Structural Geology
Third Stage Geology
Lecture 2
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Lec. 2:Force and Stress / Dr.Salim H.
Sulaiman / Dept Of Geology / Uni of
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Materials in structural geology
Structural geology deals primarily with solids, but also with
liquids and to some extent with gases.
Solid matter is important in structural geology because it forms
the earth crust.
Liquids are less important or important only when they exist in
the pore spaces and influence the mechanical behavior of rocks.
Gases present in the outer shell of the earth with petroleum and
volcanoes.
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Lec. 2:Force and Stress / Dr.Salim H.
Sulaiman / Dept Of Geology / Uni of
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Force (F):
Is a vector quantity that changes or tends to produce
a change in the motion of a body, so it is defined by its
magnitude and direction.
The force vector called traction of force.
The magnitude is the length of the vector and the way
it is pointed is its direction.
Forces have a magnitude and a direction
5N, north (up)
Magnitude: 5N
Direction: north (up)
5N
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Lec. 2:Force and Stress / Dr.Salim H.
Sulaiman / Dept Of Geology / Uni of
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Balanced and Unbalanced Forces
Forces occur in pairs and they can be either balanced or unbalanced
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Lec. 2:Force and Stress / Dr.Salim H.
Sulaiman / Dept Of Geology / Uni of
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Balanced Forces
Balanced forces do not cause change in motion
They are equal in size and opposite in direction
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Lec. 2:Force and Stress / Dr.Salim H.
Sulaiman / Dept Of Geology / Uni of
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Unbalanced Forces
An unbalanced force always causes a change in motion
=
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Lec. 2:Force and Stress / Dr.Salim H.
Sulaiman / Dept Of Geology / Uni of
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Unbalanced Forces
5 N, right + 10 N, right = 15 N, right
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Lec. 2:Force and Stress / Dr.Salim H.
Sulaiman / Dept Of Geology / Uni of
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Resultant forces :
The resultant vector is the vector that 'results' from adding two
or more vectors together.
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Lec. 2:Force and Stress / Dr.Salim H.
Sulaiman / Dept Of Geology / Uni of
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Lec. 2:Force and Stress / Dr.Salim H.
Sulaiman / Dept Of Geology / Uni of
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The Pythagorean theorem is a mathematical equation that
relates the length of the sides of a right triangle to the length of
the hypotenuse of a right triangle.
If there are normal angle between the vectors the Pythagorean
theorem can be applied
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Lec. 2:Force and Stress / Dr.Salim H.
Sulaiman / Dept Of Geology / Uni of
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50º
6N
8N
By Cosine role:
R2
= 82+ 6
2- 2 * 8 * 6 * Cos 130º = 161.707
R = 161.707
R = 12.7 N
2
Resultant of two forces
8N
50º130º
6N
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Lec. 2:Force and Stress / Dr.Salim H.
Sulaiman / Dept Of Geology / Uni of
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: A force equal to, but opposite of, the resultant sum
of vector forces; that force which balances other
forces, thus bringing an object to equilibrium.
Equilibrant
50º130º
8N
6N
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Lec. 2:Force and Stress / Dr.Salim H.
Sulaiman / Dept Of Geology / Uni of
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Differential forces :Are those forces acting on a body but they are unequal from all
sides. They act in the in a specific direction or along certain planes or
lines. These are :
1-Compression (Compressive forces) :
Compressive forces tend to compress (or squeeze) the body
toward its center. They can be represented by two arrows acting on
the same straight line and are directed toward each other (towards
body center).
Compression
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Lec. 2:Force and Stress / Dr.Salim H.
Sulaiman / Dept Of Geology / Uni of
Sulaimani
2-Tension (Tensile forces) :
Are external forces that tend to pull the body apart. They can be represented by two arrows which act on the same straight line but are directed away from each other.
Tension
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Sulaiman / Dept Of Geology / Uni of
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3-Torsion : It results from twisting, i.e.: if the two ends of a body are turned in opposite directions.
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Sulaiman / Dept Of Geology / Uni of
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4-Couple : Consist of two equal forces that act in opposite directions in the same plane but not along the same line.
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Lec. 2:Force and Stress / Dr.Salim H.
Sulaiman / Dept Of Geology / Uni of
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Stress It is directed pressure and it is expressed or defined as force per
unit area,
Stress is usually measured in megapascals. (MPa). (1 N/m2 =1Pa)
1MPa = 106 Pa
or psi = Pound per square inch. F
σ =A
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Sulaiman / Dept Of Geology / Uni of
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Third stage geology (2014 -
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Lec. 2:Force and Stress / Dr.Salim H.
Sulaiman / Dept Of Geology / Uni of
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Stress acting on a planeGiven a point within a body and a plane that passes through that
point, the stress traction on that plane at that point is the force/area
In this case we are examining stress acting on a single plane,
and thus we are looking at stress in 2-dimensions.
