On the relation between plasticity, friction, and geometryS. Barbot1, a)

Earth Observatory of Singapore, Nanyang Technological University

Plasticity refers to thermodynamically irreversible deformation associated with a change of configuration ofmaterials. Friction is a phenomenological law that describes the forces resisting sliding between two solidsor across an embedded dislocation. These two types of constitutive behaviors explain the deformation ofa wide range of engineered and natural materials. Yet, they are typically described with distinct physicallaws that cloud their inherent connexion. Here, I introduce a multiplicative form of kinematic friction thatclosely resembles the power-law flow of viscoplastic materials and that regularizes the constitutive behavior atvanishing velocity, with important implications for rupture dynamics. Using a tensor-valued state variable thatdescribes the degree of localization, I describe a constitutive framework compatible with viscoplastic theoriesthat captures the continuum between distributed and localized deformation and for which the frictionalresponse emerges when the deformed region collapses from three to two dimensions.

I. INTRODUCTION

The deformation of solid media under stress is com-plex, often nonlinear, and involves a wide range of mi-crophysical processes. Anelastic deformation representsthe thermodynamically irreversible deformation of mate-rials, implying that under this type of deformation ma-terials do not recover their initial configuration if theapplied stress is removed. Often, deformation includesboth elastic (reversible) and anelastic components. Plas-tic flow refers to the distributed anelastic deformationof materials under stress and is observed in most engi-neered and natural materials, including metals, soils, androcks. Friction describes the resistance to sliding of solidsurfaces in contact. Being confined at a solid interface,friction represents a case of localized deformation.

The rheology of plastic flow has been classically de-scribed by the von Mises model1,2 or its variants3 thatassumes no plastic flow when the stress is below a yieldsurface. Friction has also be described mathematically asa yield surface, implying that sliding may not occur un-less the shear forces reach a critical value that depends onnormal stress. These results were found experimentallyby Da Vinci, re-discovered by Amontons4, and confirmedby Coulomb5. These models afford a simple descriptionof the phenomenology but they do not describe how fastthe medium flows or breaks when placed under sufficientstress nor the complex dynamics or instabilities that candevelop near the yield surface6.

The limitations of the classic formulations have beenremedied by the development of viscoplasticity7,8 andthe Dieterich-Ruina rate-and-state form of kinematicfriction9,10. In the classic view of plastic flow and friction,anelastic deformation occurs only at the yield surface andthe stress cannot exceed the yield surface (Fig. 1). Inthe modern view, the yield surface represents a referencepoint under which deformation is very slow and abovewhich deformation is very fast. The concept of a yieldsurface is only useful if the anelastic deformation rate is

a)Electronic mail: sbarbot@ntu.edu.sg

superlinear with stress. The weak sensitivity of the fric-tional resistance to sliding velocity noted by Coulombis captured by the logarithmic sensitivity to velocity inthe modern formulation of rate-and-state friction (a widerange of slip velocity occurs for a narrow range of effec-tive stress). For plastic flow, the strength is often ratedependent, taking a linear form for viscoelastic materialsand often a power-law form for plastic materials. How-ever, there is a continuum between linear, power-law, andexponential stress-strain-rate relationships embodied byprocesses such as diffusion creep, dislocation creep11, andPeierls creep12,13. Like friction, plastic flow may also ne-cessitate state variables for phenomena associated withtransient creep14 and work-hardening in general15. Theviscoplasticity and rate-and-state friction theories haveallowed modeling multiple cycles of ruptures with com-plex deformation histories16,17.

As noted by others18,19, the similitudes in the paral-lel development of the theories of plasticity and frictionare striking. The mechanics of friction and plasticityare also deeply related. For example the micromechanicsof plasticity can be explained as the motion of internaldislocations20,21 and the rate- and state-dependence offriction can be explained by the plastic behavior of thecontact asperities regulating the real area of contact22–24.In many cases, the frictional interface consists in a finite-width shear zone made of granular, damaged, or semi-brittle material and the assumption of a simple frictionalbehavior is for mathematical or modeling convenienceonly25,26. This suggests that these theories could be com-bined in a more general, unifying framework.

Plasticity and friction represent the end-members fordistributed and localized deformation, respectively. How-ever, they cannot capture the continuum between local-ized and distributed deformation. This can be partic-ularly important when describing the inception of newfrictional surfaces by the accumulation of damage or in-creased localization. During this process, the materialfirst deforms plastically until a frictional interface hasmatured. When this point is reached, a new surface hasemerged and friction can become the dominant deforma-tion mechanism.

In geophysics, there is an impetus to combine long-

arX

iv:1

711.

