Dynamical yield criterion for granular matter from first principles

O. Coquand1, 2, ∗ and M. Sperl1, 3, †

1Institut fur Materialphysik im Weltraum, Deutsches Zentrum fur Luft- und Raumfahrt (DLR), 51170 Koln, Germany2Laboratoire Charles Coulomb (L2C), Universite de Montpellier, CNRS, 34095 Montpellier, France

3Institut fur Theoretische Physik, Universitat zu Koln, 50937 Koln, Germany

We investigate, using a recently developed model of liquid state theory describing the rheologyof dense granular flows, how a yield stress appears in granular matter at the yielding transition.Our model allows us to predict an analytical equation of the corresponding dynamical yield surface,which is compared to usual models of solid fracture. In particular, this yield surface interpolatesbetween the typical failure behaviors of soft and hard materials. This work also underlines thecentral role played by the effective friction coefficient at the yielding transition.

INTRODUCTION

Understanding the way solid objects break is a ques-tion relevant to various areas of physics. At a fundamen-tal level, the determination of the precise mechanism atthe origin of solid failure — be it via elasto-plastic mod-els [1–6], statistical methods inspired from glassy physics[7, 8], or modified elasticity theories [9–15] — as well thestill debated relationship between the brittle and ductilemodes of failure [16–27] are very active fields of research.But this topic is also ubiquitous in applied physics andengineering for the study of failure of rocks, soils, andother geomaterials [28–35], concrete [36–39], or cellularmaterials [40]. A particularly successful approach con-sists in determining the yield surface of the solid, a curvethat allows to determine whether a solid in a given stateof stress will yield. Many different constructions of suchan object have been proposed [41–51] but for amorphoussolids, for which determining the yielding point from ab-initio methods used in crystals [52] is not possible, nogenerally accepted construction of a yield criterion hasbeen determined yet.

In this paper, we propose a study of the yielding tran-sition based not on a theory of the solid, but of the liq-uid state. More precisely, using a theory developed todescribe the non-Newtonian features of granular liquidflows [53–59] in the limit of very low shear rates, we de-scribe how an internal state of stress develops into theliquid as its behavior becomes more and more solid-like.Such a determination of a so-called dynamical yield cri-terion has previously been done for the study of colloidalsuspensions close to the mode coupling glass transition[60]. Although there is no evidence suggesting a completeequivalence between the dynamical yield criterion deter-mined when approaching the yielding transition from theliquid side and the yield criteria studied in triaxial testswhen breaking solids, it is reasonable to assume thatsome properties are preserved across the transition. Tothat extent, our work brings an original insight on theyielding problem with an approach which, contrary tomany failure models, is based on fundamental principles.

The reduction of our liquid state theory to an analyt-

ically solvable model, along the lines presented in previ-ous works [58, 59] allows us to derive an analytical ex-pression of the yield surface for granular materials. Thissurface turns out to display interesting properties, suchas the existence a priori of two continuously related frac-ture modes (a soft and a hard one). Furthermore, thesolvability of the model allows us to present a criticalanalysis of some widely used yield criteria. Finally, weshow that a definition of the effective friction coefficientfrom the symmetries of the stress tensor allows us tobuild a quantity which behaves very smoothly across thetransition to the solid state, thereby pointing out a po-tentially crucial quantity to understand how solid orderbuilds up in complex liquids.

INTRODUCTION TO FRACTURE

Our aim in this section is not to give a complete reviewof the theory of fracture in solids. We present some of thefeatures of usual yield criteria which are relevant to thefollowing [61]. For sake of clarity, we restrict ourselvesto the most usual yield criteria, many more refined onesbeing simple variations around those.

The state of stress in a solid piece of material can berepresented by a stress tensor σ, which can be furtherdecomposed onto irreducible representations of the SO(3)group into a diagonal part (spin 0 representation) and atraceless, deviatoric component (spin 2 representation):

σ = P I + σ′ , (1)

where P =Tr(σ)/3 is the pressure and Tr(σ′) = 0. Thesame decomposition can be applied to the strain tensorε. In particular, for elastic materials, the deviatoric partsof both tensors are related by σ′ = 2Gε′, which definesthe shear modulus G of the solid. Since most of thefollowing work concerns shear fracture, the other elasticmoduli shall not be discussed.

Another useful representation of σ is given by its threeeigenvalues — or principal stresses — σ1, σ2 and σ3,which do not depend on the basis used to represent thestress tensor. One can then define three invariants in the

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1

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following way:

I1 = σ1 + σ2 + σ3

I2 = σ1σ2 + σ2σ3 + σ3σ1

I3 = σ1σ2σ3 .

(2)

Similar definitions can be used to define the invariantsJ2 and J3 from σ′.

Most of the yield criteria can be understood in termsof an effective friction coefficient. This coefficient can bedefined in the following ways:Definition 1: In a solid, the effective friction coefficientµ compares the strength of the spin 0 and the spin 2components of the stress tensor. It can then be expressedas: µ = σ0/P , where the shear stress is σ0 = σ : ε′/|ε′|,”:” denotes a full tensor contraction, and the norm of thedeviatoric strain is |ε′|2 = 2ε′ : ε′.Definition 2: For two pieces of a broken solid to be ableto glide onto one another, they have to overcome the solidfriction between the two blocks, as defined by Coulomb’slaw. The effective friction coefficient µ of the solid isdefined as the ratio of tangential to normal stress on agiven plane that has to be overcome for the solid to breakalong that plane. Both definitions are related to oneanother, although not equivalent [33].

