Mechanics of Earthquakes and Faulting Lecture 5, 12 Sep Fluids: Mechanical, Chemical Effective Stress Dilatancy Hardening and Stability Volumetric work and stability. Read: Frank, F. C., On dilatancy in relation to seismic sources. Rev. Geophys. 3, , 1965 Friction, contact mechanics, hardness, base-level friction coefficient Instability, Stick-slip dynamics Read Rabinowicz, 1951 Read Chapter 2 of Scholz

Cohesive zone crack model, applies to fracture and/or friction Cracked/Slipping zone w Dugdale (Barenblatt) Shear Stress Slip, displacement Breakdown (cohesive) zone Intact, locked zone oo yy ff dcdc Dislocation model, circular crack ∆  = (  o -  f )

Dislocation model for fracture and earthquake rupture Shear Stress oo yy ff  Relation between stress drop and slip for a circular dislocation (crack) with radius r For  =0.25, Chinnery (1969) Importance of slip: e.g., M o =  A u w Breakdown (cohesive) zone

 22  23  21 r r’r’ G is Energy flow to crack tip per unit new crack area Critical energy release rate G crit is a material property --the “ fracture energy ” G crit = K c 2 / E = 2 , where K c is the critical stress intensity factor (also known as the fracture toughness).

Shear Fracture Energy from Postfailure Behavior Strain Differential Stress,  1 - P c  Fracture Dilatancy: cracks forming and opening Hardening, Modulus Increase: cracks closing Brace, Paulding & Scholz, 1966; Scholz 1968.

Shear Fracture Energy from Postfailure Behavior Strain  1 - P c  Fracture Lockner et al., 1991

Shear Fracture Energy from Postfailure Behavior Wong, 1982, found that shear stress dropped ~ 0.2 GPa over a slip distance of ~50 microns. Exercise: Estimate G from this data and compare it to the values reported in Scholz (Table 1.1) and Wong, Lockner et al., 1991 Inferred shear stress vs. slip relation for slip-weakening model. (based on Wong, 1982)

Cohesive zone, slip weakening crack model for friction Cracked/Slipping zone w Shear Stress Slip, displacement Breakdown (cohesive) zone Intact, locked zone oo yy ff dcdc

Fluids Mechanical Effects Chemical Effects  effective =  n - Pp Mechanical Effects: Effective Stress Law 11 11 33 33 Pp

Fluids Mechanical Effects Chemical Effects  effective =  n - Pp Mechanical Effects: Effective Stress Law 11 11 33 33 Pp Rock properties depend on effective stress: Strength, porosity, permeability, Vp, Vs, etc. Leopold Kronecker (1823–1891)

 effective =  n - Pp 11 11 33 33 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.

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

 pp oo pp Void space filled with a fluid at pressure Pp But what if Ar ≠ A ?  For example, we expect that shear strength depends on effective stress, but perhaps not in the way envisioned by:

 pp oo pp Mechanical Effects: Effective Stress Law For brittle conditions, Ar / A ~ 0.1  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  = 0.9

 pp oo pp Effective Stress Law  Coupled Effects Applied Stress Pore Pressure Strength, Stability Dilatancy 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.

 pp oo pp Effective Stress Law  Coupled Effects Applied Stress Pore Pressure Strength, Stability Dilatancy: Shear driven volume change Pore Fluid, Pp Pp, ,  Not expected for linear elastic material (K, E, µ,, ) = Fluid delivery rate

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

Dilatancy: Pore Fluid, Pp Shear Rate

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

Dilatancy: Pore Fluid, Pp Volumetric Strain: Assume no change in solid volume Dilatancy Rate: Shear Rate Dilatancy Hardening if : or = Fluid delivery rate

Pore Fluid, Pp Shear Rate Dilatancy Hardening if : or

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

dh dx Friction mechanics of 2-D particles W is total work of shearing W =  d  =   d  Data from Knuth and Marone, 2007

dh dx Friction mechanics of 2-D particles W is total work of shearing W =  d  =   d  Data from Knuth and Marone, 2007

dh dx Friction mechanics of 2-D particles Data from Knuth and Marone, 2007

Friction mechanics of 2-D particles 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

Adhesive and Abrasive Wear: Fault gouge is wear material Chester et al., 2005 where T is gouge zone thickness,  is a wear coefficient, D is slip, and h is material hardness This describes steady-state wear. But wear rate is generally higher during a ‘ run-in ’ period. And what happens when the gouge zone thickness exceeds the surface roughness? We ’ ll come back to this when we talk about fault growth and evolution.