0195

4v2

[ph

ysic

s.ge

o-ph

] 2

1 Fe

b 20

18

2

Slip

vel

ocity

(loc

aliz

ed d

efor

mat

ion)

Shear traction on surface

classicfriction

(τ = µ0 σn)modernfriction

(multiplicative form)

Stra

in ra

te (d

istri

bute

d de

form

atio

n)

Shear stress in volume

von Mises orDrucker-Prager

plasticity Visc

oplas

ticity

Newton

ian

visco

elastic

mate

rial

linear

supe

rlinea

r

A

B

FIG. 1. Displacement fields in two-dimensional models of de-formation due to increasingly localized deformation. We con-sider an embedded dislocation buried at 20 km depth, 20 kmlong in the down-dip direction, and dipping 45◦ from the ver-tical. The left panels are for a screw dislocation and the rightones for an edge dislocation. The top, middle and bottompanels are for shear zones of 10 km, 5 km, and 200 m, respec-tively. The contour lines are every 20 cm.

term and short-term processes to allow faults to emergefrom tectonic processes and resolve rupture dynamics onthese new structures27 as important feedbacks operateat these different time scales. That dynamic ruptureson frictional interfaces and plastic deformation operateat different time scales represents a significant numericalchallenge. But there is no convenient theoretical frame-work to describe friction and plasticity in a unified way.

Towards that objective, I highlight two important re-sults. In Section II, I show that the kinematics of dis-tributed deformation and localized deformation can becaptured by a single mathematical expression28. Thishighlights a continuum of kinematic descriptions betweendistributed and localized deformation. In Section III, Ishow that the modern rheological laws that describe plas-tic flow have a striking similarity with the ones describingfrictional dynamics, despite some subtle but importantdifferences. To do so, I introduce the multiplicative form

of rate-and-state friction. In Section IV, I conjecture aconstitutive framework that reduces to viscoplasticity insome situations and to kinematic friction framework inothers. The model predicts that frictional behavior is anend-member rheological response due to the collapse ofdeformation region from three to two dimensions.

In this paper, I do not discuss a new microphysicalmodel of plasticity or friction. Such important modelsshed light on the constitutive equations of anelasticityand can be found in the work of various authors29–34.Instead, I discuss the relation between the macroscopicrepresentations of plasticity and friction in a continuumthat are directly incorporated in analytic or numericalmodels of time-dependent deformation.

II. KINEMATIC CONTINUUM BETWEENDISTRIBUTED AND LOCALIZED DEFORMATION

In this Section, I consider the kinematics of deforma-tion of plastic and brittle materials and how anelasticdeformation induces strain in the surrounding mediumby elastic coupling. The static or quasi-static equilib-rium attained in the presence of anelastic deformationcan be modeled using transformation strain15,35, whichcan be due to thermal stress, multi-phase flow, or plasticdeformation26. Steketee25,36 showed that the Schmidttensor, a particular form of transformation strain, cancapture the motion of dislocations. However, because ofthe fundamental difference in nature between distributedand localized deformation, these cases were historicallytreated distinctly.

Here, I show how various degrees of localization canbe described within a single framework. To do so, I con-sider the plastic deformation of a shear zone confined ina larger continuum and subjected to the transformationstrain components εij . The region that deforms anelasti-cally takes the form of a cuboid with dimension L1, L2,and L3. For convenience, let us imagine a reference sys-tem aligned with the cuboid so that the subscript indicesmatch those of the basis vectors. The resulting deforma-tion can be evaluated by solving the governing equationfor elasticity using equivalent body forces to incorporatethe plastic strain26,28,37. For illustration purposes, I con-sider a half space with a free surface and the cases ofin-plane and anti-plane deformation.

I subject the shear zone to increasing plastic strain ina cuboid of decreasing width L2. For the sake of simplic-ity, I adopt a two-dimensional model where L1 =∞ andI choose a grid size of 250 m for all calculations, whichalso represents the dimension of a representative volumeelement. I reduce the plastic strain to the nontrivial com-ponent ε23 for the case of in-plane strain and to the non-trivial component ε12 for the case of anti-plane strain.The cumulative displacement across the shear zone dueto plasticity alone, i.e., irreversible, is

s3 = 2ε23L2 (1)

3

0

10

20

30

Dep

th (k

m)

Dep

th (k

m)

Dep

th (k

m)

40

50

60

-30 -20 -10 0Distance (km) Distance (km)

10 20 30 -30 -20 -10 0 10 20 30

0

10

20

30

40

50

60

0

10

20

30

40

50

60

L2

L3

2ε12L2=s1

Antiplanestrain

Downward displ. (m)-0.4 0 0.4

horizontaldisplacement0.5 mDisplacement (m)

