Deformation and Strain Concepts

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Questions and Answers

What does longitudinal strain measure?

Change in angle between lines

Change in particle distance only

Change in line length (correct)

Change in volume of an object

If a belemnite has an original length of 10 cm and is now 12 cm, what is the longitudinal strain?

50%

20% (correct)

-20%

100%

Which of the following expressions represents the relationship between stretch and longitudinal strain?

s = ℓo / ℓ

s = 1 + ε (correct)

s = ℓ + ℓo

s = ε / ℓo

What is the numerical value of the strain if shortening of an object is 10%?

-0.10 Signup and view all the answers

If a linear object originally measures ℓo = 200 cm and stretches to ℓ = 400 cm, what is its stretch?

2 Signup and view all the answers

What does the quadratic elongation represent?

s^2 Signup and view all the answers

Which of the following strains is described as negative?

Negative extension Signup and view all the answers

How is volumetric strain represented mathematically?

ε = ΔV / Vo Signup and view all the answers

What strain rate corresponds to a 30% extension over a million years?

9.5 x 10-15 s-1 Signup and view all the answers

Which factor increases rock ductility by activating crystal-plastic processes?

Temperature Signup and view all the answers

Which type of rock is typically classified as incompetent?

Slate Signup and view all the answers

What happens to rocks when stress is applied rapidly?

They tend to be brittle Signup and view all the answers

Which of the following correctly defines competent rocks?

Rocks that deform only under great stresses Signup and view all the answers

What does axially symmetric extension produce in terms of strain ellipsoid shape?

Cigar shaped Signup and view all the answers

In axially symmetric shortening, what relationship holds true for the principal directions $ ilde{\lambda}$?

$\lambda_1 = \lambda_2 > 1 > \lambda_3$ Signup and view all the answers

What is the shape of the strain ellipsoid in plane strain?

Triaxial ellipsoid Signup and view all the answers

Which condition specifies that shortening occurs in one principal direction during axially symmetric shortening?

$\lambda_1 = \lambda_2 > 1 > \lambda_3$ Signup and view all the answers

In terms of shape, what do the slims resulting from axial symmetry resemble?

Hotdog Signup and view all the answers

What is the relationship between $ ilde{\lambda}_3$ and the extensions during plane strain?

$\lambda_3 < 1 < \lambda_1 = \lambda_2$ Signup and view all the answers

What describes the condition where the intermediate axis of the ellipsoid remains the same length as the initial sphere?

Plane strain Signup and view all the answers

Which type of strain leads to an oblate ellipsoid shape?

Axially symmetric shortening Signup and view all the answers

What is general strain also known as?

Triaxial strain Signup and view all the answers

What change occurs to a cube during a simple shear event?

It is transformed into a parallelogram. Signup and view all the answers

Which axis remains unchanged during pure shear?

Y-axis Signup and view all the answers

What aspect differentiates simple shear from pure shear?

Simple shear involves a change in the orientation of material lines. Signup and view all the answers

How is shear represented in the context of deformation?

By the number of units along one shear zone width. Signup and view all the answers

During homogeneous flattening, what shape does a sphere transform into?

A pancake-like shape Signup and view all the answers

Which characteristic is true regarding the area during simple shear?

Area remains the same as the initial box. Signup and view all the answers

What does the symbol $ u$ represent in the context of simple shear?

Angle of shearing Signup and view all the answers

What is the main characteristic of elastic deformation?

Temporary change that recovers when the force is removed Signup and view all the answers

What defines the elastic limit in rock deformation?

The point beyond which plastic deformation occurs Signup and view all the answers

Under which conditions do rocks exhibit ductile versus brittle behavior?

Ductile behavior occurs only at high temperatures and pressures Signup and view all the answers

Which equation represents Hooke’s Law for longitudinal strain?

$\sigma = E\epsilon$ Signup and view all the answers

Which of the following statements is true about brittle rocks?

They typically fail by fracture at low strain levels Signup and view all the answers

What is the significance of Young’s modulus in the context of rock deformation?

It represents the constant of proportionality between stress and strain Signup and view all the answers

How is strain rate defined in geological terms?

The time required to achieve a certain amount of strain Signup and view all the answers

Which statement accurately describes the nature of viscoelastic materials?

They exhibit both elastic and plastic behavior under varying conditions Signup and view all the answers

What is the term used for the mechanism of distortion when a rock body transforms into a distorted state?

Plane Strain Signup and view all the answers

Which of the following is NOT a good strain marker for measuring deformation in rocks?

Granite Signup and view all the answers

How is the angular shear strain described?

It measures the change in angles between lines. Signup and view all the answers

In the Flinn diagram, what do the axes represent?

Maximum and minimum stretches Signup and view all the answers

What must be known to use strain markers effectively in determining strain?

The original shape of the feature being measured Signup and view all the answers

Which formula expresses the relationship in the Flinn diagram?

$a = X/Y = (1 + ε1)/(1 + ε2)$ Signup and view all the answers

What is the significance of the stretch (S) and extension (ε) in describing rock deformation?

They describe the amount of lengthening and shortening in the rock. Signup and view all the answers

Which characteristic must strain markers have to be effective for comparison?

They should share the same mechanical characteristics as the original rock. Signup and view all the answers

Study Notes

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.

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Description

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.