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Stress II
Stress as a Vector - Traction
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Force has variable magnitudes in different directions (i.e.,
it’s a vector)
Area has constant magnitude with direction (a scalar):
  Stress acting on a plane is a vector
 = F/A
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 = F . 1/A
A traction is a vector quantity, and, as a result, it has both
magnitude and direction
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or
These properties allow a geologist to manipulate tractions following
the principles of vector algebra
Like traction, a force is a vector quantity and can be
manipulated following the same mathematical principals
Stress and Traction
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Stress can more accurately be termed "traction."
A traction is a force per unit area acting on a
specified surface
This more accurate and encompassing definition of
"stress" elevates stress beyond being a mere vector,
to an entity that cannot be described by a single pair
of measurements (i.e. magnitude and orientation)
"Stress" strictly speaking, refers to the whole
collection of tractions acting on each and every
plane of every conceivable orientation passing
through a discrete point in a body at a given
instant of time
Normal and Shear Force
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Many planes can pass through a point in a rock body
Force (F) across any of these planes can be resolved into two
components: Shear stress: Fs , & normal stress: Fn, where:
Fs = F sin θ
Fn = F cos θ
tan θ = Fs/Fn
Smaller θ means smaller Fs
Note that if θ =0, Fs=0 and all force is Fn
Normal and Shear Stress
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Stress on an arbitrarily-oriented plane through a point, is
not necessarily perpendicular to the that plane
The stress () acting on a plane can be resolved into two
components:
Normal stress (n)
 Component of stress perpendicular to the
plane, i.e., parallel to the normal to the plane
Shear stress (s) or 
 Components of stress parallel to the plane
Normal and Shear Stress
Stress is the intensity of force
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Stress is Force per unit area
 = lim dF/dA when dA →0
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A given force produces a large stress when
applied on a small area!
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The same force produces a small stress when
applied on a larger area
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The state of stress at a point is anisotropic:
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Stress varies on different planes with different
orientation
Geopressure Gradient dP/dz
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The average overburden pressure (i.e., lithostatic P) at the
base of a 1 km thick rock column (i.e., z = 1 km), with
density (r) of 2.5 gr/cm3 is 25 to 30 MPa
P = rgz
[ML -1T-2]
P = (2670 kg m-3)(9.81 m s-2)(103 m)
= 26192700 kg m-1s-2 (pascal)
= 26 MPa
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The geopressure gradient:
dP/dz  30 MPa/km  0.3 kb/km (kb = 100 MPa)
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i.e. P is  3 kb at a depth of 10 km
Types of Stress
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Tension: Stress acts  to and away from a plane
 pulls the rock apart
 forms special fractures called joint
 may lead to increase in volume
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Compression: stress acts  to and toward a plane
 squeezes rocks
 may decrease volume
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Shear: acts || to a surface
 leads to change in shape
Scalars
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Physical quantities, such as the density or
temperature of a body, which in no way
depend on direction
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are expressed as a single number
e.g., temperature, density, mass
only have a magnitude (i.e., are a number)
are tensors of zero-order
have 0 subscript and 20 and 30 components in
2D and 3D, respectively
Vectors
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Some physical quantities are fully specified by a
magnitude and a direction, e.g.:
Force, velocity, acceleration, and displacement
Vectors:
 relate one scalar to another scalar
 have magnitude and direction
 are tensors of the first-order
 have 1 subscript (e.g., vi) and 21 and 31
components in 2D and 3D, respectively
Tensors
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Some physical quantities require nine numbers
for their full specification (in 3D)
Stress, strain, and conductivity are examples of
tensor
Tensors:
 relate two vectors
 are tensors of second-order
 have 2 subscripts (e.g., ij); and 22 and 32
components in 2D and 3D, respectively
Stress at a Point - Tensor
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To discuss stress on a randomly oriented
plane we must consider the three-dimensional
case of stress
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The magnitudes of the n and s vary as a
function of the orientation of the plane
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In 3D, each shear stress, s is further resolved
into two components parallel to each of the
2D Cartesian coordinates in that plane
Tensors
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Tensors are vector processors
A tensor (Tij) such as strain, transforms an
input vector Ii (such as an original particle line) into
an output vector, Oi (final particle line):
Oi=Tij Ii (Cauchy’s eqn.)
