mechanics of earthquakes and faulting · friction mechanics of 2-d particles € τ = σ(µ p +...
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![Page 1: Mechanics of Earthquakes and Faulting · Friction mechanics of 2-D particles € τ = σ(µ p + dθ/ dγ) € τ = σ(µ p + dh / dx) € W = τ p dγ + σ dθ • Dilatancy rateplays](https://reader034.vdocuments.net/reader034/viewer/2022042802/5f385562ed07224f381dcd36/html5/thumbnails/1.jpg)
Mechanics of Earthquakes and Faulting
www.geosc.psu.edu/Courses/Geosc508
• Work of deformation, shear and volume strain • Importance of volume change and dilatancy rate (rate of volume
strain with shear strain)• Effective stress and energy budget for shear, Dilatancy
• Discuss• Mead, W. J., The geologic role of dilatancy. Jour. Geol. 33, 685-
698, 1925.• Frank, F. C., On dilatancy in relation to seismic sources. Rev.
Geophys. 3, 485-503, 1965
10 Sep. 2019
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Fracture Mechanics andStress intensity factors for each mode
KI, KII, KIII
Linear Elastic Fracture Mechanics•Frictionless cracks•Planar, perfectly sharp (mathematical) cuts
Crack tip stress field written in a generalized form
€
σ ij =Kn12πr
fijn θ( )
σ 22
σ 21
σ 23
%
& '
( '
)
* '
+ ' tip
≈12πr
KI
KII
KIII
%
& '
( '
)
* '
+ '
σ22
σ23
σ21
v1
2r
θ
rr’
![Page 3: Mechanics of Earthquakes and Faulting · Friction mechanics of 2-D particles € τ = σ(µ p + dθ/ dγ) € τ = σ(µ p + dh / dx) € W = τ p dγ + σ dθ • Dilatancy rateplays](https://reader034.vdocuments.net/reader034/viewer/2022042802/5f385562ed07224f381dcd36/html5/thumbnails/3.jpg)
Pore Fluid, Pp
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
dhdx
![Page 4: Mechanics of Earthquakes and Faulting · Friction mechanics of 2-D particles € τ = σ(µ p + dθ/ dγ) € τ = σ(µ p + dh / dx) € W = τ p dγ + σ dθ • Dilatancy rateplays](https://reader034.vdocuments.net/reader034/viewer/2022042802/5f385562ed07224f381dcd36/html5/thumbnails/4.jpg)
σeffective = σn - Pp
σ1σ1
σ3
σ3
Pp
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 = σ1 +σ 2( )
2+σ 1 − σ 2( )
2 cos2α
![Page 5: Mechanics of Earthquakes and Faulting · Friction mechanics of 2-D particles € τ = σ(µ p + dθ/ dγ) € τ = σ(µ p + dh / dx) € W = τ p dγ + σ dθ • Dilatancy rateplays](https://reader034.vdocuments.net/reader034/viewer/2022042802/5f385562ed07224f381dcd36/html5/thumbnails/5.jpg)
Consider the implications of dilatancy and volume change for the total work of shearing, W
![Page 6: Mechanics of Earthquakes and Faulting · Friction mechanics of 2-D particles € τ = σ(µ p + dθ/ dγ) € τ = σ(µ p + dh / dx) € W = τ p dγ + σ dθ • Dilatancy rateplays](https://reader034.vdocuments.net/reader034/viewer/2022042802/5f385562ed07224f381dcd36/html5/thumbnails/6.jpg)
Friction mechanics of 2-D particles
€
τ = σ µ p + dθ / dγ( )
€
dθ=dV /V ; dγ =dx /h
€
W = τ p dγ + σ dθ
dh
dx
Data from Knuth and Marone, 2007
![Page 7: Mechanics of Earthquakes and Faulting · Friction mechanics of 2-D particles € τ = σ(µ p + dθ/ dγ) € τ = σ(µ p + dh / dx) € W = τ p dγ + σ dθ • Dilatancy rateplays](https://reader034.vdocuments.net/reader034/viewer/2022042802/5f385562ed07224f381dcd36/html5/thumbnails/7.jpg)
Friction mechanics of 2-D particles
€
τ = σ µ p + dθ / dγ( )
€
τ = σ µ p + dh /dx( )
€
W = τ p dγ + σ dθ • Dilatancy rate plays an important role in setting the frictional strength
dh
dx
Data from Knuth and Marone, 2007
![Page 8: Mechanics of Earthquakes and Faulting · Friction mechanics of 2-D particles € τ = σ(µ p + dθ/ dγ) € τ = σ(µ p + dh / dx) € W = τ p dγ + σ dθ • Dilatancy rateplays](https://reader034.vdocuments.net/reader034/viewer/2022042802/5f385562ed07224f381dcd36/html5/thumbnails/8.jpg)
• Macroscopic variations in friction are due to variations in dilatancy rate.
• Smaller amplitude fluctuations in dilatancy rate produce smaller amplitude friction fluctuations.
