1. Force and Stress – Normal and Shear Stress Lecture 5 – Spring 2016
Structural Geology
Force and Stress –
Normal and Shear Stress
Lecture 5 – Spring 2016
The application of Plate Tectonic Theory to the earth has allowed geologists to understand many processes affecting the earth. Structural geology has been strongly influenced by plate tectonic ideas. In order to understand tectonics, we need an understanding of the branch of physics known as mechanics. Newtonian mechanics is the study of forces on rigid bodies. Many geologic structures distort when they are subjected to force. We need to use a modification of Newtonian mechanics, called continuum mechanics, to understand the behavior of geologic structures.
Continuum mechanics treats objects as if they were made of continuous media. No discontinuities affect the behavior of the system appreciably. Clearly, this is an approximation. Many rocks are composed of discrete grains of several minerals. However, in large structures, we are dealing with a statistical assemblage of a great many particles. The statistical assemblage often behaves like a homogeneous solid, and the mathematics of continuum mechanics works well. If discontinuities, like fractures, dominate the behavior, continuum mechanics no longer applies, and more elaborate mathematical descriptions must be employed. Those treatments are reserved for graduate level structural geology courses.

2. Rocks and Force Rocks constantly experience the force of gravity
They may also experience a variety of other forces, including tectonic forces and forces associated with impact
Previously, we saw that force is defined by the following equation:
F = mA
where F is the force vector, m is mass, and A is the acceleration vector

3. Responses to Force Rocks respond to applied forces in one of two ways:
Figure 3.2 in text
Rocks respond to applied forces in one of two ways:
A. Movement – Newtonian mechanics
B. Distortion – continuum mechanics

4. Types of Force Body forces Surface forces Fb ∝ m Fs ∝ area
Some forces are the result of a field which acts on every particle within the body under discussion. Such forces are called body forces. The magnitude of body forces is proportional to the mass of the body. Body forces may change the velocity of the body, which is known as acceleration
Surface forces are proportional to the surface area affected by the force.
The concept of force does not distinguish between bodies of different shape. Hitting a rock with a flat-headed hammer, a chisel, or a pointed hammer often produces quite different results. Yet, if the weight of the hammer heads is identical, and the hammers are swung with equal accelerations, the force is identical. Clearly there is need to further describe what is happening.

5. Definition of Stress
We have previously seen that stress is an internal force set up as the result of external forces acting on a body
Stress is usually represented by the Greek letter sigma, σ
σ = F/Area
Defined in this manner, stress is really an intensity of force, which measures how concentrated the force is
Subscripts are often attached to σ to add additional information

6. Stress in Different Dimensions
In two dimensional problems, stress is a vector quantity, and is sometimes called traction
In three dimensions, stress is a second-order tensor, which will be discussed shortly
If stress is present in a rock, and we wish to note the amount of stress on an arbitrary plane, we can split the stress into two components:

7. Traction
Stress in an arbitrary direction may be resolved into components
A. Normal stress, denoted σn
B. Shear stress, denoted σs or τ (tau)
Because stress is defined as force per unit area, the resolution of stress into its components is more complicated than the resolution of forces. As the direction changes, not only does the component of force change, but the area the force component is applied to also changes. This is the essential difference between force, which is a vector or first-order tensor, and stress, which is a second-order tensor.
Figure 3.3 in text

8. Resolution of the Stress Vector
Figure 3_4a illustrates the principle of stress resolution
A plane face is ABCD in the drawing
Note: The section through a cube implies a plane, i.e. two dimensions
Two-dimensional Stress
Figure 3-4 illustrates the resolution of the stress vector into component vectors. A plane face is ABCD in the drawing. (Note: The section through a cube implies a plane, i.e. two dimensions).

9. Force and Stress Figure 3.4a & b in text
A force, F, is applied along rib AB
Line EF in the drawing is the trace of a plane which makes an angle θ with the top and bottom surfaces of ABCD
The force can be resolved into components Fn perpendicular to the plane, and Fs parallel to the plane

10. Fn and Fs σ = F/AB (Note: F = σAB) Fn = F cos θ = σAB cos θ
Since AB = EF cos θ,
Fn = σEF cos2θ
Fs = F sin θ = σAB sin θ = σ EF sin θ cos θ

11. Trigonometry Identity
We can use the following trigonometric identity to simplify Fs
sin θ cos θ = ½ (sin 2θ)
Fs = σEF ½ (sin 2θ)

12. Normal and Shear Stress
σn = Fn/EF = σ cos2θ
σs = Fs/EF = σ ½ (sin 2θ)

13. Stress Vector Resolution
Thus, the stress vector acting on a plane can be resolved into vector components normal and parallel to the plane
Their magnitudes vary as a function of the orientation of the plane

14. Normal Force and Stress vs. θ
Plot of the normalized values of normal force and the normal stress versus theta
The curves have a slightly different shape, but in both cases the normalized values decrease and go to zero at θ = 90º
Figure 3.4c in text

15. Shear Force and Stress vs. θ
The curves in this case are nearly identical until θ = 25º, then the shear force increases faster than the shear stress
After 45º, the shear force continues to increase, but the shear stress again goes to zero at 90 º
Thus, the stress vector acting on a plane can be resolved into vector components normal and parallel to the plane, but their magnitudes vary as a function of the orientation of the plane.
Figure 3.4d in text

16. Three Dimensional Stress
3D analysis is more complicated than 2D, because stress in three dimensions is a second order tensor
To reduce the complexity somewhat, we can stipulate that the rock being discussed is at rest
According to Newton’s Third Law of Motion, bodies at rest are acted upon by balanced forces, otherwise they would start to move
We can consider the stress at a point within the body
A point is the intersection of an infinite number of planes, in every possible orientation
Using a point allows us to discuss force and stress components in any plane

17. Stress Ellipse
Figure 3. 5a shows a plane cut by four other planes (a through d)
The stresses on each plane are plotted, and are perpendicular to their respective planes
Since the body is at rest, every stress is opposed by an equal an opposite stress
We can connect the endpoints of the two dimensional stress vectors with a smooth curve, generating the ellipse shown

18. Stress Ellipsoid
If we were to draw similar ellipses in the two additional, mutually perpendicular, planes, we could then combine the data to generate a three dimensional ellipsoid, as shown in figure 3.5b
This is known as the stress ellipsoid

19. Principal Stresses
Ellipsoids are characterized by three principal axes
In the stress ellipsoid, these axes are known as the principal stresses, which have two properties
They are mutually perpendicular
They are perpendicular to three planes which have no shear stresses
The three planes are known as the Principal Planes of Stress
Each principal stress is a vector

20. Stress Using Cartesian Coordinates
Stress can be visualized in another manner
Using a standard three dimensional Cartesian coordinate system, and we picture a point cube, as shown in figure 3.6
We can resolve the stress acting on each face of the cube into three components
Figure 3.6 in text

21. Stress Notation The face normal to the x-axis has a component σxx
First subscript refers to the plane, in this case the one normal to the x-axis
Second subscript refers to the component along axis x
In addition, we have two shear stresses, σxy and σxz, which lie along the y and z axes within the plane under consideration

22. Table of Stress Components
Doing the same for the other principal stress axes, we generate a table
Stress on face normal to:
In the direction of: x y z Xx
σxx
σxy
σxz Yy
σyx
σyy
σyz Zz
σzx
σzy
σzz
The normal stress components are σxx, σyy, and σzz (shown in red)
The shear stress components are σxy, σxz, σyx, σyz, σzx, and σzz (shown in orange)

23. Equivalence of Shear Stress Components
Since the object is at rest, three of the six shear stress components must be equivalent to the other three (otherwise the object would move)
σxy = σyx, σxz = σzx, and σyz = σzy

24. Independent Stress Components
This leaves six independent components:
Stress on face normal to:
In the direction of: x y z Xx
σxx
σxy
σxz Yy
σyy
σyz Zz
σzz
In geology, compressive stress is considered positive, and tensile stress is negative, as we have previously seen. Note that in both physics and engineering, this sign convention is reversed, so that tensile stress is positive, and compressive stress is negative. We will use the geologic convention in this course.
We may rotate the Cartesian coordinate system to any convenient orientation. From matrix mathematics, it is known that at least one choice of axes orientation will make all of the shear stress components have the value of zero. Our table becomes:

25. Principal Stress Table
Stress on face normal to:
In the direction of: x y z Xx
σxx Yy
σyy Zz
σzz
The stresses along the axes are the principal stresses.
Note that this orientation is the mathematical equivalent of finding the eigenvalues of a 3x3 matrix.
Thus oriented, the axes are known as the principal axes of stress, and the planes perpendicular are the principal planes of stress

26. Isotropic Stress
It is possible that the three principal stresses will be equal in magnitude
If this condition is met, the stress is said to be isotropic
The stress ellipsoid becomes a sphere

27. Anisotropic Stress
When the principal stresses are unequal, they are said to be anisotropic
We then introduce another convention:
σ1  σ2  σ3
σ1 is called the maximum principal stress
σ2 is the intermediate principal stress
σ3 is the minimum principal stress
We can then recognize several types of stress conditions that might occur in geology, and describe them using the principal stress notation:

28. Types of Stress General Triaxial Stress
σ1 ≥ σ2 ≥ σ3 ≠ 0
Biaxial stress, with one axis at zero
σ1 ≥ 0 ≥ σ3 or σ1 ≥ σ2 ≥ 0
Uniaxial tension
σ1 = σ2 = 0; σ3 < 0
The General Triaxial Stress would be likely in a plate collision between two continental plates.
Uniaxial tension represents rifting with two plates pulling straight apart.

29. Uniaxial Stress Uniaxial compression
σ1 > 0; σ2 = σ3 = 0
Hydrostatic or lithostatic pressure
σ1 = σ2 = σ3
Uniaxial compression represents two plates hitting straight on
Both the ellipsoid and second order tensor representations are useful. Ellipsoids are easier to visualize, but tensors allow mathematical calculations to be made. Both are used in geology
The principal stress notation is also useful in experimental work. We can simulate various geologic situations in the laboratory, and see what response we get to imposed forces.

30. Gabriel Auguste Daubrée
Daubrée ( ) was an early experimenter in many aspects of the geological sciences
He taught mineralogy at the French School of Mines
He introduced synthesis techniques and extended these to general experimental work
More information about Daubrée can be found at (in French)
A biography, automatically translated from the French (badly) is at
Image:

31. Daubrée Experiment
For example, Figure 3-7a shows a picture of wax placed between two wooden plates, an experiment first performed by Daubrée
He reported some of his results at the first International Geological Conference in 1878 (Paris)

32. Diagram of Daubrée Experiment
Plane AB is arbitrary, and it makes an angle θ with σ3
We can make two other simplifying assumptions:
Line AB has unit length
The plane represented by AB within the block has unit area
Figure 3-7b is an interpretation of the experiment in principal stress notation.
The wax was squeezed along the vertical axis, so that is σ1. As shown, the values of σ2 = σ3 . For simplicity, we may illustrate the problem in the σ1 - σ3 plane. .

33. Forces in Balance
The forces parallel and perpendicular to AB must balance
We resolve force z AB into the component of the force z BC (parallel to σ1) along CD plus the component of the force z AC along CD (parallel to σ3)
Note that ACD = θ

34. Resolving Forces Various forces may be resolved as:
forceBC = σ1cosθ (Force = stress C area)
forceAC = σ3sinθ
AreaBC = 1 C (cos θ)
AreaAC = 1 C (sin θ)
On the AB surface, there is a normal stress, σn and a shear stress, σs

35. Normal Stress The normal stress is the same as the stress along CD:
σn = σ1cosθCcosθ + σ3sinθCsinθ = σ1cos2θ + σ3sin2θ
Since cos2θ = ½ (1 + cos2θ) and sin2θ = ½ (1 - cos2θ) we get
σn = ½ (σ1 + σ3) + ½ (σ1 - σ3) C cos2θ

36. Shear Stress
We resolve force 2 AB into the component of the force z BC along AB plus the component of the force z AC along AB
σs = σ1cosθCsinθ - σ3sinθCcosθ = (σ1 - σ3) sinθCcosθ

37. Simplification of Shear Stress
Substituting sinθCcosθ = ½ sin2θ gives σs = ½(σ1 - σ3)sin2θ
The planes of maximum normal stress are at θ = 0E relative to σ3, because cos2θ = 1 at θ = 0E
The planes of maximum shear stress are at 45E relative to σ3, because sin2θ = 1 at 45E