Let us suppose that a piece of solid breaks as soonas the elastic energy E′el accumulated due to the devia-toric strain exceeds a certain proportion of the isotropicpart of the elastic energy ETrel ∝ P . The former energycan be expressed as E′el = σ′ : ε′/2 = J2/4G. The corre-

sponding shear stress is σ0 =√J2/2G. Finally, using the

definition 1 of the effective friction coefficient, the mate-rial yields when µ exceeds a characteristic value of thematerial, µDP , which defines the Drucker-Prager yieldcriterion [41]. It can also be written in terms of invari-ants as I2

1/I2 = CDP , where CDP is a constant.The main advantage of definition 1 is that it only de-

pends on the symmetries of the stress tensor. However,determining ε′ in a given experimental situation can bechallenging. As a result, many yield criteria are basedon the second definition of µ. In that case, the main dif-ficulty is to determine the fracture plane on which thetangential and normal stresses must be compared.

One of the simplest way to proceed is to assumethat the fracture will take place in one of the principalplanes. In that case, the problem reduces to three two-dimensional problems which are very easy to solve. Thematerial yields as soon as µ > µMC on one of the threeprincipal planes, which defines the Mohr-Coulomb yieldcriterion.

This approach can be refined further by approximat-ing the fracture plane by the so-called spatially mobilizedplane [45, 48], which is a clever interpolation between thethree principal planes. More precisely, it is defined as theunique plane which projections onto the three principalplanes gives back the Mohr-Coulomb problem [45]. By

FIG. 1: Comparison of the different yield criteria presented inthis paper. The big green dots are the points used to adjustthe constants of the Mohr-Coulomb criterion (as well as theirimages by rotation of angle 2π/3). The constants of the threeother criteria are adjusted on the upper apex.

removing the possibility that the fracture plane evolvesdiscontinuously from one principal plane into another, itprovides a much smoother version of the Mohr-Coulombyield surface (see Fig. 1). It terms of invariants, this canbe written CMN = I2I1/I3, where CMN is a constant.

Finally, another popular smooth version of the Mohr-Coulomb criterion is the Lade-Duncan yield criterion[43]. Its geometrical interpretation is not as straightfor-ward as the previous ones, but it was recently showed[51] that it can be understood as a correction to theMatsuoka-Nakai criterion taking into account sublead-ing dependence in J2/P . By cumulating the expressionsof the Drucker-Prager and Matsuoka-Nakai constants, anew quantity CLD = CMNCDP = I3

1/I3 can be con-structed. The Lade-Duncan criterion can then be definedby requiring that CLD, rather than CMN and CDP sep-arately, is constant [51]. The corresponding modificationin terms of effective friction coefficient µLD is given in[51].

A comparison of the different yield criteria is presentedon Fig. 1, which presents a cut of the yield surface alongone of the deviatoric planes (planes at constant pres-sure). Remarkably, the Drucker-Prager yield surface ismuch more isotropic than all the other ones. This can berelated to the absence of dependence in I3, which con-fers it a higher degree of symmetry. The more triangularshape of the three other criteria on the other hand canbe related to the non zero value of the effective frictioncoefficient [62], thereby illustrating the inequivalence be-tween definition 1 and 2 given above (µ is obviously nonzero too for the Drucker-Prager yield surface on Fig. 1).

All in all, the most usual yield criteria used for study-ing solid fracture can be captured in terms of an effec-tive friction coefficient µ which should not exceed a given

3

value for solid order to be preserved.

RESULTS

The yielding transition from the liquid state

The model we present in this paper is based on a sim-plification of the Granular Integration Through Tran-sients (GITT) formalism. For more details about thismodel, the reader is referred to the more detailed pre-vious publications on the subject [53–59], as well as theappendix. This formalism is based on the so-called Inte-gration Through Transients (ITT) formalism [60, 63–65],which allows to compute statistical averages in a shearedfluid by relating them to averages computed in a quies-cent state where no shear is applied. We can thereforedecompose the stress tensor as: σ = σ(0) + ∆σ, whereσ(0) = P0I, because the quiescent fluid is not sheared,and ∆σ contains all the corrections depending on theshear rate γ. The pressure P0 can be computed for afluid at equilibrium [57], but can also include an isotropiccomponent imposed by the environment in many situa-tions relevant to geophysics problems, and is thus leftunspecified here. In our model, P0 does not couple withthe correction ∆σ.

The stress correction ∆σ can also be written in a formsimilar to Eq. (1):

∆σ = ∆P I + ηD , (3)

where ∆P is the correction to the pressure due to shear,η is the viscosity of the liquid, and Dij = κij + κji isthe symmetrized flow matrix, κ = ∇v being the velocitygradient. In this work, we restrict ourselves to incom-pressible flows for which Tr(κ) = 0.

The GITT equations allow to relate ∆σ to κ via aviscosity matrix Λ through ∆σab = Λabijκij . In this for-malism, the components of Λ can be written explicitlyas integrals over time and wave vectors of the densitycorrelation function 〈ρk(t)ρ−k(0)〉 [59]. Our toy modelconsists in neglecting the wave vector dependences whichplay a subleading role (details in appendix and [58, 59]).In that case, Λ reduces to integrals over time of productsof the Finger tensor, which contains information aboutthe deformation of the system, and the density correla-tion function (see the appendix for a detailed derivation).Then, the density correlation function is reduced to thefollowing expression:

〈ρk(t)ρ−k(0)〉 ∝ e−t/tΓe−γ2t2/2γ2c . (4)

The first factor accounts for the internal dynamics of theliquid, with a typical time scale tΓ. As the behavior ofthe liquid becomes more and more solid-like, tΓ becomesvery large, mostly due to the cage-effect: in dense liq-uids, particles’ ability to move tends to be reduced by

their neighbors. The second factor is a screening factoraccounting for the effect of advection: the applied stresstends to force particles to move and facilitates the es-cape from the cages. This screening is characterized bya strain scale γc, which is a constant of the material de-scribing its compliance to external stresses. Despite itssimplicity, the reduction of the contribution to the stresstensor of the particle’s dynamics by equations like Eq. (4)has proven to provide rather accurate constitutive equa-tions for dense granular liquids [58, 59].

The final element of our toy model is the sampling ofpossible flow geometries. First, we restrict ourselves tothe basis in which σ is diagonal, so that κ is diagonal too.Then, we map the space of traceless diagonal matricesby the following two parameters family of reduced flowmatrices:

κ =

A2 0 00 −A2

(1−A1

2

)0

0 0 −A2

(1+A1

2

) , (5)

where (A1, A2) define the flow geometry, and κ = κγ.For this family of flows, and in the case of stationaryflows, the toy model integrals giving the components ofΛ can be evaluated exactly. Finally, we can examine thelimit γ → 0, which allows two distinct behaviors: Forlow packing fraction systems, the ITT correction vanisheswith γ, the system remains a liquid; For denser systems,a non trivial yield stress develops in this limit, whichsignals the onset of solid-like behavior. The correctionsto the eigenvalues of the stress tensor predicted by thetoy model in the latter case can then be expressed asfollows:

∆σ1 =S0 + S1

2

√π[F(κ1γc) + F(κ2γc) + F(κ3γc)

]+ S1

√πF(κ1γc)

∆σ2 =S0 + S1

2

√π[F(κ1γc) + F(κ2γc) + F(κ3γc)

]+ S1

√πF(κ2γc)

∆σ3 =S0 + S1

2

√π[F(κ1γc) + F(κ2γc) + F(κ3γc)

]+ S1

√πF(κ3γc) ,

(6)

where F(x) = x ex2

erfc(−x), and S0 and S1 are con-stants from the toy model giving the typical strength ofthe yield stress developing in the liquid (they have verylittle influence on the geometry of the yield surface).

The yield surface equation Eq. (6) is quite remarkable.Indeed, it was shown in [50] that all the usual yield crite-ria defined above can be expressed as roots of a polyno-mial equation of degree three. Here, to the contrary, theeigenvalues of the stress tensor, defined from the functionF are highly non polynomial, which make Eq. (6) quiteunique to the best of our knowledge. As shown in theappendix, the Drucker-Prager criterion can be recovered

4

FIG. 2: Cuts of the toy model yield surface along the devi-atoric plane for various values of ∆P . The parameters areS0 = S1 = 1, γc = 0.4, ∆P = 0.05 (red), ∆P = 0.5 (orange),∆P = 1 (green), and ∆P = 2 (blue).

from a A2γc 1 expansion of Eq.(6), but the relation tothe other usual yield criteria is more involved [66].

Geometry of the yield surface

On Fig. 2, we have represented the surface Eq. (6) invarious deviatoric planes, corresponding to various val-ues of I1. More precisely, fixing a pressure amounts tofixing the pressure correction ∆P , which is our controlparameter along the hydrostatic axis since P0 does notcouple to the shear corrections to the stress. All quan-tities are dimensionless, the global scale of the axis inphysical units being fixed by the value of the parametersS0 and S1.

The yield surface presents two qualitatively differentbehaviors from which we identify two fracture modes:For ∆P ' S0, S1, the shape of the yield surface cuts istriangular, similar to that of Fig. 1, which corresponds tothe typical shape observed in the failure of sedimentaryrocks and soils captured by yield criteria from the fam-ily of Mohr-Coulomb/Matsuoka-Nakai/Lade-Duncan, wecall this mode of fracture hard ; For ∆P S0, S1 on theother hand, the yield surface cuts become isotropic, andthe yield surface has a Drucker-Prager like shape, wecall this fracture mode the soft mode. This can be usedto classify material according to their preferred mode offracture, softer materials being the ones for which theshear induced pressure component of the stress is smallcompared to the typical scale of stress involved.

The soft fracture mode was already described in theyielding of soft colloids close to the mode coupling glasstransition [60]. This picture fits well our current frame-work: close to the mode coupling glass transition, the

shear correction to the pressure is rather mild comparedto the other stress scales in the system. One of the mainnon trivial additional outcomes of our model is that thisfracture mode is only observed under certain conditions(more precisely A2γc 1, see the appendix for details),and can be continuously transformed into a hard type offracture by changing the conditions of fracture.

There are a number of reasons though why this transi-tion between both fracture modes could not be observedexperimentally: (i) While our model allows us to explorethe variations of the yield surface with respect to anyvalue of ∆P , there is no guarantee that there exists anexperimental protocol which allows to explore such a re-gion for a given material. Indeed, ∆P is itself a func-tion of the applied shear stress. (ii) Our model so faronly includes shear failure, but it is expected that addingthe possibilities of dilation and compression failure putsmaterial dependent boundaries on the hydrostatic axis,which may prevent from exploring the full variation ofthe yield surface geometry.

Finally, the analytical expression of the yield surfaceEq. (6) allows us to analyse the performance of otheryield criteria in various deviatoric planes. The resultsare displayed on Figs. 5,6 and 7 in the appendix. Whileunsurprisingly the Drucker-Prager criterion performs allthe best that ∆P is small, there is no such monotonousbehavior for the precision of the Matsuoka-Nakai andLade-Duncan criteria. It can be noted, though, thatthose latter two perform all the best that the pressureP0 is large.

The effective friction coefficient

We discussed in the first section how all usual yieldcriteria can be expressed in terms of an effective frictioncoefficient. Effective friction coefficients can also be de-fined for complex liquids. It is even a crucial quantity inthe context of the study of granular liquids for which ithas been shown that the dependence of µ on the shearrate is largely universal [67–69], and provides a usefultool to relate the rheology of granular liquids and gran-ular suspensions [58, 70–73].

