1. Structural Geology
Deformation and Strain – Homogeneous Strain, Strain Ellipsoid, Strain Path, Coaxial and Noncoaxial Strain
Lecture 7 – Spring 2016
Deformation
Deformation means the collective displacements of all points in a body. It can be separated into three distinct components:

2. Translation Translation - Movement of a body from one place to another
Figure 4.2d in text

3. Rotation Rotation - Turning around an axis
Neither translation not rotation causes a change in shape
Figure 4.2c in text

4. Distortion Distortion - Change in shape of the object
Strain is a description of the change in shape of a body acted upon by stress, so strain is a description of distortion
Figure 4.2b in text

5. Change in Shape
Sometimes deformation involves all three components, sometimes just two or even one
Figure 4.2a in text
Strain is a description of the change in shape of a body acted upon by stress, so strain is a description of distortion.

6. Deformation
Sometimes deformation involves all three components, sometimes just two or even one
The displacement is described by vectors associated with each component, and the total is described by the displacement field
The sum of the vectors depends on the order is which each operation is applied, so the displacement field cannot be represented by a vector

7. Undeformed State
Determining the translation, rotation, and dilation all depend on original knowledge of the system in question
We need to know the position to determine translation or rotation, and the volume to determine dilation
Since these are often unknown, that leaves strain

8. Undeformed Shape
Strain depends only on knowing the original shape (but not the size) of an object
Since we often know the shape, it is possible to determine strain

9. Homogeneous Strain
Strain may be either homogeneous or heterogeneous. In order to be homogeneous, one of the following conditions must apply:
Two or more straight lines remain straight
Two or more originally parallel lines remain parallel
Circles becomes ellipses, and spheres become ellipsoids

10. Homogeneous Strain Image
Figure 4.3 in text
The undeformed state is on the left, the deformed state on the right
Square goes to a parallelogram
Circle goes to an ellipse
Each card slides the same amount

11. Heterogeneous Strain
If none of these conditions exists, the strain is heterogeneous
Heterogeneous strain is harder to describe than homogeneous strain
Often, we can separate heterogeneous strain into several homogenous components, or approximate the total deformation by homogeneous components
So, for the moment, we will consider only homogeneous behavior

12. Heterogeneous Strain Image
Figure 4.3 in text
The undeformed state is on the left, the deformed state on the right
Cards at top slide more than cards at bottom, which distorts the parallelogram and ellipse

13. Material Lines
The square to the rectangle and circle to ellipse illustrate a homogeneous deformation
Four material lines are originally present, at 45° angles, with two sets each perpendicular
Two lines (solid) remain perpendicular after strain
Dashed lines represent lines which do not remain perpendicular

14. Strain Ellipsoid
In three dimensions, there will be three lines that remain perpendicular after strain, defining a strain ellipsoid
They are called the principal strain axes They are denoted X, Y, and Z, and use the convention:
X > Y > Z

15. Incremental Strain Routes
Different routes to the same final state
Figure 4-5 in text

16. Noncoaxial Strain
Principal axes of strain ellipsoid move through material during deformation
At any given incremental strain principal axes lie in a different physical part of the deforming material
Figure 4.6a in text

17. Coaxial Strain
If the same material lines remain the principal strain axes throughout each increment, then it is coaxial strain accumulation
Principal axes of strain ellipsoid do not rotate through material, therefore the deformation is "coaxial"
Figure 4.6b in text

18. Pure Shear Animation Compression along the vertical axis
An example of a coaxial strain
Animation: Rod Holcombe, Associate Professor of Structural Geology , University of Queensland
Animation:

19. Pure Shear Animation of Points
Animation: Rod Holcombe
Traces movement of material

20. Pure Shear: Random Ellipses
Animation:
Shows the evolution of fabrics from a random fabric in a pure shear zone
Animation: Rod Holcombe

21. Simple Shear Animation
Simple shear flow showing changes in orientation and length of lines
An example of a noncoaxial strain
Animation: Rod Holcombe
Animation:

22. Simple Shear Animation of Points
Animation: Rod Holcombe
Traces movement of material
Animation:

23. Simple Shear: Random Ellipses
Shows the evolution of fabrics from a random fabric in a pure shear zone
Animation: Rod Holcombe
Animation:

24. General Shear
If a combination of simple shear and pure shear exists, it is known as general shear, and is noncoaxial
Figure 4.7 shows two types of general shear
4.7a is transtension
4.7b is transpression

25. Kinematic Vorticity Internal vorticity can be quantified
A number called the kinematic vorticity, or Wk, may be defined as:
Wk = cos α
where alpha is defined by figure 4-8b, above
The lines shown are the flow lines of particles during progressive strain accumulation

