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