Structural Geology 3443 Ch' 3 Dynamic Analysis

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Structural Geology (3443)Ch. 3 Dynamic
Analysis
Department of Geology University of Texas at
Arlington
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Structural Geology (3443)Ch. 3 Dynamic
Analysis
Motion and deformation are explained and
understood in terms of forces, a property of the
universe that produces a change in motion or has
the potential to. Geologic deformation features
cannot be explained without understanding the
forces involved.
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Structural Geology (3443)Ch. 3 Dynamic
Analysis
Newton was the first to make the relation between
forces and change in motion and defined two types
of forces Body forces (gravity) G(M1M2)/D2
acting throughout space Surface Forces Ma M
dv/dt acting on a surface, either an external
surface (boundary) or an internal one.
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Structural Geology (3443)Ch. 3 Dynamic
Analysis
Although both Body forces (gravity) and Surface
forces have the same effect on a mass and seem to
be identical. Einstein defined gravitational body
forces as the result of distortions in space-time
produced by the presence of a Mass.
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Structural Geology (3443)Ch. 3 Dynamic
Analysis
Forces are vectors, one type of an entity called
tensors. Scalar quantities (a zero order tensor)
require just one number to define them, like
temperature. Vectors (a first order tensor) have
magnitude (a scalar) and direction and require
three numbers and a coordinate system to define
them. Second order tensors (strain) require 9
numbers and a coordinate system to define them.
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Structural Geology (3443)Ch. 3 Dynamic
Analysis
Internal and external forces Forces on the
boundary of a material are referred to as
External. It is only these external forces and
body forces that produce change in motion of
rigid bodies If a material is not rigid, both
external and internal forces (on real or
artificial interior surfaces) must be taken into
account because they may cause deformation in
addition to motion of the material as a whole.
Interior forces are generated by external forces
and body forces and are always present.
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Structural Geology (3443)Ch. 3 Dynamic
Analysis
Torque Torque is a vector that can produce
rotational motion. It is defined as the product
of the force vector and the perpendicular
distance between the center of mass and the
force.
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Structural Geology (3443)Ch. 3 Dynamic
Analysis
Equilibrium the condition of no change in motion.
Either the object is at rest or moves at constant
linear and angular velocity. For equilibrium to
occur, all the forces (body, external, internal
and torques) must sum to zero.
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Structural Geology (3443)Ch. 3 Dynamic
Analysis

• Forces are vectors, and surface forces are
  usually resolved into components
• A normal component perpendicular to the surface
  on which it acts
• A Shearing component parallel to the surface
• If the normal component tends to push on the
  surface, it is in compression. If it tends to
  pull, it is in tension.

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Structural Geology (3443)Ch. 3 Dynamic
Analysis
Example Calculation of a surface force at the
base of the lithosphere. g acceleration of
gravity 9.8 M/s2 r density of lithosphere
3.0 x 103 kg/M3 H depth of lithosphere 100km
105 M V volume A Area 1 M2 and 100M2 Use
SI units Newtons, meters, seconds
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Structural Geology (3443)Ch. 3 Dynamic
Analysis

• Equations
• F Ma Mg
• M/V
• V HxA
• What is the force on a horizontal internal
  surface at the base of the lithosphere 1M2 and
  100M2 in area? Why are they different?

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Structural Geology (3443)Ch. 3 Dynamic
Analysis

• Additional Questions
• Imagine a cube of lithosphere 1M on a side whose
  center of mass is at 100km. The previous
  calculation was the force on the horizontal
  surface through the center of the cube.
• Is the cube in equilibrium?
• Would the force on the top and bottom of the cube
  be more or less than the force on the surface
  through the center?
• Would the force on the sides of the cube be more
  or less than the force on the surface through the
  center?
• What would be the force on a surface inclined 45o?

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Structural Geology (3443)Ch. 3 Dynamic
Analysis
Stress, Pressure and Traction Traction The Force
acting on a surface divided by its area Stress
all the tractions acting at a point in a
material. The traction vectors form the radii of
the stress ellipse Pressure Isotropic stress
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Structural Geology (3443)Ch. 3 Dynamic
Analysis
Problem Calculate the traction on the base of
the lithosphere for the 1 and 100 M2 area.
Express your results in Pascal's, mega Pascal's
and gigapascals
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Ch. 3 Dynamic Analysis
Next problem is to determine stress the
tractions on all the surfaces that pass through a
point
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Ch. 3 Dynamic Analysis
Size of the volume is so small that it is in
equilibrium and S F 0
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Ch. 3 Dynamic Analysis
Size of the volume is so small that it is in
equilibrium and S F 0
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Structural Geology (3443)Ch. 3 Dynamic
Analysis
We have to use a labeling system to label the
surfaces and forces. Surface areas are labeled by
the direction of their normals. Those parallel to
x are labeled Ax. Those parallel to y are labeled
Ay. Others are labeled Ap.
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Ch. 3 Dynamic Analysis
Forces acting on the Ax surface in the x
direction are labeled Fxx. Tractions labeled
Txx What is Force in x direction on the Ay
surface labeled?
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Ch. 3 Dynamic Analysis
Assuming equilibrium we can now sum the forces
SFx 0 TxxAx TpxAp SFy 0 TyyAy
TpyAp But Ax -Ap Cos(q) And Ay -Ap Sin
(q) (Rule for signs) So 0 -TxxAp Cos(q)
TpxAp 0 -TyyAp Sin (q) TpyAp And Tpx
TxxCos(q) Tpy TyySin (q) T2p T2xxCos2(q)
T2yySin2(q)
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Ch. 3 Dynamic Analysis
Tp (T2xxCos2(q) T2yySin2(q))1/2
This is the equation for an ellipse (ellipsoid).
Tp are the radii Txx is the major axis and Tyy
is the minor axis. The radii (Tp) show all the
tractions acting on all the surfaces passing
through the center of the ellipse (ellipsoid).
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Ch. 3 Dynamic Analysis
Tp (T2xxCos2(q) T2yySin2(q))1/2
If Txx Tyy, then the ellipse degenerates into
a circle (sphere) and is called pressure. The
radii (Tp) still show all the tractions acting on
all the surfaces passing through the center.
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Ch. 3 Dynamic Analysis
However, the components of the traction, Tp,
parallel and perpendicular to the surface p are
of more interest mechanically. Need to find Tpn
and Tps in terms of Txx and Tyy.
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Ch. 3 Dynamic Analysis
Theta q is angle between x axis and surface
normal. So, Tpn TpxCos(q)TpySin(q) Tps
Tpxsin(q)-TpyCos(q) However, from
equilibrium, Tpx TxxCos(q) Tpy TyySin (q)
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Ch. 3 Dynamic Analysis
So, Tpn TxxCos2(q) TyySin2(q) Tps
TxxCos(q)sin(q) - TyySin(q)Cos(q)
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Ch. 3 Dynamic Analysis
These equations plot as a circle in stress space,
with Tpn on x axis and Tps on y! This leads to
the Mohr Diagram for Stress. Tpn TxxCos2(q)
TyySin2(q) Tps
TxxCos(q)sin(q) - TyySin(q)Cos(q)
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Ch. 3 Dynamic Analysis

