1. Structural Geology Ch. 5 – Rheology
4/14/2017
Structural Geology Ch. 5 – Rheology
Rheology - the functional relationship between stress and strain.
Just as the result of a force on a rigid mass is motion, so the result of a stress on a deformable mass is strain.
Rigid body mechanics is relatively easy and can be described by Force = mass x acceleration.
Rheology is not so simple, and there is no general equation that describes deformation other than s = f(e)

2. Convergent Divergent Transform
Compression, tension and shear forces stress the rocks, causing them to strain i.e. “give”
Units of Stress = Force / Area
Convergent Divergent Transform

3. Relationship Between Stress and Strain
Rubber Band
Strain can be a change in shape (a deformation) due to an applied stress

4. Relationship Between Stress and Strain at low Temps and Pressure or Sudden Stress
Ruler, Pencil

5. Relationship Between Stress and Strain under high Temps or Pressure
Chewing Gum

6. Structural Geology (3443) Ch. 5 – Rheology
4/14/2017
Structural Geology (3443) Ch. 5 – Rheology
Determination of the functional relationship s = f(e) must be done experimentally and that sub-discipline is called Rock Mechanics.
Rock mechanics is most important in engineering geology, where the stability of slopes, tunnels, mine adits, shafts drifts, and soils and foundations determines the economic viability of a project and the health of the users.
In Structural Geology and Tectonics, experimental rock deformation is important in determining the evolution of natural structures and tectonic features.

7. Structural Geology Ch. 5 – Rheology
4/14/2017
Structural Geology Ch. 5 – Rheology
Design of triaxial compression testing equipment is shown at left.
tress is increased vertically by hydraulic jack.
Confining stress on sides is produced independently by fluid pressure.
Pore pressure (fluid pressure in pore space) is produced the by using the natural fluids (water, petroleum).
Temperature is also controlled.

8. Structural Geology Ch. 5 – Rheology
4/14/2017
Structural Geology Ch. 5 – Rheology
Vertical stress on the specimen is calculated knowing the force on the piston and the area of the specimen top. Here vertical stress is usually the maximum stress (s1)
Stress on the side of the specimen is the same as the confining fluid pressure. Since the two side stresses cannot be controlled independently, s2 = s3.

9. Structural Geology Ch. 5 – Rheology
4/14/2017
Structural Geology Ch. 5 – Rheology
In order to graph the results on a 2-D stress-strain graph. “differential” stress is plotted against strain.
ds = s1 – s3
This is equivalent to the diameter of the Mohr graph. The greater the differential stress, the bigger the Mohr circle, and the greater the amount of possible shear stress.

10. Structural Geology Ch. 5 – Rheology
4/14/2017
Structural Geology Ch. 5 – Rheology
Pore pressure, due to water or petroleum in the pore spaces, greatly effects deformation because it subtracts from the loads on the rock. Deformation is produced by “effective stress”
se1 = s1 - sp
se2 = s2 - sp
se3 = s3 - sp

11. Structural Geology Ch. 5 – Rheology
4/14/2017
Structural Geology Ch. 5 – Rheology
Strain is measured by the displacement of the piston (Dl). Knowing the original length of the specimen (lo), finite (accumulated) strain is calculated.

12. Structural Geology Ch. 5 – Rheology
4/14/2017
Structural Geology Ch. 5 – Rheology
Finally, rate of strain must be controlled because it has a profound affect on the way rocks deform. So the displacement of the piston (Dl), must be timed (Dt).

13. Structural Geology Ch. 5 – Results of Rock testing
4/14/2017
Structural Geology Ch. 5 – Results of Rock testing
“Creep” tests show how strain accumulates over time under a constant load (stress).
Typically, the rock deforms rapidly and then begins to deform more slowly after the yield stress. This is called primary creep (area I, top figure).

14. Structural Geology Ch. 5 – Results of Rock testing
4/14/2017
Structural Geology Ch. 5 – Results of Rock testing
After primary creep (I), deformation continues at a constant rate (the linear portion of the curve) which is secondary creep (II).
Finally, deformation rate increases rapidly until the rock fails (fractures) in Tertiary creep (III).
If stress is removed (b), strain drops rapidly, but a permanent strain remains.

15. Ch. 5 – Results of Rock testing
Differential stress s1-s3 to strain graph for limestone at a confining pressure of 103 MPa (that’s about 4.2 km below the surface ). Specimen was at room temperature
4/14/2017
Ch. 5 – Results of Rock testing
On a stress-strain plot, the creep curve looks different.
Up to point A, the graph is linear, and if the load is removed, the strain is recovered and goes back to zero. This type of deformation is called “Elastic”.

