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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(CA), deviatoric stress(TA), uniaxial
stress, plane stress
•bulk modulus(MB),flexural rigidity(JTB)
•thermal conductivity(AD), geotherm(SE)
•Geoid(HF), Bouguer anomalies(TJH)
•Isostasy(CJ)
•diffusion and dislocation creep(AL), Byerlee’s Law(CP)
(one per student --- e-mail me your answer written in
PowerPoint slide– one illustration and two sentences …
worth 1 point for final, due Tuesday 12, September – e-
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
at which pressuer 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, 19 September) for elaborated
examples
Isostasy homework due Thursday, 21
2006
• Derive the relation between basin-floor depth and
Moho depth.
• Assuming that underneath Lake Baikal the continental
crust and mantle is homogeneous, calculate the
expected thickness of continental crust.
• Same for the continental shelf of the Gulf of Mexico
• 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
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 of Nm)
10
23
Nm is equivalent to about 34
km of elastic thickness (Te)
or moderately strong elastic
lithosphere
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.
=
+
=
+
=
+
=
+ +
+
+
+ +
+
=
+
+
+
+
+
+
+
+
+ +
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….
g ( m href  oc hoc   w hw ) 
qa  g ((hw  href )  w  oc hoc )
Point load
versus infinite (very wide) load
Now compare the case where the load is
relatively narrow and the strength of
the lithosphere becomes apparent….
Point load
versus infinite (very wide) load
hw
hoc
hm
g ( m hm  oc hoc   w hw )  strength
(weight)
-( Strength of
elastic lithosphere)
href
hoc
=
g (href  w  oc hoc )
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)
(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 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: Q   K dT
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
K- 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)
Q0  70mW / m3
A0  2.5  10-6Wm3
ar  10km
K  3W / m / C
Geotherms
Surface heat flow observations indicate that heat flow increases
linearly with the heat production of surface rocks. This is
mathematically 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
• Strain hardening and strain softening
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 spreadin
• Parsons and Sclater
Temperature and thickness versus age
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
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
Mantle viscosity
Models
• Diffusion creep
– Very Low stress
– Newtonian fluid
– Atoms diffuse
  v V ( fluid )
Viscosity depends on stress and temperature
 e
stress
1

T (º K )
Mantle viscosity
• High stress creep
• Disclocation creep
• Model for mantle plasticity
  V f
  Ae
 Qc 
 RT 


n
Power Law Creep
Q is activation energy
A is a creep mechanism parameter
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
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
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.
(TB) Definition taken from:
http://en.wikipedia.org/wiki/Flexural_rig
idity
• 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
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