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III. Strain and Stress Strain Stress Rheology Reading Suppe, Chapter 3 Twiss&Moores, chapter 15

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III. Strain and Stress Strain –Basics of Continuum Mechanics –Geological examples Additional References : Jean Salençon, 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 transformation 1- 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 transformations 1- 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 factorisation 4- 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 shear 4- 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 –Position (at t), particle path –displacement (t 0 and t), –Velocity (at time t) –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 vector Implication: Straight lines remain straight during deformation

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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 its tangent homogeneous transformation

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Tangent Homogeneous Deformation Any transformation can be approximated locally by its tangent homogeneous transformation

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Tangent Homogeneous Deformation 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 a vector

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Strain during homogeneous Deformation Stretch (or elongation) in the direction of a vector Extension (or extension ratio), relative length change

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Strain during homogeneous Deformation Change of angle between 2 initially orthogonal vectors Shear angle

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Strain during homogeneous Deformation Signification of the strain tensor components

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Strain during homogeneous Deformation An orthometric reference frame can be found in which the strain tensor is diagonal. This define the 3 principal axes of the strain tensor.

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Strain during homogeneous Deformation The Green-Lagrange strain tensor (strain tensor)

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Strain during homogeneous Deformation The Green-Lagrange strain tensor

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Rigid Body Transformation Strain during homogeneous Deformation

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Rigid Body Transformation Strain during homogeneous Deformation

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Polar factorisation Strain during homogeneous Deformation

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Polar factorisation Strain during homogeneous Deformation

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Polar factorisation Strain during homogeneous Deformation

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Pure deformation: The principal strain axes remain parallel to themselves during deformation Strain during homogeneous 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 the strain ellipsoid (a sphere with unit radius in the initial configuration)

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Some properties of homogeneous Deformation The strain tensor is uniquely characterized by the strain ellipsoid (a sphere with unit radius in the initial configuration)

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Some properties of homogeneous Deformation The Mohr Circle for finite strain

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Some properties of homogeneous Deformation The Mohr Circle for finite strain

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Note that knowing the strain tensor associated to an homogeneous transformation does not define the uniquely the transformation (the translation and the rotation terms remain undetermined) Some properties of homogeneous Deformation

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x= R S X + c x1=S Xx1=S Xx2=R x1x2=R x1 X = x 2 +c Homogeneous Transformation c

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Classification of strain The Flinn diagram characterizes the ellipticity of strain (for constant volume deformation: )

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E. Infinitesimal transformation

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Infinitesimal strain tensor

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Infinitesimal transformation

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Relation between the infinitesimal strain tensor and displacement gradient

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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

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Relation between the infinitesimal strain tensor and displacement gradient

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F. Finite, infinitesimal and progressive deformation –Finite deformation is said to be non-rotational if 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 coaxial if the principal axis of the infinitesimal strain tensor remain parallel to the principal axis of the finite strain tensor. This characterizes the deformation path.

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x= S X + c Non-rotational transformation a b A B

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Non-rotational non-coaxial progressive transformation Stage 1: Stage 2: a b A B

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If A and B are parallel to a and b respectively the deformation 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 Progressive simple shear is non coaxial

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Progressive pure shear Progressive pure shear is a type of coaxial strain

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