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:

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

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

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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: http://www.earthsciences.uq.edu.au/~rodh/animations/PureShearFoldedLines_ani.gif

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

Pure Shear: Random Ellipses Animation: http://www.earthsciences.uq.edu.au/~rodh/animations/PureShearRandomEllipses_ani.gif Shows the evolution of fabrics from a random fabric in a pure shear zone Animation: Rod Holcombe

Simple Shear Animation Simple shear flow showing changes in orientation and length of lines An example of a noncoaxial strain Animation: Rod Holcombe Animation: http://www.earthsciences.uq.edu.au/~rodh/animations/SimpleShearFoldedLines_ani.gif

Simple Shear Animation of Points Animation: Rod Holcombe Traces movement of material Animation: http://www.earthsciences.uq.edu.au/~rodh/animations/SimpleShearPoints_ani.gif

Simple Shear: Random Ellipses Shows the evolution of fabrics from a random fabric in a pure shear zone Animation: Rod Holcombe Animation: http://www.earthsciences.uq.edu.au/~rodh/animations/SimpleShearRandomEllipses_ani.gif

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

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

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 = 

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

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.

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)

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

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

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

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%.

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

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

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

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

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

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.

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)

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’

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

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

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

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:

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