Readings:

• Chapt. 9, Mohr Circle for Stress, P. 70-82 in Means, Stress and Strain, Springer Verlag. Read this carefully!
• Chapt. 27 Energy Consumed in deformation, p. 263-272 in Means, Stress and Strain, Springer Verlag. Skim this to get the basic idea.

Key terms and concepts:

• equation for shear and normal stress on a plane for 2 principle stresses
• stress plots on a Mohr diagram
• nondeviatoric vs. deviatoric components on a Mohr diagram
• components and variation of failure envelopes on a Mohr diagram
• failure on a new plane vs. slip on existing plane
• history of stress evolution on a Mohr diagram
• influence of pore pressure
• 3-D representation of stresses on a Mohr Diagram
• evolution of failure envelope with slip
• limitations of Mohr diagrams
• equation for energy of slip on an existing surface
• creep vs. stick-slip behavior
• elastic rebound model for stick-slip events
• energy associated with seismic events
• energy/work requirements (6 components) for long term fault motion
• least work hypothesis applied to thrusts
• influence of topography
• importance in fault reactivation

Demonstration by coincidence:

• Given a 2-D stress tensor, what are the equations for shear and normal stress tractions on a plane at some angle to the x axis? This is the general 2-d case.

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  The adjacent figure shows how through force balancing we can derive the desired equations using vector resolution and summation. Click on it to get an expanded (readable) version. Your book also has a version of this same exercise.

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• What are the equations if sigma one and sigma three are defined parallel to the coordinate axes and some trig identity manipulaton is done?
• What are the equations for points on circle that is centered on the x axis, and where the point on the circle is identified by a line from the center out that has an angle of 2*theta from the x axis?
• Normal Stress  = (S1 + S2)/2 + ((S1 - S2)/2) cos 2A
  Shear Stress = ((S1 - S2)/2) sin 2A, where S1 and S2 are the two principal stresses, and where A is the angle between the plane and S1 and 2A is the angle from the x axis in a counterclockwise direction of a radius of the Mohr circle.
• The exact coincidence of the two sets of equations indicates that a circle is a plot of the shear and normal stresses on specified planes (specified by their angle from sigma one) given the value of the principal stresses. It is a graphical device/space with which to 'visualize' stress states plotted as circles, or to solve stress problems.

Stress plots on Mohr diagram: useful to develop visual intuition with regard to stress states in Mohr space through the following rules.

• maximum shear stresses on planes 45 degrees from sigma-one.
• far out and big circles: nondeviatoric and deviatoric stress components in Mohr space.
• symmetry and conjugate planes in Mohr space.
• influence of fluid pressure and definition of effective stress.
• a brief introduction into 3-D Mohr space.

Now we can explore how mapping stress states in Mohr space can help understand faulting. Remember that there is a relationship between the shear and normal tractions on a plane and strength. The stronger the plane is the higher the shear traction that is needed to cause slip, and the higher the normal traction is the higher the shear traction that is needed. For many surfaces a simple linear relationship exists:

critical shear stress for slip = a strength intercept + the normal traction multiplied by a strength factor.

Given a straight line relationship the strength factor is equal to tan (Ø) where Ø describes the angle from horizontal of the straight line relationship.

You can then add the linear slip criteria (the shear and normal tractions along a plane that will cause failure) into the Mohr diagram in form of failure envelope. If the a shear stress associated with the stress state intersects the envelop then failure, slip will occur.

Diagram below is of three stress states, one where failure will not occur, one were shear failure is imminent, and one where tensile failure is imminent. Failure envelopes are not always linear.

For pristine failure: for pristine faults in homogenous rock or where layering is close to principal plane (e.g. flat lying strata), which does not influence path that rupture and shear take. .

• conjugate angles as function of slope. 90° - failure slope = conjugate angle.
• reverse approach - use field observations to define failure slope angle.
• Anderson's classification, and sigma-horizontal and sigma vertical.