Since the definition of a fracture plane is not appro-priate for liquids, µ is naturally defined from the sym-metries of the stress tensor, like in definition 1 above.More precisely, the spin 0 component of the stress ten-sor of the liquid is the total pressure P = P0 + ∆P ,and its deviatoric component defines a shear stress σl0 asσl0 = σ : κ/|κ| [59], with |κ|2 = D : D/2. This defines aneffective friction coefficient as µ = σl0/P .

In a Newtonian liquid, the deviatoric stress is σ′ = ηD(from Eq. (3)). Because η is a constant, this componentof the stress vanishes in the limit γ −→ 0. For com-plex liquids however, σ′ = η(γ)D. Hence, provided thatη ∼ 1/γ in the limit of low shear rates, this term survives

5

−3 −2 −1 0 1 2 3

A1

1.000

1.005

1.010

1.015

1.020

1.025

σs 0/σ

l 0

FIG. 3: Comparison of the definition of the shear stress fromthe solid state stress tensor σs

0 and from the liquid state stresstensor σl

0. The data is represented as a function of A1, whichvariation on the presented scale correspond to a sector of 2π/3Lode angle. The parameters are S0 = S1 = 1, γc = 0.4,∆P = 0.05 (red), ∆P = 0.5 (orange), ∆P = 1 (green), and∆P = 2 (blue).

and solid-like behavior builds up. All in all, analysing thedeviatoric component of the stress tensor in the limit oflow shear rates makes the connection between the liq-uid and solid definitions of the shear stress σ0. However,both definitions of σ0 are not necessarily equivalent toone another. Indeed, outside the regime of small strains,the deviatoric part of the deformation tensor is not pro-portional to the symmetrized velocity gradient D [74].

Within our model, it is possible to identify from thestress equations (6) the contribution of the strain tensor(see the details in the appendix), and thus to get accessto its deviatoric component. Hence, we can compare σl0,computed from the velocity gradient, to σs0 = σ : ε′/|ε′|,computed from the solid-like expression of the stress ten-sor Eq. (1). The results are displayed on Fig. 3 for variousvalues of ∆P [75]. We can see that, although some vari-ation is indeed present, both definitions yield very com-patible numerical values. Consequently, µ defined fromthe ratio of the spin 2 and spin 0 component of the stresstensor behaves very smoothly across the liquid-solid tran-sition, and therefore appears to be a particularly interest-ing quantity to study the onset of solid order in freezingliquids under shear.

Finally, our model allows us to perform a more in-depth analysis of the behavior of µ in a given deviatoricplane (data in the appendix). First, µ is not constantalong those planes, although relative variations are quitesmall. Provided that our model still holds for solid frac-ture, this means that a way to improve the existing yieldcriteria could be to allow for a such a deviation to theconstant value, a feature which is present in none of the

solid yield criteria presented in the first section of thepaper. Then, the relative variation of µ with the Lodeangle are all the bigger that the fracture mode becomesharder. Given that, as shown above, the hard fracturemode corresponds to cases very well described by criteriabased on definition 2 of the effective friction coefficient,this raises the question of a possible transition betweenthe two definitions as the isotropic stress ∆P caused byshear becomes more and more comparable to the intrinsicstress scale of the material fixed by S0 and S1: the defi-nition of µ from the liquid stress tensor (3) accounts forthe build up of solid-like behavior as the liquid’s behav-ior gets less and less Newtonian, then for the soft type ofsolid fracture mode, the approximation of µ, set by defi-nition 1, by a constant is quite accurate, but deterioratesas the fracture mode becomes harder, where, given thegood performance of yield criteria based on definition 2,a constant µ based on definition 2 could still be observed.Furthermore, since the value of µ on the fracture planeis determined by the value of the yield criterion constant[51], we can also study the variations of µ from defini-tion 2 within our toy model. Notably, there seems to bea regime where P0 ' 1 (in units of S0, S1) where therelative variation of µMN and µLD is suppressed at best.However, since the value of P0, playing a central role inthat case, is not prescribed by our model, quantitativecomparison to the relative variations with definition 1 isnot possible.

CONCLUSION

All in all, we have built an analytically solvable toymodel that allows us to predict the shape of the dynam-ical yield surface of soft materials and granular matter.Importantly, our yield criterion is based on fundamentalequations of liquid state theory rather than some phe-nomenological rule. The study of this model allowed usin particular to highlight the role of the effective frictioncoefficient µ, that appears to be a particularly interest-ing quantity to relate the properties of the liquid close tothe yielding transition, and those of the correspondingsolid state. It also predicted a non trivial shape for theyield surface that smoothly interpolates between knownshapes for soft and hard materials.

The picture emerging from this study raises a numberof questions: (i) How is the yield surface shape modifiedby the addition of the compression and dilation fracturemodes ? This question can be answered by extendingour toy model to compressible flows, in which case theliquid-like stress tensor provides both a shear and a bulkmodulus in the limit γ → 0. (ii) Is there a meaning-ful transition between definition 1 and 2 of the effectivefriction coefficient as the solid becomes harder ? Wouldan extension of our model to account for shear bandinghighlight some mechanisms of this transition ? (iii) What

6

is the influence of the dependence of µ on the Lode an-gle, and thus on the stress geometry on the definition ofthe jamming transition, where µ is supposed to reach afixed value through a power law evolution controlled bya critical exponent ?

Acknowledgements

This work was funded by the Deutscher AkademischerAustauschdienst (DAAD). We warmly thank Th. Voigt-mann for stimulating discussions and helpful suggestions.We thank O. Pouliquen for enlightening discussions.

GITT equations and toy models

In this appendix, we quickly review the foundations ofthe toy-model used in the main paper.

