**Lecture index: Forces
and pressure. / Tractions. / Stress state at a point. / Types
and components of a stress state. / Stress
fields. **

**Readings:**

- Chapts. 4 and 5 in Fossen, Structural Geology.
- Coblentz, D. D. & Richardson, R. M., 1995, Statistical trends in the intraplate stress field; JGR, v. 100, 20,245-20,255. Skim this for about a half an hour to get out the main points.

Key terms and concepts for this week:

**force****equation for slip on a plane****pressure****examples of geologic pressures****traction****shear and normal tractions****compressive vs. tensile tractions****units of traction (Kbar, Mpal, psi)****stress state at a point****sigma 1, 2, 3 and Cauchy's theorem****stress ellipsoid and stress tensor****use of stress ellipsoid to compute shear and normal tractions on a plane****varying stress states****nondeviatoric vs. deviatoric components of stress state****examples of stress fields****stress state as an approximation of really small stress fields.****conjugate geometry of initial failure in a stress field.**

**Force:**

- F= m x a
- gravitational acceleration: 9.8 m/sec2
- vector quantity: orientation and size.
- can be applied to any plane.
**normal**and**shear**components on a plane can be resolved from an oblique force (see diagram to the right).- shear component promotes slip on the plane and the normal component inhibits slip on a plane, and the ratio of the two at which slip occurs describes the 'friction' on the plane.

**Pressure within a geologic context:**

- describes multitude of forces at a point within a fluid.
- limit of F/area as area goes to 0
- fluids can not withstand a shear stress for a significant period of time, therefore if static non-flowing, all force vectors equal in size, and all must be normal vectors acting perpendicular to any given plane. Hence can be described by one number.
- pressure turns out to be a special simple stress state -
**hydrostatic stress state**. - geologic pressures: pore fluid pressures, magmatic pressures, 'rock pressure' = nondeviatoric component (see below).
- related strain? change in volume, no distortion (unless material is anisotropic with respect to mechanical properties and then??).

**Definition**: Given
__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__ required to keep that body in place,
if the material on one side of the plane were instantly removed.

One way to possibly understand a traction is to consider the simple 2-D elastic sheet depicted below that is under external forces exerted by attached springs. The springs could be pulling to different amounts in the different conditions. If you cut a small slit through a point and a line marker then the slit will open up.The marker at an angle to the slit could also show offset depending on the orientation of the slit. The slit is equivalent to the plane. The force that you would have to apply to close the slit and restore the marker would be the traction that existed within the sheet prior to making the cut.

Some traction traits:

- a stress traction can be at an angle to the given plane, and therefore, can be resolved into shear and normal components.
- there are two equal opposing vectors for any given plane.
- unlike a force, a stress traction is not
a simple vector,
**it is defined for a specific plane**and is meaningless just as a vector. - units Kbar, Mpal, psi; basic units gm cm-1 sec-2; 1 pascal = 1N/m2 = 1 kg m sec-2/ m-2

For a point repeat the above process of defining a traction for all the infinite number of planes that pass through that point. This array of forces and planes is the stress state at a point.

__Cauchy's theorem____ for the stress state at a point:__

- by necessity have three orthogonal tractions, all of which are normal tractions.
- these three are sigma 1, 2, and 3. 1 and 3 are the maximum and minimum, respectively
- sigma 1, 2, and 3 are unique in that they are pure normal tractions, all others have some shear component.

There are three ways to describe the stress state at a point we will explore: stress ellipsoids, tensors and Mohr diagrams. The later we will deal with in a subsequent lecture.

**Stress ellipsoid**:
As depicted below, the **three principal stresses are the ellipsoid
axes**, and they each operate on a perpendicular **principal
plane**. The vertical traction in this example is working on
the pink horizontal plane.

**Graphical method**
for finding stress traction on a plane given sigma 1,2,3 and using
stress ellipse (Powerpoint demonstration).

- a) draw ellipsoid and traction of interest.
- b) draw a circle that just encompasses the ellipsoid.
- c) draw a line from the traction tail, parallel to the minor axis of the ellipsoid, to the outer circle.
- d) from the resulting point of intersection draw a line to the joint ellipse-circle center.
- e) the perpendicular to that line is the plane the traction acts on.

