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Chapter2:
Lithospheric Mechanics
This presentation contains
illustrations from Allen and Allen
(2005 )
and Press et al. (2004)
Physical State of the
Lithosphere
Key Concepts
– Surface Forces
– Local Isostasy
– Flexural isostasy
– Thermal conductivity
– Thermal Expansion
– Heat transfer: A special case
– Rock Rheology (2.3)
– Relevant mantle rheological behavior
– Rheology of continental crust
• Elastic-perfectly plastic
• Strain hardening and strain softening
Key Concepts
•Lithostatic stress(G3), deviatoric stress(G3),
•uniaxial stress (G2), plane stress (G1)
•bulk modulus(G1),flexural rigidity(G4)
•thermal conductivity(G5), geotherm(G6)
•Geoid(G7), Bouguer anomalies(G8)
•Local vs. Regional Isostasy(G9)
•diffusion and dislocation creep(G10), Byerlee’s Law(G2)
(e-mail me your definitions to the above concepts as a PowerPoint
slide– one illustration and two sentences … worth 1 point toward final
curved grade; due , September 16. Originality (40%) and clarity
(40%) and citations(20%) )
Surface (not surficial!) forces in geology
We measure these forces of gravity and reaction to
gravity not in terms of Newtons but by using the
concept of stress, in Newtons per meter square, or
Pascals. (See structural geology notes).
What is atmospheric pressure?
What is the hydrostatic state of stress?
Lithostatic stress
1 cu. meter of water weighs 1000 kg x 10m/s2 or 10000
Newtons (N)
1 cu meter creates 10000N/m2 (Pa) of pressure at its
bas
10 meters of water depth produces 100000 Pa (1 atm) of
0.1 MPa, that is every 10 m you dive down,
pressure increases by 1 atm.
1000 vertically stacked 1-m-cubes of water weigh 10
million Newtons
1000 m (1 km) of stacked 1-m-cubes of water create 10
million Pascals (Pa) or 10 MPa at its base
Lithostatic stress
• If the above is true, then under 1 km of mud (2200 kg/m3) there
should be about 22 MPa of pressure then under 30 km of granite
(2670 kg/m3) there should be 801 MPa, or .8 GPa
The rule to convert density into MPa of pressure per km is to
take the density of the material in g/cc, move the decimal point
over one space and change the units to MPa
Other useful conversions to know are:
To get MPa from psi mutliply Pounds/sq in by 0.689 x 10 ^-2
To get psi from MPa multiply MPa by 145.05
To convert to MegaPascals.... Divide by 1000000 Pa per 1 MPa
Lithostatic stress
• If you think you understand the previous slide, then answer the
following question:
On Planet Zog the average density of the 10 km-thick crust is
2500 kg m^-3 . Acceleration due to gravity is 3.2 m s^-2 . What
is the pressure at the base of the crust?
A. 80 MegaPascals
B. 80 Newtons
C. 800 Newtons
D. 3 GigaPascals
E. 30 Gigapascals
F. None of the above
• Lithostatic stress is responsible for the increase of pressure
with overall depth in the earth but it is the differential stress
that creates the faults and folds.
• What is the vertical lithostatic stress gradient in
granitic crust? What is the vertical stress gradient in
the first 2 km of the ocean?
Faults can develop:
 min
 max (Side view)
 max
 min
 min
 min (Side View)
 min
 max
 max (Bird’s Eye
 max
View)
 min
Brittle faults can develop:
 min
 max (Side view)
 max
 min
 min
 min(Side View)
 min
 max
 max (Bird’s Eye
 max
View)
 min
Sea of
Galilea
Dead Sea
What is the direction of directed pressure
(maximum principal stress direction)?
How many orientations of faults can be
generated for the same directed pressure
direction??
Physical State of the
Lithosphere
Key Concepts
– Surface Forces
– Local Isostasy
– Flexural isostasy
– Thermal conductivity
– Thermal Expansion
– Heat transfer: A special case
– Rock Rheology
– Relevant mantle rheological behavior
– Rheology of continental crust
• Elastic-perfectly plastic
• Strain hardening and strain softening
Surface Forces (Pressure)LOCAL ISOSTASY
Depth of compensation
Isostasy or Archimedes’ Principle…
States that the crust, mantle can float above the
underlying material
If the crust and mantle float then there exists a depth
for which pressure above and pressure below are equal.
