STRUCTURAL GEOLOGY EXERCISES
with Glaciotectonic Examples – Part III

Plotting poles to planes

To simplify plotting planes on the stereonet, it is convenient to plot the pole to each plane. The pole is a line running through the center of the projection sphere and perpendicular to the plane—see Fig. 8-1. The pole forms a 90° angle with the strike line and a 90° angle with the dip line. Thus, the pole is always be found in the opposite quadrant of the stereonet from the dip of the plane.

Figure 8-1. Pole perpendicular to dipping plane in projection sphere (above), and the resulting stereographic projection of the plane and its pole (below).

1. Mark the plane's strike and dip compass directions on the outer edge of the stereonet.

2. Rotate the tracing so the strike mark points north and the dip mark falls on the E-W axis of the stereonet.

3. Count the dip angle outward from the center of the stereonet along the E-W axis toward the side opposite the dip, and mark that point.

4. Rotate the tracing back to its original position. The marked point represents the pole to the plane.

Constructing fold axes

Figure 8-2 illustrates a surface folded in a cylindrical style around three parallel fold axes. Poles perpendicular to the folded surface are indicated in various positions. Note that all of the poles are perpendicular to the fold axes, regardless of their positions on the folded surface. In fact, the poles define a plane which is perpendicular to the fold axes.

Figure 8-2. Poles on a series of northeast-plunging cylindrical folds (above), and the stereonet plot of those poles (below). On the stereonet, the poles fall on an arc that represents a plane perpendicular to the fold axis.

Problem

Figure 8-3. Denmark location map. The Limfjord estuary extends across the northern part of the Jylland pennisula. Stipling shows crystalline bedrock in Sweden.

Figure 8-4. Map of the Limfjord region of northern Jylland, Denmark showing zones of deformed Eocene strata (Fur Formation) in ice-pushed hills. The highest ridges exceed 70 m in elevation. Problem site is Ertebølle Hoved on east side of map.

	Large exposure of Fur Formation in Harhøj gravel pit on the island of Mors, Limfjord estuary, northwestern Denmark. Overturned syncline (left) with a core of glacial gravel leads into a sharply folded anticline (right) and another syncline. After folding, the structure was truncated by overriding ice.
	Fegge, a peninsula at the northern end of the island of Mors in the Limfjord (see fig. 8-4). Kite aerial photo (left) and ground shot (right) showing the cliff exposure on the side of the flat-topped hill. The cliff reveals folded and thrust bodies of Fur Formation.

1. Construct two stereonets. First plot all planes as arcs, and then plot each plane as a pole—see following examples—using either paper or Stereonet 11.

   	Begin by entering the 14 strike-and-dip values for a planar dataset. The planes plot as arcs on the stereonet. Notice two sets of planes representing two fold limbs. You may visually approximate the fold axis at the point where most of the planes intersect (red asterisk).
   	Next convert the strike-and-dip values in Table 8-1 into pole values. Strike - 90 = pole trend; 90 - dip = pole plunge. For example, site 1: 279 - 90 = 189 (trend) and 90 - 44 = 46 (plunge). Enter the pole values in a new linear dataset. You should get a plot with 14 points that represent poles to planes.

2. Find the plane (arc) along which these poles approximately lie. Sketch in this arc; it is normal to the fold axis.

3. Determine the trend and plunge of the pole to the plane sketched in step 2; this pole is the constructed fold axis.

4. Assuming that ice movement was more-or-less perpendicular to the fold axis, what direction of ice advance is indicated by the structural data at this locality?

5. Which late Pleistocene ice advance do you think deformed the Fur Formation at Ertebølle Hoved?

References

• Aber, J.S. and Ber, A. 2007. Glaciotectonism. Developments in Quaternary Science 6, Elsevier, Amsterdam, 246 p.
  
• Berthelsen, A. 1978. The methodology of kineto-stratigraphy as applied to glacial geology. Geological Society of Denmark, Bulletin 27, Special Issue, p. 25-38.
  
• Billings, M.P. 1972. Structural Geology. Prentice-Hall, Englewood Cliffs, New Jersey, 606 p.
  
• Gry, H. 1940. De istektoniske forhold i moleromraadet. Meddr. Dansk Geologisk Forening 9, p. 586-627.
  
