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Normal Strain
L x
Strain
In each case, a force F produces a deformation x. In engineering, weusually change this force into stress and the deformation into strainand we define these as follows:
Strain is the deformation per unit of the original length.
The symbol
Strain has no units since it is a ratio of length to length. Mostengineering materials do not stretch very mush before they becomedamages, so strain values are very small figures. It is quite normal tochange small numbers in to the exponent for 10 -6( micro strain).
called Epsilon
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Shear force is a force applied sideways on the material (transverselyloaded).
Shear stress is the force per unit area carrying the load. This means
the cross sectional area of the material being cut, the beam and pin. The sign convention for shear force and stress is based on how it
shears the materials as shown below.
AF
and symbol is called TauShear stress,
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L
x
L x
The force causes the material to deform as shown. The shear strain isdefined as the ratio of the distance deformed to the height
. Since this is a very small angle , we can say that :
( symbol called Gamma )
Shear strain
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Research for a reason.
If you take an infinitesmall volume element you can show all ofthe stress components
Research for a reason.
The first subscriptindicates the plane
perpendicular to the axisand the second subscriptindicates the direction ofthe stress component.
Stress Tensor
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Shear stress equilibrium
Consider a two- dimensionalstate of shear stress, asillustrated. Note that the twohorizontal stresscomponents would tend tocause a clockwise moment.
The two vertical componentsare necessary to supplyanticlockwise moment.But there is no rotation ofthe element.
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In tensorial notation the stress components are assembled in amatrix.
S =
For equilibrium it can be
shown that :
ij = ji for i j
xy = yx xz = zx
yz = zy
This symmetry reduces the shear stress components to three.
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Stress-strain Hooke Law Elastic constants Seismic Wave
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Hooke's law is a principle of physics that states that theforce needed to extend or compress a spring by somedistance is proportional to that distance.
A property of an ideal spring of spring constant k is thatit takes twice as much force to stretch the spring twice asfar. That is, if it is stretched a distance x, the restoring
force is given by F = - kx . The spring is then said to obeyF = - kx
An elastic medium is one in which a disturbance can beanalyzed in terms of Hookes Law forces.
Consider the propagation of a mechanical wave(disturbance) in a solid.
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A Mass-Spring System in which a mass m isattached to an idealspring of spring constantk.
A Prototype Hookes Law System
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Stretch the spring a distance A & release it:
In the absence of friction, the oscillations go on
forever. The Newtons 2 nd Law equation of motion is:F = ma = m(d 2x/dt 2) = -kx
A standard 2 nd order time dependent differential equation!
Fig. 1 Fig. 2
Fig. 3
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An Elastic Medium is defined to be one inwhich a disturbance from equilibriumobeys Hookes Law so that a localdeformation is proportional to an appliedforce.
If the applied force gets too large, Hookes
Law no longer holds. If that happens themedium is no longer elastic . This is calledthe Elastic Limit .
The Elastic Limit is the point at whichpermanent deformation occurs , that is, ifthe force is taken off the medium, it will not
return to its original size and shape.
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Stress-strain
Hooke Law Elastic moduli
Youngs modulus Shear modulus Bulk modulus Poisson's ratio
Seismic Wave
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A stress in the x direction, x , will result in astrain in the same direction given by:
Where E is the elastic constant called Young'sModulus. This is just a simple form of Hooke's
law.
E
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We need another elastic constant to tell us how easilya body will change its shape or suffer a shear strain ( )
under shear stress ( s). The shear modulus, orrigidity modulus, G does this:
s G
The rigidity of fluids and gasses is 0
Shear modulus G ( )
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What if the VOLUME of the material changes whenpressure is applied?WHAT IS PRESSURE? : equal stresses in all directions.
In this case the change in volume is related to the change inpressure by the bulk modulus,
Bulk modulus
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If =0, then the other dimensions do not change inresponse to stress, and volume change is maximum.
If = 0.5 , then the volume does not change at all.For fluids, 0.5, while for slinky 0. For mostsolid rocks, = 0.1-0.25.
= - transverse strain / longitudinal strain
Poissons ratio
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Stress-strain
Elastic constant
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Differential Forces
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Stress = Force/Area Stress is a vector and for an horizontal plane can be resolved into its
components in (x,y,z) directions For an inclined plane there are normal and tangential components
normalcomponent
tangential component
Note: There canbe two tangentialcomponents thus we generallyhave threecomponents ofstress
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or
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Consider a spring that is stretched a certain
distance by an applied force such as a weight. The distance stretched is related to the applied
force by:
F = kXwhere X is the displacement, F is the applied
force, and k is the spring constant.
thus, stress is proportional to strain. Explain.
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Isotropic
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Bulk Modulus ( K ): incompressibility
Resistance to volumetric compression
Shear Modulus ( ): rigidity
Resistance to shear deformation.
Both quantities are always greater than or equal to zero.
