Structural Geology 3443 Ch. 3 Force and Stress

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Structural Geology (3443)Ch. 3 Force and
Stress
Department of Geology University of Texas at
Arlington
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Structural Geology (3443)Ch. 3 Force and
Stress
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 Force and
Stress
Newton was the first to relate forces to 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 Force and
Stress
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 Force and
Stress
Forces are vectors, one type of an entity called
tensors. Scalar quantities (a zero rank tensor)
require just one number to define them, like
temperature. Vectors (a first rank tensor) have
magnitude (a scalar) and direction and require
three numbers and a coordinate system to define
them. Second rank tensors (stress, strain, the
indicatrix) require 9 numbers and a coordinate
system to define them.
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Structural Geology (3443)Ch. 3 Force and
Stress
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 both external
forces and body forces and are always present.
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Structural Geology (3443)Ch. 3 Force and
Stress
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 Force and
Stress
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 Force and
Stress

• 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 Force and
Stress
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 Force and
Stress

• 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 Force and
Stress

• 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 Force and
Stress
Stress, Pressure and Traction Traction The Force
acting on a surface divided by its area it is
also called stress, but we will usually use the
word traction to mean a particular force
divided by the area it acts on. Stress all the
tractions acting on all the surfaces passing
though a point in a material. The traction
vectors form the radii of the stress
ellipse Pressure Isotropic stress all the
tractions at a point are the same.
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Structural Geology (3443)Ch. 3 Force and
Stress
Problem Calculate the traction on the base of
the lithosphere for the 1 and 100 M2 area.
Express your results in Pascal's (Pa), mega
Pascal's (MPa) and gigapascals (GPa) 1Pa
1Newton/M2
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Ch. 3 Force and Stress
Next problem is to determine stress the
tractions on all the surfaces that pass through a
point. Assume that tractions on two
perpendicular surfaces are known, and the
associated forces are perpendicular to the two
surfaces (sxx syy).
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Ch. 3 Force and Stress
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 Force and
Stress
We have to label the surfaces (A), Stresses (s),
and Tractions (T). Stresses and tractions are
labeled by the surface the act upon and their
direction relative to a reference frame. The
surface is defined by the direction of its
perpendicular.
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Ch. 3 Force and Stress
Stresses acting on the x surface in the x
direction are labeled sxx. Tractions labeled Tpx
act on the p surface in the x direction. The p
surface is defined by q the angle between the x
axis and the surface normal. Areas of each
surface are labeled Ax, Ay, Ap
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Ch. 3 Force and Stress
SFx 0 sxxAx TpxAp SFy 0 syyAy
TpyAp But Ax Ap Cos(q) Ay Ap Sin
(q) Substituting, 0 sxxAp Cos(q) TpxAp 0
syyAp Sin (q) TpyAp Dividing by Ap Tpx
sxxCos(q) Tpy syySin (q) Since T2p T2px
T2py T2p s2xxCos2(q) s2yySin2(q)
Assuming equilibrium we can now sum the forces
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Ch. 3 Force and Stress
Tp (s2xxCos2(q) s2yySin2(q))1/2
This is the equation for an ellipse (ellipsoid).
Tp are the radii sxx is the major axis and syy
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 Force and Stress
Tp (T2xxCos2(q) T2yySin2(q))1/2
If sxx syy, then the ellipse degenerates into
a circle (sphere) and is called pressure. The
radii (Tp sxx syy) still show all the
tractions acting on all the surfaces passing
through the center.
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Ch. 3 Force and Stress
However, the components of the traction, Tp,
parallel and perpendicular to the surface p are
of more interest mechanically. Need to find spn
and sps in terms of sxx and syy.
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Ch. 3 Force and Stress
q is angle between x axis and the P, the surface
normal. So, spn TpxCos(q)TpySin(q) sps
Tpxsin(q)-TpyCos(q) But, equilibrium
equations from previous slide Tpx sxxCos(q)
Tpy syySin (q)
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Ch. 3 Force and Stress
So, spn sxxCos2(q) syySin2(q) sps (sxx
syy) Sin(q)Cos(q)
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Ch. 3 Force and Stress
These equations are usually written with the
following understanding spn sn sps ss sxx
s1 The maximum stress syy s2 The minimum
stress sn s1Cos2(q) s2Sin2(q) ss (s1 -
s2)Sin(q)Cos(q)
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Ch. 3 Force and Stress

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

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Ch. 3 Force and Stress

• sn 1/2(s1 s2) 1/2(s1 - s2)Cos(2q)
• ss 1/2(s1 - s2) Sin(2q)
• Equations for a circle can be written as
• X C R cos (F)
• Y R sin (F)
• So,
• C 1/2(s1 s2)
• R 1/2(s1 - s2)
• F 2q

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Ch. 3 Force and Stress
sn 1/2(s1 s2) 1/2(s1 - s2)Cos(2q) ss
1/2(s1 - s2) Sin(2q)
These equations plot as a circle in stress space,
with sn on the x axis and ss on the y axis This
is called the Mohr Diagram for Stress.
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Ch. 3 Force and Stress
Mohr Diagram for Stress Terms Mean Stress is the
center of the circle Differential Stress is the
radius of the circle Principal Stresses are the
maximum and minimum stresses where the circle
crosses the sn axis.
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Ch. 3 Force and Stress
3-D Stress
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Structural Geology (3443)Ch. 3 Force and
Stress
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 Force and
Stress

• In situ stress measurement is important for both
  academic and engineering purposes
• Earthquake prediction
• Formation of geologic structures
• Mining, tunneling, slope stability
• Methods of Stress measurement, see
  http//www.hydrofrac.com/

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Structural Geology (3443)Ch. 3 Force and
Stress
Earthquakes can also provide information on the
direction of the principle stresses. The
direction of the first motion on the seismograph
indicates whether the motion is toward
(compression) or away (tension) from the
seismograph. Compiling data from many
seismographs the possible orientations of the
fault and the principle stresses can be inferred.
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Structural Geology (3443)Ch. 3 Force and
Stress
This information is usually displayed as
stereographic projections (beach balls) which
show planes cutting the lower hemisphere of an
imaginary sphere.
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Structural Geology (3443)Ch. 3 Force and
Stress
The dark areas are in tension while the white
areas are in compression. The lines between them
are the possible faults.
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Ch. 3 Force and Stress
The area shown is from Eastern California. What
is the stress interpretation?
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Ch. 3 Force and Stress
The map of the Americas shows information on
maximum stress orientation from a variety of
sources.