z
∞
e
iωt
−∞
∞
e
−∞
iωt
Z
Z
∞
−∞
∞
−∞
Z
Z
∞
(WR~eR + WS ~eS + WT ~eT )dkx dky dω
(4.1)
ω(FR~eR + FS ~eS + FT ~eT )dkx dky dω
(4.2)
−∞
∞
−∞
where orthogonal vector harmonics (~eR , ~eS , ~eT ), first introduced by Takeuchi and Saito [110]
in dealing with surface waves in cylindrical coordinates, are defined as:
~eS = (
~eR = ei(kx x+ky y)~ez
(4.3)
ikx
iky
~ex +
~ey )ei(kx x+ky y)
kr
kr
(4.4)
27
~eT = (
iky
ikx
~ex −
~ey )ei(kx x+ky y)
kr
kr
(4.5)
in which kx and ky are respectively wave numbers in x and y directions, ω is the frequency,
p
and kr = kx2 + ky2 is the wave number in radial direction.
Variables WX and FX (X = R, S, or T ) in Eqs. (4.1) and (4.2) may also be expressed
in terms of the displacement vector ~u and traction vector ~t as:
1
WX (kx , ky , z, ω) =
(2π)3
Z∞
dte
Z∞
dte
−iωt
Z∞
−∞
−∞
Z∞
dx
−∞
−∞
1
FX (kx , ky , z, ω) =
ω(2π)3
Z∞
−iωt
dy[~u(x, y, z, t) · (e~X )∗ ],
(4.6)
−∞
dx
Z∞
dy[~t(x, y, z, t) · (e~X )∗ ],
(4.7)
−∞
where the asterisk denotes the complex conjugate and the dot between two vectors represents
a dot product.
Equations (4.1) and (4.2) imply that the 3D wave motion in the Cartesian coordinates
(~ex , ~ey , ~ez ) in the time-space domain can be projected into three new orthogonal harmonic
directions (~eR , ~eS , ~eT ) in the transformed domain. The 3D wave motion in the Cartesian
coordinates (~ex , ~ey , ~ez ) can also be projected in the Cartesian coordinates (~ex , ~ey , ~ez ) themselves in the transformed domain. For example, Eq. (4.1) may alternatively be written in
the following explicit form:
~u(x, y, z, t) =
Z∞ Z∞ Z∞
(ũx~ex + ũy~ey + ũz~ez )ei(kx x+ky y+ωt) dkx dky dω
(4.8)
−∞ −∞ −∞
Where the relations between displacement components in the frequency-wavenumber domain
ũd (d = x, y, z) and variables WX (X = R, S, or T ) , Figure 4.1, can be easily found as:
ũx (kx , ky , z, ω) = (
ikx
iky
WS +
WT )
kr
kr
28
(4.9)
ũy (kx , ky , z, ω) = (
iky
ikx
WS −
WT )
kr
kr
ũz (kx , ky , z, ω) = WR
(4.10)
(4.11)
Figure 4.1: P-wave, longitudinal, corresponds to a displacement in the direction of propagation. The S wave, cross, corresponds to a displacement perpendicular to the direction of
propagation, SH and SV decomposable components.
The traction vector may also be written similarly as:
τ̃xz (kx , ky , z, ω) = (
iky
ikx
ωFS +
ωFT )
kr
kr
29
(4.12)
τ̃yz (kx , ky , z, ω) = (
iky
ikx
ωFS +
ωFT )
kr
kr
τ̃zz (kx , ky , z, ω) = ωFR
(4.13)
(4.14)
It can be shown, Kennett [74], that WR , WS , FR and FS are associated with P-SV waves
shown in Figure 4.2, while WT and FT are associated with SH waves shown in Figure 4.3.
Therefore, the displacement responses in x and y directions are composed of both P-SV and
SH wave motions, while the displacement response in the z direction is only contributed by
the P-SV wave motion.
Figure 4.2: P-SV waves with displacement in the vertical plane.
30
Figure 4.3: SH waves with displacement in the horizontal plane.
After performing the transformation, the wave motion presented in chapter 3 in the timespace domain (x, y, z, t) in each layer may be rearranged in a general form in the transformed
domain (kx , ky , z, ω) , i.e.,
 
   
w
~
~   0

w
d
= [A(i)]
−
, i = 1, 2, ...., n
 ~  
dz  ~ 
p~
f
f
(4.15)
where for the P-SV waves
w
~ =w
~ P −SV ≡ {WR , WS }T
31
(4.16)
f~ = f~P −SV ≡ {FR , FS }T
(4.17)
p~ = p~P −SV ≡ {PR , PS }T
(4.18)
[A(i)] is the coefficient matrix and a function of the physical parameters of layer i. Here,
≡ denotes ”by definition.” Coefficient matrix [A(i)] in Eq. (4.15) for the P-SV waves is

2β 2
α2
ω
ρα2
0
kr 1 −



 −kr
0
0

[A] ≡ [AP −SV ] = 

−ρω
0
0



2
r
−
1
−k
0
ρω γk
r 1−
ω2
and for the SH waves
0




ω 
2
ρβ 


kr 



2
2β
0
α2
(4.19)
w
~ =w
~ SH ≡ WT
(4.20)
f~ = f~SH ≡ FT
(4.21)
p~ = p~SH ≡ PT
(4.22)
Coefficient matrix [A(i)] in Eq. (4.15) for the SH waves is

where

[A] ≡ [ASH ] = 
ω
ρβ 2
0
ρω
32
β 2 kr2
ω2
−1
0



(4.23)
γ = 4β
2
β2
1− 2
α
(4.24)
The subscripts P-SV and SH are used to distinguish the two kinds of wave motion:
the P - wave velocity, α , is given by
α2 =
(λ + 2µ)
,
ρ
(4.25)
µ
,
ρ
(4.26)
and the S- wave velocity, β , is given by
β2 =
Thus, the wave velocity in the solid depends on the elastic behavior which can be described either by the Lame coefficients λ and µ or the pair (E, ν), where ν is the Poisson’s
ratio and E is the Young’s modulus (also called modulus of longitudinal elasticity). The
elastic constants described above are related by simple relations summarized in Table 4.1.
Table 4.1: Relations between the different elastic constants of an isotropic solid
E
ν
λ
µ
(λ, µ)
µ(3λ+2µ)
λ+ν
λ
2(λ+µ)
λ
µ
(E, ν)
E
ν
Eν
(1+ν)(1−2ν)
E
2(1+ν)
consequently, depending on the Poisson’s ratio ν and Young’s modulus E, the velocities
are written
α2 =
E(1 − ν)
,
ρ(ν + 1)(1 − 2ν)
(4.27)
E
,
2ρ(ν + 1)
(4.28)
β2 =
33
Displacement and stress discontinuities will be supposed that do not exist at the same
time, however, in the transformed domain, displacement and stress discontinuities should be
null.
4.2
Wave Scattering Matrices in a Multi-layer Medium
The governing equations, Eq. (4.15), in the transformed domain and the continuity con-
dition on the interface can be obtained for each layer in a multi-layer medium, as discussed
in the previous chapter. In order to solve the responses effectively, one can introduce the
following transformation
 
~
w
 
Mu (i) Md (i) µ
~ u

= [D(i)]
=
 ~
 
 
µ
~d
Nu (i) Nd (i)
µ
~d
f
 
~ u
µ

(4.29)
where columns in matrix [D(i)] are the eigenvectors of matrix [A], µ
~ u and µ
~ d denote, respectively, an up- and down-going wave vectors.


Nu,d = 
kr εsi
ω
∓iqpi εpi
Mu,d = 

For P-SV wave motion,
kr εpi
ω
∓iqsi εsi

2β 2 k2
∓2iρi βi2 kr qsi εsi
ω
∓2iρi βi2 kr qpi εpi
2βi2 kr2
ω 2 −1
ρi ( ω2i−1r )εpi
ω
ρi (
and for SH wave motion,
Mu,d (i) =
εsi
,
βi
Nu,d (i) = ∓iρi βi qsi εsi
34
(4.30)

)εsi



(4.31)
(4.32)
(4.33)
qpi =
s
1
kr2
−
αi2 ω 2
(4.34)
qsi =
s
1
kr2
−
βi2 ω 2
(4.35)
1
εpi = (2ρ qpi )− 2
(4.36)
1
εsi = (2ρ qsi )− 2
(4.37)
To account for damping, the real-valued P and S wave speeds can be replaced by a pair of
complex ones, i.e., by αi [1 + i sgn(ω)/(2Qpi )] and βi [1 + i sgn(ω)/(2Qsi )] , where sgn(ω)
denotes the sign of frequency ω , Qpi and Qsi are the attenuation factors for P and S waves
in layer i. The branch cuts for the radicals in the expressions for qpi and qsi are taken to be:
Im(ωqpi ) ≥ 0
(4.38)
Im(ωqsi ) ≥ 0
(4.39)
where Im denotes imaginary part.
The inverse of matrix [D(i)], which will be useful later, may be obtained as follows:

[D(i)]−1 
−NdT (i)
MdT (i)
NdT (i)
−MdT (i)
where superscript T represents the matrix transpose.


(4.40)
The coefficient i indicates the layer number for matrices D, M and N .
Also, the wave vector µ
~ (upward or downward) in Eq. (4.29) relates only to the vertical
direction. Note the exact propagation directions of these waves are usually not referred to
35
the z direction. Their dependence on the spatial coordinates have been represented in the
transformed domain. (kx = ky = 0) Indicates that waves propagate precisely in z direction.
Substituting Eq. (4.29) into Eq. (4.15) and solving for a wave vector at zk in terms of
the wave vector at zi , one obtains


~ u (k)
µ
where for the P-SV waves

µ
~ d (i)
= [Q(i, k)]



~ d (k)
µ

µ
~ u (i)
(4.41)

[Q(i, k)] = diag[eiωqpi ∆z eiωqsi ∆z e−iωqpi ∆z e−iωqsi ∆z ]
(4.42)
and for the SH waves
[Q(i, k)] = diag[eiωqsi ∆z
e−iωqsi ∆z ]
(4.43)
In Eqs. (4.41)-(4.43), [Q] is called the wave propagator, diag indicates a diagonal matrix,
+ ,the out-going wave vectors µ
~ u (i) and µ
~ d (k) for medium bounded with
∆z = zi− − zi−1
(zi , zk ) or simply (i, k) are related to input wave vectors µ
~ d (i) and µ
~ u (k) through reflection
~ and T~ or the scattering matrix as
and transmission vectors R


~ u (i) 
µ

µ
~ d (k)


=
R(i, k)
T (i, k)
which is schematically shown in (Figure 4.4).
36


T (k, i)  µ
~ d (i) 



R(k, i)
µ
~ u (k)
(4.44)
Figure 4.4: Schematic of relation between out-going wave vectors µ
~ u (i) and µ
~ d (k) and input
~
wave vectors µ
~ d (i) and µ
~ u (k) through reflection and transmission vectors R and T~ .
As shown in Figure 4.5, µ
~ d (i − 1+ ) and µ
~ u (i− ) may be treated as incoming wave vectors
with respect to layer i, while µ
~ u (i − 1+ ) and µ
~ d (i− ) treated as outgoing wave vectors. Eq.
(4.41) may then be rearranged according to such an input-output relationship as follows:


~ u (i − 1+ )
µ

µ
~ d (i )
−


=
R(i − 1+ , i− )
T (i − 1+ , i− )


