INTRODUCTION
1 cm
1000 km
The diagram below represents an object that has been deformed. The scale of that object is not relevant and could be anything from 1cm up to 1000km. Our job is (1) to characterise the distribution of deformation within the object, and (2) to propose a model (or models) for the deformation history.
Our intuition tells us that to access deformation one should know the shape of the object before deformation. The trouble in geology is that most of the time, either we don't know the initial shape of the deformed object, or the shape of the object is not relevant, as we want to access the deformation of a small volume within the deformed object.
The aim of structural geologists is to analyse the deformation of rock masses.
What structural geologists need is a method to access and characterise deformation. This method should be scale independent, and should not require any knowledge relevant to the geometry of the object before deformation.
The objective of this first chapter is to define such a methodology...
INTRODUCTION
Finite Strain Analysis is a method to analyse deformation of relatively small volume of rocks. If one analyses the deformation in many points of a large object then one can visualise the distribution of strain within that object and get information about the history of the deformation.
The black ellipses represent circular markers that have been deformed. The shape of these ellipses and their orientation give information about the internal deformation of the object. By analysing deformed rocks at the outcrop scale, structural geologists can determine the shape and orientation of these ellipses.
The strain is here heterogeneous: it changes from location to another. In particular the strain is localized in three lineaments (in blue on the sketch on the right). One is parallel to the bottom margin of the object, and the two others defining a Y shape inside it. From the way the strain ellipses rotate or not across those lineaments, structural geologist can determine the kinematic of the displacements (arrows).
Long axis of ellipses
Trend of long axis
With a bit of imagination, and lots of wine,
one can propose the following model the deformation history of the object...
INTRODUCTION
A
BC
A
B
C
One thing we have assumed here is that during deformation imaginary circles are progressively transformed into ellipses. We will see later on that the shape of ellipses (or ellipsoids in 3D) can very easily be characterised. In other word strain can be easily quantified. However, when circles are transformed into some sort of irregular potato then we are in trouble as the shape of an irregular potato is not easy to characterise.
Consider for instance the three circles A, B and C on the sketches below. During deformation only C is transformed into a regular ellipse. The strain in C can easily be characterised by the orientation of its long axis with respect to the geographic North and the axial ratio of the ellipse (length of its long axis divided by the length of its short axis. In contrast the characterisation of the strain recorded by A and B is not an easy task...
In fact, a circle (sphere) is transformed into an ellipse (ellipsoid) only if strain is homogeneous at the scale of the marker. Finite Strain Analysis is therefore only possible if strain is considered at a scale at which it is homogenous.
ε=
σ
Elastic Plastic
σm
εp
εmY
Y:Young's Modulus
Hardening Creep
Creep: deformation increases under constant stress
The Creep is named "steady state flow" when
deformation rate ε is also constant.
ε
σ
I-1-1 ELASTIC, PLASTIC, BRITTLE, AND VISCOUS DEFORMATION
Plastic def -> Ductile behaviorFailure -> Brittle behavior
Elastic
Plastic
σ
σ
ε
σ
ε
σ
Low confiningpressure
High confiningpressure
ε
t
ε
t
1
2
3
1+2
3
1: Instantaneous stress application 2: Constant stress 3: No stress
ε constant
εm
ELASTIC σ = Y ε
ε = Α σ nPLASTIC/Creep
t
ε
Elastic- Plastic hardening
Plasticcreep
Failure
σ constant
Brittle
Brittle
CHAPTER I : STRAIN AND STRESS
I-1 ELEMENTS OF RHEOLOGY
Lf-LoLo
High-T
Low-T
Rheology is the branch of physics that study the deformation of material. On a simple experimental basis three different rheological behaviours can be defined: Elastic, plastic and viscous.