16. Ch. 5 – Results of Rock testing
4/14/2017
Ch. 5 – Results of Rock testing
The elastic limit at point A is called the yield strength, and at greater stress the curve is no longer linear. At point B, the load was removed, but the strain does not return to 0 because the elastic limit was exceeded. The specimen has about ½% permanent, or ductile, strain, about the same amount as from point A to B.
Primary creep would include the curve up to point B or C.

17. Structural Geology Ch. 5 – Results of Rock testing
4/14/2017
Structural Geology Ch. 5 – Results of Rock testing
The same specimen was reloaded assuming 0 strain at the start. The specimen again deforms elastically until about point C, which is the new yield strength. The difference between A and C is called strain hardening – previous ductile strain adds more resiliency to the rock.

18. Ch. 5 – Results of Rock testing
4/14/2017
Ch. 5 – Results of Rock testing
Continued loading produces more ductile strain from C to point D which is called the peak (or ultimate) strength. That is the highest load the rock can bear.
This portion of the curve (from C-D) would be secondary creep.

19. Ch. 5 – Results of Rock testing
4/14/2017
Ch. 5 – Results of Rock testing
After the ultimate strength is reached (D), it takes smaller and smaller loads to produce strain (or the strain rate increases if the load is kept constant) until the specimen ruptures (fractures).
This is equivalent to Tertiary Creep.
Fracturing is called brittle behavior in contrast to ductile.

20. Structural Geology Ch. 5 – Results of Rock testing
4/14/2017
Structural Geology Ch. 5 – Results of Rock testing
Changing the confining stress – the effect of burial depth.
Increasing the confining pressure and the mean stress, is like seeing how the specimen would behave at deeper depths.
For crustal rocks conversion of depth to Pascals (Pa) is
Pa ~ depth (in Meters) * 25000 or
depth = Pa/ 25000

21. Structural Geology Ch. 5 – Results of Rock testing
4/14/2017
This graph shows the effect of increasing depth without increasing temperature
Remember differential stress is s1-s3
s1 is the maximum normal stress, and s3 the minimum
s1 – s3

22. Structural Geology Ch. 5 – Results of Rock testing
4/14/2017
Homework: What is the range of depths shown by the graph? Remember 1 MPa = 1 x 106 Pa
Structural Geology Ch. 5 – Results of Rock testing
Confining Pressure MPa
s1 – s3
1 km = 25 MPa
Example for 20 MPa
What about 140 MPa?

23. Structural Geology Ch. 5 – Results of Rock testing
4/14/2017
What happens to the yield strength with increasing confining pressure?
Structural Geology Ch. 5 – Results of Rock testing
Increasing confining pressure increases the ability of the rock to flow, not break.
Analogy, clap hands under water
Larger strains can be achieved at greater depths before failure
s1 – s3

24. Structural Geology Ch. 5 – Results of Rock testing
4/14/2017
What happens to the ultimate strength with increasing confining pressure?
Ultimate Strength: max stress material can support before failure
s1 – s3

25. Structural Geology Ch. 5 – Results of Rock testing
4/14/2017
What happens to the rupture strength * with increasing confining pressure?
Structural Geology Ch. 5 – Results of Rock testing * *
s1 – s3 * * * *

26. Structural Geology Ch. 5 – Rock testing
This test is similar to the last one, but for sandstone. The confining pressure was 200 MPa. What is the simulated depth? Conversion factor 1km ~25 MPa
4/14/2017
Structural Geology Ch. 5 – Rock testing
s1 – s3

27. Structural Geology: Ch. 5 – Rock testing
4/14/2017
However, these 5 tests were run at different pore pressures sf under the same 200 MPa confining pressure sc. The effective confining pressure P*c, = sc – sf. What is the pore pressure sf in each of the 5 tests?
Sandstones and shales contain a significant fluid component
s1 – s3
Top sp = 0, bottom sp = 200 MPa

28. Structural Geology: Ch. 5 – Rock testing
4/14/2017
What happens to the ultimate (peak) strength (dots) as effective confining pressure increases? As the pore pressure increases?
s1 – s3

29. Structural Geology Ch. 5 – Rock testing
4/14/2017
Effect of temperature:
If the temperature gradient is 25oC/km, what is the simulated depth of the various temperatures if the surface temperature is 25oC?
s1 – s3
Example
(300oC – 25oC) x 1km/25oC

