Stress:

1
Stress
Force per unit area across an arbitrary plane
2
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
3
Stress Defined as a Tensor
z
txz
y
x
z
tzx

t(z)
t(y)
x
t(x)

No net rotation

• txx txy txz
• t t T tyx tyy tyz
  
• tzx tzy tzz

4
Relation Between the Traction Vector and the
Stress Tensor
z
txz
y
x
z
tzx

t(z)
t(y)
x
t(x)

No net rotation

• tx(n) txx
  txy txz nx
• t(n) t n ty(n) tyx tyy
  tyz ny
  
• tz(n) tzx tzy
  tzz nz

5
Relation Between the Traction Vector and the
Stress Tensor
That is, the stress tensor is the linear operator
that produces the traction vector from the normal
unit vector.

• tx(n) txx
  txy txz nx
• t(n) t n ty(n) tyx tyy
  tyz ny
  
• tz(n) tzx tzy
  tzz nz

6
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

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Stresses in a Fluid
If t1t2t3, the stress field is hydrostatic, and
no shear stress exists

• -P 0
  0
• t 0 -P
  0
  
• 0 0
  -P

P is the pressure
8
Pressure inside the Earth
Stress has units of force per area 1 pascal
(Pa) 1 N/m2 1 bar 105 Pa 1 kbar 108 Pa
100 MPa 1 Mbar 1011 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
9
Pressure inside the Earth
At depths gt 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 rgz, where
r 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/m3 is P 3,000 x 9.8 x 3,000 8.82
107 Pa 100 MPa 0.9 kbar
10
Mean (M) and Deviatoric (D) Stress

• txx txy txz
• t tyx tyy tyz
  
• tzx tzy tzz

M txx tyy tzz tii/3

• txx-M txy txz
• D tyx tyy-M tyz
  
• tzx tzy tzz-M

11
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
12
J can be divided up into strain (e) and rotation
(O)
is the strain tensor (eijeji)
?ux ?ux ?uy ?ux ?uz ?x
?y ?x ?z ?x
½( )
½( )
e


?uy ?ux ?uy ?uy ?uz ?x
?y ?y ?z ?y
½( )
½( )
?uz ?ux ?uz ?uy ?uz ?x
?z ?y ?z ?z
½( )
½( )
13
J can be divided up into strain (e) and rotation
(O)
?ux ?uy ?ux ?uz
?y ?x ?z ?x
0 ½( - )
½( - )
O


?uy ?ux ?uy ?uz ?x
?y ?z ?y
-½( - ) 0
½( - )
?uz ?ux ?uz ?uy ?x ?z
?y ?z
-½( - )
-½( - ) 0
is the rotation tensor (Oij-Oji)
14
Volume change (dilatation)

• 1/3 ( ) tr(e)
  div(u)
• gt 0 means volume increase
• lt 0 means volume decrease
  

?ux ?uy ?uz ?x ?y
?z
?ux ?x
?ux ?x
gt0
lt0
15
???2ui/?t2 ?j?ij fi
Equation of motion Homogeneous eom when
fi0
16

• Seismic Wave Equation (one version)
• For (discrete) homogeneous media and ray
  theoretical methods, we have
• ?
• ???2ui/?t2 (????)??u-??x?x u

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• Plane Waves
• Wave propagates in a single direction
• u(x,t) f(t?x/c) travelling along x axis
• A(?)exp-i? (t-sx)
  A(?)exp-i(?t-kx)
• where k ?s (?/c)s is the wave number

