The concept of stress is
extremely
important
to
understanding why and how
rocks deform and break.
Stresses can be analyzed using
plane geometries, Mohr circle
diagrams, tensors, and lots of
algebra. These methods can
be used separately or in
conjunction to explain how the
structures
that
geologists
observe form.
There is no reason to “stress out” over the
concept of stress. It may involve utilizing
and keeping track of up to nine numbers
algebraically, but its fundamentals are
very straightforward.
Force, Traction, Surface Stress
Force – the push or pull acting on a body
Traction – the force per unit area
Surface stress – a pair of opposite but
equal tractions acting on a surface of
specific orientations
Force
• The push or pull acting on a body.
• Forces are vector quantities having
both a magnitude and direction.
• There are internal and external
forces.
• Internal forces originate inside the
body. They include atomic forces in
crystalline lattices. Internal forces
do not cause motion but do define
material properties.
• External forces originate outside the
body and are important for stresses.
There are two types; body and
surface forces.
• Body forces act on the mass
particles themselves like gravity.
• Surface forces are caused by one
body, or at least part of a body,
acting on another through a shared
surface.
• Two components of force important
– normal (Fn) and shear (Fs). See
Figure 7.3
Traction
• The force intensity or the force per
unit area acting on a plane.
• Calculate by ! = F/A
• The larger the area, the smaller the
traction (inverse relationship).
• Traction at a point calculated by
! = dF/dA
• Traction also has normal and shear
components:
!n = Fn/A and !s = Fs/A
Surface Stress
• A material surface is defined as a
set of material points that are
constrained such that they cannot
accelerate away from the material
points surrounding it. This is
enforced by Newtwon’s second law
– a force acting on a surface must
be opposed by an equal but
opposite force. The same holds for
tractions.
• The surface stress is defined as the
pair of equal and opposite tractions
that act on a surface of specific
orientation.
• Surface stress can also be broken
into normal and shear components.
A Pair of equal but opposite shear
stresses are sometimes referred to
as shear couples.
• One of the most common sign
conventions, which is used in this
class, is the Geologic Mohr Circle
sign convention and is displayed in
the Figure 7.4.
• Compressive normal stresses are
positive and tensile normal stresses
are negative. Counterclockwise
(sinistral) shear stresses are
positive while clockwise (dextral)
shear stresses are negative. Rock is
almost always found in
compression, hence positive values
represent compression. The sign of
the shear stresses is nonunique – it
can be positive or negative
depending on the side you view it
from.
•
Calculate surface stress the same
way you do traction, but correct the
sign using the Mohr convention.
Planes of differing Orientations
• The concept of stress is further
complicated by the orientation of
the planes that it acts on.
• Do not associate stress components
with vector components that define
things like force. Stress combines
the effect of force and area which
are constantly changing according
to the orientation of the surface.
• For a force (W) acting
perpendicular on an inclined plane,
the surface stress equates to:
E’ = W/A’ = Wcos(")/A
where A’ = A/ cos(")
• It is common to find a force acting
at some angle to the inclined plane.
The normal and shear components
of the force can be determined by
F’n = Wcos(") and F’s = Wsin(")
where " is the orientation of the
plane. These are known as the
force transformation equations.
• The normal and shear stresses on
an oriented plane can then be
determined by combining the
equations above resulting in:
!'n = !ncos2(") and
!’s = !nsin(")cos(")
2-D State of Stress at a Point
• The stress ellipse represents the
state of stress at a point because it
represents the surface stresses
acting on all possible orientations
of a plane through that point.
• You can describe the shape of an
ellipse by determining the
magnitude of the major and minor
axes (the radii). These are two
surface stresses that are called the
principal stresses (!1 and !3).
• !1 #!3
• Together the principal stresses act
on the principal planes and parallel
to the principal axes.
• Each surface stress has a normal
and shear component and they are
labeled according to surface in
which they act and the coordinate
to which it is parallel. For example
EPS116 Chapter 7 Summary 2011
1
Stephen Ferencz & Andrew Trautz
Terminology for States of Stress:
Hydrostatic Pressure - simply put,
"ˆ1 = "ˆ 2 = "ˆ 3 = p . All principle
stresses are compressive and equal.
!