As previously mentioned:
Stress is a vector.
Stress can be acting in any orientation to a plane.
Stress acting on a single plane can be reduced to two vector
components these are:
The normal stress component and
The shear stress component.
Third stage geology (2014 -
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Sulaiman / Dept Of Geology / Uni of
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Stress acting on a plane
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Sulaiman / Dept Of Geology / Uni of
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The normal stress component has the symbol sigma n. It is the
stress component perpendicular to the plane.
The shear stress component has the symbol (τ) Tau or sigma s. It is
the stress component that is parallel to the axes of the plane.
τ
Axis 1 of shear stress
Axis 2 of shear stress
σ n
τ
Third stage geology (2014 -
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Sulaiman / Dept Of Geology / Uni of
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Stress at a PointThere are three ways to describe the stress state at a point :
1. Stress ellipsoid.
2. Stress tensors (Stress matrix).
3. Mohr diagram.
1. The stress ellipsoid.
A point defines the intersection of an infinite number of
planes, each with a different orientation.
The state of stress acting on a point describes all the
stresses acting on these planes.
Third stage geology (2014 -
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Sulaiman / Dept Of Geology / Uni of
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Each of these planes will have the normal and shear
stress components that have been described above.
This property can be illustrated in three-dimensions to
obtain the stress ellipsoid.
Third stage geology (2014 -
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Sulaiman / Dept Of Geology / Uni of
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If we envelope these stress vectors, we obtain an
ellipsoid. This is the stress ellipsoid. It fully describes the
state of stress at a point. The ellipsoid allows us to find the
stress for any plane.
Third stage geology (2014 -
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Lec. 2:Force and Stress / Dr.Salim H.
Sulaiman / Dept Of Geology / Uni of
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b) Properties of the Stress Ellipsoid
An ellipsoid is defined by three axes.
These axes are defined as the principal stresses.
These three principal stress axes are orthogonal to one another.
They are also perpendicular to three plane
The principal stresses are vectors. That is, they have magnitude
and direction.
We can describe the state of stress of a body by specifying the
orientation and magnitude of these axes.
2. Stress Tensor and Stress matrix
It is a mathematical description that defines
state of stressAn arbitrary stress on a plane resolved into normal
stress perpendicular to the surface. and shear stress
parallel to the surface
Shear stress itself be resolved into two components.
Axis 1 of shear stress
Axis 2 of shear stress
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Lec. 2:Force and Stress / Dr.Salim H.
Sulaiman / Dept Of Geology / Uni of
Sulaimani
Third stage geology (2014 -
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Lec. 2:Force and Stress / Dr.Salim H.
Sulaiman / Dept Of Geology / Uni of
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The orientation and magnitude of the state of stress of a body can be
defined in terms of its components in a specific Cartesian reference
frame. A Cartesian reference frame has three mutually perpendicular
coordinate axes, X, Y, and Z. Or in this image X1, X2 and X3.
We can extend this idea to three dimensions to look at stress
at a single point, which we’ll represent as a very small cube:
X3
X1 X2
σ21
σ23
σ22
σ33
σ31 σ32
σ11
σ13
σ12
The first subscript identifies
the plane by indicating
the axis which is
perpendicular to it
The second subscript shows
which axis the traction vector
is parallel to it.
σ 32
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Lec. 2:Force and Stress / Dr.Salim H.
Sulaiman / Dept Of Geology / Uni of
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Plane 1 Plane 2
Plane 3
Stress Tensor and Stress matrix
The nine components can be represented or written in a
matrix, This grouping of the nine stress components is known
as the stress tensor matrix (or stress matrix).
σ12 = σ21 , σ13 = σ31, and σ32 = σ23
If the cube in the figure, above, is in equilibrium so that it is not
rotating, then you can see that
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Lec. 2:Force and Stress / Dr.Salim H.
Sulaiman / Dept Of Geology / Uni of
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Principal StressesWhen the shear stresses on all of the face go to zero and each of the
three faces has only a normal stress on it. Then, the matrix which
represents the stress tensor reduces to
In this case the remaining components (σ1, σ2, and σ3) are
known as the principal stresses. By convention, σ1 is the largest
and σ3 is the smallest.
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Sulaiman / Dept Of Geology / Uni of
Sulaimani
Third stage geology (2014 -
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Sulaiman / Dept Of Geology / Uni of
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The Principal Stress axes
The principal stress axes are labelled
sigma1, sigma2 and sigma3
σ1 is the principal stress axis (the direction of maximum stress)
σ 2 is the intermediate principal stress axis.
σ 3 is the least principal stress axis (the direction of minimum stress)
Or, in mathematical terms, σ 1> σ 2> σ 3.