This surface is known as the compensation depth
General recommendations for local
isostatic calculations
(1) Define a surface of compensation
(2) Define a reference column of crust and mantle
(3) Compare the weight of the reference column
with the unknown
(4) Simplify algebra in terms of two unknowns
(4) Keep physical units the same
See syllabus (Tuesday, 23 September) for
elaborated examples
Isostasy homework due Thursday, 25
2008
• Derive the relation between ocean-floor depth and
Moho depth for continental crustal settings.
• Assuming that under the North Sea the continental
crust and mantle is homogeneous, calculate the
expected thickness of continental crust.
• Do the same for the continental crust under the
Grand Banks of Newfoundland
• Show all your work–type it up and e-mail it to me
Physical State of the
Lithosphere
Key Concepts
– Surface Forces
– Local Isostasy
– Flexural isostasy
– Thermal conductivity
– Thermal Expansion
– Heat transfer: A special case
– Rock Rheology
– Relevant mantle rheological behavior
– Rheology of continental crust
• Elastic-perfectly plastic
• Strain hardening and strain softening
Local/Regional Isostasy
•
•
•
•
Picture from answers.com
Local Isostasy - Different topographic
heights are accommodated by
changed in crustal thickness and rock
density.
Illus. a, b show the region
surrounding the load will be
horizontal, unaffected by the load.
Regional Isostasy – The crust acts as
an elastic plate and its inherent
rigidity spreads topographic loads
over a broader region.
Illus. c shows the surrounding regions
being deflected down toward the load.
Flexure of the lithosphere
• The outer skin of the earth down to depths where the
temperature is cool enough and rock properties permit
the earth can be visualized to be effectively elastic
(e.g., rubber ball) over long periods of time, i.e.,
hundreds of millions of years.
• A conclusion is that mountain belts will
not sag over time but will maintain their
mechanical strength indefinitely for
practical purposes. A measure of the
strength of the crust is how much it
bends to a given load. This value is
known as the flexural rigidity (D – units
23
of Nm) 10
Nm is equivalent to about 34 km of elastic
thickness (Te) or moderately strong elastic
lithosphere
(TB 2006) Definition taken from:
http://en.wikipedia.org/wiki/Flexural_rigidity
• Flexural rigidity is defined as the force couple required to bend
a rigid structure to a unit curvature.
• The thin lithosphere plates which cover the surface of the
Earth are subject to flexure, when a load or force is applied to
them. On a geological timescale, the lithosphere behaves
elastically and can therefore bend under loading by mountain
chains, volcanoes and so on.
• The flexure of the plate depends on:
• The plate thickness
• The elastic properties of the plate
• The applied load or force
One view on flexure in basins
• Use local isostasy as a reference
• Assume stationary conditions
• Deviation from this reference is a measure of
internal strength balanced against an applied load
Measure of elasticity
If the load is exceptionally narrow and small then the lithosphere
will appear (infinitely) very strong because it does not give way at
all to the load!
But, if we use the other extreme case …. the
case of a weight that is very wide… (i.e. >
1000 km)…..?????
When it is very wide the condition reaches
that of local isostasy and all the weight
pushing down is balanced by the reaction of
the mantle pushing up.
Uniaxial Stress
• When only one principal stress is acting on a body
within the earth’s crust.
• Example:
– s1 ≠ 0, s2 = s3 = 0
• This is a rare occurrence.
LITHOSTATIC AND DEVIATORIC STRESS
•Lithostatic
Stress= Equal
pressure from
all directions
•Deviatoric
Stress= Stresslithostatic
•Affects
textures
and
structures
Tension•
Deviatoric Stress
http://myweb.cwpost.liu.
edu/vdivener/notes/stre
ss-strain.htm
Boudinage
Compr
ession
Shear
•
•
Fold
Fractures
=
+
=
+
=
+
=
+ +
+
+
+ +
+
=
+
+
+
+
+
+
+
+
+ +
Finite (reasonable and not
extreme) geological load
versus infinite (very wide) load
Maximum depth of subsidence of the base of the crust
in the case that the load is very wide and that
hydrostatic compensation is local… i.e. some the elastic
lithosphere has no internal strength.
