Unified Model for Geomaterial Solid/Fluid States and the Transition in Between

Unified Model for Geomaterial Solid/Fluid Statesand the Transition in Between

N. Prime1; F. Dufour, M.ASCE2; and F. Darve3

Abstract: During mudflows, geomaterials evolve in a very specific manner, because initially they behave as solids, and then they turn toflow as viscous fluids. This paper proposes an original approach to simulate, within a single numerical framework, such a transition betweena solid-like and a fluid-like behavior. The model used is based on the association of an elastoplastic and a viscous constitutive relation todescribe both phases of the behavior. The transition between the two is assumed to correspond to a failure state, and the transition criterionconsidered is thus based on the general second-order work stability criterion. The Plasol elastoplastic model and the Bingham viscous modelare used because they are suitable to describe soils and granular suspensions, respectively. In this preliminary study, the global model istested for two homogeneous loading paths with drained and undrained conditions. For these cases, the radical transformation of the materialbehavior at failure is captured, and both the solid-to-fluid and the fluid-to-solid transitions are consistently enabled, making possible futureapplications of this approach for gravitational flowmodeling.DOI: 10.1061/(ASCE)EM.1943-7889.0000742.© 2014 American Society ofCivil Engineers.

Author keywords: Solid–fluid transition; Failure; Second-order work; Elastoplasticity; Bingham’s model.

Introduction

In situ soils have been extensively studied by both the geomech-anical and the geotechnical communities and are nowwell known toexhibit nonassociated elastoplastic behavior. Furthermore, a plasticfailure surface defines the limit stress states beyond which no staticequilibrium can be found. Elastoplastic behavior is thus classicallyused to describe the initiation of landslides by reaching a failure statein a natural slope (for a stress state that can be situated on the plasticlimit criterion or inside this criterion, see “Model for the Solid–FluidTransition” section) (Lignon et al. 2009). This failure stage usuallyleads to a numerical divergence of the computations, which is oftenconsidered an indicator for material instability (Klubertanz et al.2009).

Nevertheless, during the onset of flow-type landslides, it hasbeen observed that the solid soil can evolve into a mudflow of anamazing fluidity. For instance, landslides in Gansu (China, August7, 2010) and in Sarno and Quindici (Italy, May 5, 1998), developedthis way. These flows of soils have been shown to be well fitted bya Herschel-Bulkley viscous behavior characterized by a yield stress(Daido 1971) and nonlinear viscosity (Coussot and Boyer 1995;Malet et al. 2004), which is often characterized by a shear thinningcaused by the granular structure disorganization. This type of yieldstress viscous behavior has already been used to model the prop-agation of flow slides (Pastor et al. 2009).Moreover, experiments on

granular suspensions show that the solid–fluid transition depends onthe history of the system and on the energy supplied to it (Coussotet al. 2005). In dry granular materials, the drastic influence ofloading on slide behavior has also been studied by physicists(Jaeger and Nagel 1996), who introduced the notion of jamming tocharacterize that this material behavior can be solid-like or fluid-like. The first state is referred to as the jammed state (as the sand atrest), whereas the second state is referred to as the unjammed state(as sand avalanches).

Because the behavior of geomaterials can evolve from a solid-liketype to a fluid-like type, a natural question is the following: how canthe solid part of the behavior, the fluid part, and the transition betweenthe two be described within a single framework?

For decades, great effort has been devoted to accounting for,as accurately as possible, the transition between the two states ingranular materials. First of all, in granular suspensions, somemodelsare defined with a purely viscous relation, in which the transition ofthe behavior is expressed by a variation of viscosity betweenphysical values up to values tending to infinity (inducing jamming),depending on the conditions of the tests. For instance, Coussot et al.(2002) defined such a thixotropic constitutive relation to reproducethe flowing and the jamming of bentonite suspensions. However,this approach always considers time-dependent behavior, evenfor the solid phase, which is debatable. A second kind of con-stitutive relation defines a jamming stress criterion beyond whicha viscous flow is activated. Many models belong to this family,from very simple ones [e.g., the Bingham model, Fig. 1(a)] togreatly elaborated ones [e.g., the model proposed by Jop et al.(2006) for dense granular flows]. However, their main drawbackis that, even if the flowing domain is bounded by what can beconsidered a solid–fluid transition criterion, the solid behaviorinside this surface (in the stress space) is not defined. Finally,constitutive relations of the Perzyna type (Perzyna 1966) over-come this shortcoming by introducing, in the one-dimensional(1D) assumption, an elastic component in series to aBinghammodel[Fig. 1(b)]. However, in the hardening regime, the Perzyna con-stitutive relation leads to noninstantaneous plastic strains; this is anopen question.

