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Recall: Pressure and Volume Change

- The 3D stresses are equal in magnitude in all directions (as radii of a sphere)
- The magnitude is equal to the mean of the principal stresses. The mean stress or hydrostatic component of stress:

P = (s1 + s2 + s3 ) / 3

- Leads to dilation (+ev & -ev); i.e., no shape change.

ev = (v´-vo) / vo = v/vo [no dimension]

- Where v´ and vo are final and original volumes

Isotropic Stress …

- Fluids (liquids/gases) are stressed equally in all directions (e.g. magma). e.g.:
- Hydrostatic
- Lithostatic
- Atmospheric
- All of these are pressure due to the column of water, rock, or air:

P = rgz

- Where z is thickness, r is density; g is the acceleration due to gravity

Strain or Distortion

- A component of deformation dealing with shape and volume change
- Distance between some particles changes
- Angle between particle lines may change
- The quantity or magnitude of the strain is given by several measures based on change in:
- Length (longitudinal strain): e
- Angle (angular or shear strain):
- Volume (volumetric strain): ev

Longitudinal Strain

- Extension or Elongation, e: change in length per length

e = (l´-lo) / lo = Dl/ lo[dimensionless]

- Where l´ and lo are the final &original lengths of a linear object
- Note: Shortening is negative extension (i.e., e < 0)
- e.g., e = - 0.2 represents a shortening of 20%

Example:

- If a belemnite of an original length (lo) of 10 cm is now 12 cm (i.e., l´=12 cm), the longitudinal strain is positive, and e = (12-10)/10 * 100% which gives an extension, e = 20%

Other Measures of Longitudinal Strain

- Stretch: s = l´/lo = 1+e = l [no dimension]

X = l1 = s1

Y = l2 = s2

Z = l3 = s3

- These principal stretches represent the semi-length of the principal axes of the strain ellipsoid. For Example:

Given lo = 100 and l´ = 200

Extension: e1 = (l´-lo)/ lo = (200-100)/100 = 1 or 100%

Stretch: s1 = 1+e1 = l´/lo = 200/100 = 2

1 = S12 = 4

i.e., The line is stretched twice its original length!

Other Measures of Longitudinal Strain

- Quadratic elongation:

l = s2 =(1+e)2

- Example:

Given lo = 100 and l´ = 200, then l = s2 = 4

- Reciprocal quadratic elongation:

l´ = 1/ l

- NOTE: Although l´is used to construct the strain Mohr circle, l can be determined from the circle!

Other Measures of Longitudinal Strain

- Extension: e =(l´-lo)/lo = dl/lo
- Natural (logarithmic) strain, =Sei(l´< l <lo)

l=l´l´

= l/lo or 1/lo l

l=lolo

NOTE: 1/xx = ln x

After integration, and substituting l’ and lo, we get:

= ln l’–ln lo = ln l´/lo

= ln s = ln(1+e)= ln (l)1/2

= ½ ln l

Volumetric Strain (Dilation)

- Gives the change of volume compared with its original volume
- Given the original volume is vo, and the final volume is v´, then the volumetric stain, ev is:

ev =(v´-vo)/vo = v/vo[no dimension]

Shear Strain

- Shear strain (angular strain)g = tan
- A measure of change in angle between two lines which were originally perpendicular. g Is also dimensionless!
- The small change in angle is angular shear or

Displacement

- The change in the spatial position of a particle within a body of rock
- The shortest vector joining a point (x1,x2,x3) in the undeformed state to its corresponding point (x´1,x´2,x´3) in the deformed state
- It has three components (in 3D): u,v, and w, in the x1,x2,and x3 directions, respectively

Progressive Strain

- Any deformed rock has passed through a whole series of deformed states before it finally reached its final state of strain
- We only see the final product of this progressive deformation (finite state of strain)
- Progressive strain is the summation of small incremental distortion or infinitesimal strains

