Deformation and Strain Concepts

Deformation

• Deformation is a change in line length, angles between lines, or volume.
• Deformation involves a component of shape and volume change.
• Distance between particles changes, as well as the angle between particle lines.
• Strain is measured by changes in length, angle, or volume.

Measures of Strain

• Strain can be recognized as a change in line length, angles between lines, or volume.
• This deals with shape and volume change.
• Distance between some particles changes.
• Angle between particle lines may change.
• The quantity of strain is measured based on changes in length (ε), angle (γ), or volume (εv).
• Extension (elongation) ε: change in length per length = (l - l₀) / l₀, where l is the final length and l₀ is the original length
• Shortening is negative extension (e.g., ε < 0).
• ε = -0.2 represents 20% shortening
• Example: If a belemnite's original length (l₀) is 10 cm and its new length (l) is 12 cm, then ε = (12 - 10) / 10 = 0.2 or 20% (extension).

Other Measures of Longitudinal Strain

• Stretch (S) = l / l₀ = 1 + ε = √λ
• X = √λ₁ = S₁
• Y = √λ₂ = S₂
• Z = √λ₃ = S₃
• Principal stretches represent the semi-lengths of the principal axes of the strain ellipsoid.
• Example: If l₀ = 100 and l = 200, then ε = (200 - 100) / 100 = 1 or 100% extension and S = 200/100 = 2 (stretched twice).
• Quadratic elongation λ = s² = (1 + ε)²
• Reciprocal quadratic elongation λ' = 1 / λ

Measures of Longitudinal Strain (ε)

• ε = (l - l₀) / l₀ = Δl / l₀
• Δl = l - l₀
• Positive ε: elongation
• Negative ε: shortening

Volumetric Strain (Dilation)

• Volumetric strain (εv) gives the change in volume compared to the original volume.
• εv = (v - v₀) / v₀ = Δv / v₀ (v₀ is original volume and v is the final volume).

Lines of No Finite Elongation - Infe

• Drawing a circle on a surface, deforming it, and redrawing the circle in the deformed state creates an ellipse.
• The resulting ellipse has a long axis perpendicular to the strain ellipse.
• This is called the reciprocal strain ellipse.
• The original circle and the reciprocal strain ellipse intersect along the lines of infe.

Extension and Shortening Fields

• Fields in the strain ellipsoid, separated by the line of no incremental longitudinal strain.
• Boundaries between shortening and extension are always at 45 degrees either side of incremental principal axes of shortening and extension.
• Material lines migrate through boundaries during strain events.
• Rock particles or planes can change from shortening (folding) to extension (boudinage) without changing orientation of principal strain axes.

Strain Ratio

• Strain ratio compares the length of the long axis to the short axis of the strain ellipse.
• It's calculated as the length of the long axis divided by the length of the short axis.

Types of Homogeneous Strain at Constant Volume

• 1. Axially symmetric extension: Extension in one direction and equal shortening in perpendicular directions
  • Strain ellipsoid is a prolate spheroid (cigar-shaped).
• 2. Axially symmetric shortening: Shortening in one principal direction and equal extension in perpendicular directions
  • Strain ellipsoid is an oblate spheroid (pancake-shaped).
• 3. Plane strain: Intermediate axis of the ellipsoid has the same length as the diameter of the initial sphere. Shortening or extension occurs parallel to other principal directions.
  • Strain ellipsoid is a triaxial ellipsoid.
• 4. General strain: Extension or shortening in all principal directions.
  • Strain ellipsoid is a triaxial ellipsoid

Simple Shear

• Three-dimensional, constant-volume, plane strain.
• A single family of parallel planes remains undistorted.
• Orientation of material lines changes along two principal axes.
• Analogous to moving one side or a deck of cards.
• The top and bottom surfaces of a cube stay the same length during deformation.
• Sides of the parallelogram increase in length as deformation progresses.

Pure Shear

• Three-dimensional, constant-volume, irrotational, homogeneous flattening.
• Parallel lines initially oriented normal to the principal axes maintain their orientation during deformation.
• A sphere changes into a pancake shape, and a cube into a tablet shape.
• Shortening along one principal direction is accompanied by elongation along another perpendicular direction.

Strain Markers

• Deformed features used to measure strain in rocks.
• Have the same mechanical characteristics as the original rock type.
• Good markers include pebbles, ooids, fossils, vesicles, pillow basalts, burrows.

Line Changes When a Circle Becomes an Ellipse

• Initially circular objects become elliptical when homogeneously deformed.
• By measuring the stretch and extension of the long and short axes of the ellipse, we can determine the amount of lengthening and shortening that occurred.
• Assume no change in volume: πab = πr².
• Calculate strain and stretch based on the ratio of ellipse axes.

Angular Shear

• Angular shear (Y) measures the change of angles between lines.
• Needs knowing a line originally perpendicular to the line in question.
• Angular departure from initial perpendicular relationship, with magnitude in degrees.
• Sign convention: CW (+), CCW (-).

Calculate Angular Shear

• Calculate the angle change between intersecting lines in pictures.

Shape and Intensity - Flinn Diagram

• A plot of axial ratios used in strain analysis.
• Horizontal axis is Y/Z (intermediate stretch/minimum stretch).
• Vertical axis is X/Y (maximum stretch/intermediate stretch).
• Parameters: a = X/Y = (1+ε₁)/(1+ε₂) and b = Y/Z = (1+ε₂)/(1+ε₃).

How Rocks Deform

• Elastic deformation: temporary change in shape/size that recovers when the stress is removed.
• Elastic limit: the point where permanent deformation begins.
• Plastic deformation: permanent change in shape/size, not recoverable.
• Brittle deformation: loss of cohesion, usually along sub-planar surfaces.
• Ductile deformation: slippage of atoms without loss of cohesion.

Brittle vs. Ductile

• Brittle rocks fail by fracture at <3-5% strain.
• Ductile rocks can sustain 5-10% strain before fracturing.

Strain Rate

• The time interval needed to accumulate a certain amount of strain.
• Change in strain with time, measured as the rate of change of strain.
• Measured as (dl/l)/dt with units of 1/time, e.g., 1/second
• Rate affects whether deformation will be brittle or ductile.

Factors Affecting Deformation

• Temperature: Higher temperature promotes ductility, while lower promotes brittleness.
  • Confining Pressure: Higher pressure makes materials more ductile.
  • Strain Rate: Faster strain rates lead to more brittle behaviors.
  • Rock Type: Different rock types have different degrees of competency and therefore deform differently.

Effects of Rock Type on Deformation

• Competent rocks: Strong under high stress.
• Incompetent rocks: Weak, deform under moderate-low stresses.
• Rock competency resistance to flow.
• Examples of incompetent rocks: slate, phyllite, schist, salt, shale
• Competent examples: granite, basalt, gneiss, quartz sandstone, metaquartzite.

This quiz covers the fundamental concepts of deformation and strain, including changes in line length, angles, and volume. You will learn how to measure strain and calculate extension or shortening based on initial and final lengths. Get ready to deepen your understanding of these critical mechanical properties.