1Researcher, Dept. Architecture, Géologie, Environnement et Construc-tions (ArGEnCo), Univ. de Liège, Chemin des Chevreuils 1, 4000 Liège,Belgium (corresponding author). E-mail: [email protected]

2Professor, Grenoble-INP/Univ. Joseph Fourier/CNRSUMR5521, 3SRLaboratory, 38041 Grenoble, France.

3Professor, Grenoble-INP/Univ. Joseph Fourier/CNRSUMR5521, 3SRLaboratory, 38041 Grenoble, France.

Note. This manuscript was submitted on November 27, 2012; approvedon October 29, 2013; published online on October 31, 2013. Discussionperiod open until June 28, 2014; separate discussions must be submitted forindividual papers. This paper is part of the Journal of EngineeringMechan-ics, © ASCE, ISSN 0733-9399/04014031(10)/$25.00.

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Although geomaterials can alternatively behave as solids orfluids, no constitutive relations are suitable to describe both phasesand the transition between them. Therefore, a new three-dimensional(3D) unified model is proposed in this paper. The main originalityof this model lies in two new developments: (1) the association ofsuitable fluid and solid constitutive relations; and (2) the use of ageneral criterion to detect the first solid–fluid transition, namely thesecond-order work criterion.

This paper is organized as follows. In thefirst section, the second-order work criterion is described as a candidate for the solid–fluidtransition criterion. Then, the elastoplastic constitutive relation usedfor the prefailure behavior is briefly reviewed while the equations ofthe 3D Bingham model are established. Finally, preliminary illus-trations of the capabilities of the whole model are presented togetherwith parametric analyses. In this first step, only tests on homoge-neous single-phase materials are considered, but these fully exhibitsolid-like elastoplastic behavior before failure and fluid-like viscousbehavior after failure. Drained and undrained loading conditionson a proper soil are considered in these tests, leading to classical ornonclassical failure.

Model for the Solid–Fluid Transition

If failure is defined, in a conventional manner, as the existence oflimit stress-strain states, then it is always related to an unstable be-havior. At these limit states, if an infinitely small load increment isapplied, a large material response is induced.

Instabilities may be divided into two main classes: one related tospecific boundary conditions (the geometric instabilities); and theother to particular material properties and states (the material in-stabilities). Geotechnical issues generally stem from material insta-bilities, and therefore, this contribution focuses on this category.Material instabilities are of two types: the divergence instabilities,where the strains suddenly increase monotonously to high values;and the flutter instabilities, which correspond to cyclically varyingstrains of increasing amplitude (Bigoni and Noselli 2011). Thefailures occurring in landslides are generally linked to divergenceinstabilities. Finally, in this last group, various patterns can also bedistinguished in situ, such as localized failure by shear band for-mation (Desrues and Chambon 2002) or diffuse failure (Daouadjiet al. 2011). Regarding these different modes of failure for geo-materials, the existing theories for stability analysis are analyzedhereafter.

Elastoplastic Limit Theory

Today, the most classical criterion to define the failure state is theplastic limit criterion. With engineering notations (strain and stresstensors denoted by six component vectors), this is formalized as fol-lows: let M be the constitutive second-order tensor connecting thestress and strain increments (ds5Mdɛ); the failure condition isdet ðMÞ5 0. In other words, a bounded stress increment no longerproduces a bounded strain response, as expressed by the Lyapunovstability definition (Lyapunov 1907), but an unlimited response.This failure mode is often called classical failure. This criterion hasa general applicability for associated materials such as metals (Rice1971). However, for nonassociated materials such as soils, theplastic limit condition cannot always predict the loss of stability forlocalized [Fig. 2(a)] or diffuse failures [Fig. 2(b)] (Nicot and Darve2011). For instance, a diffuse failure is typically obtained during anundrained triaxial test on loose sand for which the material suddenlycollapses at the peak of the deviatoric stress q (if the test is driven bythe axial force), although the plastic limit criterion has not yet beenreached [Figs. 2(b and c)].

Thus, elastoplastic limit theory with stress limit states is not ableto detect all types of instability in geomaterials (Darve et al. 2004;Laouafa and Darve 2002), which is a major drawback to accuratelyensuring safety conditions in the context of geotechnical work(excavation work, tunnelling, etc.) or natural hazard prediction(landslides, mudflows, etc.).

