The stresses that cause deformation
Understand "stress calculations”
Spend some time with these calculations to convince yourself
that stress on a given plane resolves itself into a single stress
tensor.
Stress (s) = force/unit area
s = F/A
Stress
Goals
1) Interpret the stresses responsible for deformation.
2) Describe the nature of the forces that cause the
stresses.
3) Understand the relations between stress, strain and
rock strength.
Describing stress and force is a mathematical
exercise.
Responses to Stresses
1) Folding
2) Brittle faults
3) Ductile shear zones
4) Joints
Force
Force: changes in the state of rest or motion of a body.
Only a force can cause a stationary object to move or change
the motion (direction and velocity) of a moving object.
force = mass x acceleration, F = ma,
mass = density x volume, m = rV,
therefore, r = m/V,
Weight is the magnitude of the force of gravity (g) acting upon a
mass.
The newton (N) is the basic (SI) unit of force.
1 newton = 1 kg meter/sec2
1 dyne = 1g cm/sec2 so 1 N = 105 dyne
1 pascal = newton/m2
Forces as Vectors
Force is a vector - it has magnitude and direction.
Vectors can be added and subtracted using vector
algebra. We can evaluate vectors in order to
determine whether the forces on a body are in
balance.
Load
Force
Units of Stress
1 newton = 1 kg meter/sec2 = this is a unit of force
1 pascal = 1 newton/m2
= unit of stress
• 1 newton is about 0.224 809 pounds of force
•1 pascal is about 0.020 885 lb/ft2, thus pressure is measured
in kPa
• 1 kPa = 0.145 lb/in2
• 9.81 Pa is the pressure caused by a depth of 1mm of water
Stress on a 2-D plane:
 Normal stress act
perpendicular to the plane
 Shear stress act along
the plane.
 Normal and shear
stresses are perpendicular
to one another
Stress (s)
Stress is force per unit area:
s = F/A
Relations between F and s
(a) Fn and Fs and angle q with
top and bottom surface. EF is
trace of plane, ABCD is cube
with ribs of length AG.
Magnitudes of vectors Fs and
Fn is function of angle q
Fn = F cos q, Fs = F sin q
(b) The magnitude of normal
and shear stresses is function
of angle q and the area,
sn= s cos2q
ss = s sin2q
Stress ellipsoid
A point represents the intersection of
an infinite number of planes and
stresses on these planes describe
an ellipse.
In 3-dimensions, the ellipsoid is
defined by three mutually
perpendicular principal stresses
(s1]> s2 > s3).
These three axes are normal to the
principal
Stress ellipsoid
What is important about the
principal stresses (s1 > s2 > s3)?
The axes are perpendicular to each
other.
They do not contain shear stresses
The state of stress of any body is
described by the orientation and
magnitude of the principal stresses.
Components of stress
 Three normal stresses
 Components parallel shear
stresses
 Reference system x, y, z
Stresses
1) Normal stress
Geology sign conventions
Compressive stress is + (positive)
Positive or negative
Tensional stress is – (negative)
2) Shear stress
Positive or negative
Clockwise shear stress is – (negative)
Counter clockwise shear stress is + (positive)
Stress State
If the 3 principal stresses are equal in magnitude = isotropic stress
Here the state of stress is represented by a sphere, not an ellipsoid.
If the principal stress are unequal in magnitude = anisotropic stress
Here the greatest stress is called s1
The intermediate stress, s2 and minimum stress is called s3
s1 > s2 > s3
As a geologist, what is it called
if all three principal stresses are
equal?
s1 = s2 = s3
Hydrostatic Stress
If we calculate stress vectors within
a point of a hydrostatic stress field,
we find that the stress vectors have
the same value. Each stress vector
is oriented perpendicular to the
plane.
All stress vectors are normal
vectors, they have no shear stress
components.
Equal stress magnitudes in all
directions. Dive into a pool. All
stresses have the same values.
Hydrostatic stress = all principal
stresses in a plane are equal in all
directions. No shear stresses!
Lecture outline
1. Overview of stress
2. Minimum and maximum stress
3. Types of stress on a plane
a. Normal stress
b. Shear stress
3. Mean stress
4. Differential stress
5. Deviatoric stress
6. Hydrostatic state of stress
7. Stress and the Mohr circle
Problem set outline
1. Apparent dip
2. Angle between lines
3. Angle between planes
Stress on a dipping plane in
the Earth’s crust
2 components
Normal stress & Shear stress
sn = s cos2q
ss = s sin2q
Review sign conventions
for normal and shear
stresses
We resolve stress into two components
Normal stress, sn and the component that
is parallel to the plane, shear stress, ss
1) Normal compressive stresses tend to
inhibit sliding along the plane and are
considered positive if they are compressive.
2) Normal tensional stresses tend to
separate rocks along the plane and values
are considered negative.
3) Shear stresses tend to promote sliding
along the plane, labeled positive if its rightlateral shear and negative if its left-lateral
shear.
Squeeze a block of clay
between two planks of wood
AB, trace of fracture plane that
makes an angle q with s3.