!xxleftAx + !xxrightAx + !zxtopAz + !zxbottomAz = 0
•
•
The sum of the moments of the
forces also must be zero.
!xz = -!zx by definition. This
means that for a two dimensional
system, three out of four stress
components are independent (!xx,
!zz, and !xz).
3-D State of Stress at a Point
• The three-dimensional state of
stress at a point is essentially the
same as that explained for the 2-D
case above. There are only two
main differences from that
described already.
• There is an additional principal
stress, !2, such that !1# !2# !3.
• There are a total of six independent
stress components (out of nine
total). The normal stresses are
unique while the shear stresses are
equal but opposite: !xz = -!zx, !xy =
-!yx, and !yz = -!zy
The Mohr Diagram
• The relationship between plane
orientation and the values of
normal/shear stress is difficult to
determine using the stress ellipses.
Instead it is much more convenient
to use Mohr diagrams.
• The horizontal axis contains the
possible values for the normal
stresses while the vertical axis
contains the values for the shear
stresses.
• The Mohr circle completely
represents the state of stress at a
point in terms of the normal and
shear components. Furthermore,
!s = r*sin(2") = (!1 - !3)*
sin(2")/2
Example of a Mohr Circle. The figure
shows how different parts of the circle
are calculated.
•
•
•
•
•
•
•
The principal stresses, !1 and !3,
are the points of intersection of the
Mohr circle with the !n axis.
!
If you know the normal and shear
stresses on two perpendicular
planes, it is possible to plot the
Mohr circle. With the circle, the
surface stresses can be determined
for any other orientation.
Angles are doubled when
converted from physical space to
Mohr diagrams. Angles are
measured counterclockwise from
the major principal stress.
There are two points on the Mohr
circle with the same normal and
shear components, but act on
differently oriented planes.
The maximum shear stress is
located at the top of the Mohr
circle.
Scalar Invariants – the magnitude
of a state of stress is characterized
by its invariants.
Mean normal stress (center of
Mohr circle):
= (!1 + !3)/2
Maximum shear stress (radius of
Mohr circle): r = (!1 - !3)/2
Equations of the Mohr circle – the
normal and shear stresses can
easily be calculated for any
orientation.
!n =
+ r*cos(2") = (!1 + !3)/2
+ (!1 - !3) *cos(2")/2
Uniaxial Stress1) compression: "ˆ1 > "ˆ 2 = "ˆ 3 = 0 , the
only stress applied is a compressive
stress in one direction
2) tension: 0 = "ˆ1 = "ˆ 2 < "ˆ , the only
!
stress applied is a tensile stress in 1
direction.
!
Axial Compression - A uniaxial
compression of magnitude ("ˆ1 # "ˆ 3 ) !"#!
added to a state of hydrostatic stress
("ˆ 2 = "ˆ 3 ).
!
Axial Extension - "ˆ1 = "ˆ 2 > "ˆ 3 > 0 , A
uniaxial tension of magnitude ("ˆ1 # "ˆ 3 )
is added to a hydrostatic stress ("ˆ1 = "ˆ 2 ) $!!
!
Triaxial Stress - "ˆ1 > "ˆ!2 > "ˆ 3 , the
principle stresses are all
! unequal and
can be either positive or negative. The
stresses!plot on the Mohr diagram as
three separate circles.
Pure shear stress - "ˆ1 = #"ˆ 3 and "ˆ 2 = 0 ,
the maximum and minimum shear
principal
stresses are equal but opposite, the
intermediate principle stress is zero.
Also the!normal stress !
on planes of
maximum shear stress is zero.
Deviatoric stress – The deviatoric
normal stress on a plane, "(Dev )n , is
found by subtracting the mean normal
stress component, "n , from any normal
stress component $!The mean normal
!
stress can be given by either:
"ˆ + "ˆ
1) !
"n = 1 3
2
2) " = "xx + "zz !
"n
•
each point represents a different
plane orientation.
!
•
!xx means that the surface stress is
acting on the x plane in the x
direction (normal component). !xz
similarly shows that it is acting on
the x face but in the z direction
(shear component).
A principal plane is any plane in
which the shear stresses are zero.
The normal stresses that are acting
on this plane are therefore the
principal stresses.
As mentioned earlier Newton’s
second law must apply in this case.