Dynamics

The dynamics of the particles is accounted for throughthe evolution of the dynamical structure factor Φq(t) =〈ρq(t)ρ−q(0)〉 /Sq, where ρq is the density operator inFourier space, and Sq = 〈ρq(0)ρ−q(0)〉 is the static struc-ture factor. Its evolution with time is given by a modecoupling equation of motion, which has the followingschematic structure:

Φq(t) + νqΦq(t) + Ω2qΦq(t) + Ω2

q

∫ t

0

dτ mq(t, τ)Φq(τ) = 0 ,

(7)

where νq and Ωq are characteristic frequencies, and mq

is called the memory kernel. The explicit expression ofthose terms is not needed for the following argument.The qualitative behavior of the solutions to Eq. (7) is asfollows: When the memory term is small enough, Eq. (7)reduces to a linear second order differential equation,whose solutions have a decaying exponential envelope;When the memory term is large, typical solutions satu-rate to a finite plateau value at large time, signaling theonset of solid behavior.

There are no analytical expression that captures wellthe global expression of the solutions to Eq. (7), hencebuilding analytically solvable models require to do someapproximations. A finer way to capture the behaviorof Φq(t) is to use the Vineyard approximation, which de-composes it in a product of static structure factor and selfcorrelation function, and then use an exponential ansatzfor the latter term:

Φq(t) ' Sq e−q2〈∆r2(t)〉 , (8)

where⟨∆r2(t)

⟩is the mean squared displacement. It is

not difficult to check that usual evolutions of⟨∆r2(t)

⟩

yields the expected qualitative phenomenology: for sim-ple liquids, the large time behavior of

⟨∆r2(t)

⟩is given by

the law of diffusion⟨∆r2(t)

⟩∼ t, which corresponds to

the case of negligible memory effects with an exponen-tial envelope of the decay, whereas in a solid, particlesacquire a well-defined mean position,

⟨∆r2(t)

⟩reaches a

finite limit value, and Φq(t) evaluated from Eq. (8) doesindeed saturate.

From this qualitative analysis, we draw two main con-clusions: (i) the wave vector dependence is irrelevant tounderstand the time evolution of Φq(t), at least in a low-est order approximation — notice that the same type ofargument is used when using schematic mode couplingmodels, such as those used in [60, 64] — and (ii) the wholecomplexity of Eq. (7) can be captured in a very crude wayby a simple exponential ansatz Φq(t) ' exp(−Γt), whereΓ = 1/tΓ is structural relaxation rate. Of course, we donot pretend to capture all the mode coupling physics bythis very simple ansatz, but it already captures the mostrelevant physical phenomena at play (in that case theslow down of the dynamics due to the structural relax-ations in dense systems), and with only minor modifica-tions (namely the addition of a second step in the decayof Φq(t)), it provides a convincing model of the rheologyof dense granular flows [57, 58], allowing for example torecover the universal µ(I) of granular rheology.

Finally, when going to the case of a sheared complexfluid, the effect of the advection of particles by the ex-ternal shear stress has to be taken into account. Withinour approach, this is done by evaluating the dynamicalstructure factor Φ at the wave vector q(t) advected bythe shear flow, rather than simply q. The equation (8)then becomes:

Φq(t)(t) ' Sq e−q(t)2〈∆r2(t)〉 , (9)

where q(t) = F (t) · q, and F (t) = exp(−κt) is the defor-mation gradient [74]. For simple shear flows, κ is nilpo-tent, and F (t) = I − κt exactly, but this relationship isnot valid anymore for more general flows. Since q(t) istypically ever increasing, this provides a new channel ofdecay for Φ: Even when the packing fraction is so densethat

⟨∆r2(t)

⟩should saturate to a constant value, the

q(t)2 factor in the exponential guarantees that Φ decaysto zero. Physically, this corresponds to the fact that themotion imposed by the advection of particles by the ap-plied shear stress is strong enough to break the cages, sothat the systems always flow on some time scale.

Because our toy-model is so far independent of q, it isblind to the change of q to q(t). To account for advec-tion, we therefore add a screening term that introducesthe advection channel of decay. Guided by results in thecase of simple shear flows [58], and the useful proper-ties of Gaussian functions, we define the screening factoras exp(−γ2 t2/2γ2

c ), where γc is a typical strain scale.As a matter of fact, the precise form of the screening isnot really important to grasp the main properties of the

7

rheology, [60] for example use a Lorentzian profile withsimilar results.

All in all, at the level of the dynamics, our toy-model can be summarized by Φq(t)(t) ' Φtoy(t) =exp(−Γt) exp(−γ2t2/2γ2

c ). It has two parameters: thestructural relaxation rate Γ, and the strain scale γc.

Link to the rheology

The dynamics evolution is then linked to the rheol-ogy by use of the Integration Through Transients (ITT)formula [56, 60, 63–65]. Using this approach, the shearcorrection to the stress tensor is expressed as an inte-gral over the time evolution of a fictitious reference state,where the fluid is not sheared:

∆σαβ =1

2T

∫ +∞

0

dt

∫k

Vσk(−t)Φk(−t)(t)2Wσ

k,αβ , (10)

where the mode coupling vertices Vσk =∑κθωVσk,θω and

Wσk,αβ have been introduced, and

∫k

=∫d3k/(2π)3.

Without going into the details, the vertices can be ex-pressed as:

Vσk,αβ = kαkβ k∆σ + δαβσ⊥

Wσk,αβ =

1 + ε

2S2k

[kαkβ k∆σ + δαβσ⊥

],

(11)

as a function of the following reduced scalars

σ⊥ = T[Sk − S2

k

]∆σ = −TS′k ,

(12)

the restitution coefficient of the granular particles ε, S′k =

dSk/dk, and the normalized wave vector components kα.Note that the above formula can be applied to colloidalsuspensions by taking the elastic limit ε→ 1.