L2 = 200 m L2 = 200 m

L2 = 5 km L2 = 5 km

L2 = 10 km L2 = 10 km

-0.4 0 0.4

L2

L3

In-planestrain

2ε23L2=s3

A

B

C

D

E

F

FIG. 2. Displacement fields due to increasingly localized deformation. We consider the plastic deformation of a cuboidrepresentative volume element (rectangle) buried at 20 km depth, 20 km long in the down-dip direction, and dipping 45◦ fromthe vertical. The left panels are for out-of-plane plastic strain and the right ones for in-plane plastic strain. The top, middle andbottom panels are for shear zones of 10 km, 5 km, and 200 m, respectively. The contour lines are every 20 cm. With increasinglocalization of plastic deformation, the deformation converges to the cases of a screw dislocation and an edge dislocation forthe left and right panels, respectively.

4

for in-plane strain and

s1 = 2ε12L2 (2)

for anti-plane strain. In these simulations, I keep s1 ands3 constant and I reduce L2 incrementally. The resultingdeformations are shown in Fig. 2. When the width of theshear zone is lower than the grid size, the shear zone vir-tually collapses from three to two dimensions, i.e., froma volume to a surface. The solutions converge towardsthe one for an edge dislocation for in-plane strain andtowards the one for a screw dislocation for anti-planestrain. Using in-plane strain as an example, the limitcase of localized deformation is obtained for

ε23 = limL2→0

s32L2

. (3)

That is, the motion of dislocation is associated with in-finite strain. For brittle deformation, plastic strain isinfinite, the support is zero, but the product s3 = 2ε23L2

is finite and the cumulative plastic displacement is calledslip18.

These results indicate that brittle deformation can bethought of as an end-member of plasticity for infinitestrain over a zero volume, at least in terms of the kine-matics of deformation. These results have been exploitedto simulate fault slip in the framework of transformationstrain using numerical methods25,26 and to derive ana-lytic expressions for the deformation of elasto-plastic me-dia for various degrees of localization, from distributed tolocalized28,37. In reality, even the dislocations associatedwith the motion of lattice defects in crystal structureshave a finite width commensurate to the atomic scale(L2 & 1 A), so plastic strain is always bounded.

Unified kinematic models of anelastic deformationfrom distributed to localized have been explored26,28 andform a motivation to explore the possibility of a unifica-tion of the constitutive laws.

III. THE MULTIPLICATIVE FORM OF KINEMATICFRICTION AND ITS RELATION TO VISCOPLASTICITY

Viscoplasticity describes the rate-dependent deforma-tion of plastic materials under stress38. The model de-scribes the response of a representative volume element(Fig. 3a). The simplest model for viscoplasticity is a com-bination of a spring and a (potentially nonlinear) dash-pot. Broadly speaking, the rate of plastic strain can bedescribed as

ε = ε(σ, ε, ψi) , (4)

where ε is the plastic strain rate tensor, σ is the stresstensor, and the ψi are state variables following the evo-lution laws

ψi = ψi(ε,σ, ψi) . (5)

The inclusion of state variables and the cumulative strainin the constitutive behavior allows for the transient and

º

ε11

ε12

ε22

ε23ε12

ε33

ε23ε13

ε13

tractionvector t(e3)

tractionvector t(e2)

tractionvector t(e1)

º

º

º

º

º

º

ºº

Plastic flow in a volume

B

A

Frictional sliding on a surface

spring-dashpot

normalvector n

strain-ratecomponents

shear tractionvector ts

velocityvector v

spring-slider

faultsurface

normal tractionvector tn

FIG. 3. Representative elements for viscoplasticity and kine-matic friction. A) The dynamics of viscoplastic flow is rep-resented by a stress-strain-rate constitutive relationship, thesimplest one being a spring-dashpot model. B) The dynam-ics of frictional sliding relies on a velocity-stress constitutiverelationship that incorporates the orientation of the represen-tative surface element. The simplest model is a spring-slider.

work-hardening effects and makes the rate of deformationnon-conservative, i.e., dissipative and path dependent.

