Structural Geology
Force and Stress - Mohr Diagrams,
Mean and Deviatoric Stress, and the
Stress Tensor
Lecture 6 – Spring 2016
1
Graphical Analysis
• Another way of analyzing stress can be done using
graphical methods to solve for σn and σs
• The technique is called a Mohr’s Circle Diagram
for Stress
• Named after Christian Otto Mohr (1835-1918), a
German civil engineer
2
Mohr’s Circle Diagram
• A Cartesian (i.e. X-Y) plot of σs versus σn
that graphically solves the equations for
normal and shear stress acting on a plane
within a stressed body
• σn is plotted on the horizontal axis, and σs
on the vertical axis
• The following slides show the construction
of a Mohr’s Circle
3
Mohr Circle Construction 1
• The construction for a plane P
that makes an angle θ with the
σ3 direction is as follows:
 A. Lay off a distance equal to σ3
from the origin in the positive
direction of σn
• Mark the point, and label it σ3
Figure 3.8, text
4
Mohr Circle Construction 2
• B. Lay off a distance equal to σ1
in the positive direction of σn
 Mark the point, and label it σ1
• C. Construct a circle through
points σ1 and σ3
 The center of the circle is located
at ½ (σ1 + σ3), and the radius is ½
(σ1 - σ3)
Figure 3.8, text
5
Mohr Circle Construction 3
• D. Draw radius OP such that
POσ1 equals 2θ
 We plot twice the angle between
the plane and the σ3 axis, but in a
counterclockwise sense
 (Other conventions exist; be
careful if you read similar material
in other literature)
Figure 3.8, text
• The diagram is now complete
6
Normal and Shear Stress
Components
• We can read the value of
σn,P along the σn axis
(the σn is often referred
to simply as the σ axis,
and the value of σs,P
along the σs axis, which
is also called the τ axis
Figure 3.8, text
7
Information from Mohr Diagrams
• Examining the completed drawing,
figure 3_8, we see that:
 σn,P = ½ (σ1 + σ3) + ½ (σ1 - σ3) C
cos2θ and
 σs,P = ½ (σ1 - σ3) C sin2θ
• Note: cos2θ = - sin (2θ – 90º)
• sin2θ = cos (2θ - 90º)
8
More Information from Mohr
Diagrams
• Figure 3.9, text
• We can also see that
there are two planes
oriented at θ and its
complement, (90Eθ) which have equal
values of σs but
different values of σn
(Points P1 and P2)
9
More Information from Mohr
Diagrams
• There are also two
planes with equal
values of σn but
with shear stresses
of opposite sign
(Points P2 and P3)
Figure 3.9, text
10
Plane Orientations and Points
 For every orientation of a plane as defined
by the  θ there is a corresponding point on
the circle
 The coordinates of the point give the normal
and shear stresses on the plane
• If we do a tension experiment (σ1 = σ2 = 0,
and σ3 < 0) the center of the circle will be
on the negative side of the origin
11
Stress Difference
• We can also see that the shear stress will be
at a maximum when  θ = 45E
• Then 2θ = 90E, and σs = ½ (σ1 - σ3)
• We can use the term stress difference (σd)
for (σ1 - σ3)
• Thus,
 σd = 2 σs
12
3-D Mohr Diagrams
• Three dimensional Mohr circles are plotted
in a similar fashion, except now we must
plot σ1, σ2 and σ3 along the σn axis
• Figure 3-11 shows an example
• There are three individual circles, (σ1 - σ2),
(σ1 - σ3), and ( σ2 - σ3)
13
Triaxial Stress
• Figure 3_11a shows
the triaxial case, when
no value of the
principal stress equals
zero
14
Biaxial Stress
• Figure 3_11b shows
the biaxial case, when
one principal stress
value = 0
15
Uniaxial Compression
• Uniaxial compression,
where σ1 > 0, and σ2 =
σ3 = 0, is shown in
Figure 3_11c
16
Hydrostatic Stress
• The last case, figure
3_11d, is for
hydrostatic stress
• Since all three
principal stresses are
equal, the diagram
reduces to a point
17
Mean Stress
• σm = (σ1 + σ2 + σ3)/3
• σtotal = σm + σdev
• Mean stress is often referred to as hydrostatic
component of stress, since hydrostatic stress is
equal in all directions
• Another name is the hydrostatic pressure.
Hydrostatic stress is isotropic
• Deep within the earth, we use the term lithostatic
pressure, denoted Pl, for the isostatic component
18
Load Pressure Formula
• As we have seen,
 Pl = ρgh
• At depth, the lithostatic stress is usually
orders of magnitude greater than anisotropic
differential stresses
• Thus, deviatoric stress, which is anisotropic,
might seem to be of little consequence
19
Deviatoric Stress
• σtotal = σm + σdev
• Deviatoric stress deforms the body, and is
responsible for a shape and volume change
in the rock
• In structural geology, it is common to
measure the shape change of a body
• This is the strain
20
Tensor Rank
• We can represent the stress in terms of a
second-order tensor, as has previously been
indicated
• The rank indicates the number of subscripts
the quantity has
• Each subscript ranges in value from one to
three, since there are three physical
directions
21
σij
• Second-order stress tensor
22
Decomposition in Mean and
Deviatoric Stress
• This notation can be used for decomposition into
mean and deviatoric stress, as follows:
23
Deviatoric Component
• When decomposed in this way, we see that
the shear stresses appear only in the
deviatoric component
• A change in reference frame (a rotation)
will change the components of the stress
tensor, but such changes are much easier to
handle in tensor notation than using
ellipsoids
24
Stress Trajectories
• One method of representing the stress field
is to plot the position of a selected stress
vector, such as σ1, at a number of points,
and then connect the heads of the vectors
• This gives a series of lines, called stress
trajectories
• A second set of lines representing another
principal stress vector can also be drawn
25
Stress Trajectory Image
26
Determination of Stress
• Differential stress may be determined in a
number of ways
• Determining the absolute stress is much
harder
• All methods of measuring differential stress
are applicable only to the upper crust, yet
we are often interested in the lower crust, or
mantle of the earth
27
Time
• Stress measurements also apply only to today
• Determining stress states in the past (paleostress)
is limited to the analysis of fault and fracture date,
and to microstructural studies (changes in grain
size, twinning of susceptible minerals)
• The fault data is limited to restricted to the upper
crust, were we can measure fault displacements
• Microstructural data can be obtained using various
remote sensing measurements, such as electrical
currents generated by squeezing olivine crystals in
the upper mantle (piezoelectric measurements) 28
Global Stress Summary
Figure 3.15a in text
29
Plate Stress Trajectories
• Figure 3.15b in text
30
Stress Limits
• Differential stress cannot increase without
bounds
 Modern stress measurements often give results
in the 50-150 MPa range
 If differential stress increases too much, we
discover the strength of the rock
 Strength is the ability of a material to support
differential stress.
 If the differential stress exceeds the strength,
the rock will fail
31
Failure
• Failure may occur in one of two ways:
 A. Rupture (brittle behavior)
 B. Flow (plastic behavior)
32
Effect of Geothermal
Gradient
• Region of low geothermal
gradient (Precambrian
Shield)
• Region of high geothermal
gradient (continental rift
region)
33