Second-Order Work Criterion

Given this limitation of the classical plastic limit theory, other the-ories have been developed to predict failure. First, localized failureshave been characterized by the vanishing of the determinant of theacoustic tensor, according to the Rice criterion (Rudnicki and Rice1975). In addition, diffuse failures (Nicot et al. 2011b) can be ex-plained by the loss of definite positiveness of the elastoplastic matrixM, as follows (Hill 1958):

det ðMsÞ# 0 (1)

whereMs 5 symmetric part ofM. The vanishing of det ðMsÞ definesa 3D limit surface between a stability domain and a bifurcationdomain (Prunier et al. 2009) in the principal stress space. By in-troducing the so-called second-order work d2W , a sufficient con-dition for stability becomes, at the solid scale, the following:

d2W ¼ðV

dsijdɛijdV . 0 "dɛ� 0 (2)

dsij is related to dɛij by the elastoplastic constitutive relation.V is thevolume occupied by the solid being studied, strains are assumed tobe small, and geometrical effects are ignored. For a material point,a local expression is defined (Laouafa and Darve 2002), and itsnormalized form is

d2Wn ¼ dsijdɛijjdsjjdɛj . 0 "dɛ� 0 (3)

dsij is related to dɛij by the elastoplastic constitutive relation.According to this expression, d2Wn stands for the cosine of the anglebetween the stress and the strain increment directions. Thus, stabilitydepends both on stress and strain paths, in contrast to the plastic limitcriterion where a given stress state is sufficient to determine whetherfailure is met. According to Darve et al. (2004, p. 3059), if Eq. (2) isnot verified (i.e., at failure), it means that the system is still subjected

Fig. 1. (a) Bingham and (b) Perzyna models in 1D

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to strains, whereas no more energy is transferred to it (i.e., “...thedeformation process will continue ‘on one’s own’.”) This notion canbe related to loss of controllability (Nova 1994). More precisely,when the second-order work is taking negative values in at least oneloading direction, a small additional load in this direction, or moregenerally a perturbation, leads to a burst of kinetic energy (Nicotet al. 2011b) and to a transition from a quasi-static regime to adynamic one (Nicot et al. 2011a).

It is essential to underline that, according to linear algebra, theloss of definite positiveness of the elastoplastic matrix [Eq. (1)]includes, as particular cases, the plastic limit condition (vanishing ofthe determinant of the elastoplastic matrix) and the localizationcondition (vanishing of the determinant of the acoustic matrix)(Bigoni and Hueckel 1991; Nicot and Darve 2011). The particularcase where failure occurs on the plastic limit criterion correspondsto a nonzero strain increment for which the stress increment iszero, and hence, so is d2Wn. Furthermore, it has been shown morerecently that the second-order work criterion is able to describe anykind of divergence instability for nonconservative elastic systems(Challamel et al. 2010; Lerbet et al. 2012). This is therefore thegeneral criterion of failure for rate-independent nonassociated geo-materials (Daouadji et al. 2011).

The second-order work criterion has been successfully applied togeomechanical issues (Darve and Laouafa 2000; Laouafa and Darve2002; Lignon et al. 2009). It has also been extended to unsaturatedsoils submitted to instabilities caused by hydraulic conditions(Buscarnera and di Prisco 2012). At the material scale, the power ofthe second-order work is typically highlighted by the undrained tri-axial test on loose sand forwhich d2Wn changes signs at the deviatoricstress peak, without reaching the plastic limit criterion [Fig. 2(c)].

In addition to being the most general failure criterion for soils,the second-order work criterion can detect diffuse failure modes forwhich the soil is deeply and globally destructured (burst of kineticenergy, possible liquefaction, etc.). For these reasons, it has beennaturally chosen to detect the solid–fluid transition state.

General Form of the Newly Proposed Model

Based on the second-order work transition criterion, the model takesinto account a solid-like prefailure behavior (with nonassociatedelastoplasticity) and a fluid-like postfailure behavior (with viscosityand a yield stress). Themodel is formulated such that total strains arethe sum of elastoplastic and viscous strains, if any (Fig. 3). The latterare nil before the failure and are activated as soon as d2Wn # ɛ (andif the yield stress is exceeded). Viscous strain rate activation is thusthe only simple way the transition is considered. A limit value ofɛ5 1026 is arbitrarily chosen for d2Wn to activate the transition evenif the limit load is obtained asymptotically.