Given that we defined above a procedure of approxima-tion of the Φ2

k(−t)(t) term, the next step is to reduce thetensorial structure of the mode coupling vertices. Fac-toring out the κθω term embedded in the vertex Vσk , ourintegral becomes a tensor of rank four corresponding tothe viscosity tensor Λαβθω. The vertex product is thendeveloped:

Vσk,θωWσk,αβ ∝δαβδθωσ2

⊥ + δθωkαkβkσ⊥∆σ

+ δαβ kθ(−t)kω(−t)k(−t)σ⊥∆σ

+ kαkβ kθ(−t)kω(−t)kk(−t)∆σ2 .

(13)

Then, we use the deformation gradient F (t) introducedabove to extract the remaining time dependence carriedby the wave vectors by κα(t) = Fαν(t)kν . The remainingtime independent wave vector component being the onlynon isotropic terms, the spherical part of the k integralcan be performed. We use the following formula:∫

k

kikj f(k2) =δij3

∫k

f(k2) , (14)

and ∫k

kikjkakb f(k2) =Xijab

15

∫k

f(k2) , (15)

where we defined the fully symmetrized product of kro-necker symbols as:

Xijab = δijδab + δiaδjb + δibδja . (16)

Finally, Φk(−t)(t) is replaced by Φtoy(t), and the remain-ing integral over the norm of k reduces to a constantprefactor.

After all the above steps are performed, the viscos-ity tensor can be decomposed along the three followingterms:

Bcompαβθω = δαβδθω

∫ +∞

0

dtΦ2toy(t)

B0αβθω = δαβ

∫ +∞

0

dtΦ2toy(t)

[δθω + Fνθ(−t)Fνω(−t)

]B1αβθω = Xαβνι

∫ +∞

0

dtΦ2toy(t)Fνθ(−t)Fι ω(−t) .

(17)

In the above formula, Bcomp comes from the σ2⊥ term.

It never contributes to ∆σ for incompressible flows. B0

comes from the σ⊥∆σ term in Eq. (13), the constantprefactor defines S0. It includes both the effect of resti-tution coefficient and the remnants of the k integral. Thefirst term in the bracket only contributes in the case ofcompressible flows. The second term is the Finger ten-sor, the inverse of the right Cauchy-Green deformationtensor. Lastly, B1 comes from the ∆σ2 term in Eq. (13)and its prefactor defines S1. Thus, in the case of incom-pressible flows, two scaling constants — S0 and S1 — areneeded, and all the contributions to Λ have the form of atime integral of the product Φtoy(t) and two deformationgradients, with varying types of tensor contractions.

For the last step, let us place ourselves in the basisin which the flow matrix is diagonal. In that case, itis possible to show that the remaining contributions inEq. (17) only involve one component of κ at a time, sayκjj . A typical term will hence have the following form:

κjj

∫ +∞

0

dt exp

(−2Γt− γ2t2

γ2c

+ κjjt

)=

√π

2κjj exp

[γ2c (κjj + Γ/γ)2

]erfc

[γc(κjj + Γ/γ)

].

(18)

Finally, our study is restricted to the study of dense gran-ular liquids and suspensions, close to the yielding tran-sition, which Weissenberg numbers Wi are very large,hence Γ γ, so that Eq. (6) is recovered. Note thatin the mode coupling paradigm, such systems are char-acterized by Γ = 0, because they would be solid if the

8

system were not sheared. However, a weaker version ofthe argument is needed here: we only use the fact thatWi 1. This corresponds to taking a limit γ → 0 underthe constraint that Γ γ.

The Drucker-Prager limit

Let us discuss the limit of small ∆P . The value of∆P can be deduced from Eq. (6), and depends mainlyon the behavior of the function F . Hereafter are some ofits basic properties:

limx→+∞

F(x) = +∞

limx→−∞

F(x) = − 1√π.

(19)

Then, the typical size of the argument of F in Eq. (6) isset by the combination A2γc, A1 fixing the relationshipbetween the different components of κ (see Eq. (5)). Thiscombination contains both information of the strength ofthe shear flow, through A2, and the material’s response,through γc. Because of Eq. (19), and since Tr(κ) = 0,which ensures that at least one of its components is posi-tive, if A2γc 1, ∆P is large. Consequently, the regimeof small ∆P correspond to small or moderate values ofA2γc, which can be explored by an expansion in powersof A2γc.

The expansion of F around x = 0 is as follows:

F(x) = x+2√πx2 +O(x3) . (20)

Because Tr(κ) = 0, the first order term in the expressionof ∆P vanishes, leaving at leading order:

∆P = I1 − P0 =(3S0 + 5S1)(3 +A2

1)

2A2

2γ2c +O(A3

2γ3c ) .

(21)Since I1 and P0 are parameters of our model, the aboverelationship truncated at this order allows us to computeA1 as a function of A2, and the parameters of the modelon a given deviatoric plane.

The corresponding geometry of the yield surface canbe deduced by studying the evolution of the stress tensorinvariants. For example, applying the same expansion,

J2 = S21π

3 +A21

2A2

2γ2c +O(A3

2γ3c ) , (22)

so that combining it with Eq. (21) yields:

J2

I21

=S2

1π

2

I1 − P0

(3S0 + 5S1)I21

+O(A2γc) . (23)

This formula can be read as follows: (i) the cuts of theyield surface in a given deviatoric plane are circles and(ii) their radius is all the smaller that ∆P/(3S0 + 5S1)

FIG. 4: Toy model yield surface for ∆P S0, S1. The pa-rameters are S0 = S1 = 1, γc = 0.4, ∆P = 0.001 (orange),∆P = 0.005 (yellow), ∆P = 0.01 (green), ∆P = 0.03 (blue),and ∆P = 0.05 (violet).

is small. It is the behavior of the Drucker-Prager yieldsurface. Such behavior is indeed observed with the yieldsurface of Eq. (6), as can be seen on Fig. 4. As A2γcgets closer to 1, ∆P/(3S0 + 5S1) does too, and the yieldsurface cuts become less and less isotropic.

