Fig. 1 Conceptual model of a) solid-solid frictional interface and b)

(1)

Microphysical modeling of fault friction: extension to high-velocity regime

Jianye Chen , Andr R. Niemeijer, Christopher J. Spiers ( ^* é Utrecht University, The Netherlands , j.chen3@uu.nl )

Recent experimental studies show that the value of friction varies over a wide range of sliding velocities (slow sliding exhibits high friction and rate-and-state behavior, and remarkable weakening occurs as velocity approaches seismic velocity). In this study, we extend a microphysically-based friction model for granular gouges (Chen and Spiers, 2016; Niemeijer and Spiers, 2007) to the high- velocity regime, by introducing additional creep mechanism(s) activated by frictional heating. The modeling results capture all of the main features and trends seen in the experiments, including both steady state and transient aspects of the observed behavior, with reasonable quantitative agreement.

1. Chen-Niemeijer-Spiers (CNS) Microphysical Model

Fig. 1 Conceptual model of a) solid-solid frictional interface and b)

granular friction, giving physical underpinnings for classic rate-and-state friction laws and a general microphysical friction model, respectively.

1) At low velocities , it shows velocity strengthening behavior with deformation accommodated by non-dilatant plastic creep.

2) At intermediate velocities , as the velocity increases, the friction behavior will be first controlled by dilatant granular slip that is mediated by compactional contact creep, giving rise to a velocity weakening behavior; as the velocity increases further, the frictional behavior will be controlled by GBS which is inherently rate-strengthening.

3) At high velocities , dynamic friction decreases substantially due to thermal weakening effects (e.g. thermal pressurization, flash heating, superplasticity).

All the parameters fall into two categories. One is related to the fault zone configuration , such as shear zone thickness, particle size, temperature, and normal stress. The other category consists of the kinetic constants of the deformation mechanism (s), e.g. creep-related parameters.

State equation Friction “law”

Log strain rate

Frictional orflow stress

exponential flow

law

pure grain boundary friction Friction regime (dilatant granular flow) Flow regime

(non-dilatant plastic creep) Velocity strengthening

stronger than logarithmic

crossoverI crossoverII crossoverIII

Velocity weakening Velocity strengthening Dynamic weakening Flow regime (heating-enhanced creep)

Porosity φ

= φ φc

φ=φ⁰

Close packing

Close packing Close packing

creep accommodation

ψ

a_c φ σn τ

d

slip accommodation

σ_n τ σ_n τ

A unit strain by grain boundary sliding (GBS)

shear band ^σⁿ ^τ

W

(1)

(2) (3)

(5)

(6)

(9) (7)

high-Velocity series low-Velocity series

(1) (3) (2)

(6)

(7) (7)

(9) (9)

(7)

(8) (4)

(4)

(5)

(8)

10^-8 10^-7 10^-6 10^-5 10^-4 10^-3 10^-2 10^-1 10⁰ 0.0

0.2 0.4 0.6 0.8 1.0

Siman-Tov et al., 2015 Siman-Tov et al., 2015 Smith et al., 2013 Frondriest et al., 2013 Han et al., 2011 Pozzi et al., 2018 Di Toro et al., 2011

Carpenter et al., 2016 Chen et al., 2015 Verbenbe et al., 2015 Weeks & Tullis, 1985

Steady-state friction

Slip Velocity (m/s)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.0

0.2 0.4 0.6 0.8 1.0

Friction Coefficient

Displacement (m)

0.01 0.1 1

grain size (um)

SG grain size

PSZ grain size

SG thickness

0 10 20 30 40 50

SG zone thickness (um)

Friction

PSZ

SG SG

4c. Pozzi et al. (2018) : recystallization and microstructure evolution

.

Limitation: prescribed slip localization.

: (1) stability analysis; (2) earthquake cycle simulations with the extended model.

Future work

Fig 8. Friction and typical microstructures of calcite gouge sheared at 25 MPa, with Ti-alloy as host rocks.

1E-4 1E-3 0.01 0.1 1

0.0 0.2 0.4 0.6 0.8 1.0

Friction and porosity

Displacement (m)

0.0 0.5 1.0 1.5 2.0 2.5

Velocity (m/s)

Porosity Friction

0 5 10 15 20 25 30

0.0 0.1 0.2 0.3 0.4 0.5

Preslip to the onset of weakening (m)

Normal stress (MPa)

4b. Smith et al. (2015): “strengthening phase” and slip localization

(1) Extended CNS model predicts a steady-state frictional strength profile over a wide velocity range (Fig. 5).

(2) The model reproduces typical laboratory experiments (friction and compaction/dilation, Fig. 6).

(3) Dynamic weakening occurs after a “p rolonged strengthening phas e” which shortens with increasing normal stress (Fig. 7) and increasing slip rate (not shown) .

