Finite strain I Rigid body deformations Strain measurements The strain tensor Faith of material lines.

Finite strain I

• Rigid body deformations• Strain measurements• The strain tensor• Faith of material lines

Finite strain I: Finite versus infinitesimal strain

• Infinitesimal strain is a strain that is less than 2%.

• Finite strain can be thought of as a sum of many infinitesimal strains.

• Translation: movement of the body without rotation or distortion.

• Rigid body rotation: rotation of a body about a common axis.

Finite strain I: Rigid body deformations

Finite strain I: Strain (i.e., non-rigid deformation)

Distortion: Change in shape with no change in volume (or area in 2D). Examples include simple shear and pure shear.

• Simple shear:

• Pure shear:

Dilation: Volume change

Finite strain I: Measurements of strain

• Change in line length

• Change in angle

• Change in volume

Change in line length: Extension

li

lf

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e ≡Δl

li=l f − lili

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if e > 0 elongationif e = 0 no changeif e < 0 shortening

Finite strain I: Measurements of strain

Change in line length: Stretch

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s ≡l fli

=1+ e

Change in line length: Quadratic elongation

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λ ≡S2 = (1+ e)2

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if λ >1 extensionif λ =1 no changeif λ <1 shortening

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if S >1 extensionif S =1 no changeif S <1 shortening

Finite strain I: Measurements of strain

Change in line angle: Angular shear

x

y

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angular shear ≡ 90 −α =ψ

Change in line angle: Shear strain

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angular shear ≡Δx

Δy= γ

γ = tanψ

Finite strain I: Measurements of strain

Change in volume: Dilation

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≡V f −ViVi

Vi Vf

Finite strain I: Measurements of strain

Finite strain I: The strain tensor (but more precisely the deformation gradient tensor)

This tensor is used to calculate the position of a material particle(or vector) in the deformed configuration for any given material particle (or vector) in the pre-deformed configuration.

where:

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D =

∂x1

∂X1

∂x2

∂X1

∂x1

∂X2

∂x2

∂X2

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟,

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X1 and X2 are the coordinates before the deformation

x1 and x2 are the coordinates after the deformation

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DX = x

Finite strain I: A few examples of strain tensors

Rigid body rotation:

Simple shear:

Pure shear:

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D =cosα −sinα

sinα cosα

⎛

⎝ ⎜

⎞

⎠ ⎟

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D =1 1

0 1

⎛

⎝ ⎜

⎞

⎠ ⎟

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D =2 0

0 0.5

⎛

⎝ ⎜

⎞

⎠ ⎟

Finite strain I: The faith of material lines

Material lines of geological context include: dikes, sills, layers, faults, etc.

Let’s see what happens to material lines under progressive strain (show movies).

Finite strain I: The faith of material lines

Simple shear:

Conclusions simple-shear:

• Some Material Lines (ML) undergo stretching and rotation at the same time (rotated boudines).

• Some ML undergo shortening followed by stretching (boudinaged fold).

• Angular distance between ML changes progressively.

• ML parallel to the direction of shearing neither stretch nor rotate.

Finite strain I: The faith of material lines

Rotated boudines:

Finite strain I: The faith of material lines

Boudinaged fold (or folded boudinage):

Finite strain I: The faith of material lines

Finite strain I: The faith of material lines

Pure shear: