Yielding transition A dynamical perspective (energy ...zamponi/workshops/yielding...the local...

Yielding transition A dynamical perspective

(energy landscape is not everything)

Jean-Louis BarratUniversité GrenobleAlpes

Institut universitaire deFranceInstitut LaueLangevin,TheoryGroup

Outline

• Elastoplasticmodels• Meanfieldtreatments:Hébraud Lequeux,SGR• Strainlocalisation vscontinuoustransitions• Strainlocalisation ininertialsystems• Strainlocalisation ingranularsystems• Creep

Plasticresponse ofafoam (I.Cantat,O.Pitois,Phys.offluids2006)

Deformation ofamorphous systems atlow Tproceeds through wellidentified plasticevents orshear transformations(ArgonandKuo,1976)

Plasticresponse ofasimulatedLennard-Jones glass(Tanguy,Leonforte,JLB,EPJE2006)

Stress-straincurveatlowstrainrate,lowtemperature,smallsystems

Eventsareshear transformationsofEshelby type

•Plasticinstability inavery localregion ofthemedium(irreversible)under theinfluenceofthelocalstress.

Malandro,Lacks,PRL1998•Instability involves typically afewtens ofparticles andsmall shear strains (1to10%)

•Surroundings respond essentially asanhomogeneous elastic medium(incompressible).Quadrupolarsymmetry oftheresponse. Puosi,Rottler,JLB,PRE2014


Eshelby transformation:aninclusionwithin anelastic material undergoes aspontaneous changeofshape (eigenstrain):circular toelliptical.

Inanhomeogeneous,linear elastic solid,theInduced shear stressoutsidetheinclusionis proportional totheinclusiontransformationstrain andtotheEshelby propagator (response totwo forcedipoles):

Eventsareshear transformationsofEshelby typeBestseen inexperiments trough correlation patterns

Colloidal paste under simpleshear(Jensen,Weitz,Spaepen,PRE2014)

Bestseen inexperiments trough correlation patterns

Granular mediumunder uniaxial deformation(LeBouil,Amon,Crassous,PRL2014)


• Microscopic :Particle based,molecular dynamicsorathermalquasistatic deformations.Detailedinformation,limited sizes /times.

• Mesoscopic :Coarse grainandusethe« sheartransformations »aselementary events,withelastic interactionsbetween them.

• Continuum :Stress,strain rate,andother statevariables(« effectivetemperature »)treated ascontinuumfields.

Three levels ofmodelling

Mesoscopic description

PhysRevA,1991

Springnetworkwiththresholdinforce

Slope-1.4in2D

Mesoscopic description- anold idea

Outline


Rheologyofsoftglassymaterials (SGR)By:Sollich,P;Lequeux,F;Hébraud,P;Cates MEPHYSICALREVIEWLETTERS Volume:78 Pages:4657-4660 Published:JUN161997

Very popular,based onBouchaud’s trap model

Mode-couplingtheoryforthepastyrheologyofsoftglassymaterials (HL)By:Hébraud,P;Lequeux,FPHYSICALREVIEWLETTERS Volume:81 Pages:2934-2937 Published:OCT51998

Less popular,probably much morerealistic

Two proposals fordescribing this scenarioinamean field manner

13

The trap model (J-P. Bouchaud)

•Exponential distributionofenergy barriers (->glasstransition)

• l strain variable,increaseslinearly with time

Avery popular model:SoftGlassy Rheology(Sollich,Lequeux,Hébraud,Sollich,Fielding)

P(L,E,t) distributionofsystems indifferent « traps »andatdifferent strains L.

Fixed strain rateevolution

Activated escapefrom traps dueto« mechanical noise »x

Dynamical equation forthestrain distributionfunction P(E,l,t)onatypical site:

• Very successful model, describes many features of the flow o glassysystems + ageing

• glass transition at x=xg=1; power law fluid 1<x<2; Newtonian above• for x< xg : aging, yield stress sY, s=sY+A g1-x

But..Ømechanical temperature x is not defined self consistently, adjustableparameterØdoes it correspond to anything physical ?

Activated yield events

x=mechanical noisetemperatureResetstrain andenergyafter yield. G is thetotalplasticactivity.