~ d (i − 1+ )
T (i− , i − 1+ ) µ



−
+
µ
~ u (i− )
R(i , i − 1 )
(4.45)
where, the square matrix on the right hand side is known as a wave scattering matrix and
its sub-matrices R and T are known as reflection and transmission matrices, respectively.
+
The reflection and transmission matrices in layer i, which is bounded by depths zi−1
and
zi− , are.
37
R(i − 1+ , i− ) = R(i− , i − 1+ ) = 0
T (i − 1+ , i− ) = T (i− , i − 1+ ) =
(4.46)

 diag[eiωqpi ∆z eiωqpi ∆z ], (P − SV )

e
iωqpi ∆z
,
(4.47)
(SH).
Reflection and transmission matrices characterize the properties of the medium between
+
level zi−1
and level zi− in response to the incoming waves µ
~ d (i−1+ ) and µ
~ u (i− ). For example,
R(i−1+ , i− )~µd (i−1+ ) and T (i−1+ , i− )~µd (i−1+ ) represent, respectively, the reflected and the
transmitted portions of the incoming wave µ
~ d (i−1+ ). The physical meanings of T (i− , i−1+ )
and R(i− , i − 1+ ) are similar. By focusing our attention to within the layer between levels
+
−
+
zi−1
and zi− , we have excluded the layer-to-layer interfaces (zi−1
, zi−1
) and (zi− , zi+ ) from
immediate consideration. Therefore, the two reflection matrices in Eq. (4.46) are null.
Figure 4.5: The reflection and transmission matrices in layer i.
38
Next, we shall focus our attention on the interface (zi− , zi+ ). The continuity condition on
the i-th interface, i.e. Eq. (3.12), may be rewritten as:
 
~
w
=
 ~
f z−
 
~
w
(4.48)
 ~
f z+
i
i
However, the corresponding two wave vectors obtained from applying Eq. (4.29) are
different, because the transformation matrices [D(i)] and [D(i + 1)] are different. These two
wave vectors are related as


~ u (i− )
µ

where

[Q(i+ , i− )] = 
µ
~ d (i )
−
= [Q(i+ , i− )]

Q11 (i+ , i− )
+
Q21 (i , i )
−


~ d (i+ )
µ

Q12 (i+ , i− )
+
Q22 (i , i )
−
µ
~ u (i+ )
(4.49)


 = [D(i)]−1 [D(i + 1)]
(4.50)
With the aid of Eq. (4.40) and the definition of matrix [D], the sub-matrices of [Q] are
easily obtained as follows:
Q11 = −iNdT (i)Mu (i + 1) + iMdT (i)Nu (i + 1)
(4.51)
Q12 = −iNdT (i)Md (i + 1) + iMdT (i)Nd (i + 1)
(4.52)
Q21 = iNuT (i)Mu (i + 1) − iMuT (i)Nu (i + 1)
(4.53)
Q22 = iNuT (i)Md (i + 1) − iMuT (i)Nd (i + 1)
(4.54)
39
The wave vectors in Eq. (4.50) may be classified as incoming and outgoing waves with
respect to interface z = zi , shown in Figure 4.6. Therefore, Eq. (4.49) may be recast in an
input-output relationship similar to Eq. (4.45),


~ u (i− )
µ

+
µ
~ d (i )


=
R(i− , i+ )
+
T (i , i )
−


T (i+ , i− ) µ
~ d (i− )



+ −
R(i , i )
µ
~ u (i+ )
(4.55)
It can be shown that the reflection and transmission matrices at the interface zi bounded
by zi− and zi+ or (i− , i+ ) are
R(i− , i+ ) = Q12 Q−1
22
(4.56)
R(i+ , i− ) = Q−1
22 Q21
(4.57)
T (i− , i+ ) = Q−1
22
(4.58)
T (zi+ , zi− ) = Q11 − Q12 Q−1
22 Q21
(4.59)
When the two bounding depths are within the same layer and they have the same physical
properties, evidently Q12 = Q21 = 0, which would lead to no wave reflection occurs within
a uniform medium (R = 0). Clearly, Eq. (4.50) depends on the physical properties of
two layers. the reflection matrices in Eq. (4.55) are not null by chance that the physical
properties of the two layers are different, and the transmission matrices, according to the
well-known Snells law, should account for both energy dissipation and change of phase.
40
Figure 4.6: Transmission matrices at the interface.
Based on the fundamental reflection and transmission matrices within each layer, at each
interface and surface, the corresponding reflection and transmission matrices between any
two depths can be constructed using the following composite rule
R(i, k) = R(i, j) +
T (i, k) =
T (j, i)R(j, k)
T (i, j)
[I − R(j, i)R(j, k)]
T (j, k)
T (i, j)
[I − R(j, i)R(j, k)]
where (i, k) can be used reversely, i.e., (k, i), (Figure 4.7).
41
(4.60)
(4.61)
Figure 4.7: Composition Rule in locations (k,j,i)
4.3
Boundary Conditions
A reflection matrix alone can characterize the boundary conditions on the free surface.
At the free surface, the relationship between the state vector and the wave vector, Eq. (4.29)
can be written as
 
~ 0
w
~ 
f0
=
 
~ 0
w
~ 
f0 z +
i


~ u (0+ )
Mu (1) Md (1) µ

=


µ
~ d (0+ )
Nu (1) Nd (1)

where Mu (1), Md (1), Nu (1), and Nd (1) are submatrices of D1
42
(4.62)
The reflection matrix at one free surface and the other as free or fixed in Eqs. (3.14)-(3.16)
are respectively
R(0+ , 0) = −[Nd (1)]−1 Nu (1)
(4.63)
R((n − 1)− , (n − 1)+ ) = −[Nd (n − 1)]−1 Nu (n − 1)
(4.64)
R((n − 1)− , (n − 1)+ ) = −[Md (n − 1)]−1 Mu (n − 1)
(4.65)
and
or
4.4
Buried Point Source in Transformed Domain
The dislocation of a fault will be treated as a displacement or stress discontinuity. For
discontinuity source in layer k, Eqs. (3.17) and (3.18) become
 
~
w
−
 ~
f z=z+
s
 
~
w
=
 ~
f z=z−
s
 
~ s
w
(4.66)
~ 
fs
Substituting Eq. (4.29) into Eq. (4.66) and using wave reflection and matrices at two
depths, one can find

Mu (k)

Mu (k)


Nu (k)
Nu (k)


Md (k) R(s+ , n+ )~µu (s+ )

−


µ
~ d (s+ )
Nd (k)
  

~ s
µ
~ u (s− )
Md (k) 
 w

=

 ~ 
R(s− , 0)~µd (s− )
Nd (k)
fs
(4.67)
Equation (4.67) can be solved for dislocation within the k th layer
+
−
µ
~ (s− ) = [I − R(zsj
, ∞)R(zsj
, 0)]−1 [wsu − fsu ]
43
(4.68)
NdT (k) + R(s+ , n+ )NuT (k) w
~s
o
T
−
T
~
− Md (k) + R(zsj , ∞)Mu (k) fs
wsu − fsu =i
(4.69)
and for dislocation at the interface of layer k − 1 and k, equation (4.67) becomes:


Md (k + 1) R(s+ , n+ )~µu (s+ )


−


Nu (k + 1)
Nd (k + 1)
µ
~ d (s+ )
  


