Geology 5372 Stress Review

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Geology 5372Stress Review
Department of Geology University of Texas at
Arlington
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Geology 5372Stress Review
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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Geology 5372Stress Review

• 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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Geology 5372Stress Review
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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Geology 5372Stress Review
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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Geology 5372Stress Review
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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Stress Analysis
Next problem is to determine stress the
tractions on all the surfaces that pass through a
point
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Stress Analysis
Size of the volume is so small that it is in
equilibrium and S F 0
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Stress Analysis
We have to use a labeling system to label the
surfaces and forces. Tractions are labeled by the
direction of the force and the direction of the
surface normal
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Stress 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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Stress 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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Stress 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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Stress 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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Stress 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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Stress Analysis
Mohr Diagram for Stress Terms Mean Stress
Pressure component Differential Stress Shearing
potential Principal Stress
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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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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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Rock testing
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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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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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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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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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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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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Rock testing
Graph shows the effect of increasing depth
without increasing temperature
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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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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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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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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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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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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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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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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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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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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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Rock testing
Most ductile rocks do not plot as a straight
line. If we plot the data for Yule marble we get
a curved relationship