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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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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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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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Wave types Wave equation Wave velocity Wave mode Snells law


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


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



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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

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