thermodynamics of friction

Thermodynamics of friction

Noa Mitsui

1

Contents

1. Introductions ・tectonics & earthquakes ・theory of earthquakes ・relation between earthquakes and friction

2. Purpose: thermodynamic modeling of rock friction

3. Research ・introduction: rock experiments ・thermodynamic modeling

4. Summary

2

Tectonics & earthquakes in the world

• Plates move relatively at several cm/yr

• Japan locates on some boundaries

• Many earthquakes occur around Japan

Plate movements and their boundaries

Earthquake epicenter (M5.5<) 3 M= -3.2 + 0.67log10E (E: energy)

Difficulty in understanding of earthquake

(Source area= ruptured area estimated by detected seismic waves)

earthquakes distribution (seismicity) along plate boundaries

Eurasian Plate

Pacific Plate

Philippines sea Plate

N. American Plate

yr

yr

Tokyo Fukushima

Nagoya

6

• People want to know “WHEN & WHERE” earthquakes will happen

Difficulty of the forecast

Incompleteness of theory




Eurasian Plate

Pacific Plate


N. American Plate

yr

yr

Tokyo Fukushima

Nagoya

7







Eurasian Plate

Pacific Plate


N. American Plate

yr

yr

Tokyo Fukushima

Nagoya

8

2011

Cross section along the plate movement

(1964-2007 & 2011, M5<, USGS)




Incomplete theory = elasticity

It cannot explain small earthquakes (=seismicity)

occurring in the plate ↓

realistic deformation theory is necessary (= motive)

• current bases of seismology: All strain released by slip at contact surface plate boundary

||

Schematic diagram of deformation at plate boundary

trench

strain increase

tsunami

rebound 9

All strain released

Slip only at boundary

Only in theory

Main theme of today’s talk

☆ How we can obtain deformation theory of medium? (bases of the earthquakes occurrences) fracture and deformation in the earth’s crust

Research object: friction law in rock experiments ・earthquakes are compared to frictional slips along the tectonic plate boundary ・easiness to understand: rock experiment > earthquakes I will focus on the rate- and state-dependent friction law (Dieterich, 1979)

10

Sand particles in frictional layer between rocks

• Even in the small frictional layer, sand particles are deformed, crushed, compacted.

• These phenomena probably causes the complexity of friction

• The mechanism of friction can be related to that of earthquakes

11 (Mair & Marone,1999)

Before dislocation After dislocation

1mm

Sand particles in frictional layer between rocks

• Even in the small frictional layer, sand particles are deformed, crushed, compacted.

• These phenomena probably causes the complexity of friction

• The mechanism of friction can be related to that of earthquakes

12 (Mair & Marone,1999)

Before dislocation After dislocation

1mm

2011

• wide distribution along the plate boundary

• Magnitude: M9.0-M2(detectable limit)

Wideness of earthquakes

The mechanism can be related to them

2011

Cross section along the plate movement

Magnitude-frequency diagram

Gutenberg-Richter law (Power law)

freq

uenc

y

Magnitude (1924-2007, M5<, JMA)

(1964-2007 & 2011, M5<, USGS)

13

← Another model of G-R law called as “cumulative Gompertz distribution” can have the same entropy formula as that in heavy ion physics (Biro et al, 2015)

http://pubs.usgs.gov/of/2010/1083/d/

Main theme of today’s talk

☆ How we can obtain deformation theory of medium? (It causes the earthquakes occurrences) fracture and deformation in the earth’s crust

Research object: friction law in rock experiments ・earthquakes are compared to frictional slips along the tectonic plate boundary ・easiness to understand: rock experiment > earthquakes I will focus on the rate- and state-dependent friction law (Dieterich, 1979)

14

Contents

1. Introductions ・tectonics & earthquakes ・theory of earthquakes ・relation between earthquakes and friction

2. Purpose: thermodynamic modeling of rock friction

3. Research ・introduction: rock experiments ・thermodynamic modeling

4. Summary

15

Outline of rock experiments

• Static & dynamic friction are tested with various condition:

rock type, temperature, humidity, normal stress with/without sand at the interface

16

Load (velocity controlled)

stress stress

Frictional interface

rock

（Dieterich, 1981）

Servocontrolled to constant

Static friction increases with time

• Static friction ∝ log (time)

• Friction decreases to dynamic one with displacement

(Marone, 1998) 17

Feature1/3: Stable dynamic friction ∝ log(slip velocity)

• Rock samples loaded with constant velocity

• Friction weakens or strengthen with velocity depending on the tested condition

ex. Interface sand without sand: weaken with sand: strengthen

(Marone, 1998)

Dyna

mic

fric

tion

Slip velocity

Velocity weakening

18

logV dependency

Feature2/3: frictional jump ∝ log(velocity step)