26. Types of Strain
We can use the kinematic vorticity to separate different types of strain:
Type of Strain
Value of Wk
Pure shear
General Shear
0 < Wk < 1
Simple Shear
Wk = 1
Rigid-body rotation
Wk = 

27. Illustration of Strain Types
General Shear
Pure Shear
Simple Shear
Rigid-body rotation

28. Strain Path: Coaxial Superposition
I – Lines continue to be extended
II – Lines continue to be shortened
III – Initial shortening followed by extension
Figure 4.9a in text
Strain paths are created by superimposing a series of strain increments as seen in Figure 4.9.
In a single strain ellipse, we have regions of extension separated from regions of contraction by two lines of zero length change. Superimposing a series of strain ellipses, we find we get additional regions with initial contraction followed by extension, or initial extension followed by contraction. Thus deformational histories may be complex, and outcrop patterns may appear contradictory, showing both extension and contraction.

29. Strain Path: Noncoaxial Superposition
I – Lines continue to be extended
II – Lines continue to be shortened
III – Initial shortening followed by extension
IV – Initial extension followed by shortening
Figure 4.9b in text
For non-coaxial superposition, we add regions of initial extension followed by shortening (IV)

30. Quantification of Strain
We need to be able to quantify the strain we observe. There are three basic measures:
Longitudinal strain - length change
Volumetric strain - volume change
Angular strain - angular change

31. Longitudinal strain
Longitudinal strain is defined as the change in length divided by the original length, and is expressed by the elongation, e, which is dimensionless

32. Elongation
Contraction is indicated by negative numbers, and expansion by positive numbers
E = elongation
S = stretch
Figure 4.11a in text

33. Shortening and Extension
The elongations associated with each principal strain axis are labeled e1, e2, and e3, and have the numerical relationship:
e1 ≥ e2 ≥ e3
Contraction is often simply called shortening, and expansion is called extension.
Number may be given as percentages, ∣e∣ x 100%.

34. Volumetric Strain
The equation for volumetric strain (Δ) is identical in form to that for longitudinal strain:
Δ = (V-V0)/V0 = (δV)/V0

35. Angular Strain
Angular strain is defined as the change in the angular relationship between two lines that were initially perpendicular
Two quantities may be used

36. Angular and Shear Strain
Angular shear = Ψ (psi) = change in angle
Shear strain = γ = tan Ψ
Both of these are dimensionless
Figure 4.11b in text

37. Quadratic elongation
In the Mohr circle for strain, we can use a quantity called the quadratic elongation, λ, defined as:
λ = (l/l0)2 = (1 + e)2
The stretch is defined as the root of the quadratic elongation:
s = λ0.5 = l/l0 = 1 + e

38. Relation to Strain Axes
We can relate these quantities to the principal strain axes as follows:
X2 = λ1, Y2 = λ2, Z2 = λ3, where X  Y  Z
X = s1, Y = s2, Z = s3

39. Illustration of Relationships
A circle, with radius = 1, is drawn
An arbitrary line, OP, is chosen with and angle of φ (phi) between OP and X
Figure 4.11c in text
Figure 4-11c illustrates a two dimensional example of distortion relationships. In this figure, we have a circle with unit radius.
An arbitrary line, OP, is chosen with and angle of ϕ between OP and X.

40. After Distortion The circle is distorted and has new axes X and Z
The line OP is elongated to OP’ with a new angle φ’ with X
Figure 4.11d in text
The relationship between these various quantities is shown in figure 4_11c. The circle, with radius = 1, is distorted and has new axes X and Z.
Ψ (psi) is the angular shear , which is the difference between φ’ and φ (phi)

41. Trigonometric Relationships
After distortion, the line OP’ makes an angle of φ’ with the X axis. So:
tan φ’ = Y/X• tan φ = (λ2/λ1)0.5 • tan φ
This can be rearranged:
tan φ = X/Y • tan φ’ = (λ1/λ2)0.5 • tan φ’

42. Dilation
Dilation is a special type of deformation in which length changes are proportionate along different axes
We can look at a two-dimensional case:
Area of ellipse = π (X •Y)
Area of circle = π r2

43. Zero Dilation If r = 1 (unit circle), then the area of the circle is π
Assuming zero dilation on going from the circle to the ellipse,
π (X •Y) = π r2 = π (X • Y) = 1

44. Rewriting Equations
This gives Y = 1/X, and we can rewrite the earlier equations:
tan φ’ = X-2 • tan φ or
tan φ = X2 • tan φ’
If Y = 1, the three-dimensional case reduces to the two-dimensional case
Structural geologists often assume that Y = 1 for this reason

45. Definition of Natural Strain
We defined the elongation as:
e = (δl)/l0 for any given finite length change
We can rewrite this using calculus for infinitesimal elongations:

46. Ratio of Natural Logs
We can integrate this expression to get: Or