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?s ???x sin? v?t, ?t/?x sin?/v u sin?
p u slowness, p ray parameter
(apparent/horizontal slowness) rays are
perpendicular to wavefronts
?x
?
?s
wavefront at t?t
wavefront at t
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p u1 sin ?1 u2 sin ?2 u slowness, p ray
parameter (apparent/horizontal slowness)
?1
v1
?2
v2
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p u1sin ?1 u2sin ?2 u3sin ?3 Fermats
principle travel time between 2 points is
stationary (almost always minimum)
?1
v1
?2
v2
?3
v3
21
Continuous Velocity Gradients p u0sin ?0 u
sin ? constant along a single ray path
X
v
z
?
?0
?
90o, uutp
T
dT/dX p ray parameter
X
22
X(p) generally increases as p decreases -gt dX/dp
lt 0
X
v
z
?
p decreasing
?
90o, uutp
T
Prograde traveltime curve
X
23
X(p) generally increases as p decreases but not
always
X
v
z
Prograde
Retrograde
T
caustics
Prograde
X
24
Reduced Velocity
Prograde
Retrograde
T
caustics
Prograde
X
T-X/Vr

X
25
X(p) generally increases as p decreases -gt dX/dp
lt 0
Shadow zone
X
v
z
lvz
T
?
X
p
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j
Traveltime tomography
T ? 1/v(s)ds ?u(s)ds Tj ? Gij ui ?Tj ?
Gij ?ui i1 dGm GTdGTGm mg(GTG
)-1GTd
j-th ray
i1
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Earthquake location uncertainty
n i1
?2 ? ti-tip2/?i2 ?i expected standard
deviation ?2 (mbest) ? ti-tip(mbest)2/ndf mbe
st is best-fitting station ?2(m) ?
ti-tip2/?2 - contour!

n i1
????? ?????????????????????????????
n i1
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Fast location S-P times D 8 x S-P(s)

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Other sources of error Lateral velocity
variations slow fast Station
distribution




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Emean 1/2 ??A2 ?2 (higher frequencies carry
more E!) A2/A1 (?1c1/?2c2)1/2
ds2 ds1
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???1cos?1-???2cos?2 SS
???1cos?1???2cos?2 2
???1cos?1 SS ???1cos?1???2cos?2
since ??????????ucos??cos??? For vertical
incidence (?????

????1 - ????2 A1
????1 ????2 2 ????1
A2 ????1 ????2
??????????????????????1-???2
2 ???1 SSvert
SSvert ???1???2
???1???2
36
S waves vertical incidence? P waves
vertical incidence ???????????????????
?????1-???2
2 ???1 PPvert -
PPvert ???1???2
???1???2

??????????????????????1-???2
2 ???1 SSvert
SSvert ???1???2
???1???2
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E1flux 1/2 c1 ???A12 ?2 cos?1 E2flux 1/2 c2
???A22 ?2 cos?2 Anorm E2flux/E1flux1/2
A2/A1 c2??cos?2/ c1??cos?11/2
Araw
c2??cos?2/ c1??cos?11/2
???1cos?1-???2cos?2 SSnorm
SSraw
???1cos????2cos? 2
???1cos? (c2??cos?2)1/2 SSnorm
x
???1cos?1???2cos?2
(c1??cos?1)1/2
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2 ???1cos? SS
???1cos????2cos? What happens beyond ?c ?
There is no transmitted wave, 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
with Imaginary vertical slowness are called
inhomogeneous or evanescent waves.
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Phase changes Vertical incidence, free
surface S waves - no change in polarity P waves
- polarity change of ?? Vertical incidence,
impedance increases S waves - opposite
polarity P waves - no change in polarity Fig
6.4 Phase advance of ?/2 - Hilbert Transform
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Attenuation scattering and intrinsic
attenuation Scattering amplitudes reduced by
scattering off small-scale objects, integrated
energy remains constant Intrinsic
1/Q(?) -?E/2?E E is the
peak strain energy, -?e is energy loss per
cycle Q is the Quality factor
A(x)A0exp(-?x/2cQ) X is distance along
propagation distance C is velocity

41
Ray methods t ?dt/Q (
r ), A(?)A0(?)exp(-?t/2) i.e., higher
frequencies are attenuated more! pulse broadening