Modern friction describes the rate of sliding acrossa representative surface element upon application ofshear and normal tractions9,10. The simplest constitu-tive model is to combine a spring and a slider (Fig. 3b).Rate-and-state friction is a phenomenological law thatdescribes the frictional resistance to sliding includinghealing and weakening. This framework has some knownshortcomings39–41, but it has nonetheless been successfulat explaining many dynamic mechanical systems over awide range of materials and boundary conditions42–51. Afriction law can be written as follows

v = v(σ, θi ;n) , (6)

where v is the sliding velocity, the θi are state variableswith the evolution laws

θi = θi(v,σ, θi ;n) , (7)

and n is the local normal direction of the surface wherefrictional sliding occurs. The notable difference betweenplasticity and friction is the role of geometry. The con-stitutive equation for viscoplasticity applies everywherein the medium. In constrast, the traction and the slipvector cannot be established without a proper represen-

5

viscoplasticshear zone

elasticmedium

width L

velocityvector v

strain-ratetensor εº

FIG. 4. A finite shear zone model for frictional sliding. Therelative velocity across frictional interface is accommodatedby plastic strain within the shear zone.

tation of the surface in terms of its orientation and po-sition. Indeed, frictional sliding may be associated withthe transformation strain

R =1

2(v⊗ n + n⊗ v) δ (x · n) (8)

that explicitly incorporates the normal vector of the fric-tional surface. Any constitutive law for anelastic defor-mation that does not explicitly include geometry cannotfully represent friction.

A. The multiplicative and additive forms of kinematicfriction

With this important difference in mind, I now describesome similarities in form between the constitutive lawsfor viscoplasticity and kinematic friction. I consider athought experiment where the frictional interface is mod-eled by a narrow shear zone of width L undergoing plasticdeformation (Fig. 4) with the following rheology

ε

ε0=

(τ

µ0σ

)n(d0d

)m

, (9)

where ε0 is a reference strain rate, τ is the norm of theshear stress in the direction of sliding, d is the averagegrain size in a gouge layer, d0 is a reference value, andµ0σ is a reference stress. I assume that grain size is astate variable that varies with

d = V0 − 2εd , (10)

where the first term in the right-hand side gives riseto healing and the second term accounts for dynamicrecrystallization52. Classically, viscoplastic rheologies areformulated with a different nomenclature, but I make thischoice of parameterization for the sake of the argument.

The power exponent is assumed in the range n = 50−100,corresponding to low-temperature creep. The grain sizeexponent is assumed in the range m ∈ [0, 3], the boundscorresponding to negligible and significant dependence ongrain size, respectively.

The above formulation can be shown to be mathemat-ically equivalent to rate-and-state friction with the aginglaw with the following few assumptions. First, to easethe comparison, I turn the strain rate into a velocity us-ing V = 2Lε and V0 = 2Lε0. Using the state variabled = θV0, with d0 = θ0V0, I obtain

V

V0=

(τ

µ0σ

)n(θ0θ

)m

. (11)

Then, I recast the stress-strain-rate relationship in termsof frictional resistance

τ = µ0σ

(V

V0

)1/n(θ

θ0

)m/n

. (12)

As the sensitivity of stress to velocity is only weak, Iconsider a Taylor series expansion using xy = ey ln x = 1+y lnx+O(ln2 x) around x = 1, expanding the relationship(12) accordingly around V = V0. Defining friction asµ = τ/σ and keeping only the linear terms, I obtain

µ = µ0

[1 +

1

nlnV

V0+m

nln

θ

θ0

]. (13)

The correspondence with the classic nomenclature ofrate-and-state friction is immediate if we note a = µ0/n,b = ma, and d0 = L. To go further let us consider theevolution law (10). Turning the strain rate into a veloc-ity, I obtain

d = θV0 = V0 −V

LθV0 , (14)

or, simply,

θ = 1− V θ

L, (15)

which is the aging law of rate-and-state friction10. Inthis formulation the state variable is interpreted as theaverage grain size scaled by a reference sliding velocityand the reference grain size coincides with the width ofthe gouge layer, which also matches with the charac-teristic weakening distance. The direct effect coefficienta = ∂µ/∂ lnV is related to the reciprocal of the powerexponent, consistent with values of a of the order of 10−2.The b parameter is controlled by the dependence to grainsize.

I note that there is no laboratory experiment that de-scribes a combination of power-law flow with a high stressexponent with a grain-size dependence such as in (9), al-though it complies to the case of dislocation glide53 withn = 2. Nonetheless, the plastic model for friction encap-sulated in equations (9) and (10) is validated a posteriori

6

by a mathematical equivalence with rate-and-state fric-tion, an empirical law tuned to a number of laboratoryexperiments. Admittedly, the proof is only mathemati-cal, and therefore the evolution law (10) may not repre-sent grain size but any other physical property that varieswith the same rate.

I call (11) and (12) the multiplicative or power-lawform of rate-and-state friction, and the Dieterich-Ruinaformulation (13) the additive form of rate-and-state fric-tion. The multiplicative form is more appealing than theadditive one because the latter is ill-posed for vanishingvelocities. Indeed, (13) incorrectly predicts a negativenorm of the traction vector for sufficiently small veloc-ities and an infinite value for a vanishing velocity. Theviscoplastic model of frictional resistance also brings newinsight. Indeed, some laboratory measurements supportthat the critical slip distance correlates with the widthof the frictional interface54,55.

