Rheology of rocks (models for stress-strain relationships)

Alfred Wegener
"All that can be said with certainty is that the earth behaves as a solid, elastic body when acted upon by short-period forces such as seismic waves, and there is no question of plastic flow here. However, under forces applied over geological time scales, the earth must behave as a fluid: for example, this is shown by the fact that its oblateness corresponds directly to its period of rotation. But the critical point in time where elastic deformations merge into flow phenomena depends precisely on the viscosity coefficient."

The concept of rheology is needed to make sense of the earth.

Lecture index: Utility of understanding stress-strain relationships. / Five components of a constant load test in strain time space. / Three models to explain stress strain relationships. / Combination models and definition of rheid. /

Reading: Chapt. 6 in Fossen, Structural Geology

Utility of understanding stress-strain relationships

Stress-strain relationships are important because:

• elastic stress-strain relationships are crucial to understanding seismic waves.
• these are tje basis of in-situ stress measurement.
• they are fundamental for geologic engineering.
• understanding important geologic phenomena including:
• isostatic rebound.
• ductile deformation.
• mantle convection.

Five components of a constant load test in strain time space

An idealized constant load (constant stress) test result looks something like the above. Strain over time is the strain rate, and so changes in the slope of the line represents changes in strain rates. The temperature, pressure, material properties, and load amount all influence the specific shape of the curve. The curve shown here is for conditions that produce the full suite of components. Under certain conditions only limited portions of this behavior will be seen.

How can we model this behavior? First, we can divide it this history into separate components.

• elastic response: immediate and recoverable.
• primary creep: decelerating strain rate.
• secondary creep: constant strain rate (viscous behavior).
• tertiary creep: accelerating strain rate that leads to failure.
• brittle failure.
• Tertiary creep is often result of high loads needed to produce deformation under lab conditions and within human time frames. It is important engineering wise. Lower loads, hotter conditions, etc., lengthen secondary creep, permitting unlimited ductile deformation. Thus, for ductiley deforming rocks the goal is to explain up to secondary creep.
• alternative testin environment is a constant strain rate instead of a constant load.

Three models to explain stress strain relationships

These are idealized simple mechanical models, each with their own iconic representation.

• elastic:
• E=Youngs modulus = stress / axial elongation
• similar to spring stiffness constant.
• measured under uniaxial compression.
• uniaxial loading path, get mixture of permanent and recoverable deformation, recovery path used to compute E.
• There is also a Shear modulus (rigidity) = shear stress/shear strain.
• Poisson's ratio = lateral/longitudinal elongation;
• permissible values 0 -> .5.
• no compressibility then = .5, totally collapsible then = 0, rocks typically .25.
• 'sponginess'.
• as saw in previous discussion of lithostatic gradient, this parameter is important in the transmittal of stress.
• bulk modulus another elastic constant (K).
• tensor description allows for description of directional elasticity. Individual crystals are anisotropic elastically.
• iconic depiction of elastic behavior as a spring (see below).
• plastic:
• yield strength, incomplete description, doesn't specify strain rate.
• question as to fundamental strength.
• a threshold at which deformation initiates.
• iconic depiction as a friction block (see below).
• viscous:
• stress = viscosity x strain rate, or viscosity = stress/strain rate; cgs units for viscosity - poise, with 1 poise = .1 Pa sec.
• defined in stress strain rate space.
• note implied lack of fundamental strength.
• Newtonian vs. power law types of viscosity; e = (1/µ) s^n
• Newtonian viscosity, n=1
• for many rocks n=3-5 range.
• nonlinear indicates as shear strain increases that shear thinning is occurring (internal weakening).
• atomic level what is going on? Bond breakage and formation - as things get hot atom neighbor swapping accelerates.
• iconic depiction as a dashpot (see below).

Combination models and definition of a rheid

Linking these three components in different ways produces some 'emergent' behavior.

The top row depicts three basic deformational behaviors that can be combined to build models for more complex behavior. The second row depicts models of the the spring and dashpot combined in series, and then in parallel. The lower most depiction is with the two modes above combined in series, and this model reproduces elastic, primary and secondary creep behavior as depicted and described above.

Consider the 3 time and stress constant load curves above. Note that mid-way the driving stress (the load) is removed and recoveryis possible. Which one best represents the Maxwell body, which one best represents the Kelvin body, and which Carey's rheid body?

• Maxwell body:
• spring and dashpot in series.
• models secondary creep component.
• missing primary creep.
• recovery of the elastic spring component, but dashpot component permanent.
• Firmoviscous Kelvin Body:
• spring and dashpot in parallel.
• models primary creep.
• time recoverable deformation, all of it.
• can be used to model glacio-isostatic rebound and mantle viscosities. Works on time frames of thousands of years for earthly deformation conditions.
• Carey's rheid body:
• Maxwell and Kelvin body in parallel.
• models primary and secondary creep.
• e(t)=(2s/9K) + (s/3G2) + (s/3G1) - (s/3G4) * (e^-(G1t/h1)) + (s t / 3h2)
• the Gs are spring constants, K is the Bulk Modulus
• 2nd term -> spring in Maxwell body
• 3rd term -> spring in Kelvin body
• 4th term -> dashpot + spring in Kelvin body
• 5th term -> dashpot in Maxwell body

Definition of rheidity: The units are time. The rheidity of a given material is the time at which the permanent deformation = 1000 times the recoverable deformation. At this point you can consider the last term to dominate and the material is acting like a viscous fluid for those time frames. If deformation conditions persist for this amount of time then the finite deformation can be treated as viscous.