e.g., wind tensor changing the initial velocity vector
of a boat into a final velocity vector!
|O1|
|O2|
|a
= |c
b||I1|
d||I2|
Example (Oi=TijIi )
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Let Ii = (1,1) i.e, I1=1; I2=1
and the stress Tij be given by:
|1.5 0|
|-0.5 1|
The input vector Ii is transformed into the
output vector(Oi) (NOTE: Oi=TijIi)
| O1 |=| 1.5
| O2 | | -0.5
0||I1| = |1.5
1||I1|
|-0.5
0||1|
1||1|
Which gives:
O1 = 1.5I1 + 0I2 = 1.5 + 0 = 1.5
O2 = -0.5I1 + 1I2 = -0.5 +1 = 0.5
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i.e., the output vector Oi=(1.5, 0.5) or:
O1 = 1.5 or
|1.5|
O2 = 0.5
|0.5|
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Cauchy’s Law and Stress Tensor
Cauchy’s Law: Pi= σijlj (I
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& j can be 1, 2, or 3)
P1, P2, and P3 are tractions on the plane parallel to the three
coordinate axes, and
l1, l2, and l3 are equal to cosa, cosb , cosg
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direction cosines of the pole to the plane w.r.t. the coordinate
axes, respectively
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For every plane passing through a point, there is a unique
vector lj representing the unit vector perpendicular to the
plane (i.e., its normal)
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The stress tensor (ij) linearly relates or associates an output
vector pi (traction vector on a given plane) with a particular
input vector lj (i.e., with a plane of given orientation)
Stress tensor
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In the yz (or 23) plane, normal to the x (or 1) axis: the normal
stress is xx and the shear stresses are: xy and xz
In the xz (or 13) plane, normal to the y (or 2) axis: the
normal stress is yy and the shear stresses are: yx and yz
In the xy (or 12) plane, normal to the z (or 3) axis: the normal
stress is zz and the shear stresses are: zx and zy
Thus, we have a total of 9 components for a stress acting
on a extremely small cube at a point
|xx
xy
xz |
ij = |yx
yy
yz |
|zx
zy
zz |
Thus, stress is a tensor quantity
Stress tensor
Principal Stresses
The stress tensor matrix:
| 11 12
13 |
ij = | 21 22
23 |
| 31 32
33 |
 Can be simplified by choosing the coordinates so that they are
parallel to the principal axes of stress:
| 1
0
0 |
ij = | 0
2
0 |
|0
0
3 |
 In this case, the coordinate planes only carry normal stress;
i.e., the shear stresses are zero
 The  1 , 2 , and  3 are the major, intermediate, and minor
principal stress, respectively
 1>3 ; principal stresses may be tensile or compressive
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Stress Ellipse
State of Stress
Isotropic stress (Pressure)
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The 3D stresses are equal in magnitude in all directions;
like the radii of a sphere
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The magnitude of pressure is equal to the mean of the
principal stresses
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The mean stress or hydrostatic component of stress:
P = (1 + 2 + 3 ) / 3
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Pressure is positive when it is compressive, and negative
when it is tensile
Pressure Leads to Dilation
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Dilation (+ev & -ev)
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Volume change; no shape change involved
We will discuss dilation when we define strain
ev=(v´-vo)/vo = dv/vo [no dimension]
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Where v´ & vo are final & original volumes,
respectively
Isotropic Pressure
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Fluids (liquids/gases) such as magma or water, are
stressed equally in all directions
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Examples of isotropic pressure:
 hydrostatic, lithostatic, atmospheric
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All of these are pressures (P) due to the column
of water, rock, or air, with thickness z and density
r; g is the acceleration due to gravity:
P = rgz