• Gry, H. 1979. Beskrivelse til geologisk kort over Danmark, Kort bladet Løgstær, Kvartære aflejringer. Danmarks Geologisk Undersøgelse, I række, Nr. 26.

Stress and strain

A stress unit commonly used in structural geology is the kilobar (kb), which is 1000 bars or approximately 1000 atmospheres. For continental rocks of medium density, a pressure of 1 kb is achieved at a depth of 3½ to 4 km (2 to 2½ miles), and a pressure of 10 kb is reached at the base of continental crust. It was under such high-pressure conditions that plutonic and metamorphic rocks now exposed in mountain belts and shields were created.

The high pressure imposed on deeply buried rocks due to weight of the overburden is uniform and equal in all directions. A similar, uniform pressure is experienced by underwater divers. Such stress in crustal rocks is called lithostatic pressure, and it is determined solely by the thickness and density of the overburden. It is compressive or positive stress, in contrast to tension which is a negative stress.

Lithostatic pressure may produce strain in rocks due to simple compaction, but it probably cannot account for most folding and faulting. For this, differential stress is required. Differential stress may be developed by unequal normal stresses or by shear stress—see Figs. 9-1 and 9-2.

Figure 9-1. The stress ellipsoid: three normal (perpendicular) stress axes define the total stress applied to a rock body. õ1 is the maximum compressive stress, õ2 is the intermediate, and õ3 is the minimum. Normally, all three stresses are compressive; however, õ3 may be negative (tensional) in some cases. Where only lithostatic pressure is developed, the stresses are equal and the stress ellipsoid becomes a sphere.

Figure 9-2. Hard augen, such as feldspar, within a ductile schist subjected to shear stress parallel to foliation. The sense of shear on oblique microfractures of augen is opposite to overall sense of shear in the rock. T symbolizes shear stress. Adapted from Simpson and Schmid (1983, fig. 9).

           change in length
     e  =  ---------------- x 100, where e = percent strain.
           original length

The standard means of portraying the behavior of a stressed rock is with a stress-strain diagram, where strain is plotted on the horizontal axis and stress is plotted on the vertical axis—see Fig. 9-3. Strain is the percentage of shortening parallel to õ₁ and stress is the difference õ₁ - õ₃. As differential stress increases, most rocks initially behave in an elastic manner. In other words, strain is directly proportional to stress, and strain disappears when stress is removed. This results in a straight line on the graph.

Figure 9-3. Typical stress-strain diagram showing the behavior of a limestone subjected to increasing õ1 stress under a high confining pressure. Adapted from Davis (1984, fig. 5.25).

Rock failure

Figure 9-4. Fractures produced experimentally in a block of Solenhofen limestone under differential stress. Fracture sets A and B are conjugate shear fractures, C are extension fractures, and D are release fractures. Adapted from Hobbs et al. (1976, fig. 7.31).

• Shear fractures – plane of failure oriented about 30° from õ₁, 60° from õ₃, and parallel to õ₂. Most common type of fractures; two crossing sets may form.
  
• Extension fractures – plane of failure parallel to õ₁ and õ₂, and perpendicular to õ₃. Not common geologically; develop only where there is little or no confining pressure.
  
• Release fractures – plane of failure perpendicular to õ₁, and parallel to õ₂ and õ₃. Develop when stress load is released, as when deeply buried rocks are unloaded by erosion of overburden or removal of an ice sheet.

           õ1 + õ3     õ1 - õ3
     n  =  -------  -  ------- (cos 2Ð)
              2           2

           õ1 - õ3
     T  =  ------- (sin 2Ð)
              2

Figure 9-5. A - components of normal (n) and shear (T) stresses affecting a plane of potential failure within a rock. Ð is the angle between the maximum stress and the plane of failure. B - normal stress (gravity), shear stress (T), and angle of friction (ø) for a brick being dragged over a horizontal surface.

In order to drag the brick, a horizontal shear stress must be applied. The total stress operating on the brick is the shear stress plus the normal stress of gravity, and the total stress vector forms an angle (ø) with the vertical. This represents the angle of internal friction; its magnitude is a measure of frictional resistance to sliding of the brick. Now, suppose the brick is bonded to the underlying surface by mortar. An additional shear stress must be applied to break the bond before any movement could take place. This additional shear stress, labelled T_o, is the cohesive strength of the mortar.