30. Structural Geology Ch. 5 – Rock testing
4/14/2017
What happens to the ultimate (blue bar ) and yield strength (dot o) as temperature (depth) increases?
s1 – s3

31. Strain Rate s1 – s3 Effect of strain rate de/dt
4/14/2017
Fast 4 x 10-1/sec = 0.4 Dl/l0/sec
Tension (extension)
Effect of strain rate de/dt
Homework:
Look at the differential stress rupture points.
For the same rock (here Yule Marble) is the rupture point stress higher under slow or fast strain rate?
See also Table 5.7 for e’
s1 – s3
Slow 3.3 x 10-8 /sec

32. Structural Geology Ch. 5 – Rock testing
4/14/2017
High e’
Effect of strain rate.
What happens to the yield strength as the strain rate e’ decreases?
Low e’

33. Structural Geology Ch. 5 – Rock testing
4/14/2017
Example: Divergent Margin
Consider a cross section which was extended to 51 km from an initial length of 33 km.
If the deformation took 1 million years,
a.Calculate the strain rate.
b. How does that compare to the experimental strain rates shown here?

34. Structural Geology Ch. 5 – Rock testing
4/14/2017
Example:
Consider a cross section which was extended to 51 km from an initial length of 33 km.
If the deformation took 1 million years,
a.calculate the strain rate.
b. How does that compare to the experimental strain rates shown here?
e’ = – 33 km
33 km x 106 years
e’ = – 33 km
33 km x x seconds
e’ = x 10-14/seconds
A much slower strain rate than the experiments seen above.
106 years x 365 days/ 1 year x 24 hours/1 day x 60 min/1 hour x 60 sec/1 min
1 million years ~ x seconds.

35. Structural Geology Ch. 5 – Rock testing
4/14/2017
Effect of rock type
Rocks are so variable that it is hard to generalize, but siliceous rock with little pore space are generally strongest.
Competence “is a relative term that compares the resistance of rocks to flow, their viscosity.
Weakest rock salt < shale < limestone < graywacke < sandstone (arenite) < dolomite Strongest

36. Structural Geology Ch. 5 – Rock testing
4/14/2017
The linear portion of the stress-strain curve is called the elastic region where s = Ee.
E is Young’s Modulus, a “constant”, that changes with type of material. Once the material has exceeded it’s yield strength, the elastic equation doesn’t apply.

37. Structural Geology Ch. 5 – Rock testing
4/14/2017
Young’s Modulus is the slope of the stress-strain elastic line, and is a measure of stiffness, not strength. A material with a high Young’s modulus is said to be stiffer than a rock with a lower one, not stronger.

38. Structural Geology Ch. 5 – Rock testing
4/14/2017
There is a similar elastic relationship between shear stress and shear strain: t = Gg. where G is the shear modulus that depends on rock type.

39. Structural Geology Ch. 5 – Rock testing
4/14/2017
If you squeeze a specimen in one direction, it will produce a perpendicular strain just to maintain volume. The ratio of these two perpendicular strains is called Poisson’s ratio
n = eh/ev

40. Structural Geology Ch. 5 – Rock testing
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Finally, The bulk modulus K is a measure of the amount of dilation (change in volume) produced by the mean stress (pressure)
s = K x [(V – V0)/V0]

41. Structural Geology Ch. 5 – Rock testing
4/14/2017
The second general model to describe stress-strain is Plastic behavior. This describes the ductile portion of the graph where strain accumulates continuously when stress reaches a critical value. Most rocks behave as elastic-plastic materials, as does this
Crown Point Limestone:

42. Structural Geology Ch. 5 – Rock testing
4/14/2017
The material is perfectly plastic if the ductile portion is nearly horizontal. If the curve rises (more stress is needed the maintain deformation) it is called strain hardening. If the curve declines (less stress required) it is strain softening.

43. Structural Geology Ch. 5 – Rock testing
4/14/2017
The last general relationship is the Newtonian Viscous one which describes a few ductile rocks and most fluids. Here, the relationship is not between stress and strain, but stress and strain rate. Think of a fluid – if stress is applied, the fluid flows (deforms) continuously at a certain rate and continues to deform at that rate until the stress is changed.

44. Structural Geology Ch. 5 – Rock testing
4/14/2017
The viscosity is the ratio of maximum shear stress to shear strain rate, which is the slope of the line on the graph