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Continuum Mechanics for Hillslopes: Part III
Today we will focus on Deformation and Strain. Conservation Laws and Constitutive Relations on Thursday. 12/4/2018
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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) 12/4/2018
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Normal Strain Elongation of contraction of a displacement vector.
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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) 12/4/2018
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Normal Strain 12/4/2018
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Normal Strain (by definition: the normal component of strain is a change in line length) (note: strain is a dimensionless quantity) 12/4/2018
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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. 12/4/2018
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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 12/4/2018
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Normal Strain (Area) Fractional change in area 12/4/2018
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Normal Strain (Area) Calculating the area Of the final region, A1
Substituting the expression last into: 12/4/2018
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Normal Strain (Area) and because are <<1,
their product is very small. Thus, and in 3 dimensions, dilation is: 12/4/2018
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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 12/4/2018
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Shear Strain By the small angle approximation where: 12/4/2018
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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: 12/4/2018
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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. 12/4/2018
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Combined normal strain and average shear strain give a strain tensor:
Total shear strain would remove the ½’s from the off-diagonal terms. 12/4/2018
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Application: Debris Flows
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Rotation 12/4/2018
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Rotation 12/4/2018
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Rotation 12/4/2018
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Strain in Alternate Coordinate Systems
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Strain in Alternate Coordinate Systems
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Rate of Deformation 12/4/2018
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