Now compare the case where the load is relatively
narrow….
Our aim, now, is to understand the following:
Downward directed invisible
load creates space that fills
with water and adds more
vertical load
Use reference at infinity
(very far away) and pressure
at level of compensation.
At level of compensation
pressures are in equilibrium.
Level of compensation
g(h.rhom + hw. rhow + w .rhom)
Point load
versus infinite (very wide) load
Compare the case where the load is
relatively narrow and the strength of
the lithosphere becomes apparent….
Point load
versus infinite (very wide) load
(At infinity)
g(h.rhom + hw. rhow + w .rhom) =
(Under load)
qa (Point load) + g(w.rhow +
hw.rhow + h.rhom)-internal
resistance to bending
g(h.rhom + hw. rhow + w .rhom) =
qa (Point load) + g(w+hw) rhow
+ h.rhom) – internal resistance
to bending
If there is internal strength in
the lithosphere, then hw will
not be as deep as it should be
because the oceanic
lithosphere resists!
(Under load)
(At infinity)
g(h.rhom + hw. rhow + w .rhom) =
internal resistance to bending +
=
qa (Point load) + g( (w+hw).rhow +
h.rhom) – internal resistance to
bending
qa (Point load)
g (rhom-rhow) w
d 4w
d 2w
D 4  P 2   gw  qa ( x )
dx
dx
“Equation 2.28”
Equation 2.28
Physical State of the
Lithosphere
Key Concepts
– Surface Forces
– Local Isostasy
– Flexural isostasy
– Thermal conductivity
– Thermal Expansion
– Heat transfer: A special case
– Rock Rheology
– Relevant mantle rheological behavior
- Rheology of continental crust
• Elastic-perfectly plastic
• Strain hardening and strain softening
• Thermal
Conductivity
References
Byerlee’s
Law
The thermal conductivity of a material describes
• Defines
criteria
the effect of kinetic energy
passing failure
through
a
substance by intramolecular collisions.
Its value
for movement
along a
depends on several factors such
asplane.
composition,
fault
The NGS Geoid Page. (2008, April 22). Retrieved
chemical structure, temperature, and pressure of
September 15, 2008, Web
site:
•
Movement
depends
rocks. It is represented by k and measured
in: on
http://www.ngs.noaa.gov/GEOID/
Wm-1°C-1. overcoming frictional
resistance.
Allen, P. and Allen, J. (2006). Basin analysis:
• σ < 200 MPa
Principles and applications.
Malden,
MA: Blackwell.
σ = 0.85σ
• 200 MPa<σ <2000MPa
•
σ = 50 MPa + 0.6σ
From Earth Structure
n
s
n
n
s
n
(2004)
Allen, Philip A., and John R. Allen. Basin Analysis: Principles and Applications. 2nd ed. Malden, MA: Blackwell,
G5
2005. 20-60.
Equation Source:
http://en.wikipedia.org/wiki/Thermal_conductivity
Image
Source: http://www.defluteglass.co.nz/images/studio_blowin
Geotherm (G6)
A curve that
represents the
increase of
temperature with
depth.
• Because the crust
is granitic, it
contains radioactive
elements that per
capita produce a
higher heat flow
than the mantle.
•
References: Basin Analysis Principles and
Applications, www.earthsci.org
Thermal conductivity measures how
well…
for a given temperature gradient, conductive heat
transfers/ moves through rock. Heat moves from
higher temperature to areas of lower temperature.
Halite: 7 kW/m/ºK
Shale: 3 kW /m/ºK
Thermal conductivity
• The efficiency of that transfer is the thermal
conductivity. So, for a given temperature gradient –
dT/dz (continental or oceanic geotherms) the
amount of heat being passed across any given portion
of the earth’s surface (heat flux-Q) per unit time
will depend on the coefficient of thermal conductivity
(K).
