1. The stresses that cause deformation
Understand "stress calculations”
Spend some time with these calculations to convince yourself that stress on a given plane resolves itself into a single stress tensor.
Stress (s) = force/unit area
s = F/A

2. Stress Goals 1) Interpret the stresses responsible for deformation.
2) Describe the nature of the forces that cause the stresses.
3) Understand the relations between stress, strain and rock strength.
Describing stress and force is a mathematical exercise.

3. Responses to Stresses
1) Folding
2) Brittle faults
3) Ductile shear zones
4) Joints

4. Force Force: changes in the state of rest or motion of a body.
Only a force can cause a stationary object to move or change the motion (direction and velocity) of a moving object.
force = mass x acceleration, F = ma,
mass = density x volume, m = rV,
therefore, r = m/V,
Weight is the magnitude of the force of gravity (g) acting upon a mass.
The newton (N) is the basic (SI) unit of force.
1 newton = 1 kg meter/sec2
1 dyne = 1g cm/sec2 so 1 N = 105 dyne
1 pascal = newton/m2

5. Load Force Forces as Vectors
Force is a vector - it has magnitude and direction. Vectors can be added and subtracted using vector algebra. We can evaluate vectors in order to determine whether the forces on a body are in balance.
Load
Force

6. Units of Stress 1 newton = 1 kg meter/sec2 = this is a unit of force
1 pascal = 1 newton/m2 = unit of stress
1 newton is about pounds of force
1 pascal is about lb/ft2, thus pressure is measured in kPa
1 kPa = lb/in2
9.81 Pa is the pressure caused by a depth of 1mm of water

7. Stress on a 2-D plane:
Normal stress act perpendicular to the plane
Shear stress act along the plane.
Normal and shear stresses are perpendicular to one another

8. Stress (s)
Stress is force per unit area:
s = F/A

9. Forces in the Geologic World
Typically we think of the Earth as at rest - in static equilibrium, or moving very slowly.
When there are net forces, they cause accelerations that are usually one of 2 kinds:
1) slow ponderous motion of a tectonic plate that increases or decreases velocity over a very long time, or;
2) sudden, short lived, strong accelerations during fault slip accompanying earthquakes.

10. Two Types of Forces
Body forces, that act on the mass of a body (gravity, electromagnetic), and are independent of forces applied by adjacent material, and;
2) Contact forces, are pushes and pulls across real or imaginary surface of contact such as faults.
Three different type of loading due to contact forces:
1) gravitational loading - pushing on adjacent rock.
2) thermal loading - expansion or contraction.
3) displacement loading - push due to motion.

11. Relations between F and s
(a) Fn and Fs and angle q with top and bottom surface. EF is trace of plane, ABCD is cube with ribs of length AG.
Magnitudes of vectors Fs and Fn is function of angle q
Fn = F cos q, Fs = F sin q
(b) The magnitude of normal and shear stresses is function of angle q and the area,
sn= s cos2q
ss = s sin2q
(a) Fn and Fs and angle theta with top and bottom surface.. EF is trace of plane, ABCD is cube with ribs of length AG.
Magnitudes of vectors Fs and Fn is function of angle theta, Fn = F cos theta, Fs = F sin theta
(b) The magnitude of normal and shear stresses is function of angle theta and the area, normal stress = total stress cos2 theta, and shear stress = total stress sin2 theta

12. Stress ellipsoid
A point represents the intersection of an infinite number of planes and stresses on these planes describe an ellipse.
In 3-dimensions, the ellipsoid is defined by three mutually perpendicular principal stresses (s1]> s2 > s3).
These three axes are normal to the principal

13. Stress ellipsoid
What is important about the principal stresses (s1 > s2 > s3)?
The axes are perpendicular to each other.
They do not contain shear stresses
The state of stress of any body is described by the orientation and magnitude of the principal stresses.

14. Components of stress
Three normal stresses
Components parallel shear stresses
Reference system x, y, z

15. Geology sign conventions
Stresses
1) Normal stress
Positive or negative
2) Shear stress
Compressive stress is + (positive)
Tensional stress is – (negative)
Clockwise shear stress is – (negative)
Counter clockwise shear stress is + (positive)

16. Stress State s1 > s2 > s3 s1 = s2 = s3
If the 3 principal stresses are equal in magnitude = isotropic stress
Here the state of stress is represented by a sphere, not an ellipsoid.
If the principal stress are unequal in magnitude = anisotropic stress
Here the greatest stress is called s1
The intermediate stress, s2 and minimum stress is called s3
s1 > s2 > s3
As a geologist, what is it called if all three principal stresses are equal?
s1 = s2 = s3

17. Hydrostatic Stress
If we calculate stress vectors within a point of a hydrostatic stress field, we find that the stress vectors have the same value. Each stress vector is oriented perpendicular to the plane.
All stress vectors are normal vectors, they have no shear stress components.
Hydrostatic stress = all principal stresses in a plane are equal in all directions. No shear stresses!
Equal stress magnitudes in all directions. Dive into a pool. All stresses have the same values.