Incremental vs. Finite Strain

- Incremental strains are the increments of distortion that affect a body during deformation
- Finite strain represents the total strain experienced by a rock body
- If the increments of strain are a constant volume process, the overall mechanism of distortion is termed plane strain (i.e., one of the principal strains is zero; hence plane, which means 2D)
- Pure shear and simple shear are two end members of plane strain

Strain Ellipse

- Distortion during a homogeneous strain leads to changes in the relative configuration of particles
- Material lines move to new positions
- In this case, circles (spheres, in 3D) become ellipses (ellipsoids), and in general, ellipses (ellipsoids) become ellipses (ellipsoids).
- Strain ellipsoid
- Represents the finite strain at a point (i.e., strain tensor)
- Is a concept applicable to any deformation, no matter how large in magnitude, in any class of material

Strain Path

- Series of strain increments, from the original state, that result in final, finite state of strain
- A final state of "finite" strain may be reached by a variety of strain paths
- Finite strain is the final state; incremental strains represent steps along the path

Strain Ellipse

- It is always possible to find three originally mutually perpendicular material lines in the undeformed state that remain mutually perpendicular in the strained state.
- These lines, in the deformed state, are parallel to the principal axes of the strain ellipsoid, and are known as the principal axes of strain
- However, the length of the material lines parallel to the principal strains have changed during strain!
- The principal stretches: X > Y > Z
- Theprincipal quadratic elongations 1 > 2 > 3

Perpendicular lines that are originally parallel to the incremental principal axes of strain, will remain perpendicular after strain

Lines of no finite elongation - lnfe

- If we draw a circle on a deck of card, and deform the deck, the strain ellipse will intersect the original circle (if we redraw it) along the two lines of lnfe
- If we draw a new circle in the deformed state (in a different color, say red) and restore the deck to its original unstrained configuration, the red circle becomes an ellipse
- This ellipse which has long axis perpendicular to the strain ellipse is called the reciprocal strain ellipse

Extension and Shortening Fields

- Fields in the strain ellipsoid, separated by the line of no incremental (infinitestimal) longitudinal strain (lnie)
- Boundaries between the shortening and extension are always at 45 degrees either side of incremental principal axes of shortening and extension during each incremental strain event
- As ellipse flattens, material lines migrate through the boundaries separating fields of shortening and extension
- Note that rock particle lines or planes can undergo transition from shortening (folding) to extension (boudinage) without change in orientation of the principal axes of strain

Strain Ratio

- We can think of the strain ellipse as the product of strain acting on a unit circle
- A convenient representation of the shape of the strain ellipse is the strain ratio

Rs = (1+e1)/(1+e3) = S1/S3 = X/Z

- It is equal to the length of the semi-long axis over the length of the semi-short axis

Rotation of lines

- If a line parallel to the radius of a unit circle, makes a pre-deformation angle of with respect to the long axis of the strain ellipse (X), it rotates to a new angle of ´ after strain
- The coordinates of the end point of the line on the strain ellipse (x´, z´) are the coordinates before deformation (x, z) times the principal stretches (S1, S3)
- X’ = X x S = X 1
- Lf = Lo x S = Lo 1

Strain Ellipsoid

- 3D equivalent - the ellipsoid produced by deformation of a unit sphere
- The strain ellipsoids vary from axially symmetric elongated shapes – cigars and footballs - to axially shortened pancakes and cushions

Principal Axes of Strain

- The ellipse has a semi-long axis and a semi-short axis that we can designate X and Z, or sometimes X1 and X3
- Stretches are designated S1 and S3
- The shear strain along the strain axes is zero
- These are the only directions in general that have zero shear strain
- So, in 2D, the principal axes are the only two directions that remain perpendicular before and after an incremental or uniaxial strain. Note: they may not stay perpendicular at all the intermediate stages of a finite non-coaxial strain

Nine quantities needed to define the homogeneous strain matrix

|e11e12e13|

|e21e22e23|

|e31e32e33|

- eij, for i=j, represent changes in length of 3 initially perpendicular lines
- eij, for ij, represent changes in angles between lines