~ s
Mu (k)
Md (k) 
µ
~ u (s− )
 w


=

 ~ 
Nu (k)
Nd (k)
R(s− , 0)~µd (s− )
fs

Mu (k + 1)
(4.70)
By using [D(k)]−1 , equation (4.70) can be solved as:
−
µ
~ (s− ) = [I − Qif c R(zsj
, 0)]−1 [wsu − fsu ]
(4.71)
wsu = −iNdT (k + 1) + iR(s+ , n+ )NuT (k) w
~s
(4.72)
fsu = iR(s+ , n+ )MuT (k + 1) + iMdT (k + 1) f~s
(4.73)
Qif c = [Q11 R(s+ , n+ ) + Q12 ][Q21 R(s+ , n+ ) + Q22 ]−1
(4.74)
where
The wave responses at depth z = 0 in transformed domain subjected to the dislocation
can then be found as follows:
−1
w
~ = Mu (1) + Md (1)R(0+ , 0) I − R(0+ , s− )R(0+ , 0)
T (s− , 0+ )µu (s− )
(4.75)
With the knowledge of solution in the transformed domain, the response in the space
and time domains can be obtained through the inverse integral transform, Eq. (4.1).
44
CHAPTER 5
NUMERICAL EXAMPLE
Numerical examples that demonstrate the validity of the model are presented. Finite
integral transform procedure is applied to solve for wave response.
Specifically, triple
finite Fourier transforms are used between time-space domain (x, y, z, t) and frequencywavenumber domain (kx , ky , z, ω). It can be shown that the displacement and traction
vectors can be represented in a concise form with |t| ≤ tt , |x| ≤ xt and |y| ≤ yt , that
is, Eq. 4.1 can be represented as
∞
∞
∞
π π π X iωt X X
dte
~u(x, y, z, t) =
dx
dy(WR~eR + WS ~eS + WT ~eT )
tt xt yt −∞
−∞
−∞
(5.1)
in which kx = mπ/xt and ky = nπ/yt are respectively wave numbers in x and y directions,
ω = lπ/tt is the frequency.
As a limiting case, summations in Eq. 5.1 become integrals when the truncated parameters of tt , xt and yt approach infinity, indicating the conventional integral transform
approach.
5.1
Features of Wave Responses to Point Impulsive Dislocation Source
For illustration, displacement wave responses at surface locations, see Table 5.1, are
computed to a point, impulsive shear-dislocation source buried in a uniform half-space with
7000-series aluminum alloys [59].
The parameters used are P-wave velocity of 6320 (m/s), S-wave velocity of 3100 (m/s),
a density of 2700 (kg/m3 ) and the source location at xs = ys = 0, zs = 0.783 (mm). The
dislocation parameters are selected as ∆un = 0, ∆us = 0.01 (mm), θs = 0, and fault area
A = 0.001 (mm)2 . The truncated values in Eq. 5.1 are selected as xt = yt = 193.8 (mm) and
tt = 125 (µs) so that no wave interference in the responses from wave reflection at truncated
45
boundaries (xt , yt ) within the time window (tt ), i.e., 2xt /c > tt and 2yt /c > tt where c is the
slowest wave speed in the material, Figure 5.1.
Table 5.1: Observation positions at the surface
Observation
Positions
1
Surface Locations
x (mm) y (mm)
15
20
2
22.5
30
3
30
40
The truncated values correspond to the resolution of frequency and wavenumbers as
∆kx = ∆ky = 4.0537 (m−1 ), and ∆ω = 4.0537 (rad/s). For comparison, the displacements
at the surface locations are also computed for the three-layer medium given in Table 5.2 and
Figure 5.2.
Table 5.2: Physical properties of layers
Layer
1
α(m/s)
4,320
β(m/s)
2,200
Qα
25
Qβ
50
ρ(kg/m3 )
1670
z(mm)
0-0.5
2
6,320
3,180
12.5
25
2700
0.5-4.7
3
7,320
4,150
6.25
12.5
2610
4.7-∞
Figures, (Figure 5.3 - Figure 5.6), show the P-SV and SH wave response amplitudes
at surface (i.e., WS and WT ) versus the wave number in radial direction, kr at selected
frequencies 5 and 20 M (rad/s) using Eq.(4.75). The dominant peaks of P-SV wave response
(WS ) in Figure 5.3 for the uniform half space corresponds to the P- and S-wave modes at
kr /α = 791 (rad/m) and kr /β = 1572 (rad/m), while similar peaks for the three-layer
medium are found, showing the averaged P- and S-wave modes over the three layers. No
peak of SH wave response (WT ) in Figure 5.3 is observed for uniform half-space, indicating
46
that no SH-wave mode or surface waves exist for the uniform half-space. In contrast, a couple
of peaks show off for the three-layer medium in Figure 5.4, corresponding to the different
surface wave modes such as Love waves. Figures Figure 5.5 and Figure 5.6 show the similar
wave response features at different frequency.
Figures, (Figure 5.7 - Figure 5.10), present the x- and z-direction displacement amplitudes
in the wave number domain Eqs. (4.9) and (4.11) at selected frequencies for a uniform and
a layered media, indicating respectively the combined P-SV and SH wave responses for the
former and only P-SV wave responses for the latter. Figure 5.11 shows the amplitudes of
x-direction displacement in the frequency domain for a uniform and a three-layer media,
indicating more complexity of wave responses in a layered medium than the uniform half
space in addition to the similar overall frequency features.
Figure 5.12 shows the x-direction displacements at observation locations 1,2 and 3. It
can be seen from this figure that the displacement at location 1 is zero until the first P wave
signal arrives at 4.0(10−6 ) s which is measured in Figure 5.12 and also consistent with the
theoretical calculation based on the P wave speed and source-to-response distance. The S
wave signals arrive later and give rise to larger displacement than that of the P waves at
8.2(10−6 )s. The displacement at location 2 and 3 show similar wave propagation features
with later arrival times and smaller, damping-related amplitudes. Figure 5.13 also shows
the response in frequency domain at observation locations 1,2 and 3. We also compared our
finding with a special case (uniform half space) with Hamstad et al., [59] and the general
feature is consistent with each other.
For a three-layer medium, Figure 5.14 demonstrates that, in addition to the similar, first
arrival wave times, the AE signals show more wave reflection from the layers below the
source. This figure also shows that the amplitude in the x-direction is larger than those in
the y- and z-directions, attributable to the dislocation source direction. Figure 5.15 shows
the response in frequency domain in a three-layer medium.
47
Figure 5.1: Representation of wave propagation and the reflection. A source emits a signal through the medium and the boundary retransmits the signal, returned in the reverse
direction.
48
Figure 5.2: One-layer half space for numerical example. Dashed line shows the direction of
the observation positions 1, 2 and 3 on the surface.
49
Figure 5.3: P-SV and SH wave responses at surface with 5 M rad/s for a uniform media.
Figure 5.4: P-SV and SH wave responses at surface with 5 M rad/s for a layered media.
50
Figure 5.5: P-SV and SH wave responses at surface with 20 M rad/s for a uniform media.
Figure 5.6: P-SV and SH wave responses at surface with 20 M rad/s for a layered media.
51
Figure 5.7: The x- and z-direction displacement amplitudes with 5 M rad/s in the frequencywave number domain in a uniform medium.
52
Figure 5.8: The x- and z-direction displacement amplitudes with 20 M rad/s in the frequencywave number domain in a uniform medium.
53
Figure 5.9: The x- and z-direction displacement amplitudes with 5 M rad/s in the frequencywave number domain in a layered medium.
54
Figure 5.10: The x- and z-direction displacement amplitudes with 20 M rad/s in the
frequency-wave number domain in a layered medium.
55
Figure 5.11: The x-direction displacement amplitude spectra in a uniform and a three-layer
media.
56
Figure 5.12: Displacement wave responses at observation locations 1,2 and 3, in time domain.
57
Figure 5.13: Displacement wave responses at observation locations 1,2 and 3, in frequency
domain.
58
Figure 5.14: Displacements vs time for a three-layer medium in x, y and z directions.
59
Figure 5.15: Displacements vs frequency for a three-layer medium in x, y and z directions.
60
5.1.1
The Influence of Damping
In Figure 5.16, we compare the displacement response spectra at three different damping
attenuation factors for three layered medium in the x- direction taken at observation points
(x = 15 mm, y = 20 mm). The damping attenuation factors of the three layered media have
been chosen as in Table 5.3 to show the effect of damping on the response of the medium.
There is a large difference in the observed values using damping 1, and this is clearly a
distinct feature of the response spectra. In comparison, to the other results with different
damping attenuation factors, the damping parameters strongly influence the displacement
response. This figure shows that a certain layered medium with a smaller damping ratio in
a given frequency range will cause the response spectrum to have larger amplitudes in that
same frequency range in other medium.
Table 5.3: Comparison with different damping
Layer
Damping 1
(Q)
P wave S wave
Damping 2
(Q)
P wave S wave
Damping 3
(Q)
P wave S wave
1
25
50
16.67
25
12.5
16.67
2
12.5
25
10
16.67
8.33
12.5
3
6.25
12.5
6.25
12.5
6.25
10
To gain additional insights, Figure 5.17 shows the damping of the layered medium can
suppress its vibration. With the increase of the damping attenuation factors, the maximum
amplitude of the response decreases rapidly.
61
Figure 5.16: Affecting different damping ratios on the response in layered medium in frequency domain.
62
Figure 5.17: Affecting different damping ratios on the response in layered medium in time
domain.
63
5.1.2
Source at Different Depth
To provide results with a source at different depth, Figure 5.18 and Figure 5.19 display
similar views for a dislocation source located at 1.723 (mm) below the top surface. These two
figures demonstrate the fact that the signals in the x- direction have amplitude significantly
larger than those in the z- and y-directions.
Figure 5.18: Comparison of displacement signals at observation point located at (x =
15 mm, y = 20 mm) from the source centered at a depth of (1.723 mm) below the surface in frequency domain.
64
Figure 5.19: Comparison of displacement signals at observation point located at (x =
15 mm, y = 20 mm) from the source centered at a depth of (1.723 mm) below the surface in time domain.
65
5.1.3
Influences of Boundary Conditions
This part presents the model with two new boundary conditions: one is the free surface
boundary condition and the other is the fixed boundary condition at bottom. In order to
have a fixed boundary in our model, we increase the wave speed for the last layer with a huge
number compared to that of the other layers. For example, a wave propagates from the source
to the fixed boundary, which has a much higher shear modulus and bulk modulus compared
to those of wave source. The fixed boundary is equivalent to the boundary condition with
(u = 0).
Figure 5.20 is the simulated displacement in the x-, y- and z- direction at an observation
point 1. It can be seen from the figure that the displacements are zero until the first P wave
signal arrives, as explained previously. The S wave signals which arrive later give rise to a
greater response compared with those due to the P waves. The displacements gradually die
down to zero after the last wave signal arrives from the point source. Figure 5.21 shows the
magnitude of the corresponding frequency response of the surface displacement in the x-, yand z- direction.
In Figure 5.22 The effect of the fixed boundary condition on the model can be seen by
comparing its results (solid line) with those obtained at infinity boundary condition (dashed
line) in x- direction. The results show that the amplitude of wave response in half space
example is less than that the amplitude of wave response in the fixed boundary example
in frequency domain since part of the energy in the incident wave is reflected and part
undergoes transmission; that is, some of energy passes through the boundary. However, the
energy of the reflected wave is equal to the energy of the incident wave at the fixed boundary.
According to the principle of the conservation of energy, when the wave breaks up into a
reflected wave and transmitted wave at the boundary, the sum of the energies of these two
waves must equal the energy of the incident wave. Because the reflected wave contains only