The global consistency is ensured by the fact that the second-orderwork transition criterion remains the same for any rate-independentconstitutive relations. The dissociation of the two parts of the strainsmakes it possible to plug in any solid or fluid model independently.The model is thus adjustable with respect to any recent developmentsmade in solid or fluid mechanics and to a wide range of materials.

Solid and Fluid Constitutive Relations

Prefailure Behavior: Plasol Elastoplastic Model

The elastoplastic model called Plasol (Barnichon 1998) is chosenbecause it reproduces the main features of soils.

Fig. 2. (a) Localized failure in a drained triaxial test (Colliat-Dangus 1986, with permission); (b) diffuse failure in an undrained triaxial [Servant, G.,Darve, F., Desrues, J., and Georgopoulos, I. O. (2005). “Diffuse modes of failure in geomaterials.” Deformation characteristics of geomaterials,Di Benedetto et al., eds., Taylor & Francis, London, 181–200, with permission from Taylor & Francis]; (c) stress path for the undrained test [ReprintedfromComputer Methods in AppliedMechanics and Engineering, Vol. 193, No. 27–29, Darve, F., Servant, G., Laouafa, F., and Khoa, H.D.V., “Failurein geomaterials: Continuous and discrete analysis,” pp. 3057–3085, Copyright (2004), with permission from Elsevier]

Fig. 3.Diagramof the global constitutivemodel in 1D in a general form

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First, it is based on a Van Eekelen plastic criterion (Van Eekelen1980), which is close to theMohr-Coulomb plastic criterion withoutgeometric singularities. It is expressed as

F ¼ J2s þ m

�J1s 2

3ctanwc

�¼ 0 (4)

with m5 að11b sin 3uÞn and sin 3u52ffiffiffi6

p ðJ3s=J2sÞ3, where

b 5ðrc=reÞ1=n2 1

ðrc=reÞ1=n þ 1, a ¼ rc

ð1þ bÞn, rc ¼ 1ffiffiffi3

p�

2 sinwc

32 sinwc

�,

re ¼ 1ffiffiffi3

p�

2 sinwe

32 sinwe

�

J1s 5 trðsÞ, J2s 5ffiffiffiffiffiffiffiffiffiffiffitrðs2Þp

, and J3s 5ffiffiffiffiffiffiffiffiffiffiffitrðs3Þ3

p5 three invariants of

the Cauchy stress tensors (s5s2 1=3trðsÞ × 1 is the deviatoric partof s); and c 5 material cohesion. a and b depend on the materialfriction angles in compression and extension, wc and we, respectively.These two angles together with the Lode angle u modify the shape ofthe criterion in the deviatoric stress plane and determine a nonconstantradius of the trace, contrary to a Drucker-Prager criterion. n is a di-mensionless parameter whose value of20:229 has been calibrated byBarnichon (1998) to ensure the convexity of the yield surface.

Secondly, a hardening regime is described. The plastic para-meters vary—according to the equivalent plastic strain Ep

eq

5ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2=3epije

pij

q(with ep 5 ɛp 2 1=3trɛp × 1 being the deviatoric part of

plastic strain tensor) and two parameters Bw and Bc—between elasticlimit values (co, weo, wco) and plastic limit values (cf , wef , wcf ), such as

wc ¼ wc0 þ�wcf 2wc0

�Epeq

Bw þ Epeq

we ¼ we0 þ�wef 2we0

�Epeq

Bw þ Epeq

c ¼ c0 þ�cf 2 c0

�Epeq

Bc þ Epeq

(5)

Using Eq. (4), (co, weo, wco) and (cf , wef , wcf ) define the elastic limitand the plastic failure criterion, respectively. In all applications ofthis study, the final values of c and w are set strictly larger than theinitial values to avoid softening and associated mesh dependencies.In the global transition model, the plastic limit condition is includedin the more general second-order work failure criterion.

Finally, Plasol can describe a nonassociated plastic flow, i.e., theincremental plastic strain vector is not perpendicular to the surfacedefined by F [Eq. (4)], but instead to the plastic potential surface,whose equation is

G ¼ J2s þ m9

�J1s 2

3ctanwc

�¼ 0 (6)

wherem9 is defined asm [Eq. (4)] substituting the friction angleswiththe dilatancy angles, cc in compression and ce in extension. Di-latancy angles vary with the same amplitude as friction angles,according to the Taylor rule (Taylor 1948): wcf 2wc 5ccf 2cc.Consequently, only the final values of c need to be given.

From an experimental point of view, triaxial tests can directlydetermine cohesion and friction angles from the failure envelope inthe stress diagram and dilatancy angles from the volume variationmonitoring. In contrast,Bc and Bw have to be calibrated according tothese triaxial results.

