Lithospheric Mechanics

1

• Chapter2
• Lithospheric Mechanics
• This presentation contains illustrations from
  Allen and Allen (2005 )
• and Press et al. (2004)

2
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

3

• 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), Byerlees
  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-mail to me)

4
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?

5
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

6
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

7
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

8

• 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.

9

• What is the vertical lithostatic stress gradient
  in granitic crust? What is the vertical stress
  gradient in the first 2 km of the ocean?

10
Faults can develop
(Side view)
(Side View)
(Birds Eye View)
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Brittle faults can develop
(Side view)
(Side View)
(Birds Eye View)
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Sea of Galilea
Dead Sea
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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??
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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

17
Surface Forces (Pressure)- LOCAL ISOSTASY
Depth of compensation
18
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

19
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

20
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

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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

22
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.

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• 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)

Nm is equivalent to about 34 km of elastic
thickness (Te) or moderately strong elastic
lithosphere
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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

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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. gt 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.

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Finite (reasonable and not extreme) geological
load
versus infinite (very wide) load
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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.
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Point load
versus infinite (very wide) load
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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
35

-( Strength of elastic lithosphere)
(weight)
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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)
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(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
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g(h.rhom hw. rhow w .rhom)
qa (Point load) g(whw) 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!
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(Under load)
(At infinity)
qa (Point load) g( (whw).rhow h.rhom)
internal resistance to bending
g(h.rhom hw. rhow w .rhom)
internal resistance to bending g (rhom-rhow)
w
qa (Point load)
Equation 2.28
Equation 2.28
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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

41
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

42
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 earths surface (heat
  flux-Q) per unit time will depend on the
  coefficient of thermal conductivity (K).
• Fouriers Law

Q for continents is 60 mW/m2 or 60W/1000 m2 Q
for continents is 80 mW/m2
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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.

Q- heat flow K- conductivity A- internal heat
generation Z -depth
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Geotherm
Temperature
oceanic
z
continent
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Heat Production versus depth
Heat production at surface (Hs )is maximum
H eat production Hs exp (-z/ar)
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Heat production
Z Depth(km)
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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.

ar is the depth at which heat production is
halved A0 is the surface heat production
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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

49
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)
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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.

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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

52
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

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Thermal expansion

• 100,000 m 10-5 1ºK 1m/ºK

54
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

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Thermal contraction
Start (at time0)
After 200 my
1300º
1300º
O km
125 km
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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
  ..?

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Choose an answer

• (a) 2km
• (b) 4 km
• (c) 10 km
• (d) 20 km
• (e) none of the above

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Answer

• 125,000 m 650ºC 10-5 812 m

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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)

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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

61
Time-dependent heat conduction

• We observe that
• heat flow decreases away from the mid-ocean
  ridges as a function of age and
• water depth increases as a function of age

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Heat flow versus age
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Plate Model for Sea-floor spreadin

• Parsons and Sclater

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Temperature and thickness versus age
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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

66
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
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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

69
Mantle viscosityModels

• Diffusion creep
• Very Low stress
• Newtonian fluid
• Atoms diffuse

Viscosity depends on stress and temperature
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Mantle viscosity

• High stress creep
• Disclocation creep
• Model for mantle plasticity

Power Law Creep
Q is activation energy A is a creep mechanism
parameter
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Dislocation Creep

• Temperature-activated creep
• Movement of mantle by microfractures at the
  subcrystal scale and synchronous healing of these
  imperfections

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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

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Rheology of continental crust
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Byerlees Law

• Linear relation between shear stress and normal
  stress for rock strength

Shear stress
Normal stress
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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

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Elastic-Plastic model for breaking Rock
strain
stress
strain
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Strain hardening
strain
stress
strain
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Strain softening
strain
stress
strain
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Elastic-plastic
stress
strain
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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.

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Elastic-plastic
stress
strain
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Elastic-plastic
stress
strain
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Elastic-plastic
stress
strain
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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.

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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-strain.htm
86
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

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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
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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

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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
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Thermal Conductivity (AD)

• Heat transfer is achieved by processes of
• Conduction- a diffusive process in which kinetic
  energy is transferred by intermolecular
  collisions. Conduction is the primary thermal
  process in the lithosphere.
• Convection- requires motion of the medium to
  transmit heat. Convection of heat from the core
  is the principal thermal process of the mantle.
• Electromagnetic radiation- only important in
  determining surface heat budget, not the internal
  heat budget

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Fouriers Law

• Fouriers 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

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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.

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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 Earths heat loss takes place
  through the ocean floor.

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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

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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.

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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.

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(TB) Definition taken from http//en.wikipedia.o
rg/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

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North pole is up black line runs through
Greenwich

• Geoid (HF)

GEOID a surface on which the earths
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 earths actual surface.
• For more info http//www.answers.com/topic/geoid,
  http//solid_earth.ou.edu/notes/geoid/earths_geoi
  d.htm

H. FOLEY
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