part of the energy of the incident wave, its amplitude must be smaller.
66
Figure 5.20: Displacement due to point source in layered medium with free-fixed boundary
condition in frequency domain.
67
Figure 5.21: Displacement due to point source in layered medium with free-fixed boundary
condition in time domain.
68
Figure 5.22: Comparison of displacement in the x-direction for layered medium at fixed
boundary condition (solid line) to infinity boundary condition (dashed line) in frequency
domain.
69
5.2
Features of Wave Responses to General Dislocation Source
After finding a wave response at z=0 in the transformed domain, the wave response to
the individual point source can be easily obtained. Each individual point source can be
found on the fault plane. The motion at surface location (x, y, z = 0) and at time t due to
dislocation point source at (0, 0, zs ) and initiated at time τ is given by
1
gd (x, y, z, t; xs , ys , zs , τ ) =
2π
Z
∞
R̃(ω, τ )ũd (x, y, z; ω) eiωτ dω
(5.2)
−∞
where R̃(ω, τ ) is the Fourier transformation of a source time function characterizing the
dislocation growth, and ũd (= x, y, z) the displacement response to a unit-impulse, point
dislocation in the frequency domain. For the source time function selected as a ramp function,
as described in Figure 3.2, R̃(ω, τ ) is given by
R̃(ω, τ ) =
1
ω2T
(e−iωTr − 1)eiωτ
(5.3)
r
Where τ is a triggering time and Tr is the rise time.
The acceleration at ground in the x direction has also been computed for layered medium,
with the same material properties as given in table (2) and the results are shown in Figure 5.26.
Figure 5.25 describes the amplitudes of the computed frequency-domain responses, showing generally negligible response beyond w = 2.5(107 ) (rad/sec) and zero response at
w = 0.418(107 ) (rad/sec). The frequency content of the moment is given by equation (5.3)
and is plotted in Figure 5.24 , which shows R̃(w, t) = 0 at wT r = 2nπ, or w = 0.418(107 )
(rad/sec) for T r = 1.5(10−6 ) s.
70
Figure 5.23: A ramp source function with triggering time τ , rise time Tr , and slip amplitude
S.
Figure 5.24: Fourier transform of the time-variation of the moment in which described by
R(t, τ ).
71
Figure 5.25: Displacement response to general dislocation source at location 1 in x, y and z
directions in frequency domain.
72
Figure 5.26: Displacement response to general dislocation source at location 1 in x, y and z
directions in time domain.
73
CHAPTER 6
ON WAVE RESPONSE FEATURES WITH PROPAGATING DISLOCATION SOURCE
IN LAYERED MATERIALS
In the previous chapters, an analytical procedure is developed for numerical modeling of
AE Sources and wave propagation. The analysis of motion in a horizontally layered media
has been demonstrated. Application of integral transform has been used in solving the wave
equations for each layer. Wave scattering matrices have been constructed for individual
layers, for layer to layer interfaces, and for treatment of boundary conditions. Surface of
motion has been computed in the domain of frequency and wave number, and then inverted
to the time space domain. The result so obtained gives a single pulse which arrives at the
surface due to a signal point source. Since many such pulses of different amplitudes can be
superposed and arrived at different times, one can obtain a model for surface motion due to
a line source.
6.1
A Model with Propagating Source Line
This chapter presents a continuum-mechanics model for AE wave propagation in layered
materials, which is generated by time-dependent dislocation-source in the form of a line
buried within one of the layers or at the layer-to-layer interface. The materials under consideration are modeled as vertically non-homogeneous or layered media with each layer being
isotropic, homogeneous and linearly elastic, as schematically shown in Figure 6.2, while the
AE source is described as a line source, which can be discretized into a series of point sources
with separated distances controlled by rupture velocity vr starting at point A and ending at
point B. The j − th point source at (xsj , ysj , zsj ) is characterized with triggering time tsj
and rising time Trj and slip amplitude Sj of a ramp function in Figure 6.1.
The total displacement response utd (d = x, y, or z) to the (n) point sources along the
line source, in Figure 6.2, can then be found as:
74
utd (x, y, z, t)
=
n
X
Sj gd (x, y, z, t; xsj , ysj , zsj , tsj )
(6.1)
j=1
Where Aj is the area of the j − th point source, Sj the j − th slip amplitude, and gd the
displacement response in the d-direction at the observation point (x, y, z) and at time (t),
due to a unit-area point source activated at time tsj at location (xsj , ysj , zsj ).
Figure 6.1: Geometry and discretization of dislocation in an inclined-to-surface fault. The
dislocation at the j − th sub-area denoted with a star is characterized with a ramp source
function with triggering time tsj , rise time Trj , and slip amplitude Sj .
The d-direction displacement wave response gd to a time-dependent point dislocation
source can be found as:
1
gd (x, y, z, t; xsj , ysj , zsj , tsj ) =
2π
Z
∞
R̃j (ω, tsj )ũd (x, y, z; ω) eiωtsj dω
(6.2)
−∞
where R̃j (ω, tsj ) is the Fourier transformation of a source time function characterizing
the dislocation growth, and ũd (= x, y, z) the displacement response to a unit-impulse, point
75
dislocation in the frequency domain.For the source time function selected as a ramp function,
in Figure 6.1, R̃j (ω, tsj ) is given by
R̃j (ω, tsj ) =
1
ω2T
(e−iωTrj − 1)eiωtsj
(6.3)
rj
The wave response ud or ũd (= x, y, z) is obtainable by solving force-free wave equation
of motion with an impulsive dislocation rupturing at a point with a given unit fault area.
In doing so, it is also required of pertinent conditions that are the continuity conditions at
each layer-to-layer interface and boundary conditions at both surfaces.
6.2
Numerical Example
For illustration, displacement wave responses at surface locations (x = 15 mm, y =
20 mm, z = 0) are computed to the line source in the x-direction, located in the second layer
with 7000−series aluminum alloys [59], as shown in Figure 6.3.
The material parameters used can be found in Table B.1, while dislocation parameters
are selected as shown in previous chapter.The source depth zs = 0.783 (mm), source length
= 45 (mm) and the assumed rupture speed = 1000 (m/s).
Figure 6.4 and Figure 6.5 present the x- and z-direction displacement amplitudes in the
wave number domain Eqs. (4.9) and (4.11) at selected frequencies for a layered media,
indicating respectively the combined P-SV and SH wave responses for the former and only
P-SV wave responses for the latter.
Figure 6.6 shows amplitudes of frequency response characteristic in the x-, y- and zdirections. The results show that the peak of displacement in the x- direction is the largest,
due to the fact that the dislocation occurs in the x- direction. The displacements in y- and
z-directions at the same location show different wave propagation features with significantly
smaller amplitudes, damping-related amplitudes. Figure 6.7 shows the magnitude of the
corresponding time response of the surface displacement in the x-, y- and z- directions at
an observation location (x = 15 mm, y = 20 mm, z = 0). It can be seen from the figure
76
Figure 6.2: Profile of layered materials with buried line source.
77
Figure 6.3: Profile of layered materials with buried line source in x-direction.
78
that the displacement is zero until the first P wave signal arrives. The S wave signals which
arrive later give rise to a greater response compared with those due to the P waves. The
displacement gradually dies down to zero after the last wave signal arrives from the line
source.Among the three magnitudes in the three directions, the peak of displacement in the
x-direction is the largest due to the fact that the dislocation occurs in the x-direction.
Figure 6.4: The x- direction displacement amplitudes in the frequency-wave number domain
in a layered medium.
79
Figure 6.5: The z-direction displacement amplitudes in the frequency-wave number domain
in a layered medium.
80
Figure 6.6: Displacement responses in frequency domain to line source at location (x =
15, y = 20) in the x-, y- and z-directions.
81
Figure 6.7: Displacement responses in time domain to line source at location (x = 15mm, y =
20mm) in the x-, y- and z-directions.
82
CHAPTER 7
CONCLUSION
In this thesis, a three-dimensional wave-propagation model has been developed to compute AE wave motion in a layered medium. This model consists of (n) horizontal layers
separated by (n − 1) interfaces. The materials in the model have been characterized by a
layer of homogeneous, isotropic and linearly-elastic media with one surface free and the other
as free, fixed or infinite boundary. A dislocation source was buried in one layer where the
waves were originally generated and then propagated through the layers of medium.
In terms of primary contributions, we implemented a dislocation source that we described
directly with a dislocation form, which is subsequently used in the medium and interface for
finding wave motion instead of using equivalent body forces. In addition, the equivalent body
forces are valid only for describing a dislocation within a medium, which is not applicable to
generalized dislocations at the interface of two materials such as debonding damage. We also
implemented a dislocation source in AE modelling. The advantage is that the dislocation
source, compared to the traditional way, can directly be used to solve wave equations. In
addition, the dislocation source can be applied to the interface of two materials, which the
traditional way is not capable of solving.
Wave propagation theory is widely used for a layered media, mostly in other fields such as
seismology and solid mechanics. However, this theory is not applied to AE wave propagation
modeling. Our second contribution is an application of this well-known theory to the AE
wave propagation. One of the major accomplishments is that the dislocation source can be
implemented at interface with the aid of AE wave propagation model.
In the numerical model, a Cartesian coordinate formulation was used to solve the wave
equations in 3-dimension. In a uniform medium, the 3-dimensional wave motion was decomposed into two sets of wave motion, the coupled P and SV wave, and the SH wave.
83
Orthogonal vector harmonics were introduced so that the 3-dimensional wave motion could
be represented in a concise form, thus simplifying the application to more complex problems. The modeling of a general dislocation source has been described and the resulting
displacement discontinuity is validated with the traditional description. A detailed physical
explanation has been given for the conditions that must be imposed to ensure the accuracy
of the numerical results when finite integral transformations are applied.
Numerical examples were carried out to simulate AE wave propagation. Wave response
features at the surface were found to agree well with analytical solutions. Furthermore,
the AE response could also be calculated for a dislocation source at an interface. We also
compared our finding with a special case (uniform half space) with Hamstad et al., [59] and
the general feature is consistent with each other.
The continuum-mechanics model for AE wave propagation in layered materials has been
extended to treat a dislocation source in a specific layer in the form of a line. The AE source
line is described as equidistant point sources with the assumption that the rupture speed
for a propagating dislocation from one point-source position to the next is constant. With
the model, AE wave propagation features have been studied in general, and wave responses
at free surface have been calculated. Subsequently, a numerical example of the line source
model has been examined to see the extent of influence on AE wave motion in general, and
motion at surface in particular. Furthermore, the AE responses for a line source in the