ε = dεdt
Elastic deformation is reversible, Plastic deformation is permanent
Yield strength
Ideal or pure plasticity=>no hardening
Rigid plastic
Elastic plastic
t
ε
Viscous
Elastic
Plastic Failureσ = η ε
η: Viscosity coefficient
Viscous
σ
ε
σ ε
t
1+2
3
ε
t
σ constant
I II III
I: Primary flow (elastic flow)
II: Secondary flow (creep)
III: Tertiary flow (instable)
VISCOUS
σ constant
σ
ELASTO-VISCO-PLASTIC
I-1-1 ELASTIC, PLASTIC, BRITTLE, AND VISCOUS DEFORMATION
CHAPTER I : STRAIN AND STRESS
I-1 ELEMENTS OF RHEOLOGY
+ +
% liquid
εPercolation treshold
Porousimpermeable
Porouspermeable
≈35≈2-3
I-1-2 THE VISCOUS-PLASTIC TRANSITION
CHAPTER I : STRAIN AND STRESS
I-1 ELEMENTS OF RHEOLOGY
R
DISCONTINUOUS STRAIN
CONTINUOUS STRAIIN
HOMOGENEOUS STRAIN
HETEROGENEOUS STRAIN
x
y
TRANSLATION + ROTATION: RIGID TERMS OF THE DEFORMATIONDEFORMATION SENSUS STRICTO => CHANGE IN LENGTH.
TR
ε 11 ε 12 ε 13ε 21 ε 22 ε 23ε 31 ε 32 ε 33
Strain tensor
Mathematic expression Geometrical representation
λ 1
λ 3
R
I-2-1 DISCONTINUOUS, CONTINUOUS (HOMOGENEOUS- HETEROGENEOUS)
DEFORMATION
CHAPTER I : STRAIN AND STRESS
I-2 STRAIN
Finite strain ellipsoid
•Translation•Rotation•Distorsion•Volume change
DEFORMATION is the combination of:
STRAIN
Straight lines remainstraight=> no straingradient.
Straight lines becomes curved=> strain gradient
Lines are broken
λ 2
λ1/λ2 - 1
Z, λ3
Y, λ2X, λ1
X ≥ Y ≥ Z
λ1 ≥ λ2 ≥ λ3
k = λ1/λ2 - 1
λ2/λ3 - 1
D = λ1/λ2 + λ2/λ3 - 1
Principal axes of thefinite strain
εs = [ 3 / [ 3* (Log a + Log b + Log ab ) ] ]2 2 2 0.5 0.5
XY, YZ, XZ λ1 λ2, λ2 λ3, λ1 λ3
Principal planes of thefinite strain
lo
ε = L - Lo
LoPrincipal elongation(relative change of length)
λ = ( ) (1 + ε) Quadratic elongation2L
Lo=
2
√ λ = LLo
Stretch
Characterization of the Finite Strain Ellipsoid
How much strain ?
k=0, K =0 : Pancake ellipsoid
k= , K = : Cigar ellipsoid K =
λ1/λ2
λ2/λ3
ln [ ]
ln [ ]
[ ] + [ ] 2
λ2/λ3 - 12
d =
D = [ ln [ λ1/λ2 ] ] + [ ln [ λ2/λ3] ]2 2
I-2-2 THE FINITE STRAIN ELLIPSOID
CHAPTER I : STRAIN AND STRESS
I-2 STRAIN
l
At a scale at which deformation can be approximated to be continuous homogeneous the strain can be characterized an ellipsoid called the Finite Strain Ellipsoid.
Before deformation After deformation
The principal elongation, the quadratic elongation and the stretch are different ways to measure thechange in length of a line.
What is the ellipsoid shape?
2
0
1
Ln (X/Y)
Ln (Y/Z)
D=0.681
D=1.02D=1.14
D=1.36
Plane
strain
1<K< 8
0<K<1
D=0.49
Flattening strain
K=1
Con
stric
tiona
l str
ain
Ln (λ2/λ3)
Ln (λ1/λ2)
0 1 2
Flinn's diagram, modified by Ramsay
K = 8
K = 0
Uniaxial oblate
Uniaxial prolate
I-2-3 STATE OF STRAIN: THE FLINN'S DIAGRAM
CHAPTER I : STRAIN AND STRESS
I-2 STRAIN
D
K
0 1
2
0
1
SL
S
L=S
Ln (λ2/λ3) 2
Tectonites classification
LLS
Ln (
λ1/λ
2)
The exact shape of any ellipsoid is fully characterized by only two parameters K, and D. Both of them are simple functions of the ratios λ1/λ2 and λ2/λ3. By defining a graph where the first ratio is plotted against the other one can create a space where every ellipsoid can be plotted. This space is called the Flinn diagram. Flat (pancake-shaped) ellipsoids plot along the abscise axis where the parameter K is or close to 0 (flattening strain). At the other end of the spectrum long (cigar-shaped) ellipsoids plot along the ordinate axis where K is infinite where strain is said to be constrictional. The diagonal ( λ1/λ2 = λ2/λ3) defines the domain of plane strain deformation where the axis λ2 axis of finite strain field ellipsoid keeps constant during deformation (λ2 is a direction on non-deformation).
