mechanics of earthquakes and faulting · friction mechanics of 2-d particles € τ = σ(µ p +...
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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
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’
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
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
Consider the implications of dilatancy and volume change for the total work of shearing, W
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
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
• 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
€
W = τ p dγ + σ dθ
W is total work of shearing
W = τ dγ = σ µ dγ
dhdx
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:
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
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
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:
Volu
me
Stra
in, θ
Shear Strain, γ
Marone, Raleigh & Scholz, 1990
Volu
me
Stra
in, θ
Shear Strain, γ
Marone, Raleigh & Scholz, 1990
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) =
Fluids: Consider the affect on shear strength
•Mechanical Effects•Chemical Effects
σeffective = σn - PpMechanical Effects: Effective Stress Law
σ1σ1
σ3
σ3
Pp
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
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
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
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
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
σ
�p�o
�p
Effective Stress Law
σ
Coupled Effects
Applied Stress
Pore Pressure
Strength, Stability
Dilatancy
Pore Fluid, PpPore Fluid, Pp
Shear Rate
Dilatancy:
Pore Fluid, PpPore Fluid, Pp
Volumetric Strain:Assume no change in
solid volume
Dilatancy Rate:
Shear Rate
Dilatancy:
Pore Fluid, PpPore Fluid, Pp
Volumetric Strain:Assume no change in
solid volume
Dilatancy Rate:
Shear Rate
Dilatancy Hardening if : or
Undrained loading
Pore Fluid, PpPore Fluid, Pp
Shear Rate
Dilatancy Hardening if :
Pore Fluid, PpPore Fluid, Pp
Shear Rate
Dilatancy Weakening can occur if:
This is shear driven compaction