Tractions, stress state at a point and stress fields

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

Readings:


Key terms and concepts for this week:


Discussion question for end of week: Having discussed the complexity of stress fields. Now return to considering a stress state at a point, somewhere down in the crust of the earth. What geologic processes will naturally cause that stress state and hence the stress field for a rock body to evolve with time?

Forces and pressure

Force:

Pressure within a geologic context:


Tractions (stress state components)

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:



Stress state at a point.

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:

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).

Tensor description of stress at a point:


Types and components of a stress state

Types of stress states:

Deviatoric (and differential stress) vs. nondeviatoric component(s) of stress state:



Stress fields

Situations stress fields are expected in:

Examples:

Stress component changes modeled due to Hector earthquake slip. Source: Preliminary Report on the 10/16/1999 M7.1 Hector Mine, California Earthquake
Scientists from the U.S. Geological Survey, Southern California Earthquake Center, and California Division of Mines and Geology http://pasadena.wr.usgs.gov/hector/hector_srl.html

Graph showing how the maximum shear stress component grows with depth near the San Andreas. This will make even more sense when we explore Mohr diagrams. Source: USGS Arthur H. Lachenbruch and A. McGarr http://education.usgs.gov/california/pp1515/chapter10.html

Stresses associated with San Andreas fault. Source: USGS, Hauksson, E. http://erp-web.er.usgs.gov/reports/annsum/vol40/sc/g3028.htm

World stress map.

How are stress fields mapped?

Some references for further information:


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