lecture12 strain examples

8/11/2019 Lecture12 Strain Examples

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Lecture 12: Strain Examples

GEOS 655 Tectonic GeodesyJeff Freymueller


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A Worked Example Consider this case of

pure shear deformation,

and two vectors dx 1 anddx 2. How do they rotate? Well look at vector 1 first,

and go through eachcomponent of ! .

"i

= e ijk # km + $ km( )d

x j d

xm

" =

0 # 0

# 0 0

0 0 0

$

%

& & &

'

(

) ) )

* = (0,0,0)

(1) dx = (1,0,0) = d x 1(2) dx = (0,1,0) = d x 2

dx (1)

dx (2)


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A Worked Example First for i = 1

Rules for e 1jk If j or k =1, e 1jk = 0 If j = k =2 or 3, e 1jk = 0 This leaves j=2, k=3 and

j=3, k=2 Both of these terms will

result in zero because j=2,k=3: " 3m = 0 j=3,k=2: dx 3 = 0

True for both vectors

"1

= e1 jk # km + $ km( )d x j d xm

" =

0 # 0

# 0 0

0 0 0

$

%

& & &

'

(

) ) )

* = (0,0,0)

(1) dx = (1,0,0) = d x 1(2) dx = (0,1,0) = d x 2

dx (1)

dx (2)


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A Worked Example Now for i = 2

Rules for e 2jk If j or k =2, e 2jk = 0 If j = k =1 or 3, e 2jk = 0 This leaves j=1, k=3 and

j=3, k=1 Both of these terms will

result in zero because j=1,k=3: " 3m = 0 j=3,k=1: dx 3 = 0

True for both vectors

"2

= e 2 jk # km + $ km( )d x j d xm

" =

0 # 0

# 0 0

0 0 0

$

%

& & &

'

(

) ) )

* = (0,0,0)

(1) dx = (1,0,0) = d x 1(2) dx = (0,1,0) = d x 2

dx (1)

dx (2)


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A Worked Example Now for i = 3

Rules for e 3jk Only j=1, k=2 and j=2, k=1

are non-zero

Vector 1:

Vector 2:

"3

= e 3 jk # km + $ km( )d x j d xm

" =

0 # 0

# 0 0

0 0 0

$

%

& & &

'

(

) ) )

* = (0,0,0)

(1) dx = (1,0,0) = d x 1(2) dx = (0,1,0) = d x 2

dx (1)

dx (2) ( j = 1, k = 2) e312 dx1" 2 mdx m= 1 #1 # $ #1 + 0 #0 + 0 #0( )= $

( j = 2, k = 1) e321

dx2

" 1m

dxm

= %1 #0 # 0 #1 + $ #0 + 0 #0( )= 0

( j = 1, k = 2) e312 dx1" 2 m dx m= 1 # 0 # $ #0 + 0 #0 + 0 #0( )= 0

( j = 2, k = 1) e321 dx 2" 1mdx m= %1 #1 # 0 #0 + $ #1 + 0 #0( )= %$

- #

#


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Rotation of a Line Segment There is a general expression for the rotation

of a line segment. Ill outline how it is derivedwithout going into all of the details.

First, the strain part

"i

= e ijk # km + $ km( )d

x j d

xm

"i( strain )

= e ijk # km d x j d xm = e ijk d x j # km d xm( )"

i( strain )

= e ijk d x j # $ d x( )k " ( strain ) = d x % # $ d x

( )

If the strain changes theorientation of the line, thenthere is a rotation.


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Rotation of a Line Segment Here is the full equation then:

This is used for cases when you have angle

or orientation change data, or when you wantto predict orientation changes from a knownstrain and rotation.

"i

=

e ijk d

x j #

kmd

xm+ $

i % $

j d

x j ( )d

x i" = d

x & # ' d

x( )+ $ % $ ' d x( )d x


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Vertical Axis Rotation Special case of common use in tectonics

Well use a local east-north-up coordinate system All motion is horizontal (u 3 = 0)

All rotation is about a vertical axis (only % 3 is non-zero) All sites are in horizontal plane (dx 3 = 0) The expression for % gets a lot simpler

"i

# " j dx j dx i $ " i " % dx = 0( )

" 3 = # 12 e 3 ij & ij "

3= # 1

2 e312 & 12 + e321 & 21 + 0[ ]"

3= # 1

2 1 %& 12 # 1 % #& 12( )[ ]

"3

= # & 12

The vertical axisrotation is directlyrelated to the rotationtensor term


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Strain from 3 GPS Sites There is a simple,

general way to calculate

average strain+rotationfrom 3 GPS sites If you have more than 3

sites, divide networkinto triangles For example, Delaunay

triangulation asimplemented in GMT

A

B

C

u AB

u AC


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Strain from 3 GPS Sites We have seen the equations for strain

from a single baseline before

C A

B u AB

u AC

" u 1 AB

" u 2 AB#

$ % &

' ( = ) 11 " x1

AB

+ ) 12 " x 2 AB

+ * 12 " x 2 AB

) 12 " x1 AB

+ ) 22 " x 2 AB + * 12 " x1

AB#

$ % &

' (

" u 1 AB

" u 2 AB

#

$

% &

'

( =" x1

AB " x 2 AB 0 " x 2

AB

0 " x1 AB " x 2

AB +" x1 AB

#

$

% &

'

( ,

) 11

) 12

) 22

* 12

#

$

% % % %

&

'

( ( ( (

Observations = (known Model)*(unknowns)


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Strain from 3 GPS Sites Including both sites we have 4 equations

in 4 unknowns:

C A

B u AB

u AC

" u 1 AB

" u 2 AB

" u 1 BC

" u 2 BC

#

$

% % % %

&

'