Data from Knuth and Marone, 2007
![Page 9: Mechanics of Earthquakes and Faulting · Friction mechanics of 2-D particles € τ = σ(µ p + dθ/ dγ) € τ = σ(µ p + dh / dx) € W = τ p dγ + σ dθ • Dilatancy rateplays](https://reader034.vdocuments.net/reader034/viewer/2022042802/5f385562ed07224f381dcd36/html5/thumbnails/9.jpg)
€
W = τ p dγ + σ dθ
W is total work of shearing
W = τ dγ = σ µ dγ
dhdx
![Page 10: Mechanics of Earthquakes and Faulting · Friction mechanics of 2-D particles € τ = σ(µ p + dθ/ dγ) € τ = σ(µ p + dh / dx) € W = τ p dγ + σ dθ • Dilatancy rateplays](https://reader034.vdocuments.net/reader034/viewer/2022042802/5f385562ed07224f381dcd36/html5/thumbnails/10.jpg)
Mead, 1925 (Geologic Role of Dilatancy)
Shear Localization
Marone, 1998
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:
![Page 11: Mechanics of Earthquakes and Faulting · Friction mechanics of 2-D particles € τ = σ(µ p + dθ/ dγ) € τ = σ(µ p + dh / dx) € W = τ p dγ + σ dθ • Dilatancy rateplays](https://reader034.vdocuments.net/reader034/viewer/2022042802/5f385562ed07224f381dcd36/html5/thumbnails/11.jpg)
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:
€
τ = σ µ p + dθ / dγ( )
€
W = τ p dγ + σ dθ
Shear strength depends on friction and dilatancy rate
Deformation mode (degree of strain localization) minimizes dilatancy rate
![Page 12: Mechanics of Earthquakes and Faulting · Friction mechanics of 2-D particles € τ = σ(µ p + dθ/ dγ) € τ = σ(µ p + dh / dx) € W = τ p dγ + σ dθ • Dilatancy rateplays](https://reader034.vdocuments.net/reader034/viewer/2022042802/5f385562ed07224f381dcd36/html5/thumbnails/12.jpg)
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:
€
τ = σ µ p + dθ / dγ( )
€
W = τ p dγ + σ dθ
Shear strength depends on friction and dilatancy rate
Deformation mode (degree of strain localization) minimizes dilatancy rate
![Page 13: Mechanics of Earthquakes and Faulting · Friction mechanics of 2-D particles € τ = σ(µ p + dθ/ dγ) € τ = σ(µ p + dh / dx) € W = τ p dγ + σ dθ • Dilatancy rateplays](https://reader034.vdocuments.net/reader034/viewer/2022042802/5f385562ed07224f381dcd36/html5/thumbnails/13.jpg)
Frank, 1965. Volumetric work, shear localization and stability. Applies to Friction and Fracture
Volu
me
Stra
in, θShear Strain, γ
Strain hardening when:
Strain softening when:
![Page 14: Mechanics of Earthquakes and Faulting · Friction mechanics of 2-D particles € τ = σ(µ p + dθ/ dγ) € τ = σ(µ p + dh / dx) € W = τ p dγ + σ dθ • Dilatancy rateplays](https://reader034.vdocuments.net/reader034/viewer/2022042802/5f385562ed07224f381dcd36/html5/thumbnails/14.jpg)
Volu
me
Stra
in, θ
Shear Strain, γ
Marone, Raleigh & Scholz, 1990
![Page 15: Mechanics of Earthquakes and Faulting · Friction mechanics of 2-D particles € τ = σ(µ p + dθ/ dγ) € τ = σ(µ p + dh / dx) € W = τ p dγ + σ dθ • Dilatancy rateplays](https://reader034.vdocuments.net/reader034/viewer/2022042802/5f385562ed07224f381dcd36/html5/thumbnails/15.jpg)
Volu
me
Stra
in, θ
Shear Strain, γ
Marone, Raleigh & Scholz, 1990
![Page 16: Mechanics of Earthquakes and Faulting · Friction mechanics of 2-D particles € τ = σ(µ p + dθ/ dγ) € τ = σ(µ p + dh / dx) € W = τ p dγ + σ dθ • Dilatancy rateplays](https://reader034.vdocuments.net/reader034/viewer/2022042802/5f385562ed07224f381dcd36/html5/thumbnails/16.jpg)
Pop Quiz:
Derive the relations for shear and normalstress on a plane of arbitrary orientation interms of principal stresses σ1 and σ2Hint, use the Mohr Circle.
σ1
σ2
α
plane Pσ
τ
σ(α) =
τ(α) =
![Page 17: Mechanics of Earthquakes and Faulting · Friction mechanics of 2-D particles € τ = σ(µ p + dθ/ dγ) € τ = σ(µ p + dh / dx) € W = τ p dγ + σ dθ • Dilatancy rateplays](https://reader034.vdocuments.net/reader034/viewer/2022042802/5f385562ed07224f381dcd36/html5/thumbnails/17.jpg)
Fluids: Consider the affect on shear strength
•Mechanical Effects•Chemical Effects
σeffective = σn - PpMechanical Effects: Effective Stress Law
σ1σ1
σ3
σ3
Pp
![Page 18: Mechanics of Earthquakes and Faulting · Friction mechanics of 2-D particles € τ = σ(µ p + dθ/ dγ) € τ = σ(µ p + dh / dx) € W = τ p dγ + σ dθ • Dilatancy rateplays](https://reader034.vdocuments.net/reader034/viewer/2022042802/5f385562ed07224f381dcd36/html5/thumbnails/18.jpg)
σeffective = σn - PpMechanical Effects: Effective Stress Law
σ1σ1
σ3
σ3
Pp
Rock properties depend on effective stress: Strength, porosity, permeability, Vp, Vs, etc.