Postfailure Behavior: Three-DimensionalBingham Viscosity

The fluid behavior of postfailure soils is chosen to reproduce therheology of mudflows.

Main Features of MudflowsMudflows are known to follow a Herschel-Bulkley model (Fig. 4).However, given that the viscous nonlinearity is often quite in-significant, the behavior can be considered in a first approximationwith a Bingham constitutive relation (i.e., linear viscosity beyonda yield stress).

Fig. 4. Nonlinearity in the Hershel Bulkley relation (Reprinted from Coussot and Boyer 1995, with kind permission from Springer Science andBusiness Media)

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Inversion of the Three-Dimensional BinghamConstitutive RelationIn the context of a global solid–fluid transition model where elasto-plasticity is defined in 3D, theBingham constitutive relationmust alsobe expressed in 3D. Moreover, implementing this relation in a nu-merical code can require writing the strain rate tensor (see “Imple-mentation of Viscoelastoplasticity in Ellipsis” section). The key pointis to determine the stress threshold direction in 3D in a consistentmanner.

On one hand, as is well known, the Bingham 1D relation forpositive or negative shearing is [Fig. 5(a)]

if   _g� 0: t ¼ h _g þ s0 × sgnð _gÞ, else:   jtj # s0 (7)

withh5 dynamic viscosity; t5 shear stress; _g5 velocity gradient;and s0 5 stress threshold. The function x→ sgnðxÞ returns the signof the scalar x. Because t in Eq. (7) is a one-to-one relation onð2‘; 0ÞUð0; 1‘Þ, it is invertible on this interval and its inversefunction is [Fig. 5(b)]

if   jtj . s0: _g ¼ ½t2 s0 × sgnðtÞ�=h, else:   _g ¼ 0 (8)

The sign of the stress threshold is given by the sign of the shearstress.

On the other hand, the 3D Bingham model is usually expressedsuch that the deviatoric stress tensor depends on the deviatoric strainrate tensor. Various authors (Duvaut and Lions 1972; Balmforth andCraster 1999) define the stress threshold direction as the deviatoricstrain rate tensor direction ð _eij=J2 _ɛÞ

if   J2 _ɛ � 0: sij ¼ 2h _eij þ s0_eijJ2 _ɛ

, else:   J2s # s0 (9)

where s and _e5 deviatoric part of the stress and the strain rate tensors;and J2s and J2 _ɛ 5 second invariants of the stress tensor and the strainrate tensor. According to the 1D inversion, one can consider that the3D relation is also invertible, and the stress threshold direction isprovided by the stress tensor direction ðsij=J2sÞ

if J2s . s0: _eij ¼ 12h

�sij2 s0

sijJ2s

�¼ J2s 2 s0

2h×sijJ2s

,

else _eij ¼ 0 (10)

h and s0 of a mud suspension can be determined by inclined planetests with different slope angles or by rheometer tests with differentstrain rates applied (Coussot and Boyer 1995). s0 therefore has norelation with the failure stress state.

At the end, considering both Plasol and Bingham’s relations, theglobal model needs 15 parameters to be identified: two related to

elasticity (E and n), 11 to plasticity (wco, wcf , weo, wef , co, cf , cc, ce,Bw, Bc, n), and two to viscosity (h and s0).

Synthesis of Transition Management

The solid-to-fluid transition takes place when a failure state is reachedfor an initially elastoplastic material. Because the viscous strain ratesdevelop only if the stress threshold is exceeded, this condition there-fore also needs to be satisfied to enable the transition. Nevertheless,Fig. 6 shows that, for realistic parameters, the stress domain where thefailure is possible (J2s belonging to the plastic domain, between theelastic limit and the plastic limit state) is included in the domain where

Fig. 5. (a) Bingham one-to-one relation; (b) inverted relation in 1D

Fig. 6. (a) Characteristic surfaces of the global model for Plasol andBingham relations (c5 3e10 kPa, wc 5we 5 6e28�, s0 5 2 kPa); (b)trace of the surfaces in the deviatoric plane; VE 5 Van Eekelen

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the viscous flow is possible ðJ2s . s0Þ. Reaching failure would thusbe, in general, sufficient to induce an effective solid-to-fluid transition.

In the fluid phase, the behavior is still viscoelastoplastic, but theorder of magnitude of viscosity is so low with respect to elasticparameters (see the parameters in the “Description of the Model”section) that the deviatoric part of the stresses are quickly relaxed,and the behavior is almost viscous. If J2s becomes smaller than s0,the viscous strain rates are disabled (according to the Binghammodel), and the material becomes elastoplastic again.