For example the forces parallel to
the x-axis must sum to zero:
n
2
The deviatoric shear stresses remain
!
unchanged,
so the Mohr circle remains the
same
! diameter and merely shifts. For the
deviatoric stress in two dimensions, the
center of the Mohr circle is shifted to the
origin of the graph.
EPS116 Chapter 7 Summary 2011
Stephen Ferencz & Andrew Trautz
2
The Stress-Component Matrix
Differential Stress – The differential stress
is simply the difference between the
maximum and minimum principle
stresses, "(Dif ) = "ˆ1 # "ˆ 3 $!
!
Effective stress – The effective normal
(Eff )
%!is found by
!stress on a plane , " n
subtracting the pore fluid pressure in the
rock, p f , from any normal stress
component.! " n . The shear stresses for the
! stress are unchanged so the
effective
Mohr circle does not change diameter.
!
The!effective stress is a result of shifting
the Mohr circle to the left by a value equal
to the pore fluid pressure: " n (Eff ) # "n $ p f $!!
!
Mohr Diagram for 3 Dimensional Stress
!
Three-dimensional stress is plotted on a
Mohr diagram as three Mohr circles. Each
circle is a graph of the surface stress
components on sets of planes that are
parallel to one of the principle axes. All
three principle stresses are plotted as
points on the " n !axis, and each point is
common to two of the Mohr circles.
!
The stress at a point ! !is called the stress
tensor. In two dimensions the stress tensor
is represented by a 2X2 matrix composed
of four scalar components, and for three
dimensions it is represented by a 3x3
matrix composed of nine scalar
components.
! !ˆ
0 0 $
# 1
&
0 &!
Ex. # 0 !ˆ 2
#
&
#" 0 0 !ˆ 3 &%
!
For both 2x2 and 3x3 matrices the normal
stress components plot on the diagonal of
the matrix, which is referred to as the
“principle diagonal,” and the shear stress
components are plotted off the principle
diagonal.
Each row of the component matrix
contains the three components of one of
the surface stresses. For example, if you
consider the first row of a matrix,
[! 11, ! 12 , ! 13 ] %!you have the normal stress
of the coordinate surface points
in a negative direction
There is a good image on page 183 of
the text, Figure 7.14
In order to resolve the ambiguity of shear
stress sign convention in the Morh circle
in two dimensions we follow this simple
convention: The two dimensional diagram
of the stress components must be
constructed so that there is a clockwise
sense of rotation from the positive
coordinate axis parallel to the largest
normal stress component, toward the
positive coordinate axis parallel to the
smallest normal stress component (Twin
and Moore, 188)
References & Resources
Twiss Moores. Structural Geology-second
edition. 2007.151-191
http://en.wikipedia.org/wiki/Mohr’s_circle
http://en.wikipedia.org/wiki/Stress
and the two shear stresses that act on the
coordinate plane perpendicular to the x1 !
axis.
The Stress Tensor
The components of surface stresses on all
planes must lie inside the largest Mohr
circles and outside the two smaller circles
as depicted by the shaded area in the
above diagram.
With the mean normal stress being
defined by:
!n =
! xx + ! yy + ! zz !ˆ1 + !ˆ 2 + !ˆ 3
=
3
3
A tensor is defined as a mathematical
quantity that can be used to describe the
physical state or physical properties of a
material. It is important to understand that
in geology stress tensors have a different
sign convention than there traction
counterparts. For normal stresses,
compressive stresses are positive and
tensile stresses are negative just the same
as the Mohr circle. However, for shear
stresses the signs can differ from the
Mohr circle components. Positive and
negative tensor convention for shear
forces is:
1) Positive if the shear traction
component on the negative side
of the coordinate surface points
in a positive coordinate direction
2) Negative if the shear traction
component on the negative side
EPS116 Chapter 7 Summary 2011
Stephen Ferencz & Andrew Trautz
3
The concept of stress is
important in understanding
both the brittle and ductile
deformation of rock. Stresses
can be visualized using plane
geometries,
Mohr
circle
diagrams, and tensors. These
methods
can
be
used
separately or in conjunction to
explain
how
geologic
structures are formed.
composed of two tractions that
point in opposite directions.
• Keep the remaining bullets.
Planes of Differing Orientations
Edit sub-title. Change to Stress Across
Planes of Different Orientations
Remove first and second bullet.