•Parameter D (the intensity of strain) represents the distance between the origin of the graph to the location of the ellipsoid in the graph.
•Parameter K is the slope of the line joining the ellipsoid in the graph and its origin.
2
0
1
Ln (X/Y)
Ln (Y/Z)
Plane
strain
1<K< 8
0<K<1
Flattening
K=1
Constrictional
Ln (λ2/λ3)
Ln (λ1/λ2)
0 1 2
K = 8
K = 0
I-2-3 STATES OF STRAIN: THE FLINN'S DIAGRAM
CHAPTER I : STRAIN AND STRESS
I-2 STRAIN
Distribution of shortened and lengthened domains
As a sphere is progressively deformed into an ellipsoid some of its diameters become shorter, others become longer, and others do not change. This can be visualized by superposing the FSE with its original spherical shape.
Direction ofnon-deformation
In red: Lengthened diametersIn blue: Shortened diameters
γ = tg ψ
ψ
γ
α = λ1/λο
SIMPLE SHEAR
PURE SHEAR
SIMULTANEOUS SIMPLE AND PURE SHEAR
α: 1.25
λ0λ1λ2
ψ: 30º
ψ: 30º α: 1.25
STRAIN HISTORY
I-2-4 STRAIN REGIME: PURE AND SIMPLE SHEAR
CHAPTER I : STRAIN AND STRESS
I-2 STRAIN
Pure Shear Simple Shear
Simple Shear Pure Shear
Initial State Final States
+
+=
0
k0
k
-1
γ
10
1
γ = tg ψ k = x2/x1
k0
k
ψ
x
y
-1
γ (k-k )
2 ln (k)
-1
x'
y'=
x
y
x'
y'=
x
y
x'
y'
=x
y
T12
T22T21
T11x'
y'=
x
y
x' = x T11 + y T12
y' = x T21 + y T22
I-2-5 STRAIN HISTORY: THE FLOW LINES
CHAPTER I : STRAIN AND STRESS
I-2 STRAIN
θ
Kinematic vorticity number
Wk= Cosθ
Flow lines represent paths followed by material particles during deformation. Flow lines allow the visualization of displacement field associated with strain.
Simple shear is characterized by straight flow lines that keep parallel to each other's.
Pure shear is characterized by curve flow lines that organise themself into a symmetric pattern.
Pure and simple shear
+ =S
S
L
L
S L
S
SS
Incremental strainIntermediate strain Finite strain
S L
LSL
LL
LL
SSSL
S: finite shortened sector
L: finite lengthened sector
S: Instantaneous shortened sector
L: Instantaneous lengthened sector
+ =
SS
SSLLLL
SL
S
S
LLS
LS
LS
LS
L
CHAPTER I : STRAIN AND STRESS
I-2 STRAINI-2-5 STRAN HISTORY: THE INCREMENTAL STRAIN ELLIPSOID
Incremental strain is that bit of strain that is successively added and whose accumulated effect leads to the finite strain.
In the simple shear strain regime the finite strain ellipsoid keeps rotating as strain accumulates. In addition the shortened sector
of the FSE enlarges at the expense of the lengthened sector. In the mean time, the incremental strain keeps
the same orientation with the same partitioning between instantaneous lengthening and shortening throughout the strain history. This has some fundamental
consequences on the possible strain histories. In particular it can
be demonstrated that some diameters of the initial sphere will have a strain
history characterized by finite shortening but instantaneous lengthening. These
diameters are first lengthened then shortened, overall however they are still shorter than they were initially. You find that confusing? well it is...
S
S
LL
S L
SL
SL
SL SL
SL
The same hold true for pure shear strain regime though with a fundamental difference. The finite shortening and
instantaneous lengthening sectors are symmetrically distributed with respect to the principal axes of the FSE. In contrast this distribution is asymmetric in the simple shear
strain regime. We can use this difference to recognize in the field the dominant strain regime.
