2 deformation with figures

8/13/2019 2 Deformation With Figures

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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

2 deformation with figures

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