Fault Growth and Development Fault gouge is wear material Fault offset, D Gouge Zone Thickness, T ‘ run in ’ and steady-state wear rate This describes steady-state wear. But wear rate is generally higher during a ‘ run-in ’ period.

Fault Growth and Development Fault gouge is wear material Fault offset, D Gouge Zone Thickness, T ‘ run in ’ and steady-state wear rate This describes steady-state wear. But wear rate is generally higher during a ‘ run-in ’ period. 11  2 >  1

Fault Growth and Development Fault gouge is wear material Fault offset, D Gouge Zone Thickness, T ‘ run in ’ and steady-state wear rate And what happens when the gouge zone thickness exceeds the surface roughness? ? This describes steady-state wear. But wear rate is generally higher during a ‘ run-in ’ period.

Fault Growth and Development Fault gouge is wear material Scholz, 1987 ‘ run in ’ and steady-state wear rate Fault offset, D Gouge Zone Thickness, T ?

Fault Growth and Development Fault Roughness Scholz, 1990

Fault Growth and Development Scholz, 1990

Cox and Scholz, JSG, 1988 Fault Growth and Development

Tchalenko, GSA Bull., 1970 Fault Growth and DevelopmentFault zone width

Scholz, 1990 Fault zone roughness Ground (lab) surface

  Friction Base-level friction coefficient in terms of contact mechanics and hardness

Time dependent yield strength: Dieterich and Kilgore [1994] Time dependent growth of contact (acyrlic plastic)- true static contact

Friction Base-level friction coefficient in terms of contact mechanics and hardness Adhesive Theory of Fricton (Bowden and Tabor) Real contact area << nominal area Contact junctions at inelastic (plastic) yield strength Contacts grow with “age” Add: Rabinowicz’s observations of static/dynamic friction “Static” friction is higher than “Dynamic” friction because contacts are older (larger) -> implies that contact size decreases as velocity increases

Friction Base-level friction coefficient in terms of contact mechanics and hardness Adhesive Theory of Fricton (Bowden and Tabor) Real contact area << nominal area Contact junctions at inelastic (plastic) yield strength Contacts grow with “age” Add: Rabinowicz’s observations of static/dynamic friction “Static” friction is higher than “Dynamic” friction because contacts are older (larger) -> implies that contact size decreases as velocity increases

Classic theory of friction  - shear stress  n - normal stress F n - normal force F s - shear force A T - total fault area A c - the real area of contact S- contact shear strength  y - yield strength or hardness Bowden and Tabor [1960] Modified from Beeler, 2003 Friction is the ratio of shear strength to hardness This is base level friction

Karner & Marone (GRL 1998, JGR 2001) base level friction (~ 0.6 for rocks)

Time dependent yield strength: Dieterich and Kilgore [1994] Time dependent growth of contact (acyrlic plastic)- true static contact Modified from Beeler, 2003

time dependent closure (westerly granite) - approximately static contact after Dieterich [1972] Other measures of changes in ‘ static ’ friction, contact area, or strength ‘ hold ’ test Modified from Beeler, 2003

compaction/dilatancy associated with changes in sliding velocity after Marone and Kilgore [1993] Net change in dilatant volume Modified from Beeler, 2003

‘ hold ’ test Rate dependence of contact shear strength Rate dependent response Modified from Beeler, 2003

Summary of friction observations: 0. Friction is to first order a constant 1. Time dependent increase in contact area (strengthening) 2. Slip dependent decrease in contact area (weakening); equivalently increase in dilatancy 3. Slip rate dependent increase in shear resistance (non-linear viscous) Modified classic theory of friction: Discard products of second order terms: [e.g., Dieterich, 1978, 1979] Modified from Beeler, 2003

1st order term second order terms & 2. time dependenceslip dependence Rate and state equations: Dieterich [1979] Rice [1983] Ruina [1983]  is contact age Summary of friction observations: 0. Friction is to first order a constant 1. Time dependent increase in contact area (strengthening) 2. Slip dependent decrease in contact area (weakening); equivalently increase in dilatancy 3. Slip rate dependent increase in shear resistance (non-linear viscous) Modified from Beeler, 2003