chapter2: lithospheric mechanics this presentation contains illustrations from allen and allen (2005...

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-mail to me)

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:

(Side view)

(Side View)

(Bird’s Eye View)

min

max

max

min

min

max

max

max

minmin

min

Brittle faults can develop:

(Side view)

(Side View)

(Bird’s Eye View)

min

max

max

min

min

max

max

max

minmin

min

Dead Sea

Sea of Galilea

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??


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


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



•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)

2310

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

• Use local isostasy as a reference• Assume stationary conditions• Deviation from this reference is a measure of

internal strength balanced against an applied load

One view on flexure in basins

Measure of elasticity

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.

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!

+

=

+ + + + + + + + + + + + + + + +

Finite (reasonable and not extreme) geological load

versus infinite (very wide) load

Now compare the case where the load is relatively narrow….

( )m ref oc oc w wg h h h (( ) )a w ref w oc ocq g h h h

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.

Point loadversus infinite (very wide) load

Now compare the case where the load is relatively narrow and the strength of the lithosphere becomes apparent….

Point loadversus infinite (very wide) load

( )m m oc oc w wg h h h strength ( )ref w oc ocg h h

ochoch

mh

wh

(weight) -( Strength of elastic lithosphere)

=

refh

Downward directed invisible load creates space that fills with water and adds more vertical load

Level of compensationLevel of compensation

g(h.rhom + hw. rhow + g(h.rhom + hw. rhow + w .rhom)w .rhom)

Use reference at infinity (very far away) and pressure at level of compensation.

At level of compensation pressures are in equilibrium.

g(h.rhom + hw. rhow + w .rhom) =

qa (Point load) + g(w.rhow + hw.rhow + h.rhom)-internal resistance to bending

(At infinity)

(Under load)

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!

g(h.rhom + hw. rhow + w .rhom) = qa (Point load) + g( (w+hw).rhow + h.rhom) – internal resistance to bending

internal resistance to bending +

g (rhom-rhow) w

= qa (Point load) “Equation 2.28”

Equation 2.284 2

4 2( )a

d w d wD P gw q x

dx dx

(At infinity) (Under load)


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


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/ºKShale: 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:

dTQ 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.

200 2

Q AT T z z

K K

Q- heat flow

K- conductivity

A- internal heat generation

Z -depth

Geotherm

z

Temperature

oceanic

continent

Heat Production versus depth

Heat production at surface (Hs )is maximum

H eat production = Hs exp (-z/ar)

-6 30 2.5 10A Wm

10ra km

30 70 /Q mW m

3 / /K W m C

Z

Depth(km)

Heat production

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.

20 0

0 1 r

y

arQ A aT T z e

K K

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.


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


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

125 km

O km1300º 1300º

Start (at time=0)

After 200 my

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)

(1 )vfinal start T


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


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


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


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

Mantle viscosityModels

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

( )vV fluid

stresse

Viscosity depends on stress and temperature

1

(º )T K

Mantle viscosity

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

fV

cQ

nRTAe

Q is activation energy

A is a creep mechanism parameter

Power Law Creep

Dislocation Creep

• Temperature-activated creep– Movement of mantle by microfractures at the

subcrystal scale and synchronous healing of these imperfections

Rheology of continental crust

Byerlee’s Law

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

Shear stress

Normal stress

Elastic-Plastic model for breaking Rock

stress

strain

strain

Strain hardening

stress

strain

strain

Strain softening

stress

strain

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

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-strain.htm

http://myweb.cwpost.liu.edu/vdivener/notes/stress-strain.htm

• 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

Bouguer Anomalies (TJH)

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_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

http://en.wikipedia.org/wiki/Flexural_rigidity

http://en.wikipedia.org/wiki/Flexural_rigidity

http://en.wikipedia.org/wiki/Force

http://en.wikipedia.org/wiki/Couple

http://en.wikipedia.org/wiki/Rigid

http://en.wikipedia.org/wiki/Curvature

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

North pole is up & black line runs through Greenwich

H. FOLEY

http://www.answers.com/topic/geoid

http://solid_earth.ou.edu/notes/geoid/earths_geoid.htm

chapter2: lithospheric mechanics this presentation contains illustrations from allen and allen (2005...

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