(7) III-Basics of Continuum Mechanics-strain

8/11/2019 (7) III-Basics of Continuum Mechanics-strain

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III. Strain and Stress

Strain

Stress

Rheology

Reading

Suppe, Chapter 3Twiss&Moores, chapter 15


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http://www.alexstrekeisen.it/english/meta/deformedoolite.php


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Photomicrograph of ooid limestone. Grains

are 0.5-1 mm in size. Large grain in center

shows well developed concentric calcite layers.


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Thin section (transmitted light) of deformed oolitic limestone, Swiss Alps. The approximate

principal extension (red) and principal shortening directions (yellow) are indicated. The

ooids are not perfectly elliptical because the original ooids were not perfectly spherical

and because compositional differences led to heterogeneous strain. The elongated ooidsdefine a crude foliation subperpendicular to the principal shortening direction. Photo

credit: John Ramsay.

http://www.rci.rutgers.edu/~schlisch/structureslides/oolite.html


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Deformed Ordovian trilobite. Deformed samples such as thisprovide valuable strain markers.

Since we know the shape of an undeformed trilobite of this

species, we can compare that to this

deformed specimen and quantify the amount and style of strain in

the rock.


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III. Strain and Stress

Strain

Basics of Continuum Mechanics

Geological examples

Additional References :

Jean Salenon, Handbook of continuum mechanics: general concepts,

thermoelasticity, Springer, 2001

Chandrasekharaiah D.S., Debnath L. (1994) Continuum Mechanics

Publisher: Academic press, Inc.


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Deformation of a deformable body can be

discontinuous (localized on faults) or

continuous.

Strain: change of size and shape of a body


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Basics of continuum mechanics, Strain

A. Displacements, trajectories, streamlines, emission lines

1- Lagrangian parametrisation

2-Eulerian parametrisation

B Homogeneous and tangent Homogeneous transformation1- Definition of an homogeneous transformation

2- Convective transport equation during homogeneous transformation

3-Tangent homogeneous transformation

C Strain during homogeneous transformation

1-The Green strain tensor and the Green deformation tensor

Infinitesimal vs finite deformation

2- Polar factorisation

D Properties of homogeneous transformations1- Deformation of line

2- Deformation of spheres

3- The strain ellipse

4- The Mohr circle

E Infinitesimal deformation

1- Definition

2- The infinitesimal strain tensor

3-Polar factorisation4- The Mohr circle

F Progressive, finite, and infinitesimal deformation :

1- Rotational/non rotational deformation

2- Coaxial Deformation

G Case examples

1- Uniaxial strain

2- Pure shear,

3- Simple shear4- Uniform dilatational strain


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Reference frame (coordinate system): R

Reference state (initial configuration): 0

State of the medium at time t: t

Displacement (from t0to t),

Velocity (at time t)

Position (at t), particle path (=trajectory)

Strain (changes in length of lines, angles between

lines, volume)

A. Describing the transformation of a

body


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A.1 Lagrangian parametrisation

Displacements

Trajectories

Streamlines


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Volume Change

A.1 Lagrangian parametrisation


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A.2 Eulerian parametrisation

Trajectories:

Streamlines:(at time t)


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A.3 Stationary Velocity Field

Velocity is independent of time

NB: If the motion is stationary in the chosen reference frame then

trajectories=streamlines


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B. Homogeneous Tansformation

definition


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Homogeneous Transformation

Changing reference frame


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Homogeneous transformation

Convective transport of a volume


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Homogeneous transformation

Convective transport of a surface


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Tangent Homogeneous Deformation

Any transformation can be approximated locally by itstangent homogeneous transformation


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Any transformation can be approximated locally by itstangent homogeneous transformation


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Displacement field


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D. Strain during homogeneous

Deformation

The Cauchy strain tensor(or expansion tensor)


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Strain during homogeneous

Deformation

Stretch(or elongation) in the direction of avector

Extension(or extension ratio), relative lengthchange


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Deformation

Change of angle between 2 initially orthogonalvectors

Shear angle


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Deformation

Signification of the strain tensor components


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Deformation

An orthometric reference frame can be found in which the strain tensor is

diagonal. This define the 3 principal axesof the strain tensor.


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Deformation

The Green-Lagrange strain tensor (strain tensor)


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Deformation

The Green-Lagrange strain tensor


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Rigid Body Transformation


Deformation


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Rigid Body Transformation


Deformation


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Polar factorisation


Deformation


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Polar factorisation


Deformation


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Pure deformation: The principal strain

axes remain parallel to themselves during

deformation


Deformation


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D. Some properties of homogeneous

Deformation


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Some properties of homogeneous

Deformation

The strain tensor is uniquely characterized by thestrain ellipsoid(a sphere with unit radius in the initial

configuration)


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Deformation

The strain tensor is uniquely characterized by thestrain ellipsoid(a sphere with unit radius in the initial

configuration)


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Note that knowing the strain tensor

associated to an homogeneous

transformation does not define the

uniquely the transformation (the translationand the rotation terms remain

undetermined)


Deformation



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x= R S X + c

x1=S X x2=R x1 X= x2+c


c


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Classification of strain

1

2

a

2

3

b

1

1

ak

b

1 2 3 1 The Flinn diagram characterizes the ellipticity of strain

(for constant volume deformation: with1 2 3


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E. Infinitesimal transformation

Infinitesimal strain tensor


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Infinitesimal transformation

Relation between the infinitesimal


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Relation between the infinitesimalstrain tensor and displacement gradient

1 ( )2

te

The strain ellipse


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NB: The representation of principal extensions on this diagram is

correct only for infinitesimal strain only

The strain ellipse


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Rk: For an infinitesimal deformation the principal

extensions are small (typically less than 1%). The strain

ellipse are close to a circle. For visualisation the strain

ellipse is represented with some exaggeration

Relation between the infinitesimal


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Relation between the infinitesimalstrain tensor and displacement gradient

1 ( )2

te

F Fi it i fi it i l d i


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F. Finite, infinitesimal and progressive

deformation

Finite deformation is said to be non-rotationalif the principle

strain axis in the initial and final configurations are parallel. This

characterizes only how the final state relates to the initial state

Finite deformation of a body is the result of a deformation path

(progressive deformation).

There is an infinity of possible deformation paths to reach a

particular finite strain.

Generally, infinitesimal strain (or equivalently the strain rate

tensor) is used to describe incremental deformation of a body

that has experienced some finite strain

A progressive deformation is said to be coaxialif the principal

axis of the infinitesimal strain tensor remain parallel to the

principal axis of the finite strain tensor. This characterizes the

deformation path.

N t ti l t f ti


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x= S X + c

Non-rotational transformation

a

b

A

B

N t ti l i l i


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Non-rotational non-coaxial progressivetransformation

Stage 1:

Stage 2:

a

b

A

B

R i l i l d f i


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If A and B are parallel to a and b respectively thedeformation is said to be non-rotational

(This means R= 1)

Rotational vs non-rotational deformation


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NB: Uniaxial strain is a type a non-rotational

deformation

Uniaxial strain


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Pure Shear

NB: Pure shear in is a type a non-rotational

deformation (plane strain, 2=1)


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Simple Shear

NB: Simple shear is rotational (Plane strain,2=1)


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Progressive simple Shear


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Progressive simple Shear

Progressive simple shear is non coaxial

Progressive pure shear


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Progressive pure shear

Progressive pure shear is a type of coaxial strain

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(7) III-Basics of Continuum Mechanics-strain