Mohr diagram

Figure 9-6. Basic Mohr diagram showing normal stress (horizontal axis) and shear stress (vertical axis) developed on a plane at Ð = 30° (2Ð = 60°) with õ1 = 7 kb and õ3 = 2 kb. Solving the stress equations for this situation gives n = 3.25 kb and T = 2.17 kb.

A series of stress circles representing different test conditions for a particular rock may be plotted together to produce a Mohr failure envelope—see Fig.9-7. Each Mohr circle represents the õ₁ and Ð conditions of shear fracturing for a given confining pressure (õ₃). As confining pressure increases, the õ₁ stress and Ð angle at which failure occurs also increase. The points of rock failure on each circle define a line that is more-or-less straight at lower confining pressures and flattens out toward higher confining pressures. This line is the failure envelope; its intercept with the shear-stress axis gives cohesive strength (T_o), and its slope is the angle of internal friction (ø). Any point on or above the failure envelope represents stress conditions that would cause the rock to fracture.

Figure 9-7. Mohr failure envelope for Carrara Marble. To represents cohesive strength, and ø is the angle of internal friction.

ø = 90 - 2Ð, or Ð = (90 - ø)/2.

Problem

Figure 9-8. Map of Densmore Creek and vicinity showing local ice movement (arrows) along bedrock valley (dashed line) of Irondequoit Bay. Based on Andrews (1980, figs. 1 and 6).

Figure 9-9. Densmore Creek section showing thrust bedrock mass. Thrust zone is located within a brecciated shale unit. Based on Andrews (1980, fig. 2).

Genesse River gorge in western New York, near Rochester. Known as the Grand Canyon of the East, a thick sequence of upper Devonian clastic strata is exposed in the walls of the gorge.

A temperate glacier, which is separated from its bed by a film of meltwater, may develop basal shear stress up to 1 bar, and basal shear stress may reach 10 bars for glaciers frozen to the substratum (Weertman 1961). The lithostatic pressure developed at the base of a glacier is given by:

1. Intact sample-cores of the Irondequoit Limestone have an average unconfined compressive strength of approximately 1200 bars. Construct a Mohr stress circle for the Irondequoit Limestone.

2. Assume an average ø value of 36° for limestone (chemical sedimentary rock), and plot a straight-line failure envelope on the Mohr diagram. Determine the T_o and Ð values.

3. Use the stress equations to calculate T and n stress values for unconfined failure of the Irondequoit Limestone. Compare calculated values with graphical results.

4. Assume the glacier was 300 m thick over Densmore Creek when bedrock thrusting took place. Calculate the confining (lithostatic) pressure developed at the base of the ice.

5. Plot the basal stress conditions for both frozen and thawed glacier-bed situations on the Mohr diagram. Are the stresses developed for either type of glacier bed sufficient to fracture intact Irondequoit Limestone?

6. Based on your answer to question 5 and the geological setting (fig. 9-9), explain how ice shoving was able to thrust bedrock at Densmore Creek.

References

• Andrews, D.E. 1980. Glacially thrust bed-rock--An indication of late Wisconsin climate in western New York State. Geology 8, p. 97-101.
  
• Davis, G.H. 1984. Structural Geology of Rocks and Regions. J. Wiley & Sons, New York.
  
• Hobbs, B.E., Means, W.D. and Williams, P.F. 1976. An Outline of Structural Geology. J. Wiley & Sons, New York, 571 p.
  
• Jayaraman, A. 1984. The diamond-anvil high-pressure cell. Scientific American 250/4, p. 54-62.
  
• Kulhawy, F.H. 1975. Stress deformation properties of rock and rock discontinuities. Engineering Geology 9, p. 327-350.
  
• Simpson, C. and Schmid, S.M. 1983. An evaluation of criteria to deduce the sense of movement in sheared rocks. Geological Society of America, Bulletin 94, p. 1281-1288.
  
• Weertman, J. 1961. Mechanism for the formation of inner moraines found near the edge of cold ice caps and ice sheets. Journal of Glaciology 3, p. 965-978.