GG 5210

Problem Set 7

Autumn 2006

Due November 20, 2006

 

Practical Strain Measurements and Rheology Problems

 

1.  Do the problems 2-19, -20, -21, and -22 and 3-1, 3-2, 3-3, and 3-4 from Turcotte and Schubert.

 

2.  Strain Measurements Of An Active Glacier (assuming infinitesimal strain).

 

The Teton glacier flows eastward from an ice field beneath the Grand Teton along a linear valley that forms a U-shaped gorge.  In 1929 a steel post was installed on the surface of the glacier at its midpoint in the eastern end of the valley.  Four more posts were emplaced equidistant 10 meters to the NW, NE, SE and SW of the center post.

 

By 1966 the post array has moved a substantial distance down center of the valley, but the posts remain in their NW-SE and NE-SW alignments.  However, while the NW and SE posts are now 15 meters from the central post, those to the NE and SW are only 6 meters from the center.

 

a.       What is the orientation of the strain ellipse developed over the period 1929-1966?  Justify your answer.

 

b.      Calculate e1, e2, S1, S2. l1 and l2 over the ~40 year period.

 

c.       Assuming the strain field to be homogeneous with the surface of the glacier forming a principal plane of the strain ellipsoid, and that no volume change has occurred, what is the value of linear extension perpendicular to the glacier’s surface? Recall that volumetric dilatation, Dv = (1 + e 1)(1+ e 2)(1+ e 3) - 1

 

d.      Radar sounding showed the glacier to be 100 meters thick beneath the peg array in 1929.  What thickness would you expect it to have beneath the array in 1966?

 

Actual picture of the Teton Glacier in its modern recessional phases.

3. The next problem focuses on understanding the effect of composition, temperature, and strain rate, on flow and fracture of common rocks in the earth by solving the rheological equations for various rock types, temperatures and specified strain rates.

 

a.  For this problem, first read and write an abstract of the paper below (there is a copy in 706).  There are several other papers on the subject, but the program used in part 2 is based on this paper.

 

Smith, R. B. and R. L. Bruhn, 1984, Intraplate extensional tectonics of the western U. S. Cordillera: Inferences on structural style from seismic reflection data, regional tectonics and thermal-mechanical models of brittle-ductile deformation, J. Geophys. Res., 89, 5733-5762.

 

b. Using the MatLab program on the CMES system, "ELASVISCOSTRESS," based on the theory in Smith and Bruhn (1984), calculate the stress profiles for various depths, tectonic regimes, and heat flow provinces. The program and all files are accessible on the college system. 

 

To run the program:

                    /data/grizz4/cmpuskas/VESTRESS/stress

 

Explanation:  The ELASVISCOSTRESS program calculates the shear strength vs. depth for various rock properties (temperature gradient, composition, and strain rate.)  Determine the failure envelopes for three brittle failure criteria: compressional, strike-slip, and extensional regimes.

 

  1. Run the program for materials that you judge best represent:

 

                     i.   the upper crust,

                   ii.   the lower crust, and

                  iii.   the upper mantle


b.  Do the problem for three thermal regimes with heat fluxes of:

 

                     i.            a cool stable continental regime, of 50 mWm-2,

                   ii.            an extending active tectonic regime, with a heat flux of 90 mWm-2, and

                  iii.            an active volcanic system with120 mWm-  2. 

 

Plot out your results as strength vs. depth.

 

c.  For each thermal and tectonic regime, describe what the rupture and flow laws imply about long term tectonics and earthquakes.

 

4.  Consider a 2D section of isotropic, linearly elastic granite crust with a thickness = 5 km, pore pressure = 0, ν = 0.25, E = 106 bars, and ρ = 2.6 gm/cc. We are given that the crust has stretched 1 m over a distance of 30 km to 30.001 km.

  1. How much has the crust thinned (or necked) in the middle?
  2. Calculate σ22 at the base of the crust assuming σ22 is strictly gravitational.
  3. Use the equations for plane strain to evaluate σ11 at the top and bottom of the crust.