Structural Geology 3443 Ch. 5 Rheology

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Structural Geology (3443)Ch. 5 Rheology
Rheology - the functional relationship between
stress and strain. Just like 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 F Ma. Rheology is not so simple,
and there is no general equation that describes
deformation other than s f(e)
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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,
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.
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Structural Geology Ch. 5 Rheology
Design of triaxial testing equipment is shown at
left. Load (stress) 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 independently. Temperature is also
controlled.
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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. Vertical stress is usually the
maximum stress (s1) Stress on the side of the
specimen is the same as the confining fluid
pressure. It is usually the minimum and
intermediate stress since the two side stress
cannot be controlled independently (s2 s3).
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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 radius of the Mohr graph. The
greater the differential stress, the bigger the
Mohr circle, and the greater the amount of
possible shear stress.
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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
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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.
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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.
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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.
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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.
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Ch. 5 Results of Rock testing
Differential stress-strain graph for limestone at
a confining pressure of 103Mpa (thats about 3.9
km below the surface Specimen was at room
temperature
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.
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Ch. 5 Results of Rock testing
The elastic limit at point A is called the yield
strength, and 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.
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Structural Geology Ch. 5 Results of Rock
testing
The 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 is called strain
hardening previous ductile strain adds more
resiliency to the rock.
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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.
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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.
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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 Pa is Pa
depth (in Meters) 24,500 or Depth Pa/24,500
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Structural Geology Ch. 5 Results of Rock
testing
Graph shows the effect of increasing depth
without increasing temperature
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Structural Geology Ch. 5 Results of Rock
testing
What is the range of depths shown by the graph?
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Structural Geology Ch. 5 Results of Rock
testing
What happens to the yield strength with
increasing confining pressure?
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Structural Geology Ch. 5 Results of Rock
testing
What happens to the ultimate strength with
increasing confining pressure?
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Structural Geology Ch. 5 Results of Rock
testing
What happens to the rupture strength with
increasing confining pressure?
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Structural Geology Ch. 5 Results of Rock
testing
At what depth does the rock no longer behave as a
brittle material and becomes ductile?
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Structural Geology Ch. 5 Rock testing
This test is similar to the last one. The
confining pressure was 200 Mpa. And T 24o. What
is the simulated depth?
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Structural Geology Ch. 5 Rock testing
However, these 5 tests were run at different pore
pressures under the same 200 Mpa confining
pressure. The effective confining pressure Pc,
sc sp. What is the pore pressure in each of the
5 tests?
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Structural Geology Ch. 5 Rock testing
What happens to the ultimate (peak) strength as
the pore pressure increases? At what point does
the rock change from ductile to brittle?
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Structural Geology Ch. 5 Rock testing
Compare the two graphs. The right one was
conducted at confining pressures up to 140 Mpa
with zero pore pressure. The left one had a
confining pressure of 200 Mpa with pore pressures
up to 200 Mpa and a depth of 7.8km How would you
describe the effect of pore pressure on
brittle/ductile behavior of rock? Why is
effective pressure more important than confining
pressure or depth?
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Structural Geology Ch. 5 Rock testing
Effect of temperature The graph on the right
shows tests on basalt run at 5 kbar confining
pressure (1 Mpa 10 bars) while varying the
temperature. What is the simulated depth of the
tests? If the temperature gradient is 25oC/km,
what is the simulated depth of the various
temperatures if the surface temperature is 25oC?
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Structural Geology Ch. 5 Rock testing
What happens to the ultimate (peak) and yield
strength as temperature (depth) increases? At
what temperature does the rock stop being
brittle? What is the depth?
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Structural Geology Ch. 5 Rock testing
The graph on the left is limestone run at room
temperature at various confining pressures. The
graphs on the right is basalt at 500 Mpa
confining pressure at various temperatures. Why
is the differential stress so different at
similar temperatures?
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Structural Geology Ch. 5 Rock testing
What is the trend in peak and yield strength as
temp increases at constant confining pressure?
What is the same trend as confining pressure
increases at constant temp? What would be your
prediction about the same trend as depth
increases?
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Structural Geology Ch. 5 Rock testing
What is the trend in brittle behavior as temp
increases? As confining pressure increases? As
depth increases? How would increasing pore
pressure effect this trend?
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Structural Geology Ch. 5 Rock testing
Effect of strain rate. The tests shown at right
were conducted at 5000bars (500 Mpa) confining
pressure and 500oC. About what depth does this
simulate?
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Structural Geology Ch. 5 Rock testing
Effect of strain rate. What happens to the yield
strength as the strain rate decreases?
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Structural Geology Ch. 5 Rock testing
In chapter 4 you calculated the strain in a cross
section which was extended to 51 km from an
initial length of 33 km. Recalculate the
strain. If the deformation took 1 million years,
calculate the strain rate. How does that compare
to the experimental strain rates?
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Structural Geology Ch. 5 Rock testing
Effect of rock type Rocks are so variable that it
is hard to generalize, but siliceous rock with
little pore space are generally
strongest Quartzite, intrusive igneous Weakest
rocks are salts and mudstones Carbonates usually
intermediate in strength
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Structural Geology Ch. 5 Rock testing
Other terms Competent, and incompetent are
imprecise terms that usually refer to strength
and/or ductility. Avoid them and say what you
mean.
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Structural Geology Ch. 5 Rock testing
Now that we have seen how rocks behave
experimentally, its time to generalize this
stress-strain behavior into mathematical models
so it will be possible to calculate, predict and
model rock deformation.
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Structural Geology Ch. 5 Rock testing
Now that we have seen how rocks behave
experimentally, its time to generalize this
stress-strain behavior into mathematical models
so it will be possible to calculate, predict and
model rock deformation.
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Structural Geology Ch. 5 Rock testing
The linear portion of the stress-strain curve is
called the elastic region where s Ee. E is
Youngs Modulus, a constant, that changes with
type of material. Once the material has exceeded
its yield strength, the elastic equation doesnt
apply.
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Structural Geology Ch. 5 Rock testing
Youngs Modulus is the slope of the stress-strain
elastic line, and is a measure of stiffness, not
strength. A material with a high Youngs modulus
is said to be stiffer than a rock with a lower
one, not stronger.
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Structural Geology Ch. 5 Rock testing
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.
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Structural Geology Ch. 5 Rock testing
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 Poissons ratio n
eh/ev
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Structural Geology Ch. 5 Rock testing
Finally, The bulk modulus K is a measure of the
amount of dilation produced by the mean stress
(pressure) (s1s2s3)/3 K(e1e2e3)/3
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Structural Geology Ch. 5 Rock testing
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.
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Structural Geology Ch. 5 Rock testing
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.
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Structural Geology Ch. 5 Rock testing
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.
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Structural Geology Ch. 5 Rock testing
The viscosity is the ratio of maximum shear
stress to shear strain rate, which is the slope
of the line on the graph
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Structural Geology Ch. 5 Rock testing
Most ductile rocks do not plot as a straight
line. If we plot the date for Yule marble we get
a curved relationship
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Structural Geology Ch. 5 Rock testing
If we plot it on semi log paper, it becomes
nearly linear. meaning that stress is a function
of the power of strain
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Structural Geology Ch. 5 Rock testing
Most ductile rocks follow a power law
stress-strain rate relationship.
Q is an activation Energy. T is temperature R the
gas constant N an exponent A a material constant