Structural Geology
Lecture 8
Strain Markers
(Strain Analysis)

Strain in rocks is measured using objects of known initial shape.  The initial shapes can vary from round crinoid columnals to the irregular shape of breccia fragments.  In two dimensions initially round objects include scolithus tubes, crinoid columnals, reduction spots, vesicules, concretions, and oolites.  Initially elliptical objects include congolmerate pebbles.  Fossils are usually irregular in shape but some such as leaves, brachiopods and trilobites may have a bilateral symmetry.  More complicated shapes include the spiral of the ammonite, cavities of the coral, belemnites, and the branches of the graptolite.  Other markers are deposited with centers at uniform distances from their nearest neighbors.  In this lecture we are going to consider the analysis of four situations where rock strain can be inferred from the shape or position of deformed markers.
The least complicated strain analysis is the measure the elliptical shape of initially circular objects.  This approached was used to map strain over 45,000 km2 of the Appalachian Plateau.  The Devonian Catskill Delta of the Appalaachian Plateau contains many beds in which crinoid columnals parallel the bedding plane.  Their elliptical shape on pavement surfaces testifies to the deformation of the Appalachian Plateau.  The state of strain parallel to bedding in these outcrops is represented by three components:  principal strains, e1 and e2, and the orientation of the strain ellipse as measured using one of the principal strain axes relative to north.  In the case of the Plateau the strike of the long axes of the ellipse relative to north is designated as q.  Strain on the plateau may have a rotational component but this was impossible to measure.
The actual measurements consisted of two numbers: the axial ratio and the orientation of the long axes.  The axial ratio is a measure of ellipticity (R):

	R  =  (1 + e1 )/(1 + e2 )
                         
In a paper published in Geology, Engelder and Engelder (1977) presented the a map showing the strike of q and the value of R which varied between 1.1 and 1.2.  The values of e1 and e2 are not possible to measure directly nor can they be calculated from the above equation because the strain on the Plateau is a volume loss strain.
The second type of deformed object that proves to be a useful strain marker is the ellipse with an initial ellipticity Ri.  Upon deformation the shape of the final ellipse Rf is a function of the orientation and ratio of the initial ellipse ralative to the deformation (Figure 8-1).  In the deformed state the orientation of the long axis of the ellipse relative to some marker is F.  Data on Rf and F can be graphed to form Rf /F plots.  These plots can then be compared with standard Rf /F reference curves for different values fo initial ellipticity Ri and the strain ellipse Rs.  The Rf /F plots have two shapes depending on whether  Ri > Rs or Ri < Rs.  In the former case the data envelope is symmetric about the orientation of the long axis of the strain ellipse and shows maximum and minimum Rf values.  In the latter case the data envelope is closed and the data points shown a limited range of orientations defining the fluctuation F.

(Fig. 8-1)

The center-to-center technique allows the assessment of the bulk strain of a rock.  This is commonly known as the Fry technique.  Sometimes rigid inclusions such as fossils do not deform uniformly with the matrix of the rock.  The best technique for determining the  behavior of the matrix is the measure the distance between neighboring grains.  Figure 8-2 shows an example where the distance  and azimuth between various grains has been measured.  From the distance/azimuth plot the ratio of the strain ellipse and its orientation can be determined.  The Fry technique may also be used to take advantage of the characteristic center-to-center distance of the deformed matrix of a rock.

The most beautiful and, hence, popular strain markers are the trilobites and brachiopods with lines of symmetry.  If the line of symmetry lies parallel to a principal strain direction, then the angular shear strain of the principal lines in the fossil are zero.  In this orientation the deformed fossil is still in a symmetrical form.  If the initial lines of symmetry are not parallel to principal strain directions the lines of symmetry appear to shear during deformation.  Final shape of the brachiopod or trilobite is an oblique form.  The oblique forms can be either right or left handed depending upon the deflection of the symmetry axis.  The symmetrical forms can appear in a narrow or broad form depending the initial orientation of the fossil relative to the shortening direction (Figure 8-3).

(Fig. 8-2)

The initial shape of the fossil can be characterized in terms of its length to breadth:

	r0 = l0/b0                                                                             

The strain of the rock can be determined by using the ratio of the final length and breadth.  In the narrow form

	rn = ln/bn  =  (l0R)/b0                                                          

and in the broad form 
	
	rb = lb/bb  =   (l0R)/b0                                                      

 Even though the original shape ratio r‚ is unknown we can calculate the strain ellipse and r‚ from the following equations.

	R = (rn/rb)1/2

	r0  =  (rbrn)1/2.

(Fig. 8-3)

The strain ellipsoid represents strain in three dimensions where the three axes of finite strain are known as (1 + e1) > (1 + e2 )  > (1 + e3).  The longest extension is in the e1 direction  (Fig. 8-4).  The principal strain ratios are defined as
 
	Rxy = (1 + e1 )/(1 + e2)                                                           

and 

	Ryz = (1 + e1 )/(1 + e3).                                                        

The plot of Rxy versus Ryz became known as the Flinn Graph (Fig. 18-4).  Flinn suggested the parameter k to describe the position of the strain ellipsoid in the Flinn Graph.

	          Rxy  -   1
	k   =   ________ 
	          Ryz  -   1                                                                      

If k is greater than one then the strain ellipsoid is in the form of a cigar whereas if k is less than one the strain ellipsoid is in the form of a pancake.

(Fig. 8-4)