• Tpn TxxCos2(q) TyySin2(q)
• Tps (Txx-Tyy)Sin(q)Cos(q)
• Conventional notation.
• Normal stress is usually symbolized by s
• Shearing Stress by t
• Maximum stress (long ellipse axis) is s1
• Minimum Stress (short axis) is s2
• Making these substitution, the equations above
  become
• s s1Cos2(q) s2Sin2(q)
• t (s1 - s2)Sin(q)Cos(q)

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Ch. 3 Dynamic Analysis

• s s1Cos2(q) s2Sin2(q)
• t (s1 - s2)Sin(q)Cos(q)
• Finally, these equations can be put in the form
  of parametric equations for a circle substituting
  these trig identities
• ½ Sin(2q) Sin(q)Cos(q)
• ½(1Cos(2q) Cos2(q)
• ½(1-Cos(2q) Sin2(q)
• To get,
• s 1/2(s1 s2) 1/2(s1 - s2)Cos(2q)
• t 1/2(s1 - s2) Sin(2q)

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Ch. 3 Dynamic Analysis

• Mohr Diagram for Stress
• s 1/2(s1 s2) 1/2(s1 - s2)Cos(2q)
• t 1/2(s1 - s2) Sin(2q)

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Ch. 3 Dynamic Analysis
Mohr Diagram for Stress Terms Mean Stress
Pressure component Differential Stress Shearing
potential Principal Stress
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Ch. 3 Dynamic Analysis
3-D Stress
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Structural Geology (3443)Ch. 3 Dynamic
Analysis
Problem Go back to the calculation of the
traction on the horizontal surface at base of the
lithosphere. Assume that calculated vertical
traction is the maximum principal stress, s1. Let
the Minimum Principal stress, s3, be 75 of s1
and oriented E-W. Let the intermediate principal
stress, s2, be 90 of s1 and oriented N-S. Plot a
Mohr diagram for this stress state at the base of
the lithosphere in gigapascals. What is the
normal and shearing stress on a surface striking
N-S and dipping 45 to the East?
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Structural Geology (3443)Ch. 3 Rheology
Relationship between stress and strain
Rheology 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. 3 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 structures and
tectonic features.
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Structural Geology Ch. 3 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. 3 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. 3 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. 3 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. 3 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. 3 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. 3 Results of Rock
testing
Differential stress-strain graph for limestone at
a confining pressure of 103Mpa (thats about 3.9
km below the surface do the calculation)
Specimen at room temperature 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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Structural Geology Ch. 3 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.
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Structural Geology Ch. 3 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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Structural Geology Ch. 3 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. After that, is takes smaller and
smaller loads to produce strain until the
specimen ruptures (fractures) and strain will
increase with little load. Fracturing is called
brittle behavior in contrast to ductile.
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Structural Geology Ch. 3 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 megapascals to depth
is Depth (in m) megapaschals/25,480 1 Mpa 39
M 10 Mpa 390M 100 Mpa 3,900M Etc.
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Structural Geology Ch. 3 Results of Rock
testing
Graph shows the effect of increasing depth
without increasing temperature
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Structural Geology Ch. 3 Results of Rock
testing
What is the range of depths shown by the graph?
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Structural Geology Ch. 3 Results of Rock
testing
What happens to the yield strength with
increasing confining pressure?
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Structural Geology Ch. 3 Results of Rock
testing
What happens to the peak strength with increasing
confining pressure?
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Structural Geology Ch. 3 Results of Rock
testing
What happens to the rupture strength with
increasing confining pressure?
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Structural Geology Ch. 3 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. 3 Rock testing
This test is similar to the last one. The
confining pressure was 200 Mpa. What is the
simulated depth?
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Structural Geology Ch. 3 Rock testing
However, these 5 tests were run at different pore
pressures under the same 200 Mpa confining
pressure. The effective confining pressure Pc,
is from s s sp. What is the pore pressure in
each of the 5 tests?
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Structural Geology Ch. 3 Rock testing
What happens to the 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. 3 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. 3 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. 3 Rock testing
What happens to the 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. 3 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. 3 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. 3 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. 3 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. 3 Rock testing
In chapter 2 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. 3 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. 3 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. 3 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. 3 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. 3 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. 3 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. 3 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. 3 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. 3 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. 3 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. 3 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. 3 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. 3 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. 3 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. 3 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