1. Continuum Mechanics for Hillslopes: Part III
Today we will focus on Deformation and Strain.
Conservation Laws and Constitutive Relations on Thursday.
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2. Deformation Driven by both body forces and stresses
Style and rate of deformation differs based on material properties (liquids, solids, etc.)
Deformation described by a ‘displacement field’
Vectors connect positions before and after deformation
Rigid-body translation
Rigid-body rotation
Distortion (strain)
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3. Normal Strain Elongation of contraction of a displacement vector.
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4. Normal Strain Displacement of point b can be described as:
Displacement of point a PLUS
Product of the gradient of displacement and the original line length PLUS
An expansion series of higher order terms
(using Taylor’s Theorem)
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5. Normal Strain
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6. Normal Strain
(by definition: the normal component of strain is a change in line length)
(note: strain is a dimensionless quantity)
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7. Normal Strain
For infinitesimal strains, can assume only linear relationships matter.
Assumption good for strains as large as 0.1% or even 1%.
Works for large strains, if considered over short periods of time.
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8. Normal Strain By definition: positive in elongation.
Relates infinitesimal normal strain to the gradient of displacement, along a coordinate direction.
Note subscripts:
If related: normal
If unequal: shear
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9. Normal Strain (Area)
Fractional change in area
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10. Normal Strain (Area) Calculating the area Of the final region, A1
Substituting the
expression last into:
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11. Normal Strain (Area) and because are <<1,
their product is very small.
Thus,
and in 3 dimensions, dilation is:
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12. Shear Strain
The change in angle between lines that were originally perpendicular.
Rotation α1 is positive in ccw direction because produces a displacement in the + y direction.
Same for α2.
When α1 = α2, this is pure shear
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13. Shear Strain
By the small angle approximation
where:
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14. Shear Strain DEFINING:
The average angular change from the original right angle of the elemental area
(average shear strain):
Plugging in from above:
Or:
Finding components as symmetric:
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15. Shear Strain Same derivations can be done for:
Many engineering applications use the total shear strain (the sum of the angular changes, α1 + α2),
But most geological analyses use the average shear strain.
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16. Combined normal strain and average shear strain give a strain tensor:
Total shear strain would remove the ½’s from the off-diagonal terms.
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17. Application: Debris Flows
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18. Rotation
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19. Rotation
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20. Rotation
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21. Strain in Alternate Coordinate Systems
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22. Strain in Alternate Coordinate Systems
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23. Rate of Deformation
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