1. Deformation and Strain – Finite Homogeneous Strain Analysis
Structural Geology
Deformation and Strain – Finite Homogeneous Strain Analysis
Lecture 9 – Spring 2016
Many methods have been used to measure finite strain. Some of proven more useful than others, and have become commonly accepted. Computers and peripherals have greatly changed and enhanced our ability to analyze large data sets. They also allow us to make many mistakes, based on our lack of knowledge of the technique and the assumptions being made in gathering and analyzing the data. The computer adage GIGO applies very strongly here. Uncritical reliance on numbers obtained from some form of strain analysis may lead to acceptance of conclusions wholly unjustified by the data.
The object of strain analysis is to determine both the magnitude and orientation of the three principal strain axes. Absolute measurements of strain are difficult, so it is much more common to obtain strain ratios, such as X/Y, X/Z, and Y/Z. We need to ask two very important questions about our strain data (and these questions apply to many other forms of field data, as well).
A. How complete is our strain measurement? Does it represent the entire rock, or region, that we are examining? If not, in what ways it is limited?

2. How Representative is an Analysis?
B. How representative is our analysis? Did we sample all major environments in our rock? Measuring the strain in the clasts may not be at all representative of the strain in the matrix. When we analyze entire regions, the possibilities for omitting important samples increase many fold over single rock analysis.
The latter question is especially important in dealing with heterogeneous samples (most samples in geology are heterogeneous). We need to recognize two types of strain markers, passive and active.
Figure 4.18 in text

3. Passive Markers
Passive markers are those that behave in a manner indistinguishable from that of the whole body
They have no mechanical significance
The markings on the sand formed a rectangular grid prior to deformation
Drawings on a deck of cards are perfect examples of passive markers, since the drawings play no role in the deformation of the card deck.
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4. Dinstinguishable Inclusions
In nature, we look for distinguishable inclusions which are part of a matrix
Individual quartz grains in a quartzite are one example
Image:

5. Active Markers
Active strain markers are bodies that have contrasting mechanical properties from the matrix they are found in
Typically, they behave differently, sometimes much differently, than the matrix
Garnet porphyroblasts in a mica schist are a common example
Since we seek to measure changes in the shape of objects, it is useful to select as markers objects whose initial shape we believe we know.
We can split these objects in to categories, initially spherical and initially non-spherical objects.
Image:

6. Spherical Markers
The best example of spherical markers are ooids, which grow by accretion in warm, high-energy marine environments
Since they grow in a fluid, they are typically almost perfect spheres
Figure 4.19a in text

7. Spherical Ooid
Carmel Formation carbonate, exposed near Kane County, Utah, is Middle Jurassic in age
Ooids coat the exterior of the rock
The photo here shows an ooid with other smaller ooids surrounding the large one
Width of view: 1.2 mm (100x)
Image:

8. Flattened Ooid Ooids visible in this section appear to be strained
This photo shows more ooids together in a shallow marine environment
Notice the small bivalve fragments above the large ooid
Width of view: 1.2 mm (100x)
Image:
Figure 4.19 b and c
b = 25% and c = 50% shortening

9. Vesicles
Vesicles and amygdules in basalt flows are often used as spherical markers
Photo from NASA shows a vesicular basalt with nearly circular vesicles
Vesicles near the top of a flow are more apt to be circular – deeper vesicles will flatten under load pressure
Image:

10. Vesicles
However, if the basaltic lava is flowing it is common for these features to be elongated
Image:
Vesicles in this basalt are
elongated by flow

11. Amygdule
Like vesicles, amygdules may
be used as spherical markers
The oval feature in this photomicrograph is an amygdule: a formerly open vesicle which has been filled with a secondary mineral(s) precipitated from low-T ground waters which have penetrated into the rock
In this case, the amygdule is probably filled with a zeolite mineral
Image:

12. Strain from Initially Spherical Objects
Once chosen, the analysis method is straight-forward.
Three sets of mutually perpendicular sections are cut, and the ratio of long-short axes for each sectional ellipse is measured.
If we chose our sections to be perpendicular to the principal stress axes, we obtain X/Y, Y/Z, and X/Z directly. Otherwise we need to correct the results using appropriate trigonometric formulae.
Figure 4.20 in text

13. Measurements from Section
We measure and find X/Y = 2
Measurements of Y/Z give the value 1.2
Therefore we conclude that X:Y:Z = 2.4/1.2/1
We predict that X/Z should be 2.4
This should be checked by actually measuring the section

14. Assume Zero Dilation
Assuming that Δ = 0 (no dilation), we can calculate X, Y, and Z:
Δ + 1 = X • Y • Z = 1
Using X = 2Y, and Z = Y/1.2 (from our data) we get
2Y • Y •(Y/1.2) = 1.7Y3 = 1 Y = 0.8
X/Y/Z = 1.7/0.8/0.7
If a rock were to undergo 50% compression, such as slates are thought to experience in their formation from shales, what would the ratio be?

