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

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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- 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)

Brittle faults can develop: (Side view) (Side View) (Bird’s Eye View)

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) Nm is equivalent to about 34 km of elastic thickness (Te) or moderately strong elastic 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…. 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 and the strength of the lithosphere becomes apparent…. Point load versus infinite (very wide) load

(weight) -( Strength of elastic lithosphere) =

Downward directed invisible load creates space that fills with water and adds more vertical load Level of compensation g(h.rhom + hw. rhow + 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 2.28 (At infinity) (Under load)

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 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. Q- heat flow K- conductivity A- internal heat generation Z -depth

Geotherm z Temperature oceanic continent

Heat Production versus depth Heat production at surface (H s )is maximum H eat production = Hs exp (-z/a r )

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. ar is the depth at which heat production is halved A 0 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

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 km 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)

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 Viscosity depends on stress and temperature

Mantle viscosity High stress creep Disclocation creep Model for mantle plasticity 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

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 stress strain

Strain hardening stress strain

Strain softening 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-strain.htmhttp://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_rig idity 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.forcecouple rigidcurvature 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

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.htmhttp://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

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