(4) The model predicts a “ Sintering Gradient ” zone, extending beyond the PSZ, characterized by low-porosity and nearly-uniform grain size, whose grain size and thickness increase with shear displacement (Fig. 9 and 10).

As grain size increases, dislocation creep will become increasingly important (c.f. creep-accommodated GBS).

10^-4 10^-3 10^-2 10^-1 10⁰ 10¹

0.0 0.2 0.4 0.6 0.8 1.0

Steady-state friction coefficient

Slip rate (m/s)

0.0 0.2 0.4 0.6 0.8 1.0

Steady-state porosity

10 nm

100 nm

A)

200 400 600 800 1000 1200 1400 0.0

0.2 0.4 0.6 0.8 1.0

Temperature ( C)^o

0.0 0.2 0.4 0.6 0.8 1.0

Steady-state porosity

Steady-state friction coefficient

10 nm 10 nm

100 nm

B)

2. Extension to high velocity regime: frictional heating actives new creep mechanism(s)

Fig. 2 Schematic model of steady-state friction as a function of log(strain rate) for granular fault.

Fig 3. Schematic model illustrating the deformation by GBS for both i n t e r m e d i a t e - a n d h i g h - v e l o c i t y r e g i m e s . I n t h e i n t e r m e d i a t e regime, GBS is accommodated by (frictional) slip, while in the high- v e l o c i t y r e g i m e , b y a ( p l a s t i c ) creep mechanism.

Fig 5. Left: compilation of carbonate friction data at sub- to co-seismic slip rate & room-T conditions. Right: Predicted steady-state friction and porosity with respect to A) slip rate and B) slip zone temperature.

Fig 4. Deformation steps of a unit strain by two end- member and the intermediate GBS processes.

4a. De Paola et al. (2015): general fearture

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

0.0 0.2 0.4 0.6 0.8 1.0

Displacement (m) 18 MPa 12 MPa

0.0 0.5 1.0 1.5

Velocity (m/s)

low speed grain size d0 = 1 um high speed grain size d1= 2 nm

3. Steaty-state behavior

Fig 6. Evolution of friction with displacement and calculated temperature of the shear zone (Left: laboratory results, Right: model prediction)

Fig 7. Predicted evolution of friction with the logarithm of displacement and the pre-weakening strengthening phase with normal stress (Left:

laboratory results, Right: model prediction)

Fig 9. Predicted friction and microstructural parameters with displacement.

Grain size growth :

Fig 10. Evolution of a) friction, b) temperature, c) porosity and d) grain size over log(gouge thickness).

5. Discussion and future work

Prescribed evolution of shear band thickness as guided by microstructural observations

low speed : W

1

= 50 um : = 10 um high speed W

0

Prescribed nominal grain size within the shear zone following laboratory studies (De Paola et al., 2015)

= 1 um low velocity : d

0

= 10-100 nm high velocity : d

1

Steady-state temperature follows Di Toro et al. (2011)

Multi-creep mechanisms:

For carbonate gouges pressure solution

+

diffusion-accommodated grain boundary sliding (GBS)

+

dislocation creep (grain size- insensitive)

Schmid et al.(1977)

Schmid et al.(1980)

Numerical method:

Assuming a spring-slider fault system and incorporating realistic across-fault structure, the governing (ordinary)

equations are solved in combination with a FE model for calculating temperature and grain size evolution over the fault thickness. Evolution of PSZ thickness is prescribed based on laboratory microstructural observations.

These are implemented in the FE package Comsol.

Equation (1) rewrites as:

(1)

0 200 400 600 800 1000

0.0 0.2 0.4 0.6 0.8 1.0

Temperature (C) 18 MPa

PSZ is pre-scribed following exp., but allowed to evolve as slip decelerates.

Leading to continuous transition from friction to flow behavior

from dilatant to compacted gouge

Diffusion creep-accommodated GBS (zero porosity)

Frictional slip-accommodated GBS

ψ=26^o

σn τ

shift ψ=26^o

σn τ

pure shear +dilatation

pure shear +compaction

friction slip +dilatation

friction slip +compaction ψ=26^o

σn τ

pure shear pure shear

Diffusion creep-accommodated GBS (high porosity)

rotation by 13^o

(Ashby and Verral model)

Key

1000

200 50 100

20

150 200

1000

PSZ SG

0.17 0.17

0.08 0.08

0.02

0.02 0.02

0.02

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.5 1.0 1.5

Velocity (m/s)

a)

b)

c)

d)

PSZ SGPSZ SG

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Distance from SS (µm)

Displacement (m)

Distance from SS (µm)Distance from SS (µm)

calculated T at varied displacements

Fig. 1 Conceptual model of a) solid-solid frictional interface and b)

Microphysical modeling of fault friction: extension to high-velocity regime

Jianye Chen , Andr R. Niemeijer, Christopher J. Spiers ( * é Utrecht University, The Netherlands , j.chen3@uu.nl )

1. Chen-Niemeijer-Spiers (CNS) Microphysical Model

Fig. 1 Conceptual model of a) solid-solid frictional interface and b)

granular friction, giving physical underpinnings for classic rate-and-state friction laws and a general microphysical friction model, respectively.

