Stress and Strain

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



+ +

% liquid

εPercolation treshold

Porousimpermeable

Porouspermeable

≈35≈2-3

I-1-2 THE VISCOUS-PLASTIC TRANSITION



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


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


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


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


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


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


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


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


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


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


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.


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.


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.


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.


I-5.2 TECTONIC CONTOURS MAPS

I-5 FSE: THE CONCEPT IN PRACTICE IN THE FIELD

K<0.3

Stress and Strain

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