Pure shear regime
Simple shear regime
Incremental strainIntermediate strainFinite strain
Rotation of the non-deformation
lines
Rotation of amaterial lines
Rotation of the non-deformation
lines
Rotation of a material lines
CHAPTER I : STRAIN AND STRESS
I-2 STRAINI-2-5 STRAN HISTORY
Simple shear Pure shear
Ok you are confused about all this and wander about its relevance to geology. Do not worry and keep in mind that... What is good to know is hard to learn.The sketches above represent a progressive simple shear (left), and a progressive pure shear (right). It both column we can keep track of the progressive rotation of the non-deformation lines and that of a material line (thick dark lines). In both cases we can see that the physical line rotates faster that the non-deformation lines. Hence, the physical lines progressively rotate passing from the shortened sector into the lengthened sector.
What does all this mean? Well suppose that in the field you observe some mafic dikes as shown on the sketch below. These dikes seem to have been folded then boudinaged. Folding suggests shortening whereas boudinage suggests lengthening. A field geologist will interpret this structure as field evidence for a strain history involving two phases of deformation. The first phase, let's call it D1, is responsible for the shortening and folding whereas a second phase D2 is being responsible for the extension of the dike and their boudinage, Alternatively a structural geologist will suggest that folding and boudinage may well develop during a progressive deformation during which the dikes rotates across non-deformation envelop in the FSE.
Field observation Interpretation 2
Interpretation 1
D1 D2
Progressive shearing
I-3-1 THE STRESS ELLIPSOID
CHAPTER I : STRAIN AND STRESS
I-3 STRESS
Picture this... Two balloons filled with seawater float in the sea. When the sea is calm the balloons are perfect sphere. Still there are stresses acting on them. These stresses are related to the weight of the column of water above them. These stresses act perpendicularly to the balloons envelop and all have the same magnitude. The balloons are in mechanical equilibrium: no displacement/rotation and no deformation.
Now the weather has changed. Waves change the distribution of the stresses around the balloon. Depending on their exact location the balloons can enjoy either a big vertical squeeze or alternatively a big horizontal one. The stresses acting on the balloon are no longer hydrostatic. Stresses change with their direction of application. Indeed if one consider the balloon on the left, the applied stresses are larger along the vertical direction and smaller on the horizontal direction hence the balloon spread along the horizontal direction and shorten in the vertical direction. If the balloons are extremely small so that they can be approximated as material point, the stresses acting of them define an ellipsoid named the stress ellipsoid.
From this we learn two very important lessons: (1) The state of stress acting on a point is represented by an ellipsoid named the Stress Ellipsoid. (2) No strain without anisotropic stresses.
Now let consider that the balloon are made of solid material having the same density than the surrounding seawater. Furthermore they consist of two half spheres, the surface of contact acting as a potential slipping plane. The balloons no longer deform has a continuum but by slip along that surface. Stresses applied to the balloons envelop from the surrounding water give rise to two resulting stresses: each half sphere applying a stress equal but opposite on the other. This couple of stress is represented by a couple of vector acting on the centre of the sphere (i.e. the centre of the surface of contact). If the stress acting of the balloons envelop are of hydrostatic type this couple of stress act perpendicularly to the contact surface. No deformation occurs as the balloons are in mechanical equilibrium. If the stresses acting on the balloons envelop are anisotropic then the couple of resulting stress may not act perpendicularly to the contact surface. In this case slip may occur. The magnitude of the stresses and they orientation depends on the orientation of the contact surface with respect to the stresses acting on the balloons envelop.
Hydrostatic stresses => No deformation Anisotropic stresses=>deformation possibledepending on the orientation of the contactsurface.
From this we learn another very important lesson: The state of stress on a planar surface is represented by a vector. The orientation and magnitude of which is dependent on the orientation of the surface on which it acts.
