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

Readings:

• Chapt. 3 in Van Der Pluijm, B. & Marshak, S., 2004, Earth Structure, Norton Press.
• 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.

• 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.

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?

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 can be resolved from a force oblique to plane of interest.

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.
• this is a special simple stress state - hydrostatic stress state.
• geologic pressures: pore fluid pressures, magmatic pressures, 'rock pressure' = nondeviatoric component.
• related strain? change in volume, no distortion (unless material is anisotropic with respect to mechanical properties).

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. 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.

• 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 combinations of the coordinate axes. 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 is traction plane as described by axis perpendicular to it. 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 vs. nondeviatoric component(s) of stress state:

• mathematical description
  • snd = (s1 + s2 + s3)/ 3
  • s1d = s1 - snd
• relation to ellipsoid = departure from sphericity.
• geologic significance:
  • nondeviatoric controlled by crustal depth, roughly equivalent to confining pressure.
  • deviatoric by tectonic setting.
  • larger the deviatoric the greater the potential for deformation.

Situations stress fields are expected in:

• in association with topography.
• 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.
• in association with plate boundaries.

Sources of measurements from which to map stress fields:

• earthquakes and focal plane solutions.
• bore hole break out tests/measurements.
• overcoring.
• borehole deformation.
• neotectonic structures.

Examples:

World stress map.

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.