I emphasize that (11) is not formally a viscoplasticlaw because the dynamic variables include the shear andnormal traction instead of the deviatoric stress and thepressure. That is, the geometric layout of the shear zonehas been incorporated in (11) to conform to the generalcase (6). However, the similarity between (11) and thepower-law form of viscoplasticity is striking and hints atthe existence of a constitutive relationship that incorpo-rates plasticity and friction as end-members.

B. Comparison with laboratory data

The choice of a constitutive framework may have im-portant implications on the predictions of failure modes,from laboratory samples to large-scale geodynamics.Therefore, one should proceed carefully when introducinga different friction law. The additive form (13) of rate-and-state friction was derived empirically by Dieterich9

and Ruina10 based partially on hold experiments that al-lowed re-strengthening at zero slip rate56. Here, I revisitthe data collected by Dieterich56 to assess the merit ofusing (11) to explain these observations. In Fig. 5, I showthe static coefficient of friction of five laboratory exper-iments conducted at different normal stresses for a widerange of hold times. While different, the predictions ofthe additive (solid profiles) and multiplicative (dashedprofiles) forms of rate-and-state friction are both wellwithin the data scatter and have virtually equal meritin reducing the laboratory data.

The apparent linear scaling of the static coefficient offriction with the logarithm of hold time (µ0 ∼ ln t) moti-vated Dieterich for his now classic formulation. The mul-tiplicative form of rate-and-state friction (11) implies adifferent scaling, namely a linear dependence between thelogarithm of the coefficient of friction with the logarithmof hold time (lnµ0 ∼ ln t). Some broader considerationsjustify the latter as a better choice.

Tarantola57 showed that the natural distance betweentwo physical quantities τ1 and τ2 that cannot be nega-

tive, which he refers to as Jeffreys quantities, is |ln τ1/τ2|.This metric is insensitive to the choice of physical unit,such that |ln τ1/τ2| is the same whether τ1 and τ2 are ex-pressed in Pa or MPa. As Jeffreys quantities live in theinterval (0,∞), their associated random variable cannotbe normally distributed and their non-informative priorcannot be uniformly distributed, both precluding usingsimple least squares for parameter estimation. The yieldstress and the amplitude of the traction vector, which areequated through the consistency relation τ = µ(V, θ)σ,are Jeffrey parameters. Therefore, it is preferable to con-sider the metric | ln(µ1/µ2)| rather than |µ2 − µ1| to de-scribe the difference between any two friction coefficients.The same argument applies for the velocity dependenceV = V (τ, θ), as velocity and its conjugate, slowness,are also Jeffrey’s parameters. Consequently, a power-law form for the yield stress dependence on the velocityand the state variable is better suited than a logarithmicform.

C. The grain-size evolution of Hall and Parmentier

Several evolution laws have been proposed in the fieldof viscoplasticity that may inspire new developments inkinematic friction. Hall and Parmentier52 introduce anevolution law for grain size where the thermally activatedgrain growth reduces the interfacial energy associatedwith grain boundaries and grain size reduction occursby sub-grain rotation. They write

d = γ p−1 d1−p − λεd , (16)

where γ is temperature dependent and controls the heal-ing rate, λ is a non-dimensional parameter controlling therate of grain size reduction, and p is a non-dimensionalmaterial parameter in the range 1 to 4. To reduce theevolution law (16) to a reference form, I conduct the fol-lowing change of variable. First, using the geometric set-ting of Fig. 4, I approximate the strain rate by the ratioof the cumulative plastic velocity across the shear zonescaled by the width

ε =V

2T,

ε0 =V02T

,

(17)

where I used a similar scaling for the reference strainrate. Then, I introduce a new state variable that scaleslinearly with grain size

θ = d

(p

γ

)1/p

. (18)

Finally, I introduce a critical distance

L =2T

λ. (19)

7

100 101 102 103 104 105 106

Hold time (seconds)

0.68

0.70

0.72

0.74

0.76

0.78

0.80

0.82

0.84

0.86

Coe

ffici

ent o

f sta

tic fr

ictio

n

18.8 MPa4.1 MPa5.4 MPa

18.7 MPa

Additive modelMultiplicative model

2.0 MPa48.0 MPa

9.5 MPa

FIG. 5. Healing and the additive and multiplicative forms of rate-and-state friction. The static coefficient of friction data forquartz sandstone from56 for hold tests. The hold time corresponds to the duration of stick. The data are reduced with theadditive model (13) (solid lines) and the multiplicative model (12) (dashed profiles). Both models explain the data well withintheir scatter.