Fourier’s Law:
dT
Q  K
dz
Q for continents is ~ 60 mW/m^2 or 60W/1000 m^2
Q for continents is ~ 80 mW/m^2
Geotherm
Temperature variation with depth in solid crust
indicates how much heat is flows from the mantle,
and how much heat is generated within the crust.
Q0
A 2
T  T0 
z
z
K
2K
Q- heat flow (negative is upward)
K- thermal conductivity
A- internal heat generation
Z -depth
Geotherm
Temperature
oceanic
z
continent
Heat Production versus depth
Heat production at surface (Hs )is maximum
H eat production = Hs exp (-z/ar)
Heat production
Z
Depth
(km)
K  3W / m / C
Q0  70mW / m3
A0  2.5  10-6Wm3
ar  10km
Geotherms
Surface heat flow (Q0) observations indicate that heat flow
increases linearly with the heat production of surface rocks.
This is mathematically best accomplished by assuming that heat
production decreases with depth in an exponential manner.
Q0
A0 ar2 
a
T  T0 
z
1  e r
K
K 
y



ar is the depth at which heat production is ~halved
A0 is the surface heat production
Global heat production
• Continental surface heat flow comes about 50% from
the mantle (U,K,Th) and about 50% from radioactive
sources.
• Heat flow was x2 what it is now, about 3 billion years
ago
• Oceanic heat flow largely depends on thermal age of
the lithosphere and not on the radioactivity
Sampling thermal conductivity
On board R/V Joides Resolution, Leg 150 New
Jersey Margin, US Atlantic Coast, B. Hoppie
(right) (MNSU, Mankato), C. Fulthorpe(left)
(UT Austin)
Thermal conductivity
• We can measure thermal conductivity with respect to
standards as you can see in this overhead of a
thermal conductivity measurements on board Leg ODP
150 New Jersey Margin in the summer of 1993.
People are (L toR) Bryce Hoppie and Craig Fulthorpe.
These needles contain heaters and temperature
sensors. These needles measure the speed at which
the temperature changes over time to calculate the
conductivity of the material into which they are
inserted.
Physical State of the
Lithosphere
Key Concepts
– Surface Forces
– Local Isostasy
– Flexural isostasy
– Thermal conductivity
– Thermal Expansion
– Heat transfer: A special case
– Rock Rheology (2.3)
– Relevant mantle rheological behavior
– Rheology of continental crust
• Elastic-perfectly plastic
• Strain hardening and strain softening
Thermal Expansion
• At a constant pressure, the average silicate rock will
expand 1/100,000 th of its entire length for every
degree that it goes up in temperature. This of course
affects the density of the rock.
• The amount that the rock contracts or expands, at an
assumed constant pressure, for a given temperature
change is known as the thermal expansion coefficient,
or the volumetric coefficient of thermal expansion,
written as  v
Thermal expansion
• 100,000 m * 10^-5 * 1ºK = 1m/ºK
Thermal contraction
• The converse is true as well…. for every degree that
temperature drops, the lithosphere will contract
1/100,000 th of its entire length
Thermal contraction
O km
125
km
Start (at
time=0)
1300º
After 200 my
1300º
Thermal contraction
• So, a 125-km piece of mantle that is initially at, say
1300ºK, and which then cools by an average of about
650ºK will shrink by how much …..?
Choose an answer
• (a) 2km
• (b) 4 km
• (c) 10 km
• (d) 20 km
• (e) none of the above
Answer
• 125,000 m * 650ºC * 10^-5 = 812 m
Isostatic consequences of cooling mantle
• If the mantle contracts as it cools it also becomes
denser for doing so….
• Final density = original density * thermal expansion
coefficient (temperature drop)
 final   start (1   T )
v
Physical State of the
Lithosphere
Key Concepts
– Surface Forces
– Local Isostasy
– Flexural isostasy
– Thermal conductivity
– Thermal Expansion
– Heat transfer: A special case
– Rock Rheology
– Relevant mantle rheological behavior
– Rheology of continental crust
• Elastic-perfectly plastic
Time-dependent heat conduction
We observe that
(1) heat flow decreases away from the mid-ocean ridges
as a function of age and
(2) water depth increases as a function of age
Heat flow versus age
Plate Model for Sea-floor spreading
• Parsons and Sclater (1977)
Temperature (from model) and thickness
versus age
Leeds et al. (1974)
Physical State of the
Lithosphere
Key Concepts
– Surface Forces
– Local Isostasy
– Flexural isostasy
– Thermal conductivity
– Thermal Expansion
– Heat transfer: A special case
– Rock Rheology
– Relevant mantle rheological behavior
- Rheology of continental crust
• Elastic-perfectly plastic
Bulk Modulus (K) (G-1)
www.ux1.eiu.edu/.../10FldRst/Images/bulk.gif
-measures a substance's resistance to uniform
compression
-defined as the pressure increase needed to effect a
given relative decrease in volume
- is the reciprocal of a substances compressibility(β)
which is the measure of the relative volume change of a
fluid or solid as a response to a pressure (or mean K=1/β
stress) change
Plane Stress (G1)
σ3
σ2
σ
1
http://maps.unomaha.edu/Maher/GEOL3300/week4/ellipse.j
peg
-a state of plane stress exists when one of the
three principal stresses is zero
σ1≥σ2>0
σ3=0
-the other two stresses are nonzero and act on the
same plane creating plane stress
At least 6 factors control how rock
deforms
e.g. at shallow depth a rock may
fracture whereas at depth it may
flow.
Factors are:
(1) rock type
(2) Confining and directed pressure
(3) temperature
(4) Fluids
(5) Time
(6) Rate of deformation
Physical State of the
Lithosphere
Key Concepts
– Surface Forces
– Local Isostasy
– Flexural isostasy
– Thermal conductivity
– Thermal Expansion
– Heat transfer: A special case
– Rock Rheology
– Relevant mantle rheological behavior
– Rheology of continental crust
• Elastic-perfectly plastic
• Strain hardening and strain softening
Diffusion and Dislocation Creep
Andrew Sampson
David Smolkin
Creep is defined as deformation that results
from long periods of stress. It is time-dependent
with respect to stress.
Creep becomes exaggerated with greater thermal
activity and is the result of mantle convection.
Diffusion Creep
Diffusion Creep occurs as vacancies, or point defects, in
the material move in the direction of maximum stress
due to high temperatures (near the melting point).
Diffusion creep occurs at low stress and results in
crystals behaving as a Newtonian fluid (similar to water).
Its viscosity depends exponentially on pressure and
inverse absolute temperature.
Dislocation Creep
Dislocation Creep occurs as high stress levels
moves dislocations, or linear/planar defects, and
behaves as a non Newtonian fluid.
Like diffusion creep, its viscosity depends
exponentially on pressure and inverse absolute
temperature.
Occurs primarily in mantle and can occur in
lower lithosphere and is the dominant form
creep in the mantle.
Mantle viscosity
Models
• Diffusion creep
– Very Low stress
– Newtonian fluid, linear
behavior
– Atoms diffuse
e    v V ( fluid )
Mantle viscosity
• High stress creep
• Dislocation creep
• Model for mantle plasticity is non-linear
 Qc 


 RT 
Power Law Creep
e  Ae  n
Q is activation energy
A is a creep-mechanism parameter
Viscosity depends on stress and temperature
1
f




V
  e stress
T (º K )
Dislocation Creep
• Temperature-activated creep
– Movement of mantle by microfractures at the subcrystal
scale and synchronous healing of these imperfections
Physical State of the
Lithosphere
Key Concepts
– Surface Forces
– Local Isostasy
– Flexural isostasy
– Thermal conductivity
– Thermal Expansion
– Heat transfer: A special case
– Rock Rheology
– Relevant mantle rheological behavior
– Rheology of continental crust
• Elastic-perfectly plastic
• Strain hardening and strain softening
Rheology of continental crust
Byerlee’s Law
• Linear relation between shear stress and normal
stress for rock strength
Shear
stress
Normal stress
Byerlee’s Law (G2)
• Defines failure criteria
for movement along a
fault plane.