frequency and the time domain in x-, y- and z-directions were calculated and found to be
consistent with the treatment of the line source.
Future directions for research could include using the model developed in this work to
create a stochastic AE model. In this case, the strengths of AE signals emitted from discretized point sources maybe assumed to be independent and identically distributed random
variables.
84
APPENDIX A - TRADITIONAL METHOD OF DISLOCATION SOURCE
In this appendix, the equation of dislocation source is considered. In materials science,
a dislocation is an event that results in a displacement or discontinuity. The real physical
process of a shear dislocation is inelastic and inhomogeneous in the source region, which is
difficult to describe and analyze precisely.
A.1
Elastodynamic Representation Theorems
Let us consider an inhomogeneous elastic body occupying the medium around the source
region and apply the elastodynamics theory to obtain the response displacement field in
terms of Green’s functions. The Betti reciprocity theorem holds if the medium is linear and
system is conservative, And that will be touched upon briefly below.
A.1.1
Betti’s Theorem
Enrico Betti (1823-1892) was an Italian mathematician who incidentally proved that
two displacement field u1 and u2 in a volume V, generated respectively by two body force
distributions f1 and f2 , and corresponding tractions T1 and T2 on surface S encompassing
the volume satisfying the following equation
ZZZ
=
ZZZ
(f1 − ρü1 ).u2 dV +
V
(f2 − ρü2 ).u1 dV +
V
ZZ
ZZ
T1 .u2 dS
S
T2 .u1 dS.
(A.1)
S
Consider u1 , T1 , f1 at time t, and u2 , T2 , f2 at time τ − t with τ an arbitrary constant,
and integrate over time from t = −∞ to t = +∞. If we assume that there exists a time t0
before which u1 and u2 , and hence u̇1 and u̇2 , are constant and equal to 0. We can then
cancel out terms with second time-derivatives of u and v and find the time-integrated form
of Betti’s theorem, valid only for displacement fields, obtaining:
85
Z
=
+∞
dt
−∞
Z +∞
−∞
ZZZ
dt
ZZ
[u1 (t).f2 (τ − t) − u2 (τ − t).f1 (t)] dV
V
[u2 (τ − t).T1 (t) − u1 (t).T2 (τ − t)] dS.
(A.2)
S
This is an important result, since it allows the representation of the displacement due to
a system of forces by those produced by a different system, given that causality conditions
are satisfied. We can, then, represent the displacements due to a complicated system of
forces in terms of those produced by a simpler one. We can select, as the simplest system of
forces, an impulsive force in space and time.
fi (ζ, t) = δin δ(ζ − ξ)δ(τ − t)
(A.3)
where δin is Kroneckers symbol (δin = 1 if i = n, 0 otherwise).
Each displacement component ui depends on the orientation of the force fn , and thus the
displacement is given by a second order tensor Gni (ζ, t; ξ, τ ) where G is the Green’s function
of elasto-dynamics, represents the elastic displacements due to a unit impulse force in space
and time, and depends on the medium properties.
Now consider a control volume in which a unit impulsive point force is acting at location
ξ(ξ1 , ξ2 , ξ3 ) in the nth direction at time τ . The displacement response at location ζ(ζ1 , ζ2 , ζ3 )
in the ith direction at time t is then defined by the Green’s function Gin (ζ, t; ξ, τ ) subject to
the causality that Gin (ζ, t; ξ, τ ) and Ġin (ζ, t; ξ, τ ) are zeros at t < τ . To find an expression
for the displacement field due to the body force f1 , we replace f2 (ζ, τ − t) and u2 (ζ, τ − t)
in Eq.(A.2) by a point force δin δ(ζ − ξ)δ(τ − t) and a Green’s function Gin (ζ, τ − t; ξ, 0),
respectively, and obtain
86
un (ζ, τ ) =
Z
+∞
dt
−∞
Z +∞
ZZ
[fi (ζ, t)Gin (ζ, τ − t; ξ, 0)] dV (ζ)
V
[Gin (ζ, τ − t; ξ, 0)Ti (ζ, t)
∂
Gkn (ζ, τ − t; ξ, 0) dS(ζ)
−ui (ζ, t).Cijkl nj
∂ξl
+
dt
ZZZ
−∞
S
(A.4)
where un , fi , Ti are the component of vectors u, f and T, respectively, and where the following
relations has been used, i.e.
Ti = τij nj
τij = Cijkl
∂)
Gkn
∂ζl
Cijkl = λδij δkl + µ (δik δjl + δil δjk )
(A.5)
(A.6)
(A.7)
in which nj is the component in the j-th direction of a normal unit vector on surface S and
δmn is a kronecker delta. Interchanging ζ and ξ as well as t and τ , Eq.(A.4) becomes
un (ζ, t) =
Z
+∞
−∞
Z +∞
dτ
ZZ
[fi (ξ, τ )Gin (ξ, τ − t; ζ, 0)] dV (ξ)
V
[Gin (ξ, τ − t; ζ, 0)Ti (ξ, τ )
∂
Gkn (ξ, τ − t; ζ, 0) dS(ξ)
−ui (ξ, τ ).Cijkl nj
∂ζl
+
−∞
dτ
ZZZ
S
(A.8)
In this equation, Green’s function acts as a propagator, since it propagates the effects
of forces, stresses, or displacements defined on coordinates (ξ, τ ) to determine the elastic
displacements at points of the coordinates (ζ, t).
87
A.1.2
Representation Theorem
The Representation Theorem defines the elastic displacements inside a volume V with
surface S due to the sum of two double integrals over space and time. The first constituent is a
volume integral over body forces multiplied by the Green’s function of the system; the second
part is a surface integral over stress times the Green’s function minus the displacements times
the derivatives of the Green’s functions. Using reciprocal properties of Green’s functions
satisfying homogeneous boundary conditions
Gnm (ζ, τ ; ξ, 0) = Gmn (ξ, τ ; ζ, 0)
(A.9)
Then, we can rewrite Eq.(A.8) to obtain an equation which is called the representation
theorem which is valid even when surface tractions exist.
un (ζ, t) =
Z
+∞
−∞
Z +∞
dτ
ZZZ
ZZ
[fi (ξ, τ )Gni (ζ, t − τ ; ξ, 0)] dV (ξ)
V
[Gni (ζ, t − τ ; ξ, 0)Ti (ξ, τ )
∂
−ui (ξ, τ ).Cijkl nj
Gnk (ζ, t − τ ; ξ, 0) dS(ξ)
∂ζl
+
−∞
A.1.3
dτ
S
(A.10)
Representation Theorem for an Internal Surface
To obtain an expression of the Representation Theorem for an Internal Surface, we consider an elastic medium of volume V surrounded by a surface S (Figure A.1). In its interior
there is a small region of volume surrounded by a surface Σ, where fracture takes place.
88
Figure A.1: An elastic medium of volume V surrounded by a surface S
The two opposite faces of the fracture are labeled Σ+ and Σ− . If slip occurs across Σ
, then the equation of motion is no longer satisfied throughout the interior of S where the
displacement field is discontinuous. However, it is satisfied throughout the ”interior” of the
surface Σ+ + Σ− + S , and to this we can apply Eq.(A.10). we shall assume that both u and
G satisfy homogeneous boundary condition on S. then
un (ζ, t) =
Z
+∞
−∞
Z +∞
dτ
ZZZ
[fi (η, τ )Gni (ζ, t − τ ; η, 0)] dV (η)
V
ZZ ∂
Gnp (ζ, t − τ ; ξ, 0)
∂ξq
Σ
−∞
−Gnp (η, t − τ ; ξ, 0)Tp (ξ, τ )] dS(ξ)
+
dτ
ui (ξ, τ ).Cijpq νj
(A.11)
This equation uses as the general position within V, and as the general position on Σ.
Square brackets are used for the difference between values on Σ+ and Σ− .
89
A.2
Equivalent Body Forces
To determine a body forces equivalent, we apply Eq.(A.10) with an additional assumption
that u also satisfies homogenous boundary conditions on S, the displacement field in V (Aki
and Richards [66]). Using the Delta δ(η − ξ) function is a convenient way to localize the
discontinuities on Σ and so, the displacement has given by the Traction discontinuity
Z
+∞
dτ
−∞
ZZZ
V
fpT Gnp (ζ, t − τ ; η, 0) dV (η)
(A.12)
where fpT is the force body equivalent of Traction discontinuity on Σ.
fpT
=
ZZ
[Tp (ξ, τ )] δ(η − ξ)dξ
(A.13)
Σ
The point of Σ within V can be localized by using the Delta function derivative
∂
δ(η − ξ)Gnp (ζ, t − τ ; η, 0)
∂ηq
(A.14)
which has the property
∂
Gnp (ζ, t − τ ; ξ, 0) =
∂ξq
ZZZ V
∂
δ(η − ξ)
Gnp (ζ, t − τ ; η, 0) dV (η)
∂ηq
(A.15)
So that the displacement discontinuity in Eq.(A.11) contributes the displacement
Z
+∞
dτ
−∞
ZZZ
V
fpu Gnp (ζ, t − τ ; η, 0) dV (η)
(A.16)
where fpu is the force body equivalent of Displacement discontinuity on Σ.
Substitution fpT and fpu in Eq.(A.11) yields
un (ζ, t) =
Z
+∞
dτ
−∞
ZZZ
V
fp + fpu + fpT Gnp (ζ, t − τ ; η, 0)dV (η)
(A.17)
It is interesting to see from Eq.(A.17) that the total displacement field may be decomposed into three parts, generated respectively by three body forces of which f u and f T are
90
equivalent body forces due to displacement discontinuity [u] and traction discontinuity [T]
. By introducing these equivalent body forces, our problem becomes much simpler since it
is no longer necessary to deal with the complex boundary conditions at the fault. Since
tractions must be equal on the two sides of a fault, the continuity of traction implies [T]=0.
Furthermore, the slip on a fault leads to a nonzero value for [u]. Taking Σ as an artificial
surface to across is the simplest and most commonly used way of determining a clarifying
property of G on Σ, which G and its derivatives are continuous, in order that G satisfies the
equation of motion even on Σ. Therefore Eq.(A.17) considerably simplified in the case of
the absence of body forces to.
un (ζ, t) =
Z
+∞
dτ
−∞
ZZZ
V
fpu Gnp (ζ, t − τ ; η, 0)dV (η)
(A.18)
Eq.(A.18) states that knowing the displacement on the fault is enough to determine
displacement everywhere.
A.3
Estimating The Slip on The Buried Fault
To reduce the body forces equivalent from Eq.(A.17), we shall take the fault to lie in the
plane η3 = 0 and let η1 be the direction of slip. As (Figure A.2) shows, Motion on the side
Σ+ (i.e.,η3 = 0) is along the direction of η1 increasing and the side Σ− is along η1 decreasing
(Aki and Richards [66]).
Figure A.2: Afault surface Σ is lying in the ξ3 = 0 plane
91
For a homogeneous and isotropic medium, the equivalent body forces have the forms of
f1u
=−
ZZ
µ [u1 (ξ, τ )]
Σ
∂
δ(η − ξ)dΣ
∂η3
f2u = 0
f3u
A.3.1
=−
ZZ
µ [u1 (ξ, τ )]
Σ
(A.19)
(A.20)
∂
δ(η − ξ)dΣ
∂η1
(A.21)
A Mathematical Representation of Moment Tensor
It is a representation of the movement on a fault, so that, the moment is a measure of the
size of the propagation of waves based on the area of fault rupture. It describes the strength
of the source due to the shear dislocation, and is directly related to the intensity scale of the
propagation of waves.
M0 (τ ) = µ[u1 (τ )]A
(A.22)
where,
µ = shear modulus
A = Area
[u1 (τ )] = Average displacement during rupture.
In the most commonly employed approach, the fault slip may be treated as a mathematical point source to simplify analysis. Therefore, we can write fpu approximately as
fpu (η, τ )
=
ZZ
[ui (ξ, τ )].Cijpq νj dΣ
Σ
∂
δ(η − ηs )
∂ηq
(A.23)
where ηs is a point on the fault plane Σ . Furthermore, to characterize the strength, orientation, etc. of a fault source, the source moment tensor is defined as
92
Mpq =
ZZ
[ui (ξ, τ )].Cijpq νj dΣ
(A.24)
Σ
In terms of these moment tensors, fpu can be expressed as
fpu (η, τ ) = −Mpq (τ )
∂
δ(η − ηs )
∂ηq
(A.25)
For a homogeneous and isotropic medium
Mpq = A[u][λνk nk δpq + µ(νp nq + np νq )]
(A.26)
where [u] is an average slip on the fault.
As shown in (Figure A.3), it is assumed that the source is located at the origin. The
corresponding equivalent forces are a couple f1u acting parallel to the η1 axis and another
couple f3u acting parallel to the η3 axis. These are represented by
f1u (η, τ ) = −µA[u1 (τ )]δ(η1 )δ(η2 )
∂
δ(η3 )
∂η3
f2u (η, τ ) = 0
f3u (η, τ ) = −µA[u1 (τ )]δ(η2 )δ(η3 )
93
(A.27)
(A.28)
∂
δ(η1 )
∂η1
(A.29)
Figure A.3: Double couple
Taking the moment of f1u about the η2 axis and integrating over volume V yields
ZZZ
η3 f1u (η, τ )dτ = µA[u1 (τ )] = M0 (τ )
(A.30)
η1 f3u (η, τ )dτ = µA[u1 (τ )] = M0 (τ )
(A.31)
V
similarly,
ZZZ
V
Equations (A.30) and (A.31) give the same results as those of equations (A.29) and win.
94
APPENDIX B - TRANSFORMATIONS COORDINATE SYSTEMS
The relationship between two coordinate systems must be characterized at specific moment in time. In order to find the instantaneous relationship between two coordinate systems,
the transformation must be determined by taking the representation of an arbitrary vector
in one system and converting it to its representation in the other. Direction Cosines are one
approach to finding the needed transformations between two coordinate systems.
Consider two reference frames, F 1 and F 2, and a vector u whose components are known
in F 1, represented as u1