18. Lecture outline
Overview of stress
Minimum and maximum stress
Types of stress on a plane
Normal stress
Shear stress
3. Mean stress
4. Differential stress
5. Deviatoric stress
6. Hydrostatic state of stress
7. Stress and the Mohr circle
Problem set outline
Apparent dip
Angle between lines
Angle between planes

19. Stress on a dipping plane in the Earth’s crust
2 components
Normal stress & Shear stress
sn = s cos2q
ss = s sin2q
Review sign conventions for normal and shear stresses

20. We resolve stress into two components
Normal stress, sn and the component that is parallel to the plane, shear stress, ss
Normal compressive stresses tend to inhibit sliding along the plane and are considered positive if they are compressive.
Normal tensional stresses tend to separate rocks along the plane and values are considered negative.
3) Shear stresses tend to promote sliding along the plane, labeled positive if its right-lateral shear and negative if its left-lateral shear.

21. Squeeze a block of clay between two planks of wood
AB, trace of fracture plane that makes an angle q with 3.
The 2-D case is simple, since
2 = 3 (atmospheric pressure)
Important: What is angle q?

22. Mohr Stress Diagram
This give us a useful picture or diagram of the stress equations.
b) They solve stress equations on page 49 (Eqs 3.7 and eq. 3.10)
c) Plot N versus S
d) Rearrange Eqs. 3.7 and 3.10 and square them yields
[sn – ½(s1 + s2]2 + ss2 = [½ (s1 – s32 )]
form (x –a)2 + y2 = r2
Important: What is angle q?
In Mohr space, we use 2q!

23. Mohr Stress Diagram
a) Mohr circle radius = ½(s1 – s3] that is centered on ½(s1 + s3] from the origin.
b) The Mohr circle radius, ½(s1 - s2] is the maximum shear stress ss max.
c) The stress difference (s1 – s3), called differential stress is indicated by sd.

24. Mohr Stress Diagram Mohr circle: sn on x-axis ss on y-axis.
Maximum principal stress (s1) and minimum stress (s3) act on plane P that makes an angle q with the s3 direction.
In Mohr space, we plot s1 and s3 on sn-axis
These principal stress values are plotted on the sn-axes because they are the normal stresses acting on plane P.
The principal stresses always have zero shear stress values (ss = 0).

25. Mohr Stress Diagram
Remember,
sn,p = ½(s1 + s3] + ½(s1 - s3] cos 2q
ss,p = ½(s1 - s3] sin 2q
Construct a circle thought points s1 and s3 with 0, the midpoint, at ½(s1 + s3) as the center with radius, ½(s1 - s3].
Now draw a line OP, so that angle POs1 is equal to 2q – confusing step, plot twice the angle q, which is the angle between the plane and s3.
Remember we measure 2q from the s1 side on the sn-axis.
We can read the values of sn,p along the sn-axis, and ss,p along ss-axis for our plane P.

26. Mohr Stress Diagram
When the principal stress magnitudes change w/o differential stress, the Mohr circle moves along the sn-axis without changing ss
How is this achieved?
Suggest geologic examples?

27. Mohr Stress Diagram
When the principal stress magnitudes change w/o differential stress, the Mohr circle moves along the sn-axis without changing ss
Change confining pressure (Pc). Increase air pressure on our clay experiment, or carry the experiment underwater.
Burial of rocks changes confining pressure. Which way along the sn-axis?
Exhumation of rock changes confining pressure. Again, in what direction along the sn-axis?

28. Problem set #1.
Handouts in class and go online for additional graph paper in Mohr space

29. Various states of stress
Uniaxial compression, two of the three principal stresses are zero.
Hydrostatic stress, a single point on the Mohr circle that lies on the x-axis. All normal stresses are the same, and no shear stresses.

30. Various states of stress
Triaxial stress, all three principal stresses are different.
Biaxial stress, all three principal stresses are non-zero, but two of the principal stresses have the same value. Typical stress ellipse (plane stress).