• Loading velocity step causes instantaneous jump of dynamic friction

• Following the jump, the friction transitions to the stable one

• Both jump and relaxation depend on ln(velocity step)

velocity (µm/s)

2.5

19

∆F<0

Velocity weakening

Velocity weakening case

25

fric

tion

Feature2/3: frictional jump ∝ log(velocity step)


• Loading velocity step causes instantaneous jump of dynamic friction

• Following the jump, the friction transitions to the stable one

• Both jump and relaxation depend on ln(velocity step)

Dc

velocity V1

20


V2

fric

tion

bln(V2/V1) aln(V2/V1)

Feature3/3: symmetric relaxation with constant displacement


• Both case of velocity increase and decrease has frictional jump and relaxation with constant displacement (Dc)

vel. increase: jump up

vel. decrease: jump down

fric

tion

Dc

velocity (µm/s)

2.5 2.5 25

21


Dc

Empirical equation of the friction law

μ: shear stress t: time V: slip velocity μ0:reference friction for V0 Dc: characteristic slip distance θ: state variable a,b: positive parameters

Empirical equations of rate- and state-dependent friction law

(Dieterich, 1979)

• Neither of them can reproduce both (static & dynamic) friction. ↓ • We will reproduce the experimental result with thermodynamics.

Simulated frictional change with displacement

V2=10µm/s V1=1µm/s V1=1µm/s

22

aln(V2/V1) bln(V2/V1)

−=

LV

LV

dtd θθθ lnor

(Ruina, 1983)

friction = dissipative event -> thermodynamic issue

: Static friction Dynamic friction

① ②

Previous thermodynamic model (Mitsui and Van, 2014)

Rock sample

external force

Start point

permanent displacement

Rock sample

xm

23

damping force

( )

amics thermodynof law 2nd0)( :balanceentropy

22:entropy

:energy total

:motion ofequation

22

≥−=

−−=

==

−=

rkrVFUST

krmVEUS

VFxFE

FFxm

d

ee

de

↑ internal energy

↑ elastic modulus

Result of linear model (Mitsui and Van, 2014)

fric

tion

velocity (µm/s) 2.5 2.5 25


Improvement of thermodynamic model • Important feature ・introduced elements 1. instantaneous jump 1. partial steady state aln(V2/V1)

2. following relaxation bln(V2/V1) 3. symmetric relaxation with disp. 2. logarithmic increment Dc

3. factor of slip reduction in elastic displacement

24

③

② ①

rVr

rxddr

dtdr

αα −=

−→

)(

( ) 22

22

2200

22

krVVmEU

krmVEU

−−−=

↓

−−=

( ) )/ln( 00 VVVV µµ ∆→−∆

(Van, Mitsui, Hatano, 2015, arXiv:1501.04608v1)

based on thermal activation theory (chemical reaction: Eyring, 1936; rock friction: Nakatani, 2001)

Improved constitutive equations

25

( )rlVVlrlVVlVr

rlVVl

κακ

κµ

2021

2021

1201

)/ln(1)/ln(

)/ln(

−+−

=

−=∆↓


( ) 0/ln)(

0)( :balanceentropy

000n

n

≥−

−∆+=

↓

≥−=

rVrrVVV

AUST

rrVAUST

ακµµ

σ

κµσ

1 2 3

1. partial steady state 2. logarithmic increment 3. factor of slip reduction

Improvements

constitutive equations:

Vr

dxdr

dtdr

=→

cf. Ruina law (1983)

Results

Data comparison

26

( )rlVVlrlVVlVr

rlVVl

κακ

κµ

2021

2021

1201

)/ln(1)/ln(

)/ln(

−+−

=

−=∆

mLllblalµ

κα

20/102.0,015.0

8.0,1

221

121

=======

==


Dieterich law

Ruina law (α=0, κ=1)

(α=8, κ=0.85)

α→larger: asymmetric relaxation

Static friction

• log(t)-like increase in static friction ⊿μ is reproduced • Both dynamic & static friction are reproduced with

one set of constitutive equations

27

(Marone, 1998)

002.0=κ

074.0=κ

Data ●： ■： 074.0

002.0==

κκ


Summary • First motive: to understand mechanism of earthquakes ↓ • Current motive: deformation theory of medium under stress

object: static & dynamic friction of rocks (quasi-thermostatic condition) Results: 1. friction at reference velocity can be explained by using partial steady-state condition of internal energy 2. logV dependencies of friction are reproduced by the logarithmic increments of entropy production 3. symmetric relaxation can be explained by introducing a factor of slip reduction in elastic evolution 4. Above three elements reproduce log-t increase of static friction Advantage: simple model can be applied to geophysics with upscaling Future plan: physical bases of assumptions in this model, etc. 28

thermodynamics of friction

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