Geol 456/656 - Lithospheric Deformation

Geol 456/656
Lithospheric Deformation

Stress and Strain

Imposing a stress on a material will produce a strain. Stress is a force per unit area, while a strain is a deformation. Stresses have the SI unit pascal (Pa); equal to one kilogram per meter per second squared. This is very slight pressure (32 pounds per square meter), so rock stresses have to be expressed in megapascal (MPa). Strains are unitless and for rocks often expressed in terms of microstrain, which is a deformation of 0.000001 proportionately, or 0.0001%.

Question: one can measure three kinds of elastic strain with simple laboratory experiments. What are those, or the names of the associated elastic parameters?

Stress at any point will have both an isotropic component, called the pressure p, and a deviatoric component that varies with direction. The variation usually simplifies to a symmetric ellipsoid having three principal and mutually orthogonal axes. The values of these axes are called the principal stresses. A deviatoric stress will have a maximum principal stress sigma max

pointing in and thus compressive and positive. There is an intermediate principal stress sigma interm

, and a minimum principal stress sigma min

pointing out and thus tensional and negative.

For rocks the mathematical relation between stress and strain can vary enormously depending on composition, temperature, strain rate, and strain history. This relation is called a constitutive relation. A simple linear proportional relation between the two, as with a spring constant, is a linear elastic constitutive relation. There are also non-linear, exponential relations for inelastic, plastic or viscous solids and fluids.

Two very different constitutive relations may operate on the same material, with the strain rate determining which will control the mode of deformation. Most solids will react elastically and with great strength when hit with a sledgehammer to induce a very high rate of strain, producing fracturing. The same solids (e.g.: silly putty, soil, asphalt, glass, steel, quartz, olivine) will exhibit very little strength and flow viscously or creep plastically when stressed over very long time periods.

Brittle Fracture

Pressure increases with depth in the crust at a rate of 33 MPa/km, from the weight of the overburden. Virtually all fractures and other porosity close before 3 km depth. Earthquakes are a result of brittle fracture, and typically initiate at somewhat deeper depths, yet must result from some tensional failure. Rocks can resist compression very well, with granite having a compressive strength around 140 MPa. But their much smaller tensile strength (only 4 MPa for granite versus nearly 100 MPa for steel) allows fracturing to occur when deviatoric stresses become only a few percent as large as the overburden pressure. The topographic deviatoric stresses created by mountain ranges easily exceed the tensile strength of crustal rocks.

fault types sketch (J. Louie)
Principal stress directions for the three fault types:
X Y Z

Normal sigma min sigma interm sigma max

Thrust

Strike-Slip

The strength of brittle rocks increases linearly with pressure, and thus very often with depth too.

Question: are single crystals or rocks stronger? Why?

$fracture sketch$ (from Kearey & Vine, copyright Blackwell Sci. Publ.)

Brittle fracturing, or cataclasis, occurs in this microscopic or thin-section view when tensional stresses become great enough to break across mineral grains, or sever the bonds between them and cause rows of them to roll. Shear fracture will take place on complementary planes oriented about 60 degrees from the direction of sigma max .

Elastic deformation is temporary and reversible until fracture is achieved.

Ductile Flow

Ductile deformation in rocks takes three forms: plastic flow; power-law creep; and diffusion creep, defined by metallurgical work. Each of these mechanisms has an exponential constitutive relation in which strain increases exponentially with applied stress. Ductile constitutive relations are more sensitive to temperature and composition than to other factors.

plastic flow sketch (from Kearey & Vine, copyright Blackwell Sci. Publ.)

Plastic Flow produces limited, permanent strains at high stresses when the yield strength of mineral grains is exceeded and they deform by gliding along internal dislocation planes or grain boundaries. Such crystal-structure dislocations, as in this microscopic view, can heal quickly and may actually strengthen the rock (work hardening).

You hammer on metal to exceed its yield strength (still much less than its elastic tensile strength) to plastically and permanently deform it.

power law creep sketch (from Kearey & Vine, copyright Blackwell Sci. Publ.)

Power-Law Creep takes place only at high temperature, at least 55% of the melting temperature. Most minerals melt at temperatures between 400 and 1800 C, so power-law creep takes place in the lower crust or deeper. For this mechanism the constitutive relation gives the strain rate with time as proportional to an exponential power of the applied stress, with the exponent being 3 or greater. A newtonian viscous fluid has strain rate proportional to the stress with an exponent of one.

Internal gliding dislocation, as with plastic flow, is the main mechanism of deformation for power-law creep. However, the high temperature allows diffusion of atoms and recrystallization, so creep can continue to strains in excess of 100% at relatively low stresses.

Power-law creep is likely the dominant mode of deformation in the mantle.

diffusion creep sketch (from Kearey & Vine, copyright Blackwell Sci. Publ.)

Diffusion Creep occurs when temperatures exceed 85% of the melting temperature. Such temperatures lead to rapid diffusion and atom migration along stress gradients, promoting continual recrystallization.

Diffusion creep has a newtonian constitutive relation with strain rate proportional to the first power of stress, so such materials flow like fluids. This mechanism should be operating in regions just below the solidus temperature, like the LVZ or Asthenosphere that underlies the oceanic lithosphere.

Question: think of a situation involving power-law or diffusion creep of a rock formation at surface temperatures and pressures.

Faulting and Deformation

The existence of very different mechanisms for deformation leads to some complication in finding the reaction of any part of the earth to an applied stress. Constitutive relations vary between layers of the earth, since temperatures and compositions also change with depth.

strength envelope sketch
(from Kearey & Vine, copyright Blackwell Sci. Publ.) Given a constant composition, as with the olivine making up most of the oceanic lithosphere, one can compute a strength envelope, as above, by figuring out what mechanism will make the rock weakest at any given depth. To depths of 30 km or so temperatures are low enough for highly-refractory olivine that brittle fracture is the easiest mode of deformation. Thus to that depth the strength envelope is linearly increasing. Below 30 km the temperature is high enough that the power-law creep mechanism, exponentially decreasing strength with increasing depth and temperature, takes over as the dominant weakening mechanism. Thus lithospheric strength peaks near 30 km, where the curves intersect. Most earthquakes nucleate near this depth, and of course break only through the brittle lithosphere above.

In the continental lithosphere the strength envelope is complicated by the high quartz content of the crust. Quartz has a lower melting temperature than olivine, so power-law creep takes over and the strength envelope peaks in the continents at 15 km depth. This in fact is the maximum depth of most earthquakes, and their depth of nucleation. The continental strength envelope has a second peak in the olivine just below the Moho.

Question: how can earthquakes occur in some places to depths of 670 km?

(J. Louie) Glacial rebound, or any imposition of a load on the lithosphere, produces a classic example of the interaction of multiple constitutive relations. The load deforms the elastic lithosphere at strains much less than 1%, according to a fourth-order differential equation for plate flexing that surrounds the load with a moat and peripheral uplift. The extent of these features is controlled by the thickness and rigidity of the lithosphere.

Question: where should the lithosphere be relatively thin? relatively thick?

The asthenosphere below responds as a viscous fluid, possibly with strains in excess of 100%, since it is weaker and serves to absorb most of the deformation. As a viscous fluid with a first-order differential equation of motion it damps out the wavy shape of the lithospheric flex.

If the deformation were all elastic, removing the load would immediately remove the deformation in one big earthquake. But the viscous asthenosphere has to flow back into place below, which takes an exponential time constant related to the viscosity.

Question: describe three different ways to load the lithosphere and observe flexure, and name an example of each.