Continuum Mechanics for Hillslopes: Part III

1. Continuum Mechanics for Hillslopes: Part III • Today we will focus on Deformation and Strain. • Conservation Laws and Constitutive Relations on Thursday.

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)

3. Normal Strain Elongation of contraction of a displacement vector.

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)

5. Normal Strain

6. Normal Strain (by definition: the normal component of strain is a change in line length) (note: strain is a dimensionless quantity)

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.

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

9. Normal Strain (Area) Fractional change in area

10. Normal Strain (Area) Calculating the area Of the final region, A1 Substituting the expression last into:

11. Normal Strain (Area) and because are <<1, their product is very small. Thus, and in 3 dimensions, dilation is:

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

13. Shear Strain By the small angle approximation where:

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:

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.

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.

17. Application: Debris Flows

18. Rotation

19. Rotation

20. Rotation

21. Strain in Alternate Coordinate Systems

22. Strain in Alternate Coordinate Systems

23. Rate of Deformation