Interestingly, the fact that the expression of the yieldsurface is known analytically also means that we canstudy the behavior of the constants associated with theother usual yield criteria in the same limit, namely theDrucker-Prager, Matsuoka-Nakai and Lade-Duncan ones.

Let us first have a look at the Drucker-Prager constantCDP = I2

1/I2. In order to have an idea of its typicalvalues, it is instructive to have a look at the value ofJ2 at the point of Lode angle equal to 0, which in ourconventions corresponds to the uniaxial extension flowκxx = 1, κyy = −1/2, κzz = −1/2. Indeed, this pointis generally used to adjust the constants of various yieldcriteria, since it corresponds to the point of maximumJ2, or, said otherwise, the point furthest to the hydro-static axis σ1 = σ2 = σ3. In that point, J2 can be ex-pressed as a function of the Mohr-Coulomb friction angleφMC =arctan(µMC) as:

J2 =4 I2

1 sin2(φMC)

3(3− sin(φMC))2, (24)

so that, since 0 6 µMC 6 +∞, 0 6 J2/I21 6 1/3, and

consequently, 3 6 CDP 6 +∞. From Eq. (6), CDP canbe expanded as:

CDP3

= 1 + 3X[1 +

3

4(A2γc)

π − 8√π

]+O(A3

2γ3c ) , (25)

9

where we introduced the combination

X =(I1 − P0)π3/2S2

1

P 20

[3S0 + (3 + 4

√π)S1

] , (26)

which appears in all the following expansion. X is allthe smaller that ∆P is, and is constant in a given devia-toric plane. From Eq. (25), we can check that CDP doesconverge toward 1/3 when A2γc is small, and becomestypically bigger than 1/3 when A2γc grows.

A similar analysis can be performed in the cases of theMatsuoka-Nakai and Lade-Duncan criteria. First, noticethat the ratio of tangential and normal stresses on thespatially mobilized plane is given by:

µ2MN =

CMN

9− 1 , (27)

which holds independently of the precision of theMatsuoka-Nakai criterion (µMN is simply more or lessconstant). Since this ratio is positive by definition,9 6 CMN 6 +∞. The expansion of this coefficient inthe limit A2γc 1 is given by:

CMN

9= 1+6X

[1 +

3

4(A2γc)

P0(π − 8) + 6πS1

P0√π

]+O(A3

2γ3c ) ,

(28)Again, this is consistent with the known limiting behaviorof this coefficient. Note also that in the limit where thehydrostatic pressure dominates the intrinsic strength ofthe material P0 S1, the second term in the bracketbecomes similar to that of Eq. (25).

Finally, using CLD = CMNCDP , 27 6 CLD 6 +∞[51]. The expansion of this coefficient in the limit A2γc 1 is given by:

CLD27

= 1+9X[1 +

3

4(A2γc)

P0(π − 8) + 4πS1

P0√π

]+O(A3

2γ3c ) ,

(29)which is consistent with the boundaries.

Analysis of the usual yield criteria

Most yield criteria can be represented by a certain com-bination of invariants of the stress tensor being constant.This type of assertion is easy to test within the realmof our toy model. For a given yield criteria of equationfx(I1, I2, I3) = Cx, where Cx is a constant, we can evalu-ate fx on a given deviatoric plane (in most cases, modelsare tested for a given value of I1), compute its averageon the whole yield surface cut in that plane, 〈fx〉, andlook at the relative variation (fx−〈fx〉)/ 〈fx〉. Finally, ina given cut of the yield surface along a deviatoric plane,the data can be represented as a function of the Lodeangle, or equivalently as a function of the parameter A1

of our surface parametrisation. In this way, we get thefigures 5, 6 and 7.

−3 −2 −1 0 1 2 3

A1

−0.125

−0.100

−0.075

−0.050

−0.025

0.000

0.025

(CDP−〈C

DP〉)/〈C

DP〉

FIG. 5: Relative variation of the Drucker-Prager constant forvarious values of ∆P . The parameters are S0 = S1 = 1,P0 = 1, γc = 0.4, ∆P = 0.05 (red), ∆P = 0.5 (orange),∆P = 1 (green), and ∆P = 2 (blue).

FIG. 6: Relative variation of the Matsuoka-Nakai (diamond)and Lade-Duncan (triangles) constants for various values of∆P . The parameters are S0 = S1 = 1, P0 = 1, γc = 0.4,∆P = 0.05 (red), ∆P = 0.42 (orange), ∆P = 1 (green), and∆P = 2 (blue).

More precisely, the determination of I1, I2 and I3 de-pend on P0, which is fixed by the environment and thusnot prescribed in our model. As can be expected, as P0

becomes larger and larger compared to the values of theintrinsic stress scales S0 and S1, the approximation ofCLD and CMN by constant values is of better and betterquality.

The effective friction coefficient

The effective friction coefficient is defined as a functionof the total pressure P = P0 + ∆P . However, there is

10

FIG. 7: Relative variation of the Matsuoka-Nakai (diamonds)and Lade-Duncan (triangles) constants for various values of∆P . The parameters are S0 = S1 = 1, P0 = 0.5, γc = 0.4,∆P = 0.05 (red), ∆P = 0.5 (orange), ∆P = 1 (green), and∆P = 2 (blue).

−3 −2 −1 0 1 2 3

A1

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

∆µ

FIG. 8: Evolution of ∆µ = σ0/∆P as a function of A1 forσ0 = σs

0 (triangles) and σ0 = σl0 (circles). The parameters are

S0 = S1 = 1, γc = 0.4, ∆P = 0.05 (red), ∆P = 0.5 (orange),∆P = 1 (green), and ∆P = 2 (blue).

no prescription for P0 in our formalism. The evolution ofeffective friction can still be observed through the studyof the reduced friction ∆µ = σ0/∆P . Its evolution as afunction of the Lode angle (represented by A1) for var-ious values of ∆P is displayed on Fig. 8, while Fig. 9represents its relative variations.

