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Fossen Chapter 2
Deformation
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Components of deformation,
displacement field, and particle paths
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Displacement
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Deformation Matrix
|D11 D12D13|
Dij=|D21 D22D23|
|D31 D32D33|
Linear Transformation:
x=Dx or x=D-1x
|D11 D12D13| |x1| = |x1|
|D21 D22D23| |x2| = |x2|
|D31 D32D33| |x3| = |x3|
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Nine quantities needed to define the
homogeneous strain matrix
|e11 e12e13|
|e21 e22e23|
|e31 e32e33|
eij, for i=j, represent changes in length of 3
initially perpendicular lines
eij, for ij, represent changes in anglesbetween lines
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Total deformationof an object
(a) displacement vectorsconnecting initial to final
particle position
(b)-(e)particle paths
(b), (c) Displacement field
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Homogeneous deforamation
Pure and simple shear deformation of brachiopods,
ammonites, and dikes
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Homogeneity depends on scale
The overall strain is heterogeneous.
In some domains, strain is homogeneous
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Discrete or discontinuous deformation
can be viewed as continuous or homogeneous
depending on the scale of observation
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Extension by faulting
is the same as stretch for extensional
basins!
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PURE SHEAR: Constant volume, coaxial,
plane (i.e., 2D) strain
Shorteningin one direction (ky) is
balanced by extensionin the other (kx)
Deformation matrix(diagonal)
|Kx 0 |
|0 ky | where ky= 1/kx
SIMPLE SHEAR: Constant volume, non-
coaxial, plane strain (i.e., 2D)
i.e., ez=0 across the page!
Has two circular sections: xz (slip
plane) and yz
Lines parallel to the principal axes
rotate with progressive deformation
Deformation matrix(triangular)
| 1 g|
|0 1| where gis the shear strain
Involves a change in orientation of
material lines along two of the
principal axes (here: 1 and2 )
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Shear Strain
Shear strain (angular strain)g = tan measure of change in anglebetween two lines which were
originally perpendicular. g Is also dimensionless! The small change in angle is angular shear or
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Rotation of Lines
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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
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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
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Coaxial Strain
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Non-coaxial Strain
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Angular shear strain gis the change inangle between two initially
perpendicular lines A & B
CW is +
CCW is -
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Classification of
strain
ellipse
Field 2
is used
since
XY
No
ellipse
1+e1=1
= 1+e1
=
1+e2
1+e2=1
e1& e2 =0
e1>0e2
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Graphic representation of strain ellipse
Point A (1,1) represents an undeformed circle (1= 2= 1) Because by definition, 1>2, all strain ellipses fall below
or on a line of unit slope drawn through the origin
All dilations fall on the 1= 2line through the origin All other strain ellipses fall into one of three fields:
1. Above the 2=1line where both principal extensions are +2. To the left of the 1=1where both principal extensions are3. Between two fields where one is (+) and the other (-)
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Shapes of the Strain Ellipse
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S1 = 1; S3 < 1.0
S1 > 1; S3 = 1.0
S1S3 > 1.0
S1 > 1.0
S3 < 1.0
plane strain (S1S3 = 1.0) is
special case in this field
from: Davis and Reynolds, 1996
S1S3 < 1.0
S3=3
S1=3
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Structures depend on the orientation of the layer
relative to the principal stretches and value of s2
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Flinn Diagram
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a. Flinn diagram
b. Hsu diagram
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Flinn Diagramb =1
Y/Z = 1
Y=Z
V l h Fli Di
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Volume change on Flinn Diagram
Recall: S=1+e = l'/lo andev= v/vo =(v-vo)/vo An original cube of sides 1 (i.e., lo=1), gives vo=1
Since stretch S=l'/lo, and lo=1, then S=l'
The deformed volume is therefore:v'=l'. l'. l'
Orienting the cube along the principal axes
V' =S1.S2.S3= (1+e1)(1+e2)(1+e3)Since v =(v-vo), for vo=1 we get:
v =(1+e1)(1+e2)(1+e3)-1
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Given vo=1, since ev= v/vo, thenev = v =(1+e1)(1+e2)(1+e3) -1
1+ev
=(1+e1
)(1+e2
)(1+e3
)
If volumetric strain, v = ev = 0, then:(1+e1)(1+e2)(1+e3) = 1 i.e., XYZ=1
Express 1+ev =(1+e1)(1+e2)(1+e3) in e & take log:
ln(1+ev) = e1+e2+e3
Rearrange: (e1-e2)=(e2-e3)-3e2+ln(1+ev) Plane strain (e2=0) leads to:
(e1-e2)=(e2-e3)+ln(1+ev)
[straight line: y=mx+b; with slope, m=1]
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Ramsay Diagram
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Ramsay Diagram Small strains are near the origin
Equal increments of progressive strain (i.e., strain path) plotalong straight lines
Unequal increments plot as curved plots
If v=evis thevolumetric strain, then: 1+v =(1+e1)(1+e2)(1+e3) = lnS=ln(1+e)
It is easier to examine von this plotTake log from both sides and substitute forln(1+e) ln(v+1)=1+ 2+3
If v>0, the lines intersect the ordinate If v
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Compaction
involves strain
(can be viewedas shrinkage
and strain.
The order is notimportant.
Final cases are
the same!
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States of strain:
Uniaxial, planar,
and 3D
General Strain:
Involves extensionor shortening in
each of the
principal directions
of strain
1 >2> 3 all 1
Extension along
X compensated
with equal
shortening along
Y and Z
shortening along Z
compensated with equal
extension along Y and Z
Z
X
shortening along Z
compensated with equal
extension along Y and Ze2= 0
X
Z
Y
X
YYZ
ZX
YZ
Strain ellipse: Prolate spheroid or cigar shaped Strain ellipse:oblate spheroid or pancake shaped
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The strain
ellipsoid
Note: error on
the figure!
|1+e1|= X=S1
|1+e2|= Y=S2|1+e3|= Z=S3
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Strain ellipse
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Isotropic volume change
(involves no strains)
Anisotropic
volume change
by uniaxial
shortening
(compaction)
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Compaction
reduces the
dips of bothlayers and
fault
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The most important
deformation
parameters:
Boundary conditionscontrol the flow
parameters, which
over time produce
strain
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Particle paths (green) and flow apophyses (blue)
describing flow patterns for planar deformations.
The apophyses are orthogonal for pure shear, oblique forsubsimple shear, and coincident for simple shear
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Progressive strain during simple shear
and pure shear
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Simple
shearing of
three sets
of lines
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Lines in thecontraction and
extension fields
experience a
history of
contraction and
extension,
respectively.
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Pure shearing
of three sets
of orthogonal
lines
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Subsimple
shearing of
three sets oforthogonal
lines
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Restored and current profile across the North Sea
rift. Locally, it is modeled as simple shear, but is
better treated as pure shear on larger scale