σ = F / S
σ= Stress = lim F/SS->0
σ 1
σ 2
Uniaxial stress: σi ≠ 0; σj, k = 0
Biaxial stress: σi,j ≠ 0; σk = 0
Triaxial stress: σi,j,k ≠ 0
σ − σi11σ − σi22σ − σi33
Principal differential stresses
σ 11 σ 12 σ 13σ 21 σ 22 σ 23σ 31 σ 32 σ 33
Stress tensor
σ 11 + σ 22 + σ 33 Invariant
σ ij = σ ji
[σij]= =
σ i 0 0 0 σ i 0 0 0 σ i
+
σ − σi11 σ 12 σ 13σ 21 σ − σi 22 σ 23σ 31 σ32 σ − σi 33
Isotrope component Deviator
σ + σ + σ321
3σi =
Isotropic stressΣ Principal differential stresses = 0
I-3-1 THE STRESS ELLIPSOID
CHAPTER I : STRAIN AND STRESS
I-3 STRESS
O
σ
σ
σn n
nO
OO
An ellipsoid (yep another one...) is the geometrical representation of the state of stress at a point.σ 3
Mathematical expression of the state of stress at a point
The same reasoning applied to balloon is seawater can be made on small volume of rock in the earth. The state of stress at a point in the Earth can be represented by an ellipsoid: The Stress Ellipsoid. It can be demonstrated that a vector can represent the state of stress on a planar surface. Its magnitude and orientation is dependent on the orientation of the surface.
Some clarification aboutstress and force:
σ; Stress vector
n; normal to the surface
σ1
PURE SHEAR
No shear stressacting on the facesof the cube.
CHAPTER I : STRAIN AND STRESS
I-4 STRESS- STRAIN RELATIONSHIPS
σ1
σ3
The principal stress axes are parallelto the axes of the incremental strain ellipsoid...
σ3
σ1
σ3
...and remain parallel to the finite strainaxes during the deformation.
We are now concerned with the relationship between the strain ellipsoids (incremental and finite) and the stress ellipsoid. We see here that this relationship depends on the strain regime. We start this the simple case of pure shear...
We therefore conclude that if one knows the orientation of the principal axes of the ISE or that of the FSE then we can derive from them the orientation of the axes of the associated stress ellipsoid.
σ1 σ3
σ3 σ1
SIMPLE SHEAR (plane strain)
Shear stresses aremaximum on the facesof the cube.
CHAPTER I : STRAIN AND STRESS
I-4 STRESS- STRAIN RELATIONSHIPS
σ1 σ3
σ3 σ1
σ1 σ3
σ3 σ1
The principal stress axes are parallelto the axes of the incremental strain ellipsoid...
...but they aren't parallel to the axesof the finite strain ellipsoid.
The relationship between strain ellipsoids and stress ellipsoid is a bit more complicated when on considers the simple shear strain regime....
We therefore conclude that if one knows the orientation of the principal axes of the ISE then we can derive from them the orientation of the axes of the associated stress ellipsoid no matter the strain regime. However when dealing with simple shear, the orientation of the stress ellipsoid cannot be derived from the FSE.
CHAPTER I : STRAIN AND STRESS
λ1λ2 plane=flattening plane=foliation plane
λ1λ2trajectories
λ1 axis=
stretching lineationλ1
trajectories
λ1 λ2 plane // schistosity plane (=foliation=flatenning plane)
λ1 axis // mineral and stretching lineation
K approximation: L, LS, S=L, SL, S tectonites
Strain gradient can be estimated using strain analysis on similar objects
Dominant strain regime-> symmetric, asymmetric patterns and trends
Kinematics deduced from comparison between ISE, FSE(t), FSE(t+∆t)
INTEGRATION IN MAPS => TECTONICS
FIELD APPROXIMATIONS
I-5.1 TECTONIC TRAJECTORIES MAPS
I-5 FSE: THE CONCEPT IN PRACTICE IN THE FIELD
2<D<10
D<2
0.3<K<0.50.5<K<1K>1
0.1<Wk<0.50.5<Wk<0.80.8<Wk<1
D>2010<D<20
If one can have access to the ratio λ1/λ2 and λ2/λ3, then contour maps can be constructed for D and K. More difficult is the determination of the kinematic vorticity number but in some favourable circumstances Wk can be determined and mapped.
CHAPTER I : STRAIN AND STRESS
I-5.2 TECTONIC CONTOURS MAPS
I-5 FSE: THE CONCEPT IN PRACTICE IN THE FIELD
K<0.3