finite strain i rigid body deformations strain measurements the strain tensor faith of material...
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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: