Structural Geology

Deformation

And

The concept of strain

-

Frédéric Flerit

SouthCalifornia

San Andreas Big Bend San Andreas south

Los AngelesLos Angeles

Les Matelles

Courtesy of J.P. Petit, Montpellier

We need a tool to Describe the deformation process

A

A A

A A

Initial final

Rotation

+

Translation

Deformation

Rigid Body : motion

Already understood

Three level to understands the earth processes

1) Geometry - position

2) kinematics - displacements

3) Dynamics – forces,

The level of the kinematics

1) Rigid body motion:- translation - Rotation

2) deformation

Deformation

Can be :

1) Continous / not continuous

()

2) Homogeneous / not homogeneous

(identical troughout the material)

Two type of elementary deformation

1) Change in length : longitudinal strain

L = Linit

Lfinal

xx = Lfinal - Linit / Linit = L / L

XL

Two type of elementary deformation

2) Change in angle : shear strain

Lx

Ly

xy = Ly / Lx

X

Y

xx yx

xy yy

The Matrix notation

( )

NOTE

yx = xy

The strain matrix is symetric

xx

yy

( )

Deformation of the vector P ?

xx

yy Matrix:

P = Px

Py P

(< given)

(given > )

The matrix product allow to resolve the components of the strain For a given direction P

xx

xx

That is Deformation of the vector P :

pxx Px + Py

Px + yy Py

Px

Py.P =

Along x

Along y

Exercices :deform the above square and

circle using the following strainssupposed uniform

1 xx = 0.5

2 yy = 0.2

3 = 0.5

4 = -0.5

1 xx = 0.5 and yy=-0.5

2 yy = 0.2 and = 0.2

3 = -0.2 and xx = 0.5 and

yy=-0.5

Displacement -

Velocities

Strain

-

Strain rates

To measure the rigid motion of the plates or

of individual points we use the concepts of :

To measure the deformation of the crust or of the lithosphere

we use the concepts of :

The math object associated is a vector

The math object associated is a matrix

REMEMBER

Question

Define the volume change associated with a strain

Given :

To be defined :V/V = ?

xx

xx

Question for next course:

Diagonalize the strain matrix (2D)

xx

yy

1

2

(V1, V2)

Given :

To be defined : The base :

SouthCalifornia

San Andreas Big Bend San Andreas south

Los AngelesLos Angeles

The transformation Matrix

14[1/yr]

xx

yy

PA

NA x

y

A

B’

C’

D

The matrix of the deformation

-

Note : for the deformation on earth : 141

This matrix is not symetric so this is not a strain matrix

The strain matrix

Deformation = strain + other deformation ???

2

2

2

2

-

Simple shear = pure shear + rotation

2

2

2

2

Symetric

=

STRAIN

AntiSymetric

=

ROTATION

Surface change : S/S

1) Simple Shear :

2) Pure Shear :

S

S’

S/S =

Surface change : S/S (continued)

(if e<<1)

Let’s diagonalize

2

2

y

x

Same strain , different Coordinate System

1

2

Y X

A

In (A,x,y) the shear strain is maximum

y

x

2

2

A

In (A,X,Y) the longitudinal strain are maximum

y

x

1

2Y XD’

B’

AY = B’D’ : dir of max extension

AX = A’C’ : dir of max shortening

AY and AY are also called the principal direction

And are perpendicular AY

A

Calculation of the principal strain 1and2

Eigenvalues : ( 1,2)

y

x

1

2

Y XD’

B’

Eigenvectors : (X,Y)

Calculation of the principal strain 1and2

Eigenvalues : ( 1,2)

y

x

1

2Y XD’

B’

2

2

Eigenvectors : (X,Y)

Convention : lengthening < 0

A’

AS geoscientist we would like to have a representation wherewe have at the same time :

Maximum shear and its direction, maximum lonitudinal strain And its direction …

the mohr diagram

It represents the state of strain (shear vs longitudinal)

at a given point A, for all the coordinate system (A,x,y)

y

x

y X

A

+

Shortening in direction AX

Shear in dir. AX

We know the state of strain in

(Ax,y) and in (A,X,Y) lets plot it

y

x

1

2

Y XD’

B’

The Mohr circleof the San Andreas fault (pure shear)

+

2

+

B’

Max shortening

Max shear

Max lengthening

+1

The signification of angle in the mohr circle

y

x

y X

A

2

+

2

+

+

4

2

34B’

+1

Properties of the mohr circle

Symetric with respect to the axe of the extension-compression

The diameter measures the max shear stress.

The position of the centre of the cercle correspond to (e1+e2)/2 which is the

Average strain and corespond to ½ of the Relative surface change.

SouthCalifornia

plot the directions

-of shortening

-of lenghtening

-of shear

does it make sense ?

San Andreas Big Bend San Andreas south

Los AngelesLos Angeles

Analysis of the strain pattern on a piece of Rock

Where are veins and solution surfaces

Surface solution

or stylolite

Fissures filled with

calcite

Stylolites are created

by calcite removal

Effect -> overall shortening

Fissures are created

and filled by precipitation

of calcite

Effect -> overall lengthening

The mechanism of creation of veins and surface solutions :Dissolution recristalisation

Shortening

Leng

then

ing

--

+

+

1. Draw veins and solution surfaces (is the deformation Homogenous?)

2 Define the principal direction of extension3 Define the principal direction of compression4 define the principal direction of shear strain5 conclude

1- Draw the veins

2- Define the principal direction of extension

3- Define the principal direction of compression

4- Draw the solution surfaces

5- Define the max. direction of shear strain

Evaluate the average lenghtening strain ?

And comment

3D state of strain…

IN 3D the strain matrix becomes in the princpal reference frame:

xxxy xz

yx yy yz

zxzyzz

y

x

z

IN 3D the strain matrix becomes in the princpal reference frame:

1

2

2

y

x

z

The mohr diagrams becomes

1

2

2

y

x

z

123

define : 121323

1

2

2

y

x

z

23

123

13

12

Exercices

D

efo

rmae

d

geo

met

ryThe circle becomes an ellipsoid

define the coresponding

strain matrixn

( convention for earth science

shortening positive)

z

x y

D

efo

rmae

d

geo

met

ryThe circle becomes an ellipsoid

define the coresponding

mohr diagrams

z

x y

REMEMBERThe strain in the lithosphere are small

(10-9-10-4) because around 10-4 the eartquake localise the deformation

You can decompose the deformation – principe of superposition-

deformation = strain + rigid body motion

One can always find a coordinate system where the strain matrix is diagonal, the direction are called principal direction of the strain and the strain are called principal strain

1 Geometry : Introduce the proper structures

2 Kinematics : Quantify the deformation

3 Mechanics : Find the Causes

The mathematics of the deformation

We look for the equation of the transformation T :

T : M -> N

N (x’,y’)

M (x,y)

TInitial rock

final rock

Linear approximation for T

X’ = a X

Y’ = b YT

Initial rock

final rock

The Matrix of the transformation

TT = a 0

0 b

ON = T.OM

N (x’,y’)

M (x,y)

T

O (0,0)

Evaluate the parameters of T

X’ = a X

Y’ = b YT

Initial rock

final rock

The extensional component

Position :

X’ = a X

Displacement :

r = X’-X

Deformation :

e = dr/dx

Initial final

X’

X

Deformation is the first derivative of the displacement

Deformation :

e = dr/dx

(slope - 1)

X’

X

Slope > 1 => e>0 => extension

Slope = 1 : no deformation

Slope < 1 => e < 0 => compression

Displacement :

r = X’-X

Slope

= 1

Ext

ensi

on

Compression

The forces responsible for the extension (linear approximation)

Deformation :

e = a

Compare with the max shear strain in California

X’

X

Initial final

HYPOTHESE

force proportional to deformation

f = k eE

xten

sion

fissures stylolites Localise the deformation

X’

XSlope infinite

Slope 1

What about the forces ?

X’

X

NO ROCK ANY MORE

=> k = 0

Fissures Damage the rock

SAFE ROCK

=> k

Evaluate k after the fissuration

Damage modify the rigidity of the rocks

K=k0 .(h -d)/h

h d

SCENARIO…

Stage 2 : load is applied

=> ELASTIC deformation

Stage 1 : initial stateX’

X

X’

X

… Scenario

X’

X

Stage 3 : fissures DAMAGE the rock the load f is relaxed

Stage 4 : the fissures are filled

with calcite and the

deformation become

permanent

PLASTICITY

X’

X

The behavior of brittle rocks

Stage 4 : PlasticityStage 2 : Elasticity

Scenario

Stage 2 : Elasticity Stage 4 : Plasticity

following the damage

of the rocks

Cycles

How many time do you predict thas this scenario is cycled if the max. strain of this

rock has a standard value 10- 3 or 10-4

A big problem :

Looking at the final deformation of rock one see the overal process and not the evolution.

For instance compare the strain calculated from the rocks and the maximal strain calculated for the San Andreas

The total deformation is obtainned

after a unknown number of cycles

Remember

Elasticity : linear relationship Sollicitation / Deformation

- elastic modulus

Damage : decrease of the elastic modulus

Here when fissures form

Plasticity : permanent deformation (not recoverable)

Here when fissures are filled by calcite

Rh

eo

log

y o

f R

oc

ks

Position, Displacement, Deformation

Sollicitation : here traction/compression

Effect : here extension/shortening,

And the characterisic structures : stylolites and veins

The Mechanics origine of veins and solution surfaces

Sollicitation

1) Traction

Or pull-apart

<- ->

2) Compression

Or Push-together

-> <-

Where are veins and solution surfaces