( ( ( (

=

" x1 AB

" x 2 AB

0 " x 2 AB

0 " x1 AB " x 2

AB )" x1 AB

" x1 BC " x 2

BC 0 " x 2 BC

0 " x1 BC " x 2

BC )" x1 BC

#

$

% % % %

&

'

( ( ( ( *

+ 11

+ 12

+ 22

, 12

#

$

% % % %

&

'

( ( ( (

+ 11

+ 12

+ 22

, 12

#

$

% % % %

&

'

( ( ( (

=

" x1 AB

" x 2 AB

0 " x 2 AB

0 " x1 AB " x 2

AB )" x1 AB

" x1 BC " x 2

BC 0 " x 2 BC

0 " x1 BC " x 2

BC )" x1 BC

#

$

% % % %

&

'

( ( ( (

) 1

*" u 1

AB

" u 2 AB

" u 1 BC

" u 2 BC

#

$

% % % %

&

'

( ( ( (


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Strain varies with space In general, strain varies with space, so you cant necessarilyassume uniform strain over a large area For a strike slip fault, the strain varies with distance from the fault x

(slip rate V, and locking depth D)

You can ignore variations and use a uniform approximation (see noevil)

Fit a mathematical model for ( x) and ' (x)

You can try sub-regions, as we did in Tibet in the Chen et al. (2004)paper

Map strain by looking at each triangle Or you can map out variations in strain in a more continuous fashion

This is a more powerful tool in general

" 12 =VD

2 # x 2 + D

2( )$ 1


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Southern California

Feigl et al. (1993, JGR)


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Strain Rates


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Rotations


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Model strain rates as continuousfunctions using a modified least-

squares method.

Uniqueness of the method: Requires no assumptions of stationary of

deformation field and uniform variance ofthe data that many other methods do;

Implements the degree of smoothingbased on in situ data strength. Method developed by Z. Shen, UCLA


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!!!!!!!!!

"

#

$$$$$$$$$

%

&

n

n

Vy

Vx

VyVx

Vy

Vx

...

2

2

1

1

=

!!!!!!!

!!

"

#

$$$$$$$

$$

%

&

'('''''

'('' '''

'('''''

nnn

nnn

x y x

y y x

x y x y y x

x y x

y y x

010

001

..................

010

001

010

001

222

222

111

111

Ux

Uy"

xx

" xy

" yy

#

$

%

& & & & & & &

'

(

) ) ) ) ) ) )

+

e x1

e y1

e x

2

e y 2

...

e xn

e yn

"

#

$ $ $ $ $ $ $ $ $

%

&

' ' ' ' ' ' ' ' '

Ux, Uy : on spot velocity components

xx, ! xy, ! yy : strain rate components

" : rotation rate

V i V j

Vk

xi x j

xk R

At each location point R , assuming a uniform strain rate field, the strainrates and the geodetic data can be linked by a linear relationship:

d = A m + e


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d = A m + e

Bd = BAm + Be

where B is a diagonal matrix whose i -th diagonal termis exp (-! x i 2 /D2 ) and e ~ N(0, E), that is, the errors are

Assumed to be normally distributed. D is a smoothingdistance.

reconstitute the inverse problem with a weighting matrix B:

m = (A t B E -1 B A) -1 At B E -1 B d

This result comes from standard least squares estimationmethods.

The question is how to make a proper assignment of D?


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W = " i exp (-! x i 2 /D 2 )

Trade-off between total weight W and strain rate uncertainty (

small D Average oversmall area

large D Average overlarge area


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Tradeoff of resolution anduncertainty

If you average over a small area, you willimprove your spatial resolution for variations instrain But your uncertainty in strain estimate will be higher

because you use less data = more noise

If you average over a large area, you will reducethe uncertainty in your estimates by using more

data But your ability to resolve spatial variations in strain

will be reduced = possible over-smoothing

Need to find a balance that reflects the actualvariations in strain.


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Example: Strain rate estimation from SCEC CMM3 (Post-Landers)


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Post-Landers Principal Strain Rate and Dilatation Rate


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Post-Landers Rotation Rate


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Post-Landers Maximum Strain Rate and Earthquakes of M>5.0 1950-2000


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Global Strain Rate Map


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Large-Scale Strain Maps The Global Strain Rate Map makes use of some

important properties of strain as a function of space: Compatibility equations

The strain tensors at two nearby points have to be related toeach other if there are no gaps or overlaps in the material

Equations relate spatial derivatives of various strain tensorcomponents

If we know strain everywhere, we can determine rotationfrom variations in shear strain (because of compatibility eqs)

Relationship between strain and variations in angularvelocity, so that you can represent all motion in terms ofspatially-variable angular velocity

Kostrov summation of earthquake moment tensors or faultslip rate estimates

Allows the combination of geodetic and geologic/seismic data


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Geodetic Velocities Used


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Second Invariant of Strain


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Invariants of Strain Tensor The components of the strain tensor depend

on the coordinate system For example, tensor is diagonal when principal

axes are used to define coordinates, not diagonalotherwise

There are combinations of tensorcomponents that are invariant to coordinaterotations Correspond to physical things that do not change

when coordinates are changed Dilatation is first invariant (volume change does

not depend on orientation of coordinate axes)


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Three Invariants (Symmetric) First Invariant dilatation

Trace of strain tensor is invariant I1 = & = trace( " ) = " ii = " 11 + " 22 + " 33

Second Invariant Magnitude

Third Invariant determinant

Determinent does not depend on coordinates I3 = det( " )

There is also a mathematical relationshipbetween the three invariants

I 2

= 12

trace " ( )2 # trace (" $" )[ ] I

2= "

11$"

22+ "

22$"

33+ "

11$"

33 # "

122 # "

232 # "

132


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lecture12 strain examples

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