With these changes, the coupled constitutive equations(9) and (16) become

V = V0

(τ

τ0

)n(θ0θ

)m

,

θ = θ1−p − V θ

L.

(20)

This formulation closely resembles the power-law form ofrate-and-state friction. Indeed, the grain-size evolutionlaw (16) takes the form of the aging law (15) of rate-and-state friction for p = 1. However, if the steady-state grainsize obeys the piezometric relationship58, then grain-sizeevolution only leads to velocity-strengthening behavior(e.g., 0 ≤ m ≤ 1).

Because the mechanisms of grain-size evolution duringplastic flow and the evolution of effective contact areaduring frictional sliding may be different, the similar-ities between these evolution laws indicate that multi-ple mechanisms of material reconfiguration at the mi-croscopic level may take a similar form mathematicallyand sometimes produce the same macroscopic constitu-

tive behavior.

D. The grain-size evolution of Rozel, Ricard and Bercovici

Rozel, Ricard and Bercovici59 employ basic non-equilibrium thermodynamics to propose a general equa-tion for the mean grain size evolution under the assump-tion that the whole grain size distribution remains self-similar. They obtain the evolution law of the form

d = γ p−1 d1−p − λd2

do

τ

τ0ε , (21)

where the healing term is similar to the formulation ofHall and Parmentier52, but the rate of grain size reduc-tion is proportional to the shear stress. Assuming thegeometrical setting of Fig. 4 and the constitutive equa-tion (9), we have τ ∼ τ0 (recall the weak dependenceof frictional strength on sliding velocity). Therefore, the

8

evolution law (21) can be simplified to

d = γ p−1 d1−p − λ(d

do

)q

εd (22)

in the context of a shear zone, where q = 1 for the lawof Rozel, Ricard and Bercovici59 and q = 0 for the lawof Hall and Parmentier52. The weakening term is nowsimilar to the one in (16), except for a factor (d/d0)q.To reduce the evolution law (22) to a reference form,I use the state variable (18) and introduce the scalingparameter

L =2T

λdo

q

(p

γ

)q/p

. (23)

With these changes, the coupled constitutive equations(12) and (22) become

V = V0

(τ

τ0

)n(θ0θ

)m

,

θ = θ1−p − V θ1+q

L.

(24)

Because of the choice of parameterization, eq. (24) re-duces to the evolution law of Hall and Parmentier52 andthe aging law of rate-and-state friction for specific valuesof p and q. For example, the formulation simplifies to theaging law of rate-and-state friction for p = 1 and q = 0.

These two examples demonstrate that largely differentassumptions about the microphysics of the state evolu-tion may lead to similar mathematical expressions for theevolution law. This may explain why the rate-and-stateframework has been so successful at capturing deforma-tion in the laboratory for so many different types of ma-terials and stress regimes.

E. Implications for rupture dynamics

The dynamics of frictional instabilities depends onthe details of the constitutive framework. To illustratethe importance of understanding the frictional resistancenear zero slip speed, I explore the predictions of the ad-ditive and multiplicative forms of rate-and-state frictionwith numerical models of mode III ruptures in anti-planestrain. Particularly, I show that the rupture modes (thepeak velocity, stress drop, stability) strongly depend onthe static coefficient of friction when the multiplicativeform of rate-and-state friction is assumed.

I consider a vertical frictional interface embedded inan elastic layer with a free surface. Instead of solving forthe inertial contribution directly, I use radiation damp-ing as an approximation within the boundary integralmethod, as in previous work16,60,61. I consider a 35 kmvertical fault that is velocity weakening between 5 and15 km depth with a−b = −4×10−3 and velocity strength-ening elsewhere with a−b = 4×10−3, all under a uniformnormal stress of σ = 100 MPa. The system is loaded at

a uniform rate of 10−9 m/s for 2 × 1010 s and the initialcondition is chosen for steady state at that slip velocity.

The results shown in Fig. 6 compare dynamic rupturesimulations assuming the additive (top panel) and themultiplicative (three bottom panels) forms of rate-and-state friction. For a strong fault with µ0 = 0.6, the resultsfrom both forms are similar in terms of recurrence times,overall number of instabilities, peak velocities and stressdrops. As the static coefficient of friction decreases, thepeak velocity and stress drop decrease and the details ofthe rupture evolution change significantly, except for thefirst event in the simulation that is strongly influenced bythe initial condition. The importance of the static coeffi-cient of friction comes about from the relative amplitudeof the yield stress and the stress drop during failure andthis effect should be exacerbated by seismic radiations.