Rotational and Irrotational Strain

- If the strain axes have the same orientation in the deformed as in undeformed state we describe the strain as a non-rotational (or irrotational) strain
- If the strain axes end up in a rotated position, then the strain is rotational

Examples

- An example of a non-rotational strain is pure shear - it's a pure strain with no dilation of the area of the plane
- An example of a rotational strain is a simple shear

Rotational and Irrotational Strain

- In theory, any rotational strain can be decomposed into two parts - a pure strain, and a rotation without distortion (rigid body rotation).
- In principle, if we only have the initial and final states we cannot tell the difference between a rotational strain like simple shear, and a pure shear followed by a rotation
- In practice, if we look at the fabric development in the rock, there may be differences in fabric development depending on whether the rotation was part of the strain history, or if it was applied later
- Notice that in theory, theory and practice are the same; in practice, however, practice and theory are different!
- A pure strain can be decomposed into a constant volume distortion plus a dilation

Types of Homogeneous Strain at Constant Volume

1. Axially symmetric extension

- Extension in one principal direction (1) and equal shortening in all directions at right angles (2 and 3)

1 >2 = 3 < 1

- The strain ellipsoid is prolate spheroid or cigar shaped

S1S3 > 1.0

S1 > 1; S3 = 1.0

S1 = 1; S3 < 1.0

S1 > 1.0

S3 < 1.0

plane strain (S1S3 = 1.0) is

special case in this field

S1S3 < 1.0

S1=3

from: Davis and Reynolds, 1996

Types of Homogeneous Strain at Constant Volume …

2. Axially symmetric shortening

- This involves shortening in one principal direction (3) and equal extension in all directions at right angles (1 and2).

1 =2 > 1 > 3

- Strain ellipsoid is oblate spheroid or pancake-shaped

Plane Strain

- The intermediate axis of the ellipsoid has the same length as the diameter of the initial sphere, i.e.:

e2= 0, or 2=(1+e2)2 =1

- Shortening, 3 =(1+e3)2 and extension, 1=(1+e1)2, respectively, occur parallel to the other two principal directions
- The strain ellipsoid is a triaxial ellipsoid (i.e., it has different semi-axes)

1 >2 = 1 > 3

General Strain

- Involves extension or shortening in each of the principal directions of strain

1 >2 > 3 all 1

Simple shear

- Simple shear is a three-dimensional constant-volume, plane strain
- A single family of parallel planes is undistorted in the deformed state, and parallel the same family of planes in the undeformed state
- Involves a change in orientation of material lines along two of the principal axes (1 and3 )

Simple Shear …

- Simple shear is analogous to the process that occurs when we place a deck of cards on a table and then gently press down on the top of the deck, moving your hands to the right or left
- A cube subjected to a simple shear is converted into a parallelogram resulting in a rotation of the finite strain axes
- The sides of the parallelogram will progressively lengthen as deformation proceeds but the top and bottom surfaces neither stretch nor shorten. Instead they maintain their original length, which is the length of the edge of the original cube

Displacement of Points During Simple Shear

Note that the magnitude of vector v is the distance that the point was translated parallel to the y-axis, while the magnitude of vector u is the distance the point was translated parallel to the x-axis. Vector h then is the resultant of these two vector components

Pure Shear (note: pure means lack of rotation)

- In contrast to simple shear, pure shear is a three-dimensional constant-volume, irrotational, homogeneous flattening, which involves either plane strain or general strain.
- Lines of particles that are parallel to the principal axes of the strain ellipsoid have the same orientation before and after deformation
- It does not mean that the principal axes coincided in all increments!
- During homogeneous flattening a sphere is transformed into a pancake-like shape and a box is changed into a tablet or book-like form.

Pure Shear

- During pure shear the sides of the cube that are parallel to the z-axis are shortened, while the lengths of the sides that are parallel to the x-axis increase. In contrast, the lengths of the sides of the cube that are parallel to the y-axis remain unchanged.
- When such geometrical changes occur during the transformation of a rock body to a distorted state then the mechanism of distortion is termed plane strain.

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