External drive

Thechallenger:Hébraud Lequeux model:Stressdiffusionduetomechanical noise+selfconsistency

⇧tP(⌅, t) = �G0�̇(t) ⇧�P +DHL(t) ⇧2�P � ⇤HL(⌅,⌅c)P + �(t) ⇥(⌅)

External drive Yield ifs>scStressdiffusion Resettozeroafter yield

P(s,t)probability distributionofstressonatypical site(nodisorder,singlelocalyield stress)

⇥HL(⇤,⇤c) ⇥1

⌅�(⇤ � ⇤c)

�(t) =1

⌅

Z

�0>�c

d⇤0 P(⇤0, t)DHL(t) = ��(t)

Nonlinear feedbackYield rule andplasticactivity

Thechallenger:Hébraud Lequeux model:Stressdiffusionduetomechanical noise+selfconsistency

• � > �c = 2 Newtonian behaviour⇤ � ⇥̇

• � < �c = 2 Herschel Bulkley law with exponent 1/2:⇤ = ⇤Y +A⇥̇1/2

Maindifference between thetwo models:descriptionoftherandom process that triggerstheyield event.

• Solve for a fixed value of D (linear equation, P (�, D, �̇) is piecewise expo-nential).

• Obtain �(D, �̇) and enforce self consistency condition D = ↵�(D, �̇) )D(�̇)

• Obtain < � >=Rd��P (�, D(�̇), �̇)

Mechanical noiseis different from thermalnoise!

A. Nicolas,K.Martens,JLB,EPL2014E.Agoritsas etal,EPJE2015

• Thermalnoiseactsonstrainvariablel inafixedlandscapebiasedbythestress

• Mechanicalnoiseactsadiffusiveprocessonthestressbiasitself

=>Verydifferentescapetimes(Arrheniusvsdiffusive)

PotentialEnergyLandscapePictureforasmallregion(STZ):

Outline


�̇ / (� � �yield)� “Secondorder”critical

behaviour,monotonousflowcurve.Avalanchebehavioratvanishingstrainrates,analogiesanddifferenceswithdepinningproblems.

Natureofthe«yield»(arrested->flow)transition?

Coexistenceofflowingandnonflowing regionsatthesamevalueofthestressisalsocommonlyobserved=>possibilityof“firstorder”transition,knownas“strainlocalisation”or“shearbanding”.”Spinodal”instabilityuponincreasingstrain->Procaccia etal.Herefocusonstationarystate,beyondyield.

Strain localisation/Shear banding

Coexistenceofflowingregionsandsolidregionsatthesamevalueofthestress

Granular pastes(Barentin et al., 2003)

Bubble Rafts(Dennin et al., 2004)

Chocolate (Coussot et al.)

4.5 5.0 5.5 6.0

0.000

0.002

0.004

4.75 5.00 5.25 5.50

0.0

2.0x10-4

4.0x10-4

Velo

city

(m/s

)

Distance (cm)

Velo

city

(m/s

)

Distance (cm)

Lennard-Jones glass(Simulation, Varnik, Bocquet, JLB, 2004)

« Explained » by static vs dynamic yield stress


Divoux,Fardin,Manneville,Lerouge,Annualreviewfluidmechanics2016

Flowprofile(cylindricalCouette)

Flowcurve

Examplesystem

Many different possiblemicroscopicmechanisms can leadtopermanentlocalisationofdeformation…Three examples here:longrecovery time(transient damage),inertia,friction



Apossiblemechanismfromamesoscopic viewpoint(Coussot andOvarlez,Martensetal):longplasticevents(large“healingtime”)

Coussot andOvarlez meanfieldanalysis(EPJE2010)

Constitutivecurvebecomesnonmonotonicatlargetres

< ⇤ >= ⇥�̇ +⇤c

1 + �̇⌅res/�c



Lifecycleofasingleblock Flowcurves

KMartens,L.Bocquet,JLB,SoftMatter2012

Assemblyofelastoplastic blocksinteractingviaelasticpropagator.Healingtimetres beforeelasticrecoveryvaries.



Cumulatedplasticactivity

Martens,.Bocquet,JLB,SoftMatter2012Tyukodi,Patinet,Roux,Vandembroucq 2016“softmodesinthedepinning transition”

Elasticpropagatorreplacedbyshortrangeinteraction

Whylinearstructure?

=0

Outsideanhomogeneousplasticband(softmodeoftheelasticpropagator)

Outline


Strain localisationininertial systems

(SalernoandRobbins2014,NicolasRottler BarratPRL2015,Karimi Barrat2016)

Backtoamicroscopicmodel

Lennard-Jones particles, 2d systemDamping ⇥, mass m, stress scale �0 = �/⇤2 (in 2d).