While looking at Fig. 8, it should be kept in mind thatwhat we represent is only the reduced friction coefficient,and not the total one. Therefore, relatively large values of∆µ must not be deemed too surprising, they are reducedby the total confining pressure P0.

We chose to represent ∆µ for both definitions of theshear stress σ0, from the solid and the liquid stress ten-

−3 −2 −1 0 1 2 3

A1

−0.15

−0.10

−0.05

0.00

0.05

(∆µ−〈∆µ〉)/〈∆µ〉

FIG. 9: Evolution of the relative variation of ∆µ = σ0/∆P asa function of A1 for σ0 = σs

0 (triangles) and σ0 = σl0 (circles).

The parameters are S0 = S1 = 1, γc = 0.4, ∆P = 0.05 (red),∆P = 0.5 (orange), ∆P = 1 (green), and ∆P = 2 (blue).

sors respectively, in order to illustrate the very close prox-imity of the predictions in both frameworks. Indeed, thisdifference appears to be visible only for its relative vari-ation, and only for the largest values of ∆P (in units ofS0, S1).

Then, we can study the variation of the effective fric-tion coefficients as prescribed by the Matsuoka-Nakai andLade-Duncan criteria, and corresponding to definition2 of the effective friction coefficient. The relations be-tween µMN (respectively µLD) and CMN (respectivelyCLD) can be found in [51]. Doing so requires to choose avalue of P0. Results for a sample of values are shown onFigs. 10, 11 and 12. It should be kept in mind that com-paring the precision to that of the toy-model of this fig-ure makes no sense since Fig. 9 presents only the reducedfriction coefficient ∆µ which necessarily varies more thanµ.

As for the case of the effective friction from the toymodel, at least for high enough value of P0, the effec-tive friction is not constant in a given deviatoric plane,and even more so that ∆P is large (or equivalently thatthe yield surface cut’s shape becomes more and more tri-angular). At high values of P0, both effective frictioncoefficient variations collapse onto each other. Perhapsmore surprising, it seems to be an optimal value of mildP0 where the variations are reduced at best.

Identification of the strain tensor

In order to get a deeper understanding of the physicalcontent of our toy model, let us push it a bit further andreplace the screening factor with a Heaviside functionΘ(1− γt/γc). Moreover, since we mainly study the high

11

FIG. 10: Relative variation of the Matsuoka-Nakai (dia-monds) and Lade-Duncan (triangles) effective friction co-efficients for various values of ∆P . The parameters areS0 = S1 = 1, P0 = 0.5, γc = 0.4, ∆P = 0.05 (red), ∆P = 0.5(orange), ∆P = 1 (green), and ∆P = 2 (blue).

FIG. 11: Relative variation of the Matsuoka-Nakai (dia-monds) and Lade-Duncan (triangles) effective friction co-efficients for various values of ∆P . The parameters areS0 = S1 = 1, P0 = 1, γc = 0.4, ∆P = 0.05 (red), ∆P = 0.5(orange), ∆P = 1 (green), and ∆P = 2 (blue).

Weissenberg number regime Wi 1, the decay of thestructural relaxation term exp(−Γt) happens outside ofthe Heaviside window, our dynamical ansatz reduces toΦq(t)(t) ' Θ(1− γt/γc).

Then, using Eq. (17) and the results of the correspond-ing appendix, and the fact that the flow is incompressible,the correction to the jth component of the stress tensoris given by:

∆σjj ' 2S1κjj

∫ +∞

0

γdt exp(2κjj γt

)Θ(1− γt/γc

)2= S1

[exp(2κjjγc)− 1

].

(30)

FIG. 12: Relative variation of the Matsuoka-Nakai (dia-monds) and Lade-Duncan (triangles) effective friction co-efficients for various values of ∆P . The parameters areS0 = S1 = 1, P0 = 10, γc = 0.4, ∆P = 0.05 (red), ∆P = 0.5(orange), ∆P = 1 (green), and ∆P = 2 (blue).

Let us have a close look at the term inside the brackets.The first term has a form of a Finger tensor for a straingiven by the typical strain scale γc, and the strength ofthe flow from which the strain originates, κjj . The terminside the brackets can therefore be understood as thedifference between the metric in the deformed state, andthe unit metric, which is by definition the strain tensor(or more precisely here its jth component). Eq. (30) canthus be straightforwardly read as a Hooke’s law, with ashear modulus S1 and a strain tensor exp(2κjjγc)− 1.

The incompressibility condition Tr(κ) = 0 ensures thatthe determinant of the deformation tensor is equal to 1,or here that the determinant of the Finger tensor is 1.In the limit of small deformations, which correspondsto κjjγc 1 — or more precisely A2γc 1, see ap-pendix on the Drucker-Prager limit — one recovers thewell known result that ε is fully deviatoric (Tr(ε) = 0).However, pay attention to the fact that this result onlyholds in this limit.

This model is of course too simple to be really ac-curate, but it shows how ε can be identified in our setof equations. When the Heaviside profile is replaced bythe Gaussian one used in the paper, the procedure staysthe same, although the results are less obvious to inter-pret since ε is now expressed as a function of F , whichdoes not let the difference between the strained and un-strained metrics appear explicitly. As it turns out, goingfrom the sharp Heaviside profile to the smoother Gaus-sian one amounts to accounting for the fact that advec-tion is not an instantaneous process, and as a result, thefinal expression of the strain is smoothened. The use ofa Lorentzian profile, such as the one used in [60] wouldprobably lead to the identification of the same type offunction, although in that case, the integral cannot be

12

performed exactly.

∗ Electronic address: oliver.coquand@umontpellier.fr† Electronic address: matthias.sperl@dlr.de

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