Leopold Kronecker (1823–1891)
Fluids: Consider the affects on shear strength
•Mechanical Effects•Chemical Effects
![Page 19: Mechanics of Earthquakes and Faulting · Friction mechanics of 2-D particles € τ = σ(µ p + dθ/ dγ) € τ = σ(µ p + dh / dx) € W = τ p dγ + σ dθ • Dilatancy rateplays](https://reader034.vdocuments.net/reader034/viewer/2022042802/5f385562ed07224f381dcd36/html5/thumbnails/19.jpg)
σeffective = σn - Pp
σ1σ1
σ3
σ3
Pp
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 = σ1 +σ 2( )
2+σ 1 − σ 2( )
2 cos2α
![Page 20: Mechanics of Earthquakes and Faulting · Friction mechanics of 2-D particles € τ = σ(µ p + dθ/ dγ) € τ = σ(µ p + dh / dx) € W = τ p dγ + σ dθ • Dilatancy rateplays](https://reader034.vdocuments.net/reader034/viewer/2022042802/5f385562ed07224f381dcd36/html5/thumbnails/20.jpg)
σ
�p�o
�p
Fluids play a role by opposing the normal stress
Void space filled with a fluid at pressure Pp
But what if Ar ≠ A ?
σ
![Page 21: Mechanics of Earthquakes and Faulting · Friction mechanics of 2-D particles € τ = σ(µ p + dθ/ dγ) € τ = σ(µ p + dh / dx) € W = τ p dγ + σ dθ • Dilatancy rateplays](https://reader034.vdocuments.net/reader034/viewer/2022042802/5f385562ed07224f381dcd36/html5/thumbnails/21.jpg)
σ
�p�o
�p
Mechanical Effects: Effective Stress Law
For brittle conditions, Ar / A ~ 0.1
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
![Page 22: Mechanics of Earthquakes and Faulting · Friction mechanics of 2-D particles € τ = σ(µ p + dθ/ dγ) € τ = σ(µ p + dh / dx) € W = τ p dγ + σ dθ • Dilatancy rateplays](https://reader034.vdocuments.net/reader034/viewer/2022042802/5f385562ed07224f381dcd36/html5/thumbnails/22.jpg)
σ
�p�o
�p
Effective Stress Law
σ
Coupled Effects
Applied Stress
Pore Pressure
Strength, Stability
Exercise: Make the dilatancy demo described by Mead (1925) on pages 687-688. 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.
Dilatancy: Shear driven volume change
![Page 23: Mechanics of Earthquakes and Faulting · Friction mechanics of 2-D particles € τ = σ(µ p + dθ/ dγ) € τ = σ(µ p + dh / dx) € W = τ p dγ + σ dθ • Dilatancy rateplays](https://reader034.vdocuments.net/reader034/viewer/2022042802/5f385562ed07224f381dcd36/html5/thumbnails/23.jpg)
σ
�p�o
�p
Effective Stress Law
σ
Coupled Effects
Applied Stress
Pore Pressure
Strength, Stability
Dilatancy
Pore Fluid, PpPore Fluid, Pp
Shear Rate
![Page 24: Mechanics of Earthquakes and Faulting · Friction mechanics of 2-D particles € τ = σ(µ p + dθ/ dγ) € τ = σ(µ p + dh / dx) € W = τ p dγ + σ dθ • Dilatancy rateplays](https://reader034.vdocuments.net/reader034/viewer/2022042802/5f385562ed07224f381dcd36/html5/thumbnails/24.jpg)
Dilatancy:
Pore Fluid, PpPore Fluid, Pp
Volumetric Strain:Assume no change in
solid volume
Dilatancy Rate:
Shear Rate
![Page 25: Mechanics of Earthquakes and Faulting · Friction mechanics of 2-D particles € τ = σ(µ p + dθ/ dγ) € τ = σ(µ p + dh / dx) € W = τ p dγ + σ dθ • Dilatancy rateplays](https://reader034.vdocuments.net/reader034/viewer/2022042802/5f385562ed07224f381dcd36/html5/thumbnails/25.jpg)
Dilatancy:
Pore Fluid, PpPore Fluid, Pp
Volumetric Strain:Assume no change in
solid volume
Dilatancy Rate:
Shear Rate
Dilatancy Hardening if : or
Undrained loading
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Pore Fluid, PpPore Fluid, Pp
Shear Rate
Dilatancy Hardening if :
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Pore Fluid, PpPore Fluid, Pp
Shear Rate
Dilatancy Weakening can occur if:
This is shear driven compaction