u x′
{u}1 = uy′ 
uz ′

(B.1)
We wish to determine the representation of the same vector in F 2, or u2 :
 
ux

{u}2 = uy 
uz
(B.2)
These are linear spaces, so the transformation of the vector from F 1 to F 2 is simply a
matrix multiplication which we will denote [T2,1 ], such that u2 = [T2,1 ]u1 Transformations
such as [T2,1 ] are called similarity transformations.
The order of subscripts of [T2,1 ] is such that the left subscript goes with the system of the
vector on the left side of the equation and the right subscript with the vector on the right.
For the matrix multiplication to be conformal [T2,1 ] must be a 3 ∗ 3 matrix:

t11 t12 t13
[T2,1 ] = t21 t22 t23 
t31 t32 t33

(B.3)
The whole point of the approach to be presented is to figure out how to evaluate the
numbers [tij ] . It is important to note that for a fixed orientation between two coordinate
95
systems, the numbers [tij ] are the same quantities no matter how they are evaluated.
Consider our two reference frames F 1 and F 2, and the vector u, shown in (Figure B.1).
We claim to know the representation of u in F 1. The vector u is the vector sum of the three
components ux′ ex′ , uy′ ey′ and uz′ ez′ so we may replace the vector by those three components,
shown in (Figure B.1)
Figure B.1: Transformation from Local coordinate system to Global coordinate system
Now, the projection of u onto x is the same as the vector sum of the projections of each
of its components ux′ ex′ , uy′ ey′ and uz′ ez′ onto x, and similarly for y and z.
′
Define the angle generated in going from x to x as Θxx′ . The projection of ux′ ex′ onto
x therefore has magnitude ux′ cos Θxx′ and direction ex , as shown if figure 3.3.
96
To the vector ux′ cos Θxx′ ex must be added the other two projections, uy′ cos Θyx′ ex and
uz′ cos Θzx′ ex . Similar projections onto y and z result in:
ux = ux′ cos Θxx′ + uy′ cos Θxy′ + uz′ cos Θxz′ .
(B.4)
uy = ux′ cos Θyx′ + uy′ cos Θyy′ + uz′ cos Θyz′ .
(B.5)
uz = ux′ cos Θzx′ + uy′ cos Θzy′ + uz′ cos Θzz′ .
(B.6)
In vector-matrix notation, this may be written
 