The 2-D case is simple, since
s2 = s3 (atmospheric pressure)
Important: What is angle q?
Mohr Stress Diagram
a)This give us a useful picture or
diagram of the stress equations.
b) They solve stress equations on
page 49 (Eqs 3.7 and eq. 3.10)
c) Plot sN versus sS
d) Rearrange Eqs. 3.7 and 3.10 and
square them yields
[sn – ½(s1 + s2]2 + ss2 = [½ (s1 – s32 )]
form (x –a)2 + y2 = r2
Important: What is angle q?
In Mohr space, we use 2q!
Mohr Stress Diagram
a) Mohr circle radius = ½(s1 – s3] that is centered on ½(s1 + s3] from the
origin.
b) The Mohr circle radius, ½(s1 - s2] is the maximum shear stress ss max.
c) The stress difference (s1 – s3), called differential stress is indicated by sd.
Mohr Stress Diagram
Mohr circle:
sn on x-axis
ss on y-axis.
Maximum principal stress (s1) and minimum stress (s3) act on plane P that
makes an angle q with the s3 direction.
In Mohr space, we plot s1 and s3 on sn-axis
These principal stress values are plotted on the sn-axes because they are
the normal stresses acting on plane P.
The principal stresses always have zero shear stress values (ss = 0).
Mohr Stress Diagram
Remember,
sn,p = ½(s1 + s3] + ½(s1 - s3] cos 2q
ss,p = ½(s1 - s3] sin 2q
Construct a circle thought points s1 and s3 with 0, the midpoint, at ½(s1 + s3)
as the center with radius, ½(s1 - s3].
Now draw a line OP, so that angle POs1 is equal to 2q – confusing step, plot
twice the angle q, which is the angle between the plane and s3.
Remember we measure 2q from the s1 side on the sn-axis.
We can read the values of sn,p along the sn-axis, and ss,p along ss-axis for
our plane P.
Mohr Stress Diagram
When the principal stress magnitudes
change w/o differential stress, the
Mohr circle moves along the sn-axis
without changing ss
How is this achieved?
Suggest geologic examples?
Mohr Stress Diagram
When the principal stress magnitudes
change w/o differential stress, the
Mohr circle moves along the sn-axis
without changing ss
1) Change confining pressure (Pc). Increase air pressure on our clay experiment,
or carry the experiment underwater.
2) Burial of rocks changes confining pressure. Which way along the sn-axis?
3) Exhumation of rock changes confining pressure. Again, in what direction along
the sn-axis?
Problem set #1.
Handouts in class and go online
for additional graph paper in
Mohr space
Various states of stress
Uniaxial compression, two of the three principal stresses are zero.
Hydrostatic stress, a single point on the Mohr circle that lies on the x-axis. All
normal stresses are the same, and no shear stresses.
Various states of stress
Triaxial stress, all three principal stresses are different.
Biaxial stress, all three principal stresses are non-zero, but two of the principal
stresses have the same value. Typical stress ellipse (plane stress).
Mean stress and deviatoric stress
Because a body’s response to
stress, we subdivide the stress
into two components, mean and
deviatoric stress.
Mean stress = [s1 + s2 + s3]/3 or
sm
In 2-D, [s1 + s3]/2
Deviatoric stress is the
difference between the mean
stress and total stress. stotal =
smean + sdev
smean is often called the
hydrostatic component (s1 =
s2 = s3 )
Lithostatic
pressure (Pl).
For rocks at depth, we use lithostatic pressure.
Consider a rock at 3 km depth. Lithostatic pressure F (weight of rock of overlying
column).
Pl = r x g x h if r (density) = 2700 km/m3, g (gravity) = 9.8 m/s2 and h (depth) is 3000 m,
we get:
Pl = 2700 x 9.8 x 3000 = 79.4 x 206 Pa ~ 80 Mpa
For every km in the Earth’s crust, the lithostatic pressure increases 27 Mpa.
The lithostatic pressure is equal in all directions (isotropic stress),
[s1 = s2 = s32 ]
Lithostatic
pressure (Pl).
So we divide the rocks state of stress into an isotropic (lithostatic/hydrostatic)
and an anisotropic (deviatoric).
Isotropic stresses act equally on all directions, resulting in a volume change of
the rock – increase water pressure on a human, or air pressure on take-off
or landing.
Deviatoric stress, changes the shape of the body. The difference between
isotropic stress and additional stress from tectonic forcing.
Measuring Stress
Present day stress
Difficult to measure
 EQ focal mechanisms
 Bore-hole breakouts
 in situ measurements
in situ borehole measurements of sd
(s1 –s3) with depth.
Stress in the Earth
World stress map and topography showing maximum horizontal stress.
Stress in the Earth
Generalized pattern based on stress trajectories for individual plates.
Stress and strength
at depth
Strength – the ability of a material to
support different stress
Maximum stress before a rock fails
Strength curves: differential stress
magnitude versus depth.
A. Regional with low geothermal gradient
B. Regional high geothermal gradients
Give some geologic examples?
This is important and will be on
exam 1!
Stress and strength
at depth
Stress and strength
at depth