1) At low velocities , it shows velocity strengthening behavior with deformation accommodated by non-dilatant plastic creep.

3) At high velocities , dynamic friction decreases substantially due to thermal weakening effects (e.g. thermal pressurization, flash heating, superplasticity).

 All the parameters fall into two categories. One is related to the fault zone configuration , such as shear zone thickness, particle size, temperature, and normal stress. The other category consists of the kinetic constants of the deformation mechanism (s), e.g. creep-related parameters.

State equation Friction “law”

A unit strain by grain boundary sliding (GBS)

PSZ

SG SG

4c. Pozzi et al. (2018) : recystallization and microstructure evolution

Limitation: prescribed slip localization.

: (1) stability analysis; (2) earthquake cycle simulations with the extended model.

Future work

Fig 8. Friction and typical microstructures of calcite gouge sheared at 25 MPa, with Ti-alloy as host rocks.

4b. Smith et al. (2015): “strengthening phase” and slip localization

(1) Extended CNS model predicts a steady-state frictional strength profile over a wide velocity range (Fig. 5).

(2) The model reproduces typical laboratory experiments (friction and compaction/dilation, Fig. 6).

(3) Dynamic weakening occurs after a “p rolonged strengthening phas e” which shortens with increasing normal stress (Fig. 7) and increasing slip rate (not shown) .

(4) The model predicts a “ Sintering Gradient ” zone, extending beyond the PSZ, characterized by low-porosity and nearly-uniform grain size, whose grain size and thickness increase with shear displacement (Fig. 9 and 10).

As grain size increases, dislocation creep will become increasingly important (c.f. creep-accommodated GBS).

A)

B)

2. Extension to high velocity regime: frictional heating actives new creep mechanism(s)

Fig. 2 Schematic model of steady-state friction as a function of log(strain rate) for granular fault.

Fig 5. Left: compilation of carbonate friction data at sub- to co-seismic slip rate & room-T conditions. Right: Predicted steady-state friction and porosity with respect to A) slip rate and B) slip zone temperature.

Fig 4. Deformation steps of a unit strain by two end- member and the intermediate GBS processes.

4a. De Paola et al. (2015): general fearture

3. Steaty-state behavior

Fig 6. Evolution of friction with displacement and calculated temperature of the shear zone (Left: laboratory results, Right: model prediction)

Fig 7. Predicted evolution of friction with the logarithm of displacement and the pre-weakening strengthening phase with normal stress (Left:

laboratory results, Right: model prediction)

Fig 9. Predicted friction and microstructural parameters with displacement.

Grain size growth :

Fig 10. Evolution of a) friction, b) temperature, c) porosity and d) grain size over log(gouge thickness).

5. Discussion and future work

Prescribed evolution of shear band thickness as guided by microstructural observations

low speed : W

= 50 um : = 10 um high speed W

Prescribed nominal grain size within the shear zone following laboratory studies (De Paola et al., 2015)

= 1 um low velocity : d

= 10-100 nm high velocity : d

Steady-state temperature follows Di Toro et al. (2011)

Multi-creep mechanisms:

For carbonate gouges pressure solution

+

diffusion-accommodated grain boundary sliding (GBS)

+

dislocation creep (grain size- insensitive)

Schmid et al.(1977)

Schmid et al.(1980)

Numerical method:

Assuming a spring-slider fault system and incorporating realistic across-fault structure, the governing (ordinary)

equations are solved in combination with a FE model for calculating temperature and grain size evolution over the fault thickness. Evolution of PSZ thickness is prescribed based on laboratory microstructural observations.

These are implemented in the FE package Comsol.

Equation (1) rewrites as:

(1)

PSZ is pre-scribed following exp., but allowed to evolve as slip decelerates.

Leading to continuous transition from friction to flow behavior

from dilatant to compacted gouge

Diffusion creep-accommodated GBS (zero porosity)

Frictional slip-accommodated GBS

Diffusion creep-accommodated GBS (high porosity)

Key

a)

b)

c)

d)

Jianye Chen , Andr R. Niemeijer, Christopher J. Spiers ( ^* é Utrecht University, The Netherlands , j.chen3@uu.nl )

All the parameters fall into two categories. One is related to the fault zone configuration , such as shear zone thickness, particle size, temperature, and normal stress. The other category consists of the kinetic constants of the deformation mechanism (s), e.g. creep-related parameters.