Rheology

Rheology: The relationship between stress and strain. Note that you book defines this term more narrowly to refer to flow.

Strength: Stress that material can support before failure (in some cases on geologic time scale--e.g. ductile flow in lower crust)

Strain Rate: Strain Rate equals strain/time or, e/t = (l-lo/lo)/t. Common "geologic" strain rates 10-12 to 10-15/sec. Typical of strain accumulation around San Andreas Fault. But certain geologic processes at much higher rates, like meteorite impact, volcanism, fault slip during an earthquake.

Simplifying Rock's Response to Stress:

Bending, Breaking or Flow

Bending: Elastic Behavior: Recoverable strain:

     

     

    s = Ee

  • Young's modulus (E): Proportionality constant between stress (s) and longitudinal strain (e) .
    Basically this is Hooke's law.

Breaking: Brittle Failure:

 

 

t = C + s m

Coulomb or other criteria controlling failure: Relate stress (t =Shear Stress, s = normal stress, and m coefficient of friction) to onset of failure and sliding. C = cohesion, remove from equation for pure frictional sliding

Flow: Viscous Behavior

 

 

Simple Viscous Behavior:

 

s= h (e/t)

Stress is proportional to strain rate. Newtonian Viscosity:

Stress (s) has a linear relationship (the viscosity h) with strain rate (e/t) .

As a reality check note that water is about 20 orders of magnitude less viscous than most rocks. (Units of viscosity are in Pascal-seconds).

 

Pressure solution, and grain boundary and volume diffusion are linearly viscous processes.

Nonlinear Behavior: Effective viscosity: not a material property but a description of behavior at specified stress, strain rate, and temperature. Most rocks follow nonlinear behavior and people spend lots of time trying to determine flow laws for these various rock types. Generally we know that in terms of creep threshold: Salt < granite < basalt-gabbro < olivine. So strength generally increases as you go from crust into mantle, from granitic dominated lithologies to ultramafic (olivine).

Creep: Slow viscous flow that occurs at differential stresses well below the rupture strength of the rock.

Example of creep.

 

Flow Laws: Describe nonlinear viscous behavior of real rocks.

General Form: e/t = A sn exp (-E*/RT), rearranged as s = ((e/t)/A)1/nexp (-E*/RT)

where e/t is strain rate, A, n, and E* a constants for a material that can be experimentally determined, s is stress, R is gas constant, and T is temperature. Strain rate increases with increasing stress and temperature and constants for various rocks distinguish their behavior. Rewritten equation allows calculation of strength for geologic strain rate of 10-14 and for a particular thermal gradient resulting in:

For example salt is much weaker than olivine. Quartz rich rocks weaker than ones richer in plagioclase (remember the mylonite photomicro where the quartz flowed). Variations of this equation appear in Chapters 5 and 9 to describe both flow associated with diffusional and crystal plastic deformation mechanisms.

Strength of the Crust and Lithosphere:

 

Series of stress measurements going down through crust. Rock at failure and showing a coefficient of friction of about 0.65.

 

What equation should be use to describe upper linear strength profile?

What equation or relationship should be used to describe strength profile below brittle-plastic transition?

 

Construct a more complete lithospheric profile. What controls the various segments in the strength profile?

Lithospheric Strength Profiles:

  • Elastic and Brittle Failure dominate upper crust. Use coefficient of internal friction or coefficient of sliding friction (on existing fractures). Coulomb failure criterion controls strength profile until brittle-ductile transition.
  • Non Linear Viscous Flow Laws describe ductile deformation and show marked decreases in strength with depth, in a constant lithology.
  • Transition to ductile lower crust in most areas and then return higher strenght behavior at crust-mantle contact because mantle peridotite is stronger than lower crust. Ultimately mantle becomes ductile with viscous behavior at base of lithosphere. These transitions from brittle to ductile may occur repeatedly in a crust of complex lithology, as shown in Figure 5.20.

Triaxial Experiments with Real Rocks: Stress Strain Curves:

Elastic Behavior, Yield Stress, Ultimate Stress, Failure Viscous Yielding, Perfect Plasticity, Work (Strain) Hardening, Work (Strain) Softening. Applications in applied world of soils and rock mechanics--Dam construction, etc. Geologists take it farther.

Triaxial Experiments with Real Rocks: Controls on Behavior

Fig. 5.12b: Weakening of a fine-grained limestone with increasing temperature.Vertical Axis is stress in MPa.
  • Increasing Temperature Weakens Rocks: In earth both temperature and confining pressure increase with depth and temperature overcomes strengthening effect of confining pressure resulting in generally ductile behavior at depths
  • Confining stress: Rock strength increases with confining strength. This effect is much more pronounced at low temperatures < 100 o where frictional processes dominate and diminishes at higher > 350 o temperatures were ductile deformation processes that are temperature dominated are less influenced by pressure.

 

Effect of varying pore pressure on a sample with a fixed confining pressure of 70 MPa.
  • High fluid pressure weakens rocks because it reduces effective stress. se = sn - p
  • Water Weakens Rocks, by affecting bonding of materials.

Strain Rate: Rocks are weaker at lower strain rates. Slow deformation allows diffusional crystal-plastic processes to more closely keep up with applied stress.