Quality factor: Q = ⌅damp/⌅vib

Overdamped: Q ⌧ 1 underdamped Q � 1

⇥damp = m/� ; ⇥vib =p

m/�0

mv̇i = ��vi + F (xi) + ⇥i(t)


Overdamped system, zero temperature:

�(�̇, T = 0) = �0(0.72 + 2pW )

with W = ⇥�̇/�0

Highertemperature(T=0.2)Stressdecreases

Zerotemperature



Underdamped systems,zerotemperature:nonmonotonic flowcurves!

Ei = �̇p

m/�0 = �̇⇥vib

Grains in 3dQ ' 0.1a

p⇥�0/�



Interpretation:inertialvibrationsata“bathtemperature”T=0actasafinitetemperature

=>DataatlargeQcanbeobtainedfromdataatsmallerQandhighertemperature.

�(Ei,Q, T0) = � (Ei, 1, TK(Q,Ei, T0))

Rateweakeningeffectcompensatedatlargestrainratesbystandardincreasewithstrainrate

Energy dissipation proportional to ��̇

TK � C|�̇|+ T0


Nonmonotonic flowcurve – Shear bands?

Nonmonotonic flowcurve – Shear bands?=>Stability analysis ofhomogeneous flow(K.Martens,V.Venkatesh,work inprogress).Assumemonotonous constitutiverelation:


ForceBalance

Temperature diffusion


Homogeneous flowbecomes linearly unstable ifthesystemis larger than acritical size

Below this length scale heat diffusionis too fast andtheshear bandsdonotpersist intime.

Confirmed quantitatively bylargescale molecular dynamics simulations.

Outline

• Elastoplasticmodels• Meanfieldtreatments:Hébraud Lequeux,SGR• Strainlocalisation vscontinuoustransitions• Strainlocalisation ininertialsystems• Strainlocalisation ingranularsystems(workinprogresswithK.Karimi)

• Creep

Experiments byAmon,Crassousetal

A.Amon,Habilitationthesis

Strain localisationingranular systems


Correlationangle53°

Failureanglecloseto60°

Why arethetwo anglesdifferent ?


• Biaxial testofgranular medium• Decorrelation ofspeckle patterngives accesstolocalplasticactivity (near thesurface).

• Correlation maps ofplasticactivity reportedduring deformation


Stressredistribution+Failure criterion

Shear transformationaligned with x,yaxisandlocated attheorigin generates changesinthethree stresscomponents:

Stresstensor in2d


Stressredistribution


Failure criterionStrain localisationingranular systems

Changeinyield function inresponse tosheartransformation


CorrelationsStrain localisationingranular systems

Shear band?• Plasticactivity localized inside alinearregion

• Picturebandasalinear array ofsheartransformations

• Ifactivity homogeneous ontheline,stressredistributionis zero everywhere outside

• Proposed criterion forselecting thebandorientation:maximise-dfy inside thebanditself


Shear bandorientation

Different from correlation angleunless frictionangleis zero!


Macroscopic friction

Finite element grid.Each elementhaselastoviscoplastic behavior,.

Localfailure criterion with MohrCoulombcriterion.Localcohesion cdrawn from anexponentialdistribution,uniform localfrictionanglef.

Numerical test

Simpleshear loading triggersfirstyield event

Solve forlocaldisplacement andstresses(overdamped propagation)

Triggernewevents iflocalfailure criterion isreached

Numerical test

Loading curve

Smallstrains:Transientfluctuations

Largestrains:stable bands

Correlations inplasticactivity

Correlations intheshear banded regime

Correlations inplasticactivity

Correlations atsmall strain

Sliding average inthestraininterval 0– yield strain

Outline


Siegenbürger etal,PRL2012Creep inacolloidal glass

Strain response:different stresslevels,different waiting times Fluidization timebehaves asa

powerlaw ofthedistancetoyield stress

Divoux etal,SoftMatter 2011Carbopol microgel

Creep:Apply afixed stresss andmeasure thestrain g (t)

Stresscontrolled versionofelastoplastic models (mean field version)ChenLiu,KirstenMartens,JLBarxiv:1705.06912

Results qualitatively similar toexperiments;very strong dependence ontheinitialconditionfortheprobability distributionfunction

Sd:decreases when systemages

Fluidization timefollows apowerlaw with thestatic yield stressasareference (can be identified with overshoot instressstrain curve).

Exponent is notuniversal,depends onsystemage.

Acknowledgements

KirstenMartens

AlexandreNicolas

EzequielFerrero

ElisabethAgoritsas

LydéricBocquet

ChenLiuFrancescoPuosi

KamranKarimi

JörgRottler

VishwasVenkatesh

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