**Tensor description of stress at a point:**

- the appropriate tensor in this case is a 3 by 3 array of numbers.
- first define a coordinate system with x, y, z axes. z by convention is often vertical, x and y are cardinal directions.
- the tensor describes the tractions operating on the three planes defined by the combinations of the coordinate axes (x and y, y and z, and z and x). Each plane can have a normal and two shear tractions. For example, the plane defined by the x axis being perpendicular to it can have a normal traction in the x direction, and a shear traction in the y direction, and a shear traction in the z direction. Describe all 9 tractions for the given coordinate system and the stress state at a point is completely described. The diagonal shear components are equal (i.e. the tensor is symmetric) so only 6 different numbers need to be specified.
__traction subscript nomenclature__: The first subscript refers to the traction plane as described by axis perpendicular to it. In the diagram below the upper face of the small cube is the z plane. The second is the traction direction. If the subscripts are same then it is a normal traction (it acts perpendicular to plane), if not a shear traction (it acts parallel to the plane).- a traction sign convention: if both outward pointing normal to plane and the stress traction are same sign, then the traction is positive. If different then negative. Compressive tractions are always negative in this convention. Different contexts, different sign conventions.
- Some of the tensor stress components on this small cube in x,y,z space are shown. For practice you can fill in the labels for the other tensor components on the z face, and you can draw and label the shear tensors on the x face. Note that normal tensors are shown in red, and shear tensors in blue.

**Types of stress states:**

__uniaxial__- test conditions approximate if have unconfined rock cyclinder.__biaxial__: where encountered?__triaxial__, found within earth:- axial prolate: sigma one > sigma two = sigma three. e.g. compaction and sedimentary burial.
- axial oblate; sigma one = sigma two > sigma one, e.g. internal to a rising salt diapir (constrictive pipe flow).
- hydrostatic.

__Deviatoric (and differential stress)__
vs. __nondeviatoric__ component(s) of stress state:

- mathematical description
**nondeviatoric component**= sigma.nd = (sigma one+ sigma two + sigma 3)/ 3**deviatoric stress**is a tensor where the nondeviatoric component is subtracted from each of the three principal stresses**differential stress**= sigma one - sigma three.

- what is the deviatoric stress component for a hydrostatic stress state?
- for differential stress there is a relation to the ellipsoid shape = greater departure from sphericity as deviatioric and/or differential component increases.
- the nondeviatoric component captures something about the "size" of ellipsoid.
- geologic significance:
- nondeviatoric controlled partly by crustal depth and rock density, roughly equivalent to confining pressure.
- deviatoric stress and differential stress controlled by tectonic setting.
- larger the deviatoric and/or the differential stress the greater the associated shear stresses and the potential for deformation.

- description by Terry Engelder of distinction: http://geophysics.ou.edu/geomechanics/readings/deviatoric.html

**Situations stress fields are expected in:**

- in association
with topography.
- e.g. exfoliation joints.

- lithostatic gradient for vertical traction.
- concentration of stresses at tip of a crack or a fault.
- in association with a large density anomaly (e.g. rift 'pillow').
- in association with point intrusions or impacts (radial and concentric geometry).
- in association with plate boundaries.

**Examples:**

**How are stress fields mapped?**

- earthquakes - first motion studies and focal mechanisms
- explanation at USGS web site.
- IRIS YouTube video on - http://www.youtube.com/watch?v=MomVOkyDdLo .

- borehole deformation (more information on - (http://astro.temple.edu/~davatzes/Methods/Methods/Borehole_Analysis.html).
- overcoring - YouTube video describing how one borehole instrument for stress measurement by overcoring works.
- in-situ hydrofracking.
- neotectonic structures.

**Some references for further information: **

- Barth et al., Tectonic stress field in rift systems – a comparison of Rhinegraben, Baikal Rift and East African Rift
- Tingay et al., 2010, Present-day stress field of Southeast Asia, Tectonophysics, 482, 92-104.
- Zoback, M. L., and Zoback, M. D., 1989, Tectonic stress field of the continental United States, in Pakiser, L. C., and Mooney, W. D., Geophysical framework of the continental United States: Boulder, Colorado, Geological Society of America Memoir 172.

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.