15. 50% Compression Now Δ = -0.5, so 2Y • Y •(Y/1.2) = 1.7Y3 = 0.5
= X • Y • Z
2Y • Y •(Y/1.2) = 1.7Y3 = 0.5
Y3 = 0.29 and Y = 0.67
X/Y/Z = 1.3/0.7/0.5
We can see that the strain ratios remain the same, regardless of volume change.
The magnitude of the individual axes lessens as volume decreases, and becomes greater as volume increases, but this has no effect on observed values of k, K, i, or I.
We cannot get any indication of volume change from our markers.
Initially non-spherical objects are easiest to analyze if they are ellipsoids.

16. Cobble Clast
Eocene age Stadium Conglomerate, San Diego county, California
Cobbles are distinctive purplish-pink rhyolite clasts sometimes referred to as Poway clasts
The source area for these cobbles is believed to have been in Mexico
Clasts in a conglomerate can often be approximated by an ellipsoid. We have seen that ellipsoids are represented by second-order tensors.
Thus, a clast can be associated with a second-order tensor. Our finite strain is also represented by a second-order tensor.
It is a property of tensors that tensors of the same rank can simply be added together by adding each element to its corresponding element in the other tensor. Thus, if we know the original tensor and the finite strain tensor, it is straight-forward to see what shape changes will occur.
How do we get the original tensor?
Two methods of estimating the initial shape of objects are available. They are:
Center-to-Center Method
Rf/φ Method
Image:

17. Example
Distance between object centers is measured as a function of the angle with a reference line
Figure 4.23a in text
The Center-to-Center method relies on the principle that the distances between centers of deformed objects are related in a systematic way to the orientation of the superimposed stain ellipsoid.
In the direction of extension, the X axis, the centers will spread. In the direction of shortening, the Y or Z axes, the centers will get closer together.
If we measure the center distances in a number of arbitrary directions, we find a maximum value and a minimum value. The maximum corresponds to the X direction, and the minimum to Y (in a two dimensional ellipse). Computers have greatly enhanced our ability to employ this method.

18. Plot for Ooids Empty area is an ellipse with axes in ratio X/Y
The angle of the long axis with the reference line, φ, can be measured
The results can be normalized by dividing the center-to-center distance between two objects by the sum of their mean radii
The Rf/φ depends on systematic shape changes that objects undergo in response to stress.
We use a given reference section, and measure the length of the long and short axes of each sectional ellipse in the section.

19. Calculations
Calculate the ratio of the long axis/short axis, denoting the result Rf
Plot Rf versus φ, defined as the angle between an arbitrary reference line and the long axis of each sectional ellipse

20. Example Plot We obtain a cluster of points, as shown
We can relate Rf to the initial ratio of objects, denoted R0, and the strain ratio
The strain ratio is denoted Rs, and is equal in two dimensions to X/Y

21. If R0 > Rs Assuming R0 > Rs, we get:
Rf max = Rs • R0
Rf min = R0/Rs
Rs = ( Rf max/ Rf min)½
The maximum initial ratio comes from:
R0 = (Rf max • Rf min )½

22. If Rs > R0 Rf max = Rs • R0 (No change) Rf min = Rs/R0
Rs = (Rf max • Rf min )½
R0 = (Rf max/Rf min )½

23. Locating X and Y
To analyze a three-dimensional case using either method, we must use at least two perpendicular sections. Thus strain analysis is quite time consuming, although computers have greatly eased the computational and plotting time.
A change from a sphere to an ellipsoid involves change in both angular relationships and lengths. Methods that can measure either quantity are useful.
Angular changes can most easily be measured with objects possessing two easily identified lines at right angles.
The position of Rf max relative to the arbitrary reference line locates the X axis, with the Y axis perpendicular to X
Figure 4.24b in text

24. Ordovician Brachiopod
The two shells of brachiopods are of unequal size, which distinguishes them from bivalve mollusks
Brachiopods flourished during the Paleozoic, and became extinct at the end of the era
Some fossils are very well suited for this purpose. Fossils must contain easily recognizable elements the are originally perpendicular. Brachiopods and trilobites are two good examples.
When several deformed objects that preserve naturally symmetric geometric relationships are available, two pieces of information can be obtained:
Orientation of the strain axes
Strain ratios
Image:

25. Brachiopods on Bedding Plane
Figure 4.25a shows a number of brachiopods on a bedding plane
An arbitrary reference line is drawn
The angle of the hinge line of each fossil with the reference is denoted angle alpha
Then the angle between the second direction and the hinge is measured - the amount it differs from 90º is angle ψ (angular shear)
Independent of the size of fossils and presence of different species, as long as the symmetry elements remain constant, since only angular changes in the objects are measured.