• Movement depends on
overcoming frictional
resistance.
• σ < 200 MPa
σ = 0.85σ
• 200 MPa<σ <2000MPa
σ = 50 MPa + 0.6σ
n
s
n
n
From Earth Structure
(2004)
s
n
Physical State of the
Lithosphere
Key Concepts
– Surface Forces
– Local Isostasy
– Flexural isostasy
– Thermal conductivity
– Thermal Expansion
– Heat transfer: A special case
– Rock Rheology
– Relevant mantle rheological behavior
– Rheology of continental crust
• Elastic-perfectly plastic
• Strain hardening and strain softening
Elastic-Plastic model for breaking rock
strain
stress
strain
Elastic-Plastic model for breaking rock
strain
stress
strain
Strain hardening
strain
stress
strain
Strain softening
strain
stress
strain
Elastic-plastic
stress
strain
Dislocation Creep (AL)• Thermally activated deformation that occurs at
relatively higher shear stress than diffusion creep.
Diffusion creep happens at very small scales (atomic
and molecular), and the crystalline solid flows as a
Newtonian fluid. Dislocation creep happens at larger
scales and causes the solid to exhibit non-Newtonian
behavior because of the higher shear stress.
Elastic-plastic
stress
strain
Elastic-plastic
stress
strain
Elastic-plastic
stress
strain
Diffusion Creep (RR)
• Diffusion is the propagation of cracks in a crystal
structure in response to stress where the parting
goes from an area of high stress to low stress.
Diffusion Creep is the movement of atoms along
partings from areas of high stress to low stress
creating foliations.
Lithostatic Stress – C.A.
The stress applied to a rock in equal directions due to
the weight of an overlying rock column. At the
surface of the earth the lithostatic stress would be
zero, but as you move further below the earth's
surface the weight of the overlying rock causes an
increase in stress.
Source: http://myweb.cwpost.liu.edu/vdivener/notes/stress-st
Bouguer Anomalies (TJH)
• The difference between measurements of gravity
based on the value used by a theoretical model of what
it should be at that latitudinal position, and a different
value that compensate for latitude, elevation, free-air
corrections, and *Bouguer correction.
• *Developed be Pierre Bouguer proved that gravity
differs with elevation
Bulk Modulus (K) (MB)
The ratio of pressure change (P) to volume change (V)
K = P/ V
This describes a materials ability to resist changes in volume
Deviatoric Stress (TA)
•
•
A condition in which the stress components operating at a point in a
body are not the same in every direction.
Is the difference between the mean stress (Sum of stress in three
directions divided by 3) and total stress
Geotherm (SE):
The variation of temperature with depth.
•
Major Influences
– Thermal Conductivity
– Concentration of Radiogenic Elements
– Temperature at Surface
– Proximity to Magma or other Heat Sources
Eugene Island Field
Gulf of Mexico
Thermal Conductivity (AD)
Heat transfer is achieved by processes of:
1) Conduction- a diffusive process in which kinetic energy is
transferred by intermolecular collisions. Conduction is the
primary thermal process in the lithosphere.
2) Convection- requires motion of the medium to transmit
heat. Convection of heat from the core is the principal
thermal process of the mantle.
3) Electromagnetic radiation- only important in determining
surface heat budget, not the internal heat budget
Fourier’s Law
• Fourier’s Law is the central
relation for conductive heat
transport
• It states that the heat flux
Q is directly proportional to
the temperature gradient
• Q = -K (dT / dy)
K = coefficient of thermal
conductivity
T = temperature at a given point
in the medium
y = coordinate in the direction of
the temperature variation
Continental Crust
•
•
Generally, regions of high heat
flow correspond to active
volcanic zones or regions of
extensional tectonics.
Areas of continental collision are
related to low or normal surface
heat flows.
Oceanic Crust
•
•
•
The surface heat flow of the
oceans is related to the age of
the seafloor rather than the
concentration of radioisotopes.
Newly created oceanic crust
cools by conduction as it travels
away from the mid-ocean ridge.