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Almandine .28 Magnetite .28 Halite .26 Pyrite .16
Spinel .27 Apatite .26 Biotite .27 Muscovite .25 Calcite .31
Dolomite .29 Quartz .08 Hematite .14 Anhydrite .27 Barite .32
Olivine .24 Augite .25 Diopside .26 Hornblende .29 Feldspars .28-.29
Significant conclusions?
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Elastic modulirelate P and Swave velocities todensity, .
= modulusof rigidity
= - (2/3) ,where is the
bulk modulus.
Yilmaz, 2001
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Body waves
Surface Waves
Ground Roll Rayleigh
Love
P
S
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The first kind of body wave is the P wave .This is the fastest kind of seismic wave.The P wave can move through solid rock
and fluids, like water or the liquid layers ofthe earth.It pushes and pulls the rock it moves
through just like sound waves push and pullthe air.
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Particle motion consists of alternating compression and dilation. Particlemotion is parallel to the direction of propagation (longitudinal). Materialreturns to its original shape after wave passes.
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Particle motion consists of alternating transverse motion. Particle motion isperpendicular to the direction of propagation (transverse). Transverseparticle motion shown here is vertical but can be in any direction. Material
returns to its original shape after wave passes.
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The first kind of surface wave is called a Lovewave , named after A.E.H. Love , a Britishmathematician who worked out the mathematicalmodel for this kind of wave in 1911.
Book A Treatise on the Mathematical Theoryof Elasticity
It's the fastest surface wave and moves theground from side-to-side.
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The other kind of surface wave is the Rayleigh wave , named for
John William Strutt, Lord Rayleigh, who mathematically predictedthe existence of this kind of wave in 1885.
A Rayleigh wave rolls along the ground just like a wave rolls acrossa lake or an ocean.
Because it rolls, it moves the ground up and down, and side-to-sidein the same direction that the wave is moving.
Most of the shaking felt from an earthquake is due to the Rayleighwave, which can be much larger than the other waves.
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Particle motion consists of elliptical motions (generally retrograde elliptical) inthe vertical plane and parallel to the direction of propagation. Amplitudedecreases with depth. Material returns to its original shape after wave passes.
Wave Equation
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Wave Equation
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Stress-strain Hooke Law Elastic constants Seismic Wave
Wave types Wave equation Wave velocity Wave mode Snells law Reflection and transmission coefficient Fresnel Zone Huygens principle
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Particle motion consists of alternating compression and dilation. Particlemotion is parallel to the direction of propagation (longitudinal). Materialreturns to its original shape after wave passes.
Wave Equation
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So, consider seismic waves propagating in a solid, whentheir wavelength is very long , so that the solid may betreated as a continous medium. Such waves are referred
to as elastic waves .
At the point x the elasticdisplacement (or change inlength) is U(x) & the straine is defined as the changein length per unit length.
Consider P-Wave Propagation in a Solid Bar
x x+dx
Wave Equation
Wave Equation
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In general, a Stress S at a
point in space is defined asthe force per unit area atthat point .
C Young s Modulus
Hookes Law tells us that, at point x & time t in the bar, the stress S produced by an elastic wave propagation
is proportional to the strain e . That is:
x x+dx
Wave Equation
Wave Equation
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To analyze the dynamics of the bar, choose an arbitrary
segment of length dx as shownabove. Use Newtons 2nd Law to write for the motion of thissegment,
2
2( ) ( ) ( )u
Adx S x dx S x At
C Young s Modulus
Mass Acceleration = Net Force resulting from stress
x x+dx
Wave Equation
due
Wave Equation
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.S C e
e
dx
2
2
.
.
