Stress: Force per unit area across an arbitrary plane.

Stress:

Force per unit area across an arbitrary plane

Stress Defined as a Vector

^N = unit vector normal to plane

t(n) = (tx,ty,tz) = traction vector

^

The part of t that is perpendicular to the plane is normal stress

The part of t that is parallel to the plane is shear stress

Stress Defined as a Tensor

^

t(x)

xx xy xz

= T = yx yy yz

zx zy zz

x

y

z

^

^

t(z)t(y)zx

xz

No net rotation

x

z

Relation Between the Traction Vector and the Stress Tensor

^

t(x)

txnxx xy xz nx

t(n) = n = ty(n) = yx yy yz ny

tz(n) zx zy zz nz

x

y

z

^

^

t(z)t(y)zx

xz

No net rotation

x

z

^

^

^

^

^

^

^ ^

Relation Between the Traction Vector and the Stress Tensor

txnxx xy xz nx

t(n) = n = ty(n) = yx yy yz ny

tz(n) zx zy zz nz

^

^

^

^

^

^

^ ^

That is, the stress tensor is the linear operator that produces the traction vector from the normal unit vector….

Principal Stresses

• Most surfaces has both normal and tangential (shear) traction components.

• However, some surfaces are oriented so that the shear traction = 0.

• These surfaces are characterized by their normal vector, called principal stress axes

• The normal stress on these surfaces are called principal stresses

• Principal stresses are important for source mechanisms

Stresses in a Fluid

P

= P

P

If 1=2=3, the stress field is hydrostatic, and no shear stress exists

P is the pressure

Pressure inside the Earth

Stress has units of force per area:

1 pascal (Pa) = 1 N/m^2

1 bar = 10^5 Pa

1 kbar = 10^8 Pa = 100 MPa

1 Mbar = 10^11 Pa = 100 GPa

Hydrostatic pressures in the Earth are on the order of GPa

Shear stresses in the crust are on the order of 10-100 MPa

Pressure inside the Earth

At depths > a few km, lithostatic stress is assumed, meaning that the normal stresses are equal to minus the pressure (since pressure causes compression) of the overlying material and the deviatoric stresses are 0.

The weight of the overlying material can be estimated as gz, where is the density, g is the acceleration of gravity, and z is the height of the overlying material.

For example, the pressure at a depth of 3 km beneath of rock with average density of 3,000 kg/m^3 is

P = 3,000 x 9.8 x 3,000 ~ 8.82 10^7 Pa ~ 100 MPa ~ 0.9 kbar

Mean (M) and Deviatoric (D) Stress

xx xy xz

= yx yy yz

zx zy zzM = xx + yy + zz = ii/3

xx-M xy xz

D = yx yy-M yz

zx zy zz-M

Strain:

Measure of relative changes in position (as opposed to absolute changes measured by the displacement)

U(ro)=r-ro

E.g., 1% extensional strain of a 100m long string

Creates displacements of 0-1 m along string

J can be divided up into strain (e) and rotation (Ω)

is the strain tensor (eij=eji)

uy ux uy uy uz

x y y z y uz ux uz uy uz

x z y z z

ux ux uy ux uz

x y x z x

e =

½( + ) ½( + )

½( + )

½( + ) ½( + )

½( + )

J can be divided up into strain (e) and rotation (Ω)

uy ux uy uz

x y z y uz ux uz uy x z y z

ux uy ux uz

y x z x

Ω=

0 ½( - ) ½( - )

-½( - ) 0

-½( - ) -½( - ) 0

½( - )

is the rotation tensor (Ωij=-Ωji)

Volume change (dilatation)

ux uy uz

x y z

= 1/3 ( + + ) = tr(e) = div(u)

> 0 means volume increase

< 0 means volume decrease

ux x

ux x

>0 <0

∂2ui/∂t2 = ∂jij + fi

= Equation of motion

= Homogeneous eom when fi=0

Seismic Wave Equation (one version)

For (discrete) homogeneous media and ray theoretical methods, we have

∂2ui/∂t2 = ()·u-xx u

Plane Waves:

Wave propagates in a single direction

u(x,t) = f(tx/c) travelling along x axis

= A()exp[-i(t-s•x)] = A()exp[-i(t-k•x)]

where k= s = (/c)s is the wave number

^

s x sin = vt, t/x = sin/v = u sin = p

u = slowness, p = ray parameter (apparent/horizontal slowness)

rays are perpendicular to wavefronts

s

x

wavefront at t+t

wavefront at t

p = u1 sin 1 = u2 sin 2

u = slowness, p = ray parameter (apparent/horizontal slowness)

2

1

v1

v2

p = u1sin 1= u2sin 2= u3sin 3

Fermat’s principle: travel time between 2 points is stationary (almost always minimum)

2

1 v1

v2

3 v3

Continuous Velocity Gradients

p = u0sin 0= u sin = constant along a single ray path

v

z

=90o, u=utp

dT/dX = p = ray parameter

X

X

T

0

X(p) generally increases as p decreases -> dX/dp < 0

v

z

=90o, u=utp

Prograde traveltime curve

X

X

T

p decreasing

X(p) generally increases as p decreases but not always

v

z

Prograde

X

X

T Retrograde Prograde

caustics

Reduced Velocity

Prograde

X

T Retrograde Prograde

caustics

T-X/Vr

X

X(p) generally increases as p decreases -> dX/dp < 0

v

z

X

X

T

Shadow zone

lvz

p

T = ∫ 1/v(s)ds = ∫u(s)ds

Tj = ∑ Gij ui

Tj = ∑ Gij ui

i=1

d=Gm

GTd=GTGm

mg=(GTG)-1GTd

i=1

Traveltime tomographyj

j-th ray

2 = ∑ [ti-tip]2/i

2

i expected standard deviation

2 (mbest)= ∑ [ti-tip(mbest)]2/ndf

mbest is best-fitting station

2(m) = ∑ [ti-tip]2/2 - contour!

Earthquake location uncertainty

n

i=1

n

i=1

n

i=1

Fast location:

S-P times: D ~ 8 x S-P(s)

Other sources of error:

Lateral velocity variations

slow fast

Station distribution

Emean = 1/2 A2 2

(higher frequencies carry more E!)

A2/A1= (1c1/2c2)1/2

ds2

ds1

1 - 2 A1’’= 1 + 2

2 1 A2’ = 1 + 2

1cos1-2cos2 S’S’’= 1cos1+2cos2

2 1cos1 S’S’ = 1cos1+2cos2

since ucoscos

For vertical incidence (

1-2 2 1 S’S’’vert= S’S’vert= 1+2 1+2

S waves vertical incidence

P waves vertical incidence:

1-2 2 1 P’P’’vert= - P’P’vert= 1+2 1+2

1-2 2 1 S’S’’vert= S’S’vert= 1+2 1+2

E1flux = 1/2 c1 A1

2 2 cos1

E2flux = 1/2 c2 A2

2 2 cos2

Anorm = [E2flux/E1

flux]1/2 = A2/A1 [c2cos2/ c1cos1]1/2

= Araw [c2cos2/ c1cos1]1/2

1cos1-2cos2 S’S’’norm= = S’S’’raw

1cos+2cos

2 1cos (c2cos2)1/2

S’S’norm = x 1cos1+2cos2 (c1cos1)1/2

2 1cos S’S’ = 1cos+2cos

What happens beyond c ? There is no transmittedwave, and cos = (1-p2c2)1/2 becomes imaginary.No energy is transmitted to the underlying layer, we have total internal reflection. The vertical slowness =(u2-p2)1/2 becomes imaginary as well. Waves withImaginary vertical slowness are called inhomogeneousor evanescent waves.

Phase changes:

Vertical incidence, free surface:

S waves - no change in polarityP waves - polarity change of

Vertical incidence, impedance increases:

S waves - opposite polarityP waves - no change in polarity

Fig 6.4

Phase advance of /2 - Hilbert Transform

Attenuation: scattering and intrinsic attenuation

Scattering: amplitudes reduced by scattering off small-scaleobjects, integrated energy remains constant

Intrinsic:

1/Q() = -E/2E

E is the peak strain energy, -e is energy loss per cycleQ is the Quality factor

A(x)=A0exp(-x/2cQ)

X is distance along propagation distanceC is velocity

Ray methods:

t* = ∫dt/Q ( r ), A()=A0()exp(-t*/2)

i.e., higher frequencies are attenuated more!pulse broadening