The additive form of rate-and-state friction has beeninterpreted in terms of statistical physics43,62 as a prob-ability for forward jumps of the microasperities formingthe real area of contact proportional to exp(τ). Whenbackward jumps are taken into account, the probabil-ity of forward jumps becomes proportional to exp(τ) −exp(−τ), leading to a term in arcsinh instead of ln, asin43,62

τ = aσ arcsinh

[V

V0exp

(µ0 + b ln(V0θ/L)

a

)]. (25)

The multiplicative form (12) assumes the probability offorward jumps of the form exp(α ln τ), i.e., a type ofpower-law distribution. Further laboratory experimentsmay be needed to constrain the constitutive behavior atsmall slip velocity.

IV. A UNIFYING CONSTITUTIVE FRAMEWORK FORPLASTIC FLOW AND FRICTION

The unified kinematic description of distributed to lo-calized deformation described in Section II and the simi-larities between the power-law viscoplastic laws and rate-and-state friction discussed in Section III hint at the ex-istence of a constitutive framework that would simplifyto plasticity in some situations and to friction in others.

In this section, I describe a unifying constitutive frame-work where the continuum between plastic flow and fric-tion is only controlled by geometrical factors. When fric-tion is interpreted as localized plastic flow, the consti-tutive relationship is anisotropic. Therefore, I introducea heuristic that controls plastic anisotropy based on thegeometry of internal deformation within a representativevolume element. As internal tractions are modulated bythe area on which they are applied, I consider the ratioof internal surface areas

A =1

maxiAi

A1 0 00 A2 00 0 A3

, (26)

9

FIG. 6. Earthquake cycle simulations with the additive (13) and multiplicative (12) forms of rate-and-state friction. A)Simulations with the additive form, which is insensitive to the value of the static friction coefficient. B, C, D) Simulations withthe multiplicative form of rate-and-state friction with a static friction coefficient of µ0 = 0.6, 0.2, and 0.1, respectively. Rupturedynamics with the multiplicative form is sensitive to the strength of the fault, which among other characteristics affects thenucleation, the source-time function, and the static and dynamic stress drops of each event.

as a dimensionless second-order tensor state variable ofthe representative volume element. In order to keep as-pect ratios free variables, I consider a cuboid element ofdimension L1, L2, and L3 with face areas A1 = L2L3,A2 = L1L3, and A3 = L1L2 (Fig. 7a). I define the

weighted moment density tensor

M = A · σ . (27)

Each element of M is proportional to a component of theforce acting in opposite directions on faces ei and −ei. I

10

stresstensor σ

stresstensor σ

surfacetraction t

L3

L3

L1

A1

A2

A3L2

L1

L2 = 0

A2 = A

L1 = L2 = L3

A

B

C

General case

Plasticity (deformation distributed in a volume)

Friction (deformation localized on a surface)

velocityvector v

velocitygradient L

ºstrain ratetensor εij

geometry(A1, A2, A3)

A

FIG. 7. A constitutive framework for plastic flow that incor-porates the role of geometry. A) The spatial velocity gradi-ent L describes how the plastic velocity v changes spatially.The velocity gradient is controlled by the stress tensor σand the geometry of internal deformation, here modeled asa cuboid of dimension L1, L2, and L3 with face areas A1,A2, and A3. B) The constitutive laws of viscoplasticity, cor-responding to a stress-strain-rate relationship, are obtainedfor A1 = A2 = A3. C) The constitutive laws for kinematicfriction is obtained when the deformed region collapses froma volume to a surface, here with L2 = 0, leading to a velocity-traction relationship.

then define the weighted mean moment density

σ =∑i

AiMii

/∑i

Ai (28)

and the weighted mean moment density tensor as

P =1∑iAi

A1 0 00 A2 00 0 A3

Mkk . (29)

Most naturally, the definition of the deviatoric weighted

moment density tensor follows as

M′ = M−P , (30)

with the apparent shear stress

τ = ||M′|| . (31)

Both plastic flow and kinematic friction can be formu-lated with a single constitutive law at the spatial scale ofthe representative volume element following

L = ε0

(τ

σ

)n(θ

θ0

)mM′

τ, (32)

where Lij = vj,i is the spatial anelastic velocity gradient,v is the anelastic velocity field, and the anelastic strainrate is given by

ε =1

2

(L + LT

). (33)

The change of scale to the dimension of internal deforma-tion requires the condition number of the state variabletensor

κ(A) = maxiLi

/miniLi , (34)

that quantifies the degree of localization within the rep-resentative volume element.