uS C
x S u
C x x
2
2( ) ( ) ( )u Adx S x dx S x At
( ) ( ) S S x dx S x dx x
2 2
2 2( ) u u
Adx C Adxt x
2 2
2 2
u uC
t x
( )i kx t u Ae
Cancelling common terms in Adx gives:This is the wave equation a planewave solution which gives the
P-wave velocity vp:Plane wave solution:
So, this becomes:
k = wave number = (2 /
),
= frequency, A = amplitude
Wave Equation
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34
K V p
sV
Wave Equation
( )( , ) i kx t u x t Ae
Plane wave solution
3D
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Stress-strain Hooke Law Elastic constants Seismic Wave
Wave types Wave equation Wave velocity Wave mode Snells law
Reflection and transmission coefficient Fresnel Zone Huygens principle
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- +0
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Period: T (second), is the duration of one cycleWavelength: ( meter), is the spatial period of the waveVelocity: V (meter/s), is the speed of wave propagation
T
Inverse of period is frequency fInverse of wavelength is wavenumber k
Peak
TroughZero cross
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Period(T, s)
Frequency(f, Hz)
Angularfrequency( , rad/s)
WaveLength( ,m )
Velocity(V, m/s)
Wavenumber(k, 1/m)
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( , ) ( , ) ( , )
Period(T, s) f=1/T =2 /T =VT V= /T k=T/V
Frequency
(f, Hz) =2 f
=V/f V=f k=f/V
Angularfrequency( , rad/s)
=V /2 V= /2 k= /2 V
Wavelength( , m)
V=f k=1/
Velocity(V, m/s) k=f/V
Wavenumber
(k, 1/m)
Wave Velocity
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Material P wave Velocity (m/s) S wave Velocity (m/s)
Air 332
Water 1400-1500
Petroleum 1300-1400
Steel 6100 3500
Concrete 3600 2000
Granite 5500-5900 2800-3000
Basalt 6400 3200
Sandstone 1400-4300 700-2800
Limestone 5900-6100 2800-3000
Sand (Unsaturated) 200-1000 80-400
Sand (Saturated) 800-2200 320-880
Clay 1000-2500 400-1000
Glacial Till (Saturated) 1500-2500 600-1000
y
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P-Wave Velocity Distributions
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y
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Gardenerequation
=0.23V 0.25
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Velocity
F = ma
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V p = K + 4/3 V s =
Moduli: , ,
Lithology
Age/Depth of Burial
Effective Pressure
Pore Pressure Overburden Pressure
Closing/Openingof microcracks
Temperature
Porosity ( )
Degree ofLithification
PoreCharacteristics
GrainCha racteristics
FluidSaturation
Anisotropy
Shale/Clay Content
Shape ( ) Size Distribution
Cementation Dolotomization
Sorting Shape Size
Fluid Type
Rock Structure
Rock/Fluid Material
Degree ofSaturation
Rock Frame
Mixed Lithologies
Temperat ure
Chemical Effects Viscosity
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Stress-strain Hooke Law Elastic constants Seismic Wave
Wave types Wave equation Wave velocity Wave mode Snells law
Reflection and transmission coefficient Fresnel Zone Huygens principle
f fl d ( h l f
Wave Mode
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So, at any interface, some energy is reflected (at the angle ofincidence) and some is refracted (according to Snells Law).Lets look at a simple model and just watch what happens to theP -wave energy...
Wave Mode
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Wave Mode
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Wave Mode
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Wave Mode
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Even for models with no noise, identifying phases may be
challenging However you can train your eye/brain to learn
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challenging. However, you can train your eye/brain to learnhow to do this.
For a two-layer model with a flat interface, the three main P-wavearrivals are:(1) Direct Wave
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(1) Direct Wave,(2) Refracted Wave (Head Wave), and(3) Reflected Wave.
Even for models with no noise, identifying phases may be
challenging However you can train your eye/brain to learn
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challenging. However, you can train your eye/brain to learnhow to do this.
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Multilayer Model
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Multilayer Model
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Multilayer Model
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Multilayer Model
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Distance
Time
Multilayer Model
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Distance
Time
Direct Wave
1st Layer Refraction2nd Refraction
2nd Reflection1st Reflection
Stress-strain
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Stress strain Hooke Law Elastic constants Seismic Wave
Wave types Wave equation
Wave velocity Wave mode Snells law
Reflection and transmission coefficient Fresnel Zone Huygens principle
What happens at a (flat) material discontinuity?
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Medium 1: v1
Medium 2: v2
i1
i2
2
1
2
1
sinsin
vv
ii
But how much is reflected, how much transmitted?
Willebrord Snellius(1580-1626)
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If V 2 >V 1 , then as i increases, r increases faster
r approaches 90 o as i increases
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approaches 90 as increases
Stress-strain
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Hooke Law Elastic constants Seismic Wave
Wave types Wave equation
Wave velocity Wave mode Snells law
Reflection and transmission coefficient Fresnel Zone Huygens principle
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Medium 1: r1,v1
Medium 2: r2,v2
T
A R
1122
1122
A
R
1122
112
AT
At oblique angles conversions from S-P, P-S have to beconsidered.
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Zoeppritz equations:
1881-1908
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Aki&Richard
Shuey
Fatti
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A P wave is incident at the free surface ...
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P PSV
i j
The reflected amplitudes can be described by thescattering matrix S
PP
r SVr
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At a solid-fluid interface there is no conversion to SV inthe lower medium.
Pt
Critical angle : The angle of incidence when i2=90 .
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11
2sin
v
i v
112
arcsin( )v
i a v
V1
V2
i1V2>V1
i2=90
Limit of seismic resolution usually makes us wonder, how thin a bed can we see?
Yet seismic data is subject to a horizontal as well as a vertical dimension ofresolution. The horizontal dimension of seismic resolution is described by theFresnel Zone
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Fresnel Zone.
Every point on a wavefront can be regarded as thesource of a subsequent wave.
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source of a subsequent wave.
Christiaan Huygens(1629-1695)
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Wave Equation
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h 0