About 60% of the Earth’s heat
loss takes place through the
ocean floor.
One-Dimensional Heat Conduction
•
•
Temperature change of a piece
of lithosphere has 3
components:
These components are a basal
heat flow term, an internal
heat generation term, and an
advective term
Advective Heat Flow
• Advective heat flow can be one of two things.
• It can be movement towards the surface associated
with downcutting action of erosion, or the velocity of
deposition.
Uniaxial stress(MS)
• Uniaxial stress is stress in only one direction and zero
stress in the perpendicular direction. This XYZ graph
shows that there is only stress in the Y direction,
both X and Z directions show a stress of Zero.
North pole is up & black line runs through Greenwich
Geoid (HF)
GEOID – a surface on which the earth’s gravitational forces
are equal everywhere and coincides with mean sea-level.
Based on these concepts:
- sea covered the earth
- no disturbing forces like winds, tides, ocean currents, ect.
- the force of gravity is perpendicular to the geoid everywhere.
- Ellipsoid represents the bulk shape of the earth.
-Geoid departs above or below the ellipsoid
resulting in a smoother representation of the
earth’s actual surface.
For more info: http://www.answers.com/topic/geoid,
http://solid_earth.ou.edu/notes/geoid/earths_geoid.htm
H. FOLEY
Geoid- equipotential reference
surface of Earth based on gravity
anomalies; best fits mean sea level.
• Vertical datum of the Earth
• Where we see high topography, we do not
necessarily see a high variation in geoid due to
isostatic equilibrium
• Find the geoid near you by going to
http://www.ngs.noaa.gov/cgibin/GEOID_STUFF/geoid03_prompt1.prl
Byerlee’s Law
• Defines failure criteria
for movement along a
fault plane.
• Movement depends on
overcoming frictional
resistance.
• σ < 200 MPa
σ = 0.85σ
• 200 MPa<σ <2000MPa
σ = 50 MPa + 0.6σ
n
s
n
n
From Earth Structure
(2004)
s
n
Bulk Modulus (K)
www.ux1.eiu.edu/.../10FldRst/Images/bulk.gif
-measures a substance's resistance to uniform
compression
-defined as the pressure increase needed to
effect a given relative decrease in volume
- is the reciprocal of a substances
compressibility(β) which is the measure of the
relative volume change of a fluid or solid as a
response to a pressure (or mean stress)
K=1/β
Plane Stress
σ3
σ2
σ
1
http://maps.unomaha.edu/Maher/GEOL3300/week4/ellipse.j
peg
-a state of plane stress exists when one of the
three principal stresses is zero
σ1≥σ2>0
σ3=0
-the other two stresses are nonzero and act on the
same plane creating plane stress
Uniaxial Stress
• When only one principal stress is acting on a body
within the earth’s crust.
• Example:
– s1 ≠ 0, s2 = s3 = 0
• This is a rare occurrence.
LITHOSTATIC AND
DEVIATORIC
STRESS
•Lithostatic
Stress= Equal
pressure from all
directions
•Deviatoric Stress=
Unequal pressure in
different directions
-Three different
Deviatoric Stress
types
Tension
• Boudinage
Compression
• Folds
http://myweb.cwpost.liu.
edu/vdivener/notes/stre
ss-strain.htm
•Affects
textures
and
structures
Shear
• Fractures
Flexural Rigidity
• “Is the resistance to
bending (flexure) of a
material”
• For instance, in regards to
the lithosphere: “is the
uppermost rheologic layer
of the earth. . .behaves
rigidly on geologic
timescales. If you place a
load on it, the lithosphere
either supports the load
or bends – it does not
simply flow out of the
way.”
Marshak, Stephen, and Ben A. van
hoc
hw
href
hoc
hm
g ( m hm  oc hoc   w hw )  strength  point load
(weight)
=
g (href  w  oc hoc )  large load
Discovered by Pierre Bouguer
in the 1700s
the difference between what
gravity is at a particular
location(absolute observed
gravity) and what gravity is
supposed to be at that
location(theoretical gravity
value). Bouguer anomaly =
theoretical gravity value absolute observed gravity
value.