Mechanics of Earthquakes and Faulting

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1 Mechanics of Earthquakes and Faulting 5 Sep Work of deformation, shear and volume strain Importance of volume change and diltancy rate (rate of volume strain with shear strain) Effective stress and energy budget for shear, Dilatancy For next class meeting, read: Mead, W. J., The geologic role of dilatancy. Jour. Geol. 33, , Frank, F. C., On dilatancy in relation to seismic sources. Rev. Geophys. 3, , 1965

2 Consider the implications of dilatancy and volume change for the total work of shearing, W

3 Data from Knuth and Marone, 2007 Friction mechanics of 2-D particles dx dh W = τ p dγ + σ dθ τ = σ ( µ p + dθ / dγ ) dθ =dv / V ; dγ =dx / h

4 Data from Knuth and Marone, 2007 Friction mechanics of 2-D particles dx dh W = τ p dγ + σ dθ τ = σ ( µ p + dθ / dγ ) τ = σ ( µ p + dh / dx) Dilatancy rate plays an important role in setting the frictional strength

5 Data from Knuth and Marone, 2007 Macroscopic variations in friction are due to variations in dilatancy rate. Smaller amplitude fluctuations in dilatancy rate produce smaller amplitude friction fluctuations.

6 Consider the implications of dilatancy and volume change for the total work (per unit volume) of shearing, W W = τ p dγ + σ dθ W is total work of shearing W = τ dγ = σ µ dγ dh dx Pore Fluid, Pp

7 W = τ p dγ + σ dθ W is total work of shearing W = τ dγ = σ µ dγ dh dx

8 Mead, 1925 (Geologic Role of Dilatancy) Shear Localization Strain homogeneity depends on whether dilatancy is restricted Homogeneous strain if dilatancy is not opposed Strain localization if deformed under finite confining pressure Shear Bands Form if: Marone, 1998

9 Mead, 1925 (Geologic Role of Dilatancy) Shear Localization Strain homogeneity depends on whether dilatancy is restricted Homogeneous strain if dilatancy is not opposed Strain localization if deformed under finite confining pressure W = τ p dγ + σ dθ τ = σ ( µ p + dθ / dγ ) Shear strength depends on friction and dilatancy rate Shear Bands Form if: Deformation mode (degree of strain localization) minimizes dilatancy rate

10 Mead, 1925 (Geologic Role of Dilatancy) Shear Localization Strain homogeneity depends on whether dilatancy is restricted Homogeneous strain if dilatancy is not opposed Strain localization if deformed under finite confining pressure W = τ p dγ + σ dθ τ = σ ( µ p + dθ / dγ ) Shear strength depends on friction and dilatancy rate Deformation mode (degree of strain localization) minimizes dilatancy rate Shear Bands Form if:

11 Frank, Volumetric work, shear localization and stability. Applies to Friction and Fracture Volume Strain, θ Strain hardening when: Shear Strain, γ Strain softening when:

12 Volume Strain, θ Shear Strain, γ Marone, Raleigh & Scholz, 1990

13 Volume Strain, θ Shear Strain, γ Marone, Raleigh & Scholz, 1990

14 Pop Quiz: σ(α) = τ(α) = Derive the relations for shear and normal stress on a plane of arbitrary orientation in terms of principal stresses σ 1 and σ 2 Hint, use the Mohr Circle. τ σ 1 α σ 2 σ plane P

15 Fluids: Consider the affect on shear strength Mechanical Effects Chemical Effects Mechanical Effects: Effective Stress Law σ effective = σ n - Pp σ 3 σ 1 Pp σ 1 σ 3

16 Fluids: Consider the affects on shear strength Mechanical Effects Chemical Effects Mechanical Effects: Effective Stress Law σ effective = σ n - Pp σ 3 σ 1 Pp σ 1 σ 3 Leopold Kronecker ( ) Rock properties depend on effective stress: Strength, porosity, permeability, Vp, Vs, etc.

17 σ effective = σ n - Pp σ 3 σ 1 Pp σ 1 σ 3 Exercise: Follow through the implications of Kronecker s delta to see that pore pressure only influences normal stresses and not shear stresses. Hint: see the equations for stress transformation that led to Mohr s circle. σ = ( σ 1 + σ 2 ) 2 ( + σ σ 1 2) 2 cos2α

18 Void space filled with a fluid at pressure Pp σ But what if Ar A? p o p σ Fluids play a role by opposing the normal stress

19 Mechanical Effects: Effective Stress Law σ p o For brittle conditions, Ar / A ~ 0.1 p σ Exercise: Consider how a change in applied stress would differ from a change in Pp in terms of its effect on Coulomb shear strength. Take α = 0.9

20 Effective Stress Law Coupled Effects σ Applied Stress p o Pore Pressure Strength, Stability p Dilatancy: Shear driven volume change σ Exercise: Make the dilatancy demo described by Mead (1925) on pages You can use a balllon, but a plastic bottle with a tube works better. Bring to class to show us. Feel free to work in groups of two.

21 Effective Stress Law Coupled Effects σ Applied Stress p o Pore Pressure Strength, Stability p Dilatancy Shear Rate σ Pore Fluid, Pp Pore Fluid, Pp

22 Dilatancy: Volumetric Strain: Assume no change in solid volume Dilatancy Rate: Shear Rate Pore Fluid, Pp Pore Fluid, Pp

23 Dilatancy: Undrained loading Volumetric Strain: Assume no change in solid volume Dilatancy Rate: Dilatancy Hardening if : or Shear Rate Pore Fluid, Pp Pore Fluid, Pp

24 Dilatancy Hardening if : Shear Rate Pore Fluid, Pp Pore Fluid, Pp

25 Dilatancy Weakening can occur if: This is shear driven compaction Shear Rate Pore Fluid, Pp Pore Fluid, Pp

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