31. Mean stress and deviatoric stress
Because a body’s response to stress, we subdivide the stress into two components, mean and deviatoric stress.
Mean stress = [s1 + s2 + s3]/3 or sm
In 2-D, [s1 + s3]/2
Deviatoric stress is the difference between the mean stress and total stress. stotal = smean + sdev
smean is often called the hydrostatic component (s1 = s2 = s3)

32. Lithostatic pressure (Pl).
For rocks at depth, we use lithostatic pressure.
Consider a rock at 3 km depth. Lithostatic pressure F (weight of rock of overlying column).
Pl = r x g x h if r (density) = 2700 km/m3, g (gravity) = 9.8 m/s2 and h (depth) is 3000 m, we get:
Pl = 2700 x 9.8 x 3000 = 79.4 x 206 Pa ~ 80 Mpa
For every km in the Earth’s crust, the lithostatic pressure increases 27 Mpa.
The lithostatic pressure is equal in all directions (isotropic stress),
[s1 = s2 = s32 ]

33. Lithostatic pressure (Pl).
So we divide the rocks state of stress into an isotropic (lithostatic/hydrostatic) and an anisotropic (deviatoric).
Isotropic stresses act equally on all directions, resulting in a volume change of the rock – increase water pressure on a human, or air pressure on take-off or landing.
Deviatoric stress, changes the shape of the body. The difference between isotropic stress and additional stress from tectonic forcing.

34. Measuring Stress Present day stress Difficult to measure
EQ focal mechanisms
Bore-hole breakouts
in situ measurements
EQ focal mechanisms, fault planes are parallel to maximum shear stresses. Several focal mechanisms might constrain regional stresses. Magnitude of stress based on energy released
Bore-hole breakouts – orientation and distortion of vertical petroleum/natural gas wells.
in situ measurements – hydraulic fracturing: cap a well and overpressure with water. Fracturing releases fluid pressure that reflects local stress field.
in situ borehole measurements of sd (s1 –s3) with depth.

35. Stress in the Earth
EQ focal mechanisms, fault planes are parallel to maximum shear stresses. Several focal mechanisms might constrain regional stresses. Magnitude of stress based on energy released
Bore-hole breakouts – orientation and distortion of vertical petroleum/natural gas wells.
in situ measurements – hydraulic fracturing: cap a well and overpressure with water. Fracturing releases fluid pressure that reflects local stress field.
World stress map and topography showing maximum horizontal stress.

36. Stress in the Earth
EQ focal mechanisms, fault planes are parallel to maximum shear stresses. Several focal mechanisms might constrain regional stresses. Magnitude of stress based on energy released
Bore-hole breakouts – orientation and distortion of vertical petroleum/natural gas wells.
in situ measurements – hydraulic fracturing: cap a well and overpressure with water. Fracturing releases fluid pressure that reflects local stress field.
Generalized pattern based on stress trajectories for individual plates.

37. Stress and strength at depth
Strength – the ability of a material to support different stress
Maximum stress before a rock fails
Strength curves: differential stress magnitude versus depth.
Regional with low geothermal gradient
Regional high geothermal gradients
The idea of a weak and highly fluid lower crust dominated tectonics research for the last 15 years.
There is now a school of thought that continental lithosphere consists of weak middle crust and a stronger lower crust, which in places may exceed underlying mantle strengths.
These arguments are based on inferences from earthquake distributions and estimates of elastic thickness of the lithosphere using topographic loads. These kinds of arguments are indirect and carry with them assumptions that have yet to be corroborated.
Nonetheless, they highlight how little we know about the composite rheology of the continents and the presence of weak layers that interrupt stronger load-bearing layers. Resolution of the rheological structure of continental regions will significantly improve our understanding of continental orogenesis in a wide range of tectonic settings.
Give some geologic examples?
This is important and will be on exam 1!

38. Stress and strength at depth
Strength envelopes with depth for continental crust and mantle illustrating competing interpretations for the relative strength of the lower crust and underlying mantle.
In all cases, upper crust strengths are represented by Byerlee's frictional strength and a thermally activated flow law for wet quartz.
Lower crust strengths are predicted by wet and dry rheologies for diabase (MD) and granulite (WC). Mantle strengths are given by wet and dry olivine rheologies.

39. Stress and strength at depth
Strength envelopes with depth for continental crust and mantle illustrating competing interpretations for the relative strength of the lower crust and underlying mantle.
In all cases, upper crust strengths are represented by Byerlee's frictional strength and a thermally activated flow law for wet quartz.
Lower crust strengths are predicted by wet and dry rheologies for diabase (MD) and granulite (WC). Mantle strengths are given by wet and dry olivine rheologies.