structural geology third stage geology lecture 2

Structural Geology

Third Stage Geology

Lecture 2

Third stage geology (2014 -

2015) 1

Lec. 2:Force and Stress / Dr.Salim H.

Sulaiman / Dept Of Geology / Uni of

Sulaimani

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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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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Balanced and Unbalanced Forces

Forces occur in pairs and they can be either balanced or unbalanced


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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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Unbalanced Forces

An unbalanced force always causes a change in motion

=


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Unbalanced Forces

5 N, right + 10 N, right = 15 N, right


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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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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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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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: 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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http://twitter.com/home?status=http://www.allwords.com/word-equilibrant.html

http://www.facebook.com/sharer.php?u=http://www.allwords.com/word-equilibrant.html&t=Definition of equilibrant

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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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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3-Torsion : It results from twisting, i.e.: if the two ends of a body are turned in opposite directions.


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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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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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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.


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Stress acting on a plane


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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


σ n

τ

http://courses.eas.ualberta.ca/eas421/lecturepages/introslides.html


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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.


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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.


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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.


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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.




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http://courses.eas.ualberta.ca/eas421/lecturepages/introslides.html


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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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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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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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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.


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Again, this can be visualized as the stress ellipsoid


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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.


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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.


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Deviatoric stress has the symbol sigma dev.

So, sigma total = sigma m + sigma dev


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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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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.


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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.


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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.


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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.


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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


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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


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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

structural geology third stage geology lecture 2

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