As an example, using the framework of equations (26-34), I consider two end-member cases. First, I assumethat the cuboid is a cube (Fig. 7b), such as L1 = L2 = L3

and A1 = A2 = A3 = A. The state variable becomesA = I and the weighted moment density tensor M = σ.The weighted mean moment density reduces to σ = 1

3σkkand the weighted mean moment density tensor simplifiesto

P =1

3σkk I , (35)

i.e., the isotropic pressure tensor. Finally, the deviatoricmoment tensor reduces to

M′ = σ′ , (36)

i.e., the deviatoric stress tensor, which plays a centralrole in the theory of viscoplasticity. With L1 = L2 = L3,the proposed constitutive law (32) reduces to the typicalviscoplastic rheology

ε = ε0

(τ

p

)n−1(θ

θ0

)m

σ′ , (37)

as the symmetry of the deviatoric stress σ′ implies ε = Lin this case.

Now I consider a second case where the cuboid flattensto a surface (Fig. 7c) with L2 = 0, leading to A1 = A3 =0 and A2 = A. In practice, L2 may be finite, but Iconsider the limit case. The state variable simplifies to

A =

0 0 00 1 00 0 0

(38)

11

and the weighted moment density tensor follows as

M =

0 0 0σ21 σ22 σ230 0 0

, (39)

i.e., only the components corresponding to the tractionvector t(e2) are non zero and the tensor is not sym-metric. The weighted mean moment density reduces toσ = σ22 = σn, where σn is the normal stress to thesurface A2. The weighted mean moment density tensorbecomes

P =

0 0 00 σ22 00 0 0

, (40)

where σ22 is the normal stress σn in this case. Finally,the deviatoric moment density tensor reads

M′ =

0 0 0σ21 0 σ230 0 0

. (41)

The norm of the deviatoric moment density tensor, τ =√σ221 + σ2

23, corresponds to the norm of the shear trac-tions that are parallel to the surface A2, as is requiredto describe the local frictional resistance. The deviatoricmoment density tensor is not symmetric either, allowingfor simple shear. For the case where the cuboid regioncollapses to a surface with L2 = 0, only two nontrivialcomponents of the constitutive relationship remain

v1,2 = ε0

(τ

σn

)n(θ

θ0

)m

cosα

v3,2 = ε0

(τ

σn

)n(θ

θ0

)m

sinα

(42)

that correspond to the strike-slip and dip-slip componentof the velocity gradient. Here, α is the rake angle, withcosα = σ21/τ and sinα = σ23/τ . The spatial velocitygradient is not symmetric, creating simple shear, compat-ible with fault slip. Both v1,2 and v3,2 are finite becausethe internal deformation is averaged over the dimensionof the representative volume element. The internal de-formation is given by κ(A)L, which approaches infinitywith increasing localization. As described in Section III,the formulation (42) corresponds to the multiplicativeform of rate-and-state friction in three dimensions.

In the proposed constitutive framework, whether thedeformation simplifies to plasticity (distributed) or fric-tion (localized) depends on a pair of geometrical factorsin three dimensions, either the pair L1/L3 and L2/L3,the pair L1/L2 and L3/L2, or the pair L2/L1 and L3/L1.In two dimensions, if L1 is infinite, the behavior will becontrolled by the aspect ratio L2/L3.

The comparison of these two end-members, one corre-sponding to the distributed macroscopic stress and theother to the tractions on a surface, shows that frictioncan emerge from anisotropic plasticity in the special caseof large aspect ratios in the internal geometry of the rep-resentative element.

V. CONCLUSIONS

The kinematics of plastic deformation in a contin-uum can incorporate all degrees of localization, from dis-tributed to localized with a single mathematical expres-sion. The constitutive law for kinematic friction closelyresembles the power laws used in viscoplastic theories, inparticular when the multiplicative form of rate-and-statefriction is employed.

A unifying constitutive framework can be identified,in which viscoplasticity and kinematic friction representend-members of the rheological behavior for distributedand localized deformation, respectively. The behavior iscontrolled by the aspect ratio of the deformation region.It is also likely that different microphysical processes be-come activated depending of the aspect ratio.

While I hope that this discussion will provide novelinsight into the place of friction in the general frameworkof anelastic deformation, I have left out some importantissues such as the physical origin of strain localization,in particular, what controls the thickness of a shear zoneor fault gouge, and the true nature of the state variableused in (11) and (12), which is left to future work.

ACKNOWLEDGMENTS

This work comprises Earth Observatory of Singaporecontribution no. 174. This research is supported bythe National Research Foundation of Singapore underthe NRF Fellowship scheme (National Research FellowAwards No. NRF-NRFF2013-04) and by the Earth Ob-servatory of Singapore, the National Research Founda-tion, and the Singapore Ministry of Education under theResearch Centres of Excellence initiative.

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