  
cos Θxx′ cos Θxy′ cos Θxz′
u x′
ux
uy  = cos Θyx′ cos Θyy′ cos Θyz′  uy′ 
uz
uz ′
cos Θzx′ cos Θzy′ cos Θzz′
(B.7)
Clearly this is u2 = [T2,1 ] u1 , which defines the coordinate transformation matrix [T2,1 ]
from u1 to u2 coordinates in terms of the dot products of unit vectors. The relationship
between local coordinate system and global coordinate system is given by direction cosine
table, which summarized as follows:
Table B.1: The relationship between local and global coordinate system
x
= cos Θxx′
y
= cos Θxy′
z
= cos Θxz′
′
nxx′
′
nyx′ = cos Θyx′
nyy′ = cos Θyy′
nyz′ = cos Θyz′
′
nzx′ = cos Θzx′
nzy′ = cos Θzy′
nzz′ = cos Θzz′
x =
y =
z =
nxy′
97
nxz′
APPENDIX C - NUMERICAL SIMULATION OF SURFACE MOTION FOR LAYERED
HALF SPACE
This program computes the displacement at the free surface of a layered half space, with
the source either located in one layer or in interface.
% Numerical s i m u l a t i o n o f s u r f a c e motion f o r l a y e r e d h a l f s p a c e
% This program computes t h e d i s p l a c e m e n t a t t h e f r e e s u r f a c e o f a
l a y e r e d h a l f space , with t h e s o u r c e e i t h e r l o c a t e d i n one l a y e r s o r
at the h a l f space or in i n t e r f a c e .
clear all ;
clc ;
X=15∗10ˆ −3;
Y=20∗10ˆ −3;
%
O b s e r v a t i o n P oin t
Tr =1.5∗10ˆ −6;
T t =0;
%
%
The r i s e time
T r i g g e r i n g time
% ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗
%
LAYER INFORMATION:
% ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗
Z=[.0005 . 0 0 4 7 ] ;
ZS = . 0 0 1 7 2 3 ;
DZ1=Z ( 1 ) ;
DZ2=ZS−Z ( 1 ) ;
DZ3=Z ( 2 )−ZS ;
DZ4=Z ( 2 )−Z ( 1 ) ;
% Z1 , Z2 :
Depth Of I n t e r f a c e In Meter
% Depth Of S o u r c e In Meter
% D i f f e r e n c e Of Depth
DZ=[DZ1 DZ2 DZ3 DZ4 ] ;
RHO=[1670 2700 2 6 1 0 ] ;
% RHO ( kg /mˆ 3 ) :
D e n s i t y Of Medium
AF =[4320 6320 7 3 2 0 ] ;
BA =[2200 3180 4 1 5 0 ] ;
% AF (m/ s ) :
% BA (m/ s ) :
P Wave Speed
S Wave Speed
DA = [ . 0 2 . 0 4 . 0 8 ] ;
DB = [ . 0 1 . 0 2 . 0 4 ] ;
% DA:
% DB:
Damping F a c t o r For P Wave
Damping F a c t o r For S Wave
98
PAI=4∗atan ( 1 ) ;
ALF=z e r o s ( 1 , 3 ) ;
BETA=z e r o s ( 1 , 3 ) ;
%
%
%
( pi ) = 3.1416
C r e a t e Array Of A l l Z e r o s
C r e a t e Array Of A l l Z e r o s
f o r i =1:3;
ALF( i ) =AF( i ) ∗ complex ( 1 , DA( i ) ) ;
BETA( i )=BA( i ) ∗ complex ( 1 , DB( i ) ) ;
end
SI ( 1 , 1 )=complex ( 1 , 0 ) ;
SI ( 1 , 2 )=complex ( 0 , 0 ) ;
SI ( 2 , 1 )=complex ( 0 , 0 ) ;
SI ( 2 , 2 )=complex ( 1 , 0 ) ;
% I d e n t i t y Matrix
%∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗
%
MAIN PROGRAM
%∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗
%
Epsilo :
Small Quantity S c a l e In S c a t t e r i n g Wave F i e l d s
t F i n a l =125∗10ˆ −6;
dtime = . 0 0 1 ;
c = BA( 1 ) ;
t = tFinal ;
Xu=( t ∗ c ) / 2 ;
IRX=Xu ;
IRY=Xu ;
% (m/ s ) :
% time
%m
% (m)
% (m)
IRX :
IRY :
S Wave Speed
Truncated D i s t a n c e In X D i r e c t i o n
Truncated D i s t a n c e In Y D i r e c t i o n
deltaKX=(PAI ) /IRX ;
deltaKY=(PAI ) /IRY ;
WK=
;
%
Frequency
Uxx=z e r o s ( 1 , l e n g t h (WK) ) ;
Uyy=z e r o s ( 1 , l e n g t h (WK) ) ;
Uzz=z e r o s ( 1 , l e n g t h (WK) ) ;
for
%
C r e a t e Array Of A l l Z e r o s
IWK= 1 : l e n g t h (WK) ;
K MAX = wsmall (IWK) / c ;
K Normal = K MAX∗ K f a c t o r (IWK) ;
N = K Normal/ deltaKX ;
99
NI=−N;
NM=N;
XYK=NI : deltaKX :NM;
XK=z e r o s ( 1 , l e n g t h (XYK) ) ;
YK=z e r o s ( 1 , l e n g t h (XYK) ) ;
RK=z e r o s ( l e n g t h (XYK) , l e n g t h (XYK) ) ;
f o r I =1: l e n g t h (XYK) ;
XK( I )=XYK( I ) /IRX ;
for
J =1: l e n g t h (XYK) ;
YK( J )=XYK( J ) /IRY ;
RK( I , J )=s q r t (XK( I ) ∗XK( I )+YK( J ) ∗YK( J ) ) ;
i f (RK( I , J )==0)
RK( I , J )=RK( I , J−1) ;
end
end
end
AW=WK;
XU=complex ( 0 , 1 ) ;
% The Imaginary Number i
UXM=z e r o s ( l e n g t h (XK) , l e n g t h (YK) ) ;
Ux Bar Dash=z e r o s ( l e n g t h (XK) , l e n g t h (YK) ) ;
UYM=z e r o s ( l e n g t h (XK) , l e n g t h (YK) ) ;
Uy Bar Dash=z e r o s ( l e n g t h (XK) , l e n g t h (YK) ) ;
UZM=z e r o s ( l e n g t h (XK) , l e n g t h (YK) ) ;
Uz Bar Dash=z e r o s ( l e n g t h (XK) , l e n g t h (YK) ) ;
for
IXK= 1 : l e n g t h (XK) ;
f o r IYK= 1 : l e n g t h (YK) ;
UTS( 1 )=complex ( 0 , 0 ) ;
UTS( 2 )=−XU∗XK(IXK) /RK(IXK , IYK) ; %
UTS( 3 )=complex ( 0 , 0 ) ;
UTS( 4 )=complex ( 0 , 0 ) ;
100
Displacement Source
% P−SV s u b r o u t i n e
[WP0, E2 ] = PS V (XK(IXK) ,YK(IYK) , AW(IWK) , UTS, DZ, RHO, ALF, BETA) ;
% SH s u b r o u t i n e
UTH( 1 )=−XU∗YK(IYK) /RK(IXK , IYK) ;
UTH( 2 )=complex ( 0 , 0 ) ;
[WH0] = S H (XK(IXK) ,YK(IYK) , AW(IWK) , UTH, DZ, RHO, ALF, BETA) ;
UXM(IXK , IYK)=XU/RK(IXK , IYK) ∗ (XK(IXK) ∗WP0( 2 )+YK(IYK) ∗WH0) ;
UYM(IXK , IYK)=XU/RK(IXK , IYK) ∗ (YK(IYK) ∗WP0( 2 )−XK(IXK) ∗WH0) ;
UZM(IXK , IYK)=WP0( 1 ) ;
Ux Bar Dash (IXK , IYK)=UXM(IXK , IYK) ∗ exp(−XU∗XK(IXK) ∗X−XU∗YK(IYK) ∗Y) ;
Uy Bar Dash (IXK , IYK)=UYM(IXK , IYK) ∗ exp(−XU∗XK(IXK) ∗X−XU∗YK(IYK) ∗Y) ;
Uz Bar Dash (IXK , IYK)=UZM(IXK , IYK) ∗ exp(−XU∗XK(IXK) ∗X−XU∗YK(IYK) ∗Y) ;
end
end
Uxx (IWK)=sum ( sum ( Ux Bar Dash ) ) ;
Uyy (IWK)=sum ( sum ( Uy Bar Dash ) ) ;
Uzz (IWK)=sum ( sum ( Uz Bar Dash ) ) ;
end
% ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗
%
SOLUTION OF DISPLACEMENT IN FREQUENCY−
%
WAVE NUMBER DOMAIN FOR P−SV WAVES
% ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗
f u n c t i o n [WP0, E1 ] = PS V (XK,YK, AW, UTS, DZ, RHO, ALF, BETA)
% XK,YK:
% AW:
Wave Numbers In X And Y D i r e c t i o n s
Frequency
% RHO= [ 1 . 6 7 2 . 2 8 2 . 5 8 ] ∗ 1 0 ˆ 3 ; % RHO ( kg /mˆ 3 ) :
%
101
D e n s i t y Of Medium
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
AF = [ 3 . 1 5 6 ] ∗ 1 0 ˆ 3 ;
%
BA = [ 1 . 7 9 2 . 8 8 7 3 . 4 6 4 ] ∗ 1 0 ˆ 3 ; %
AF (m/ s ) :
BA (m/ s ) :
P Wave Speed
S Wave Speed
DA = [ . 0 8 . 0 4 . 0 2 ] ;
DB = [ . 0 4 . 0 2 . 0 1 ] ;
%
%
DA:
DB:
PAI=4∗atan ( 1 ) ;
ALF=z e r o s ( 1 , 3 ) ;
BETA=z e r o s ( 1 , 3 ) ;
%
%
%
( pi ) = 3.1416
C r e a t e Array Of A l l Z e r o s
C r e a t e Array Of A l l Z e r o s
Damping F a c t o r For P Wave
Damping F a c t o r For S Wave
f o r i =1:3;
ALF( i ) =AF( i ) ∗ complex ( 1 , DA( i ) ) ;
BETA( i )=BA( i ) ∗ complex ( 1 , DB( i ) ) ;
end
QF11 : Q M a t r i c e s On The F i r s t I n t e r f a c e (O/ I )
PMU1: E i g e n v e c t o r M a t r i c e s In F i r s t Layer (O/ I )
WP0:
S o l u t i o n Of S u r f a c e D i s p l a c e m e n t (O)
XU=complex ( 0 , 1 ) ;
%
The Imaginary Number i
SI ( 1 , 1 )=complex ( 1 , 0 ) ;
SI ( 1 , 2 )=complex ( 0 , 0 ) ;
SI ( 2 , 1 )=complex ( 0 , 0 ) ;
SI ( 2 , 2 )=complex ( 1 , 0 ) ;
%
I d e n t i t y Matrix
RK=s q r t (XK∗XK+YK∗YK) ;
[PMU1,PMD1, PNU1, PND1, E1]=DP(RK,AW,RHO( 1 ) ,ALF( 1 ) ,BETA( 1 ) ) ;
[PMU2,PMD2, PNU2, PND2, E2]=DP(RK,AW,RHO( 2 ) ,ALF( 2 ) ,BETA( 2 ) ) ;
[PMU3,PMD3, PNU3, PND3, E3]=DP(RK,AW,RHO( 3 ) ,ALF( 3 ) ,BETA( 3 ) ) ;
[ QF11 , QF12 , QF21 , QF22 , R1I12 , R1I21 , T1I12 , T1I21 ]=INTER1(PMU1,PMD1, PNU1,
PND1,PMU2,PMD2, PNU2, PND2) ;
[ QS11 , QS12 , QS21 , QS22 , R2I12 , R2I21 , T2I12 , T2I21 ]=INTER1(PMU2,PMD2, PNU2,
PND2,PMU3,PMD3, PNU3, PND3) ;
[ R1L12 , R1L21 , T1L12 , T1L21]=LAYER1(DZ( 1 )
[ R2L12 , R2L21 , T2L12 , T2L21]=LAYER1(DZ( 2 )
[ R3L12 , R3L21 , T3L12 , T3L21]=LAYER1(DZ( 3 )
[ R4L12 , R4L21 , T4L12 , T4L21]=LAYER1(DZ( 4 )
%
Find
RF, R0J , TJ0 :
102
, E1 ) ;
, E2 ) ;
, E2 ) ;
, E2 ) ;
[ A1]= FINV(PND1) ;
[ A2]= MUL(A1 , PNU1, 2 , 2 , 2 ) ;
f o r I =1:2;
f o r J =1:2;
RF( I , J )=−A2( I , J ) ;
R0J ( I , J )=R1L12 ( I , J ) ;
TJ0 ( I , J )=T1L21 ( I , J ) ;
end
end
%
Find
RSF :
[ R01 , R10 , T01 , T10]= ASSEM1( R1L12 , R1L21 , T1L12 , T1L21 , R1I12 , R1I21 , T1I12 ,
T1I21 ) ;
[ R0S , RS0 , T0S , TS0]= ASSEM1( R01 , R10 , T01 , T10 , R2L12 , R2L21 , T2L12 , T2L21 ) ;
[ RSF]= SURF1 (RF, R0S , RS0 , T0S , TS0 ) ;
%
FIND RSN :
[ RSN, A1 , A2 , A3]= ASSEM1( R3L12 , R3L21 , T3L12 , T3L21 , R2I12 , R2I21 , T2I12 ,
T2I21 ) ;
%
Find
RJN
[ RJN, A1 , A2 , A3]= ASSEM1( R4L12 , R4L21 , T4L12 , T4L21 , R2I12 , R2I21 , T2I12 ,
T2I21 ) ;
%
Find
RJF :
[ RJF]= SURF1 (RF, R1L12 , R1L21 , T1L12 , T1L21 ) ;
%
FIND MU U( Z Sˆ−)
At S o u r c e L o c a t i o n
[ A1]= TRANSP(PNU2) ;
[ A2]= TRANSP(PND2) ;
[ A3]= MUL(RSN, A1 , 2 , 2 , 2 ) ;
[ B1]= MUL(RSN, RSF, 2 , 2 , 2 ) ;
f o r I =1:2;
103
f o r J =1:2;
A4( I , J )=XU∗ (A2( I , J )+A3( I , J ) ) ;
B2 ( I , J )=SI ( I , J )−B1 ( I , J ) ;
end
end
[ B3]= FINV( B2 ) ;
WSS=[UTS( 1 ) ; UTS( 2 ) ] ;
[ B4]= MUL( B3 , A4 , 2 , 2 , 2 ) ;
[SMUU]= MUL( B4 ,WSS, 2 , 2 , 1 ) ;
%
Find WP0 On The S u r f a c e
[ A1]= MUL(PMD1, RF, 2 , 2 , 2 ) ;
[ B1]= MUL( R0S , RF, 2 , 2 , 2 ) ;
f o r I =1:2;
f o r J =1:2;
C1( I , J )=PMU1( I , J )+A1( I , J ) ;
B2 ( I , J )=SI ( I , J )−B1 ( I , J ) ;
end
end
[ B3]= FINV( B2 ) ;
[ C2]= MUL( C1 , B3 , 2 , 2 , 2 ) ;
[ C3]= MUL( C2 , TS0 , 2 , 2 , 2 ) ;
[WP0]= MUL( C3 ,SMUU, 2 , 2 , 1 ) ;
% ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗
%
SOLUTION OF DISPLACEMENT IN FREQUENCY−
%
WAVE NUMBER DOMAIN FOR SH WAVES
% ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗
f u n c t i o n [WP0] = S H (XK,YK, AW, UTS, DZ, RHO, ALF, BETA)
% XK,YK:
% AW:
Wave Numbers In X And Y D i r e c t i o n s
Frequency
% RHO= [ 1 . 6 7 2 . 2 8 2 . 5 8 ] ∗ 1 0 ˆ 3 ;
%
RHO ( kg /mˆ 3 ) : D e n s i t y Of Medium
104
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
AF = [ 3 . 1 5 6 ] ∗ 1 0 ˆ 3 ;
BA = [ 1 . 7 9 2 . 8 8 7 3 . 4 6 4 ] ∗ 1 0 ˆ 3 ;
%
%
AF (m/ s ) :
BA (m/ s ) :
P Wave Speed
S Wave Speed
DA = [ . 0 8 . 0 4 . 0 2 ] ;
DB = [ . 0 4 . 0 2 . 0 1 ] ;
%
%
DA:
DB:
PAI=4∗atan ( 1 ) ;
ALF=z e r o s ( 1 , 3 ) ;
BETA=z e r o s ( 1 , 3 ) ;
%
%
%
( pi ) = 3.1416
