1Structural Geology Ch. 5 – Rheology 4/14/2017Structural Geology Ch. 5 – RheologyRheology - 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)
2Convergent Divergent Transform Compression, tension and shear forces stress the rocks, causing them to strain i.e. “give”Units of Stress = Force / AreaConvergent Divergent Transform
3Relationship Between Stress and Strain Rubber BandStrain can be a change in shape (a deformation) due to an applied stress
4Relationship Between Stress and Strain at low Temps and Pressure or Sudden Stress Ruler, Pencil
5Relationship Between Stress and Strain under high Temps or Pressure Chewing Gum
6Structural Geology (3443) Ch. 5 – Rheology 4/14/2017Structural Geology (3443) Ch. 5 – RheologyDetermination 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.
7Structural Geology Ch. 5 – Rheology 4/14/2017Structural Geology Ch. 5 – RheologyDesign 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.
8Structural Geology Ch. 5 – Rheology 4/14/2017Structural Geology Ch. 5 – RheologyVertical 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.
9Structural Geology Ch. 5 – Rheology 4/14/2017Structural Geology Ch. 5 – RheologyIn order to graph the results on a 2-D stress-strain graph. “differential” stress is plotted against strain.ds = s1 – s3This 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.
10Structural Geology Ch. 5 – Rheology 4/14/2017Structural Geology Ch. 5 – RheologyPore 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 - spse2 = s2 - spse3 = s3 - sp
11Structural Geology Ch. 5 – Rheology 4/14/2017Structural Geology Ch. 5 – RheologyStrain is measured by the displacement of the piston (Dl). Knowing the original length of the specimen (lo), finite (accumulated) strain is calculated.
12Structural Geology Ch. 5 – Rheology 4/14/2017Structural Geology Ch. 5 – RheologyFinally, 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).
13Structural Geology Ch. 5 – Results of Rock testing 4/14/2017Structural 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).
14Structural Geology Ch. 5 – Results of Rock testing 4/14/2017Structural Geology Ch. 5 – Results of Rock testingAfter 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.
15Ch. 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 temperature4/14/2017Ch. 5 – Results of Rock testingOn 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”.
16Ch. 5 – Results of Rock testing 4/14/2017Ch. 5 – Results of Rock testingThe 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.
17Structural Geology Ch. 5 – Results of Rock testing 4/14/2017Structural Geology Ch. 5 – Results of Rock testingThe 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.
18Ch. 5 – Results of Rock testing 4/14/2017Ch. 5 – Results of Rock testingContinued 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.
19Ch. 5 – Results of Rock testing 4/14/2017Ch. 5 – Results of Rock testingAfter 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.
20Structural Geology Ch. 5 – Results of Rock testing 4/14/2017Structural Geology Ch. 5 – Results of Rock testingChanging 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) isPa ~ depth (in Meters) * 25000ordepth = Pa/ 25000
21Structural Geology Ch. 5 – Results of Rock testing 4/14/2017This graph shows the effect of increasing depth without increasing temperatureRemember differential stress is s1-s3s1 is the maximum normal stress, and s3 the minimums1 – s3
22Structural Geology Ch. 5 – Results of Rock testing 4/14/2017Homework: What is the range of depths shown by the graph? Remember 1 MPa = 1 x 106 PaStructural Geology Ch. 5 – Results of Rock testingConfining Pressure MPas1 – s31 km = 25 MPaExample for 20 MPaWhat about 140 MPa?
23Structural Geology Ch. 5 – Results of Rock testing 4/14/2017What happens to the yield strength with increasing confining pressure?Structural Geology Ch. 5 – Results of Rock testingIncreasing confining pressure increases the ability of the rock to flow, not break.Analogy, clap hands under waterLarger strains can be achieved at greater depths before failures1 – s3
24Structural Geology Ch. 5 – Results of Rock testing 4/14/2017What happens to the ultimate strength with increasing confining pressure?Ultimate Strength: max stress material can support before failures1 – s3
25Structural Geology Ch. 5 – Results of Rock testing 4/14/2017What happens to the rupture strength * with increasing confining pressure?Structural Geology Ch. 5 – Results of Rock testing**s1 – s3****
26Structural 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 MPa4/14/2017Structural Geology Ch. 5 – Rock testings1 – s3
27Structural Geology: Ch. 5 – Rock testing 4/14/2017However, 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 components1 – s3Top sp = 0, bottom sp = 200 MPa
28Structural Geology: Ch. 5 – Rock testing 4/14/2017What happens to the ultimate (peak) strength (dots) as effective confining pressure increases? As the pore pressure increases?s1 – s3
29Structural Geology Ch. 5 – Rock testing 4/14/2017Effect 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 – s3Example(300oC – 25oC) x 1km/25oC
30Structural Geology Ch. 5 – Rock testing 4/14/2017What happens to the ultimate (blue bar ) and yield strength (dot o) as temperature (depth) increases?s1 – s3
31Strain Rate s1 – s3 Effect of strain rate de/dt 4/14/2017Fast 4 x 10-1/sec = 0.4 Dl/l0/secTension (extension)Effect of strain rate de/dtHomework: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 – s3Slow 3.3 x 10-8 /sec
32Structural Geology Ch. 5 – Rock testing 4/14/2017High e’Effect of strain rate.What happens to the yield strength as the strain rate e’ decreases?Low e’
33Structural Geology Ch. 5 – Rock testing 4/14/2017Example: Divergent MarginConsider 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?
34Structural Geology Ch. 5 – Rock testing 4/14/2017Example: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 km33 km x 106 yearse’ = – 33 km33 km x x secondse’ = x 10-14/secondsA 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 min1 million years ~ x seconds.
35Structural Geology Ch. 5 – Rock testing 4/14/2017Effect of rock typeRocks 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
36Structural Geology Ch. 5 – Rock testing 4/14/2017The 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.
37Structural Geology Ch. 5 – Rock testing 4/14/2017Young’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.
38Structural Geology Ch. 5 – Rock testing 4/14/2017There is a similar elastic relationship between shear stress and shear strain: t = Gg. where G is the shear modulus that depends on rock type.
39Structural Geology Ch. 5 – Rock testing 4/14/2017If 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 ration = eh/ev
40Structural Geology Ch. 5 – Rock testing 4/14/2017Finally, 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]
41Structural Geology Ch. 5 – Rock testing 4/14/2017The 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 thisCrown Point Limestone:
42Structural Geology Ch. 5 – Rock testing 4/14/2017The 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.
43Structural Geology Ch. 5 – Rock testing 4/14/2017The 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.
44Structural Geology Ch. 5 – Rock testing 4/14/2017The viscosity is the ratio of maximum shear stress to shear strain rate, which is the slope of the line on the graph