26. Breddin Curve Angular shear versus angle alpha
Named after Hans Breddin, Professor of Structural Geology and Tectonics, Aachen University
Angular shear versus angle alpha
This line will have two places where the ψ value is zero, corresponding to the position of two principal strain axes

27. Position After Deformation
The equation relating the original position of a line and its position after deformation is related to the strain ratio:
tan φ = X/Y • tan φ'
X/Y = tan φ/ tan φ'
φ is the angle between the original line and the reference
φ' is the same angle after deformation

28. Deformation at Maximum Angular Shear
The maximum angular shear occurs at φ = 45 º
Substituting tan φ = 1 (at 45 º) in the previous equation gives
Y/X = tan φ' max
We measure ϕ’ from the Breedin plot, and take its tangent to determine Y/X

29. Direct Determination
Fossils that do not show any angular shear may be used to directly determine the orientation of the principle strain axis
Which fossils here would be useful?
Symmetry elements coincide with principle stain axis
In the drawing, fossils A and B meet the criteria

30. Length Changes
Length changes are the easiest to determine, but require special conditions, so that they cannot often be applied
The most common usage is with deformed belemnites

31. Belemnite Photo
Part of a stretched belemnite with quartz (more translucent) and calcite (closer to the belemnite pieces) precipitated infill
Photo from the root zone of the Morcles nappe in the Rhone valley, Switzerland by Martin Casey
Image: belpart500x336.jpg
Source:

32. Stretch and Quadratic Elongation
Two equations, for stretch and quadratic elongation, can be used:
s = l/l0
λ = (l/l0)2
Figure 4.26 in text

33. Measurement Technique
For the two fossils, we measure φ to a reference line (one of the principal strain axes, if known)
Or we measure at least three fossils, measuring l1..li where li is the last measurable section
For the two fossil case, we need some indication of the orientation of one of the principal strain axes within the plane being measured
Figure 4.27 in text

34. Simultaneous Equations
We then set up simultaneous equations:
λ' = λ1' cos 2φ' + λ2'sin2 φ' , one equation for each element
λ'A = (l/(l1 + l2 + l3 +l4 + l5 + l6))2
Solving the simultaneous equations we can get X/Y, and Δ, from Δ + 1 = X • Y
Remember X2 = λ1 and Y2 = λ2

35. Calcite Twinning
Mechanical twinning takes place when the resolved shear stress acting on the future twin boundary exceeds a ctical value. During twinning, the crystal lattice rotates in the direction that produces the smallest linear displacement of atoms.
The method is called the Calcite Strain-Gauge method. There is more information in chapter 9, pages
We can utilize x-ray crystallography or magnetic susceptibility measurements to rapidly determine the predominant orientation of the grain axes. We can then model the data, making some assumptions.
The final result depends on the assumptions made, so care it is needed. Careful work can give insight into the shape of the strain ellipsoid, and we can then attempt to verify this against other data.
Diagram:
Thin section:
Rhombohedral cleavage in calcite, often due to deformation
This type of twin results from deformation

36. Calcite Twin Data
Plot of the e1 strain orientation, along with associated magnitudes, from calcite crystals in eastern North America
Figure 4.29 in edition 1 of the text
There is an obvious correlation with the Appalachian-Ouachita thrust zone. Even well inland from this front, the correlation is still evident, although the magnitudes are much less.

37. Incremental Strain History
Other methods involve studying various indicators, such as:
Orientation of foliation in shear zones
Fibrous over-growths on grains
Fibrous vein-filling
These methods provide information about the incremental strain history, as opposed to data about the finite strain history, which all previous methods gave us

38. Information from Strain Studies
Strain studies can provide several important types of information
Recognition of regions of high strain, which are often shear zones
Information, in the form of constraints, on the strain necessary to form different types of geologic structures, such as folds, foliations, and lineations
Information on the duration over which strain accumulates, which allows us to determine the strain rate, 
 = e/t