Modeling different failure histories:

• sigma-1 constant and sigma-3 decreased could micmic crustal extension.
• sigma-1 growing and sigma-3 constant could mimic crustal thrusting.
• could build in fluid pressure increase and consequent shifting of sigma-1 and sigma-3 and failure.
• erosion or sedimentation other factors that should effect the vertical principal traction.
• stress drop and reversal of principal stresses if sigma one and sigma two are close.
• note that rocks are stronger at depth (require larger deviatoric stress) with respect to brittle failure.

For slip on pre-existing anistropy:

• stable vs. unstable orientations for slip on anistropy.
• rotation of faults.

Different failure envelope geometries:

• granite, average slope 49, n = 2
• sandstone, average slope = 35.5, n = 4
• shale, average slope = 15, n = 3
• chalk, avearge slope = 0, n = 1
• importance is in variation.
• curved slopes for some rock types (e.g. chalk)

Tensile cutoff:

• rocks much weaker in tension (easier to split than crush).
• how to get there (in tensile range): surface environments or fluids.

Kink bands - on the down side of the failure envelope.

• conjugate kink band geometry and kinematics
• representation on Mohr diagram and extended failure environment.
• potential as paleostress indicators, and paleo-environment (e.g. at transition).

Limitation of Mohr diagrams.

• describes stress state at a point. What if have inhomogeneity along surface? Neighbor relationships?
• slider model for earthquakes and this way to chaos.

It is useful to discuss faulting from an energy perspective. It can give insight into why certain fault geometries may be more common. It is useful in modeling structural evolution. It links seismicity and faulting.

Basic physics:

• work = force X distance
• units: 1 Newton-m = 1 joule, 1 dyne-cm = 1 erg, 1 joule = 10E7 ergs
• energy is potential to produce work
• kinetic energy = K = .5 x mass x v2
• energy is conserved.
• least path work path is favored by nature (?), but there is contingency (prior history).

Slip on a fault, seismic perspective:

• elastic energy released as seismic waves.
  • log-log scale - USGS site on earthquakes.
  • USGS site describing how much energy is released.
  • USGS site on formulas for measuring earthquake magnitudes.
• energy required for overcoming surface friction. Work done results in heat.
• energy of slip = product of slip area * distance moved * shear stress.
• surprising lack of thermal signature associated with big faults - 2 possible explanations:
  • low-strength surfaces?
  • fluid advection?
• energy associated with change in potential energy (uplift or subsidence).

Faulting, a long term (structural) perspective:

• Work requirements:
  • initiation and propagation of rupture.
  • sliding along surface (creep versus stick-slip).
  • potential energy (can be gained or lost).
  • fragmentation and formation of fault rocks
  • hanging-wall deformation.
• There are trade offs; e.g., for a thrust fault a low dip will decrease the amount of uplift, but it will increase the fault surface area.

Much potential in this line of research!

• Elliott, D., 1976, The energy balance and deformation mechanisms of thrust sheets; Phil. Trans. R. Soc. London, A, vol 283, p. 289-312. As one of the initial papers to discuss the energetics of faulting in detail it provides some of the basic concepts and formula.
• Mitra, G. and S. E. Boyer, 1986, Energy balance and deformation mechanisms of duplexes; Journal of Structural Geology, vol. 8, p. 291-304. This paper very nicely uses energy considerations to explain why certain thrust duplex geometries and development histories are more common than others. To set the stage the authors discuss the various energy sinks in duplex formation and make quantitative estimates of their relative demands.
• Hatcher, R. D. and R. T. Williams, 1986, Mechanical model for single thrust sheets Part I: Taxonomy of crystalline thrust sheets and their relationships of the mechanical behavior of orogenic belts; Geological Society of America Bulletin, vol. 97, p. 975-985.. As part of their analysis the authors consider various energy sources available for thrust sheet emplacement, tectonic compression and gravity forces due to topographic and density gradients. They conclude that for most major overthrusts the former is necessary and the most signficant although, the later can contribute.

Copyright Harmon D. Maher Jr., This may be used for non-profit educational purposes as long as proper attribution is given. Otherwise, please contact me. Thank you.