- 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
- loading paths
- 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

- 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.
*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.*

- 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?
- Can either designate the orientation of the plane of interest by: a) the angle between sigma one and the pole to that plane (using the pole to designate the plane in a similar way as to when defining stress tensor components), or, b) the angle between sigma one and the plane in the 2-D plane working in. For a given plane and sigma one and sigma-three, the second angle equals 90 degrees minus the first angle. The first is the most common convention.
**Normal Stress = (S1 + S2)/2 + ((S1 - S2)/2) cos 2A**(using A as defined in option a) above)**.**For option b) above**Normal Stress = (S1 + S2)/2 - ((S1 - S2)/2) cos 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.

Shear Stress = ((S1 - S2)/2) sin 2A,- 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.

*This diagram defines the position of the plane using option b above. *

Applet to plot Mohr diagrams given basic input.

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

- maximum shear stresses on any plane for a given stress state is always a plane at 45 degrees from sigma-one (however, this is not typically the plane of failure).
- the larger the circle the larger the deviatioric and differential stress components of the stress state.
- the farther to the right the circle the larger the non-deviatoric component.
- fluid pore pressure determines the effective stress, and increasing pore pressures cause the Mohr circle to shift to the left (reduces the nondeviatoric component), while the deviatoric component remains constant.

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 frictional strength. Stronger planes require
higher the shear tractions to slip, and
higher normal tractions require higher shear traction to slip. For many surfaces a simple linear relationship exists, which is known as the **failure envelope**:

**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. For many geologic materials Ø is around 30 degrees. Given that both shear stress and normal stress are the crucial variables it is possible to plot this envelope on the Mohr diagram. It is also useful to know that some materials can have non-linear failure envelopes.

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 shear stress associated with the stress state intersects or exceeds those associated with the envelope 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.* *Note that in this diagram, the lower failure envelope associated with negative shear stresses is not shown, but it should be there. *

**For pristine failure**:
given an intact homogenous rock or where layering is close
to principal stress plane (e.g. flat lying strata often are in this position) slip can occur on a newly formed plane (and not follow a pre-existing weakness. For this situation the following is the case:

- theoretically, the stress state can never rise above the failure envelope because failure will occur.
- failure alters the stress state, relieving some of the stress (in an amount known as the
**stress drop**). Thus, the circle shifts so that all of its points are below the failure envelope. - conjugate angles are a function of the failure envelope slope:conjugate angle = 90 - failure envelope slope.
- can use the reverse approach - use field observations to define failure slope angle.
- Anderson's classification of faults is consistent with vertical and horizonal principal stresses and pristine failure.
- note that rocks are stronger at depth (require larger deviatoric stress) with respect to brittle failure.

**Modeling different failure histories** **(loading paths)**:

- keeping sigma-1 constant (and vertical) and sigma-3 decreasing with time could micmic crustal extension.
- having sigma-1 growing and sigma-3 constant (and vertical) could mimic crustal thrusting.
- one can build in fluid pressure increase and consequent shifting of sigma-1 and sigma-3 and failure (this is why injection of fluids into the ground can trigger earthquakes).
- erosion or sedimentation other factors that should effect the vertical principal traction can be incorporated into your loading path, as can thermal stress components.
- considering things in 3-D, a stress drop can cause reversal of principal stresses (and produce two directions of faulting) if sigma one and sigma two are close.
*The diagram to the right models one of the loading paths described above, given a progression forward in time from case 1 to case 4 (this diagram was produced with the Excel sheet that can be found at the link above, and this sheet can be used for modeling other loading paths). Can you pick out which loading path it represents?*

**For slip on pre-existing anistropy**:

- Consider the Mohr diagram below with the red lines representing a failure envelopes, but a
**failure envelope for a specific plane or anisotropy in the rock**(e.g. a joint surface or slatey cleavage). In this case the stress circle can have portions above the failure envelope, because the orientation of the plane relative to the principal stresses is critical. Consider, for example, the plane represented by the green line on the plot below. The associated shear and normal stress tractions associated with this plane (on the circle) are well below the failure envelope and so it is stable. If the plane was instead oriented in the range between the two orange lines (labeled as the unstable range) the shear stress would be above the failure envelope and slip should have occurred. - faults can rotate with time and thus can move from the unstable to the stable range (the stresses remaining in the same orientation).
- this is probably the more typical situation.

**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, which can be considerable within the class of one rock type.
- 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 (into the tensile range): surface environments or fluid pressure.

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

*Photo to the right is of a kink band in Hecla Hoek phyllites of Midterhuken, Svalbard. It shows a dextral sense of offset.*- conjugate kink band geometry and kinematics.
- representation on Mohr diagram and extended failure environment.
- above is a sketch showing one proposal for how the failure envelope changes at the brittle-ductile transition. In this case the conjugate angle of the failure planes (kink bands) would be around 120 degrees.
- 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.
- 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.