Add in:
•
•
Force
An influence that causes an
object to move. It is a vector quantity and
are classified into two types, internal and
external forces. Internal forces determine
the material properties and external forces
produce motion and deformation that are
preserved which then allow geo-scientist
to study.
The two types of external forces
are body forces and surface forces. Body
forces only act on the particle of mass
regardless to its surrounding, such us
gravity. Surface forces is a result from an
action of a body on another body on a
shared surface. In this chapter, we will
focus on surface forces.
Traction
•
•
•
Because surface stress comes in
a pair, if we only know traction
on one surface, we automatically
know the traction on another
surface which together gives us
the stress.
Traction and stress have a
different sign convection system.
This is because stress is
It is important for structural
geologists to know the surface
stress on any plane of any
orientation. This will allow us to
determine which plane(s) is/are
mostly stressed and thus will
become the fracture plane.
Although force and stress are
similar, they are very different in
a sense that stress takes area into
account while force has nothing
associated with area.
Keep the remaining bullets.
2-D State of Stress at a Point
Replace first, second, third and forth
bullet by:
•
A stress ellipse is composed
when surface stresses on all
planes of all possible orientations
through a point are plotted
because the ends of all the
arrows will trace out the ellipse
and the arrows themselves are
the radii.
•
The shape of the stress ellipse
can be described by the
magnitude of principal stresses
which are the major and minor
axes.
No edits.
Surface Stress
Keep first, second and third bullet.
Add after third bullet
Add after the second bullet:
•
Principal stresses (σ1 and σ3).
are defined to be the maximum
and minimum of the surface
stresses acting on any plane
through the point with σ1≥ σ3.
Principal stresses are always
perpendicular to each other.
They act on the principal planes
and are parallel to the principal
axes.
• Keep the remaining bullets.
3-D State of Stress at a Point
•
With σ1 being the maximum
principal stress and σ3 being the
minimum principal stress, σ2 is
called the intermediate principal
stress. All three principal stresses
are perpendicular to each other.
•
Instead of a stress ellipse, in 3-D
we have a stress ellipsoid.
The Mohr Diagram
Edit the first bullet:
It is easier to understand and more
convenient to use Mohr diagrams
than stress ellipses.
Add in:
•
It is important to distinguish the
Mohr diagram from a diagram of
physical space; they are not the
same.
Leave the second and third bullet, as well
as the Mohr Diagram figure.
Edit the fourth bullet:
•
The principal stresses, σ1 and σ3,
are the points of intersection of the
Mohr circle with the σn axis. At
these points, σs = 0.
Remove the fifth bullet.
Leave the sixth and seventh bullet.
Edit the eighth bullet:
•
The maximum shear stress at the
top and bottom of the Mohr circle.
Edit the ninth bullet:
•
Scalar invariants are values that
remain the same for any set of
components (σxx, σxz), (σzz, σzx) that
define the same stress.
Include the remaining equations in the
ninth bullet.
Leave the tenth bullet.
•
Terminology for States of Stress
Add to hydrostatic pressure definition:
•
No shear stress present. The Mohr
diagram reduces to a point on the
σn axis.
EPS116 Chapter Summary 2013
1
Frances-Julianna Leiva and Wingyee Lee
Add to uniaxial stress definition:
In this case, the Mohr circle is
tangent to the σs plane.
Leave axial compression, axial extension,
triaxial stress, and pure shear.
Edit definition for deviatoric stress:
•
Deviatoric stress is found by
subtracting the mean normal stress
component from each of the
normal stress components.
Include the equations and second
paragraph in the definition for
deviatoric stress.
Leave the definitions for differential stress
and effective stress.
•
Mohr Diagram for 3-Dimensional Stress
Leave this section alone.
The Stress-Component Matrix
Replace figure with:
The Stress Tensor
Add:
•
Stress is a second rank tensor with
9 components in three dimensional
space.
•
Since σ12=σ21, σ13=σ31, σ23=σ32, we
say that tensor is symmetric,
leaving only 6 independent
components.
References & Resources
Twiss and Moores. Structural Geology,
2nd ed. 2007. 151-191.
EPS116 Chapter Summary 2013
Frances-Julianna Leiva and Wingyee Lee
2