C r e a t e Array Of A l l Z e r o s
C r e a t e Array Of A l l Z e r o s
Damping F a c t o r For P Wave
Damping F a c t o r For S Wave
f o r i =1:3;
ALF( i ) =AF( i ) ∗ complex ( 1 , DA( i ) ) ;
BETA( i )=BA( i ) ∗ complex ( 1 , DB( i ) ) ;
end
QF11 : Q M a t r i c e s On The F i r s t I n t e r f a c e (O/ I )
PMU1: E i g e n v e c t o r M a t r i c e s In F i r s t Layer (O/ I )
WP0:
S o l u t i o n Of S u r f a c e D i s p l a c e m e n t (O)
XU=complex ( 0 , 1 ) ;
%
The Imaginary Number i
RK=s q r t (XK∗XK+YK∗YK) ;
[PMU1,PMD1, PNU1, PND1, E1]= DS(RK,AW,RHO( 1 ) ,BETA( 1 ) ) ;
[PMU2,PMD2, PNU2, PND2, E2]= DS(RK,AW,RHO( 2 ) ,BETA( 2 ) ) ;
[PMU3,PMD3, PNU3, PND3, E3]= DS(RK,AW,RHO( 3 ) ,BETA( 3 ) ) ;
QF11=XU∗ (PMD1∗PNU2−PND1∗PMU2) ;
QF12=XU∗ (PMD1∗PND2−PND1∗PMD2) ;
QF21=XU∗ (PNU1∗PMU2−PMU1∗PNU2) ;
QF22=XU∗ (PNU1∗PMD2−PMU1∗PND2) ;
T1I12=1/QF22 ;
R1I12=QF12∗ T1I12 ;
R1I21=−T1I12 ∗QF21 ;
T1I21=QF11+QF12∗ R1I21 ;
QS11=XU∗ (PMD2∗PNU3−PND2∗PMU3) ;
QS12=XU∗ (PMD2∗PND3−PND2∗PMD3) ;
QS21=XU∗ (PNU2∗PMU3−PMU2∗PNU3) ;
QS22=XU∗ (PNU2∗PMD3−PMU2∗PND3) ;
T2I12=1/QS22 ;
R2I12=QS12∗ T2I12 ;
R2I21=−T2I12 ∗QS21 ;
105
T2I21=QS11+QS12∗ R2I21 ;
T1L12=exp ( E1∗DZ( 1 ) ) ;
T2L12=exp ( E2∗DZ( 2 ) ) ;
T3L12=exp ( E2∗DZ( 3 ) ) ;
T4L12=exp ( E2∗DZ( 4 ) ) ;
T1L21=T1L12 ;
T2L21=T2L12 ;
T3L21=T3L12 ;
T4L21=T4L12 ;
R1L12=0;
R2L12=0;
R3L12=0;
R4L12=0;
R1L21=0;
R2L21=0;
R3L21=0;
R4L21=0;
%
Find
%
%
R0J , TJ0 :
FR=1.0D0
R0J=R1L12 ;
TJ0=T1L21 ;
FIND
RSF :
[ R01 , R10 , T01 , T10]= ASSEM0 ( R1L12 , R1L21 , T1L12 , T1L21 , R1I12 , R1I21 ,
T1I12 , T1I21 ) ;
[ R0S , RS0 , T0S , TS0]= ASSEM0 ( R01 , R10 , T01 , T10 , R2L12 , R2L21 , T2L12 ,
T2L21 ) ;
RSF=RS0+T0S/(1−R0S ) ∗TS0 ;
%
Find
RSN :
[ RSN, A1 , A2 , A3]= ASSEM0 ( R3L12 , R3L21 , T3L12 , T3L21 , R2I12 , R2I21 , T2I12
, T2I21 ) ;
%
Find
RJN
106
[ RJN, A1 , A2 , A3]= ASSEM0 ( R4L12 , R4L21 , T4L12 , T4L21 , R2I12 , R2I21 , T2I12
, T2I21 ) ;
%
Find
RJF :
%
RJF=RJF=R1L12+T1L12 / ( 1 . 0 D0−R1L12 ) ∗T1L21
RJF=T1L12∗T1L21 ;
%
Find
MU ( Z Sˆ−) AT SOURCE LOCATION
SMUU=XU∗ (PND2+RSN∗PNU2) ∗UTS( 1 ) /(1−RSN∗RSF) ;
%
Find
WP0 ON THE SURFACE
WP0=(PMU1+PMD1) /(1−R0S ) ∗TS0∗SMUU;
% ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗
%
WAVE SCATTERING MATRIX FOR P−SV WAVES ON INTERFACE
% ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗
f u n c t i o n [ Q11 , Q12 , Q21 , Q22 , R12 , R21 , T12 , T21]=INTER1(SMU1, SMD1, SNU1 , SND1 ,
SMU2, SMD2, SNU2 , SND2)
%
%
Q11 , Q12 , Q21 , Q22 :
R12 , R21 , T12 , T21 :
XU=complex ( 0 , 1 ) ;
[TMU1]=
[TMD1]=
[TNU1]=
[TND1]=
S u b m a t r i c e s Of Matrix D 1ˆ−1 D 2 (O)
Wave S c a t t e r i n g Matrix (O)
%
The Imaginary Number i
TRANSP(SMU1) ;
TRANSP(SMD1) ;
TRANSP(SNU1) ;
TRANSP(SND1) ;
f o r I =1:2;
f o r J =1:2;
D1( I , J )=−XU∗TND1( I , J ) ;
D1( I , J+2)=XU∗TMD1( I , J ) ;
D1( I +2,J )=XU∗TNU1( I , J ) ;
D1( I +2,J+2)=−XU∗TMU1( I , J ) ;
D2( I , J )=SMU2( I , J ) ;
D2( I , J+2)=SMD2( I , J ) ;
D2( I +2,J )=SNU2( I , J ) ;
D2( I +2,J+2)=SND2( I , J ) ;
107
end
end
[Q]= MUL(D1 , D2 , 4 , 4 , 4 ) ;
f o r I =1:2;
f o r J =1:2;
Q11 ( I
Q12 ( I
Q21 ( I
Q22 ( I
, J )=Q( I , J ) ;
, J )=Q( I , J+2) ;
, J )=Q( I +2,J ) ;
, J )=Q( I +2,J+2) ;
end
end
%
I n t e r f a c e Between Two L a y e r s
[ T12]= FINV( Q22 ) ;
[ R12]= MUL( Q12 , T12 , 2 , 2 , 2 ) ;
[A]= MUL( T12 , Q21 , 2 , 2 , 2 ) ;
[ B]= MUL( Q12 , A, 2 , 2 , 2 ) ;
f o r I =1:2;
f o r J =1:2;
R21 ( I , J )=−A( I , J ) ;
T21 ( I , J )=Q11 ( I , J )−B( I , J ) ;
end
end
% ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗
%
WAVE SCATTERING MATRIX FOR P−SV WAVES IN ONE LAYER
% ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗
f u n c t i o n [ R12 , R21 , T12 , T21]=LAYER1(DZ, E)
%
%
%
DZ :
E:
R12 , R21 , T12 , T21 :
A b s o l u t e Depth Of This Layer ( I )
Eigenvalue ( I )
Wave S c a t t e r i n g Matrix (O)
108
f o r I =1:2;
f o r J =1:2;
R12 ( I
R21 ( I
T12 ( I
T21 ( I
, J )=complex ( 0 , 0 ) ;
, J )=complex ( 0 , 0 ) ;
, J )=complex ( 0 , 0 ) ;
, J )=complex ( 0 , 0 ) ;
end
T12 ( I , I )=exp (E( I ) ∗DZ) ;
T21 ( I , I )=T12 ( I , I ) ;
end
% ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗
%
WAVE SCATTERING MATRIX FOR P−SV WAVES ON SURFACE
% ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗
f u n c t i o n [ R20]= SURF1(R0 , R12 , R21 , T12 , T21 )
%
%
%
R0 :
R12 , R21 , T12 , T21 :
R20 :
SI ( 1 , 1 )=complex ( 1 , 0 ) ;
SI ( 1 , 2 )=complex ( 0 , 0 ) ;
SI ( 2 , 1 )=complex ( 0 , 0 ) ;
SI ( 2 , 2 )=complex ( 1 , 0 ) ;
R e f l e c t i o n Matrix On Free S u r f a c e ( I )
Wave S c a t t e r i n g Matrix ( I )
Compound R e f l e c t i o n Matrix (O)
%
I d e n t i t y Matrix
[ A1]= MUL( R12 , R0 , 2 , 2 , 2 ) ;
f o r I =1:2;
f o r J =1:2;
A2( I , J )=SI ( I , J )−A1( I , J ) ;
end
end
%
C a l l FINV(A2 , A3)
[ A3]= FINV(A2) ;
[ A1]= MUL( T12 , R0 , 2 , 2 , 2 ) ;
109
[ A2]= MUL(A1 , A3 , 2 , 2 , 2 ) ;
[ A4]= MUL(A2 , T21 , 2 , 2 , 2 ) ;
f o r I =1:2;
f o r J =1:2;
R20 ( I , J )=R21 ( I , J )+A4( I , J ) ;
end
end
% ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗
%
EIGENVECTOR MATRICE AND EIGENVALUE FOR P−SV WAVES
% ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗
f u n c t i o n [SMU,SMD, SNU, SND, E]= DP(AK,AW,RO, ALF,BAT)
%
%
%
%
%
%
%
%
%
%
AK:
AW:
SMU:
SMD:
SNU:
SND:
RO:
ALF :
BAT:
E:
Wave Number In R D i r e c t i o n
Frequency
M u (O)
M d (O)
N u (O)
N d (O)
Density
P Wave Speed
S Wave Speed
E i g e n v a l u e (O)
XU=complex ( 0 , 1 ) ;
%
The Imaginary Number i
P=AK/AW;
XA=XU∗AW;
QA=s q r t ( 1 / (ALF∗ALF)−P∗P) ;
SS=−XA∗QA;
i f ( SS<0)
QA=−QA;
end
E( 1 )=XA∗QA;
QB=s q r t ( 1 / (BAT∗BAT)−P∗P) ;
SS=−XA∗QB;
i f ( SS<0)
QB=−QB;
end
E( 2 )=XA∗QB;
110
GA=RO∗ ( 2 ∗ (P∗BAT) ˆ2−1) ;
EA=1/( s q r t ( 2 ∗RO∗QA) ) ;
EB=1/( s q r t ( 2 ∗RO∗QB) ) ;
SMU( 1 , 1 )=−XU∗QA∗EA;
SMU( 1 , 2 )=P∗EB;
SMU( 2 , 1 )=P∗EA;
SMU( 2 , 2 )=−XU∗QB∗EB;
SMD( 1 , 1 )=−SMU( 1 , 1 ) ;
SMD( 1 , 2 )=SMU( 1 , 2 ) ;
SMD( 2 , 1 )=SMU( 2 , 1 ) ;
SMD( 2 , 2 )=−SMU( 2 , 2 ) ;
S3=−2∗XU∗RO∗BAT∗BAT∗P ;
SNU( 1 , 1 )=GA∗EA;
SNU( 1 , 2 )=S3∗QB∗EB;
SNU( 2 , 1 )=S3∗QA∗EA;
SNU( 2 , 2 )=GA∗EB;
SND( 1 , 1 )=SNU( 1 , 1 ) ;
SND( 1 , 2 )=−SNU( 1 , 2 ) ;
SND( 2 , 1 )=−SNU( 2 , 1 ) ;
SND( 2 , 2 )=SNU( 2 , 2 ) ;
% ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗
%
EIGENVECTOR MATRICE AND EIGENVALUE FOR SH WAVES
% ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗
function
[SMU,SMD, SNU, SND, E]= DS(AK,AW,RO,BAT)
%
%
%
%
%
%
%
%
%
Wave Number In R D i r e c t i o n ( I )
Frequency ( I )
M u (O)
M d (O)
N u (O)
N d (O)
Density ( I )
S Wave Speed ( I )
E i g e n v a l u e (O)
AK:
AW:
SMU:
SMD:
SNU:
SND:
RO:
BAT:
E:
XU=complex ( 0 , 1 ) ;
%
The Imaginary Number i
P=AK/AW;
111
QB=s q r t ( 1 / (BAT∗BAT)−P∗P) ;
SS=−XU∗AW∗QB;
i f ( SS < 0 )
QB=−QB;
end
E=XU∗AW∗QB;
EB=1/ s q r t ( 2 ∗RO∗QB) ;
SMU=EB/BAT;
SMD=SMU;
SNU=−XU∗RO∗BAT∗QB∗EB;
SND=−SNU;
% ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗
%
MULTIPLICATION OF TWO COMPLEX MATRICES
% ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗
f u n c t i o n [ C]= MUL(A, B,M, N,K)
%
%
%
A:
B:
C:
M X N Matrix ( I )
N X K Matrix ( I )
M X K Matrix (O)
f o r I =1:M;
f o r KK=1:K;
C( I ,KK)=complex ( 0 , 0 ) ;
f o r J =1:N;
C( I ,KK)=C( I ,KK)+A( I , J ) ∗B( J ,KK) ;
end
end
end
% ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗
%
TRANSPOSE 2 x 2 MATRIX
% ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗
f u n c t i o n [ B]= TRANSP(A)
%
%
A:
B:
2X2 Matrix ( I )
2X2 Transpose Matrix (O)
112
B( 2 , 1 )=A( 1 , 2 ) ;
B( 1 , 2 )=A( 2 , 1 ) ;
B( 1 , 1 )=A( 1 , 1 ) ;
B( 2 , 2 )=A( 2 , 2 ) ;
% ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗
%
INVERSE 2 x 2 MATRIX
% ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗
f u n c t i o n [ B]= FINV(A)
%
%
A:
B:
2X2 Matrix ( I )
2X2 Matrix (O)
D=1/(A( 1 , 1 ) ∗A( 2 , 2 )−A( 1 , 2 ) ∗A( 2 , 1 ) ) ;
B( 1 , 1 )=A( 2 , 2 ) ∗D;
B( 2 , 2 )=A( 1 , 1 ) ∗D;
B( 1 , 2 )=−A( 1 , 2 ) ∗D;
B( 2 , 1 )=−A( 2 , 1 ) ∗D;
% ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗
%
COMPOSTION RULE FOR P−SV WAVES
% ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗
f u n c t i o n [ R13 , R31 , T13 , T31]= ASSEM1( R12 , R21 , T12 , T21 , R23 , R32 , T23 , T32 )
%
%
R13 , R31 , T13 , T31 :
R12 , R21 , T12 , T21 ; R23 , R32 , T23 , T32 :
SI ( 1 , 1 )=complex ( 1 , 0 ) ;
SI ( 1 , 2 )=complex ( 0 , 0 ) ;
SI ( 2 , 1 )=complex ( 0 , 0 ) ;
SI ( 2 , 2 )=complex ( 1 , 0 ) ;
%
I d e n t i t y Matrix
[A]= MUL( R21 , R23 , 2 , 2 , 2 ) ;
f o r I =1:2;
f o r J =1:2;
C( I , J )=SI ( I , J )−A( I , J ) ;
113
(O)
(I)
end
end
[ B]= FINV(C) ;
[A]= MUL( T21 , R23 , 2 , 2 , 2 ) ;
[ C]= MUL(A, B, 2 , 2 , 2 ) ;
[A]= MUL(C, T12 , 2 , 2 , 2 ) ;
f o r I =1:2;
f o r J =1:2;
R13 ( I , J )=R12 ( I , J )+A( I , J ) ;
end
end
[ C]= MUL( T23 , B, 2 , 2 , 2 ) ;
[ T13]= MUL(C, T12 , 2 , 2 , 2 ) ;
[A]= MUL( R23 , R21 , 2 , 2 , 2 ) ;
f o r I =1:2;
f o r J =1:2;
C( I , J )=SI ( I , J )−A( I , J ) ;
end
end
[ B]= FINV(C) ;
[A]= MUL( T23 , R21 , 2 , 2 , 2 ) ;
[ C]= MUL(A, B, 2 , 2 , 2 ) ;
[A]= MUL(C, T32 , 2 , 2 , 2 ) ;
f o r I =1:2;
f o r J =1:2;
R31 ( I , J )=R32 ( I , J )+A( I , J ) ;
end
end
[ C]= MUL( T21 , B, 2 , 2 , 2 ) ;
[ T31]= MUL(C, T32 , 2 , 2 , 2 ) ;
114
% ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗
%
COMPOSTION RULE FOR SH WAVES
% ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗
f u n c t i o n [ R13 , R31 , T13 , T31]= ASSEM0( R12 , R21 , T12 , T21 , R23 , R32 , T23 , T32 )
%
%
R13 , R31 , T13 , T31 :
R12 , R21 , T12 , T21 ; R23 , R32 , T23 , T32 :
C1=1/(1−R23∗R21 ) ;
R31=R32+T23∗R21∗C1∗T32 ;
T31=T21∗C1∗T32 ;
C2=1/(1−R21∗R23 ) ;
R13=R12+T21∗R23∗C2∗T12 ;
T13=T23∗C2∗T12 ;
115
(O)
(I)
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