Third stage geology (2014 -
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Sulaiman / Dept Of Geology / Uni of
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Again, this can be visualized as the stress ellipsoid
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Sulaiman / Dept Of Geology / Uni of
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Mean Stress & Deviatoric stress
Mean Stress The mean stress is simply the average
of the three principal stresses.
m=(sigma1+sigma2+sigma3)/3.
Deviatoric stress
is the part of the total stress that is left after the mean
stress is removed. Deviatoric stress is equivalent to
tectonic stress and is the stress responsible for
deformation.
Third stage geology (2014 -
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Lec. 2:Force and Stress / Dr.Salim H.
Sulaiman / Dept Of Geology / Uni of
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we can break the stress tensor down into two components.
The first part or isotropic component is the mean stress, and
is responsible for the type of deformation mechanism, as
well as dilation. The second component is the Deviatoric
stress and is what actually causes distortion of the body.
When considering the deviatoric stress, the maximum is
always positive, representing compression, and the
minimum is always negative, representing tensional.
Third stage geology (2014 -
2015)
Lec. 2:Force and Stress / Dr.Salim H.
Sulaiman / Dept Of Geology / Uni of
Sulaimani
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Third stage geology (2014 -
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Lec. 2:Force and Stress / Dr.Salim H.
Sulaiman / Dept Of Geology / Uni of
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Deviatoric stress has the symbol sigma dev.
So, sigma total = sigma m + sigma dev
Third stage geology (2014 -
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Sulaiman / Dept Of Geology / Uni of
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Deviatoric Stress Example:
Given the following stress tensor
The hydrostatic or lithostatic stress is
which can be written as
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Sulaiman / Dept Of Geology / Uni of
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Subtracting the hydrostatic stress tensor from the total stress gives
The maximum is always positive, representing compression, and
the minimum is always negative, representing tensional.
Third stage geology (2014 -
2015)
Lec. 2:Force and Stress / Dr.Salim H.
Sulaiman / Dept Of Geology / Uni of
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Differential stress: Directed Stress
If stress is not equal from all directions then we say that the
stress is a differential stress or Directed Stress.
Three kinds of differential stress occur.
1- Compressive stress : It is directed pressure that tends
to compress a body.
2- Tensile stress : It is directed pressure that tends to pull
the body apart.
3-Shearing stress : Act as a couple on both sides of a surface
and tend to move each side in opposite direction.
Third stage geology (2014 -
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Lec. 2:Force and Stress / Dr.Salim H.
Sulaiman / Dept Of Geology / Uni of
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Lec. 2:Force and Stress / Dr.Salim H.
Sulaiman / Dept Of Geology / Uni of
Sulaimani
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Sulaiman / Dept Of Geology / Uni of
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When the magnitude of stress is different in different directions it is
a directed stress. If the magnitude of the directed stress exceeds an
object's strength, the shape of the object will be deformed. The
volume of the object may also be changed by the application of
directed stress.
These types of directed stresses are very common in the Earth's
surface and produce a wide variety of geologic structures. The
latter are important in the formation of oil and gas reservoirs and
ore bodies. The shape of geologic structures often provide
important clues as to the past stress regime in the crust as well as
the direction in which the stresses were acting.
Third stage geology (2014 -
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Sulaiman / Dept Of Geology / Uni of
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d) The stress states:
There are several common states of stress that can be defined by the
relationships of the principal stresses.
These stress states are:
1. isotropic
2. anisotropic
Types of anisotropic states of stress are:
uniaxial, biaxial and triaxial.
Third stage geology (2014 -
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Sulaiman / Dept Of Geology / Uni of
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Isotropic state of stress is where all three principal stresses are
equal in magnitude. The ellipsoid is actually a sphere
in this case.
sigma1=sigma2=sigma3
Such as Hydrostatic stress lithostatic stress
It is non deviatoric stress
1. No change in shape (no shear stress; no shear strain)
2. causes volume (decrease) changes and increases density
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Anisotropic is a stress state where at least one axis has a different
magnitude to the other axes. This describes an ellipsoid
it is Deviatoric Stress
Third stage geology (2014 -
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Sulaiman / Dept Of Geology / Uni of
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Uniaxial stress can be in tension or compression.
The sign convention for tension in geology is
negative, and compression is positive .
The ellipsoid is 'needle-like'.
Uniaxial tension: sigma1= sigma2= 0; sigma3<0
Uniaxial compression:
sigma2= sigma3= 0; sigma1>0.
Uniaxial compression
Third stage geology (2014 -
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Sulaiman / Dept Of Geology / Uni of
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Biaxial stress is where one axis equals zero.
For example, sigma 1>0>sigma 3
Triaxial the general triaxial state of stress is where none of the
three principle stress axis can be zero. That is,
sigma 1>sigma 2>sigma 3 ≠ 0