Analysis of the Model on Homogeneous Tests

Two tests are simulated with the aim of testing the constitutivemodel both at the material point level and at a continuum scale. Forthe latter task, the finite-element (FE) code Ellipsis 2012 based onthe advanced FEM with Lagrangian Integration Points (FEMLIP)(Moresi et al. 2003) was used. As in a standard Eulerian FE code influid mechanics, the spatial domain of study is discretized by a fixedcomputational grid on which nodal unknowns of the problem arecomputed through an integration scheme over elements using in-tegration points (that would be Gaussian points in standard methods).Contrary to standard FEMs, thematerial domain is discretized by a setof Lagrangian points (particles) carrying all material properties andvariables, including history-dependent variables. In a given materialconfiguration, these points are used as integration points, instead ofGaussian points, after being remapped to the natural element theybelong to. At each step, the numerical weight of each integration pointneeds to be updated to satisfy the conditions of the Gaussian quad-rature, such that the integration is the most accurate possible for theaimedpolynomial degree (Moresi et al. 2003).Once thenodal velocityfield is computed, the particle velocity is interpolated from the nodeswith FE shape functions, and the particles aremoved accordingly. Thefixed computational grid confers no limit in deformation magnitude,whereas the moving Lagrangian particles make it possible to trackinternal variables involved in complex behaviors. Ellipsis has alreadybeen successfully applied, for example, to geophysical viscousconvection problems in 3D (O’Neill et al. 2006), viscoelastic con-vection (Moresi et al. 2003), viscoelastic folding problems (Mühlhauset al. 2002a, b), and concrete flow in slump tests (Dufour andPijaudier-Cabot 2005; Roussel et al. 2007). This numerical tool hasalso been recently used to analyze natural slope stability based on anelastoplastic constitutive model (Cuomo et al. 2013).

Drained and undrained triaxial tests, which can lead to differentfailure patterns, are considered here. Because a drained triaxial test-mayexhibit a strain localization [Fig. 2(a)], it is better tomodel it at thematerial point level to prevent the loss of homogeneity of the strainfield [although Cosserat regularization is implemented in Ellipsis(Mühlhaus et al. 2002a)], whereas the undrained test is simulatedwithEllipsis, considering the plane strain conditions imposed by the code.Aparametric analysis is led on theviscous parameters in theundrainedcase.

Undrained Loading Path

Implementation of Viscoelastoplasticity in EllipsisIn Ellipsis, Stokes’ equations are solved and nodal unknowns arevelocities and pressure. Thus, the resolution matrix is made from theviscous parameters. Because Ellipsis is based on a mixed formu-lation to deal with incompressible materials, all tensorial equationsare split into deviatoric and isotropic parts. For better clarity, onlythe deviatoric equations are developed hereafter [see Cuomo et al.(2013) to see both of them].

First, the deviatoric strain rate is the sum of the viscous, elastic,and plastic strain rates, which gives

_etot 2 _epl ¼ _ee þ _ev ¼ ~s2m

þ s2h

(11)

with index tot, v, e, and pl, respectively, referring to the total tensor,and its viscous, elastic, and plastic parts. ~s is the Jaumann derivativeof the deviatoric stress tensor s, and m is the Lamé shear coefficient.

The time discretization of~s on a time step dte makes the rotationtensor v appear, giving

stþdte ¼ 2heff

�_etþdtetot 2 _etþdte

pl

�þ heff

st

mdteþ heff

vt × st 2 st ×vt

m

(12)

The effective viscosity heff 5 h½dte=ðdte 1TÞ� depends on the re-laxation time T 5 h=m. For the isotropic behavior, the effectiveviscous bulk modulus is Kv×eff 5Kv½dte=ðdte 1 TÞ�, assuming thesame relaxation time as for the deviatoric part.

Developing the mechanical equilibrium equation ð= × s1=p1 fext 5 0Þ gives, using Eq. (12)

2heff= ×�_etþdtetot 2 _etþdte

pl

�þ heff

= × st

mdteþ = × ðvtst 2 stvtÞ

m

þ Kv×eff=

tr�_ɛtþdtetot 2_ɛtþdte

pl

�þ pt

Kdte

þ f tþdte

ext ¼ 0 (13)

where K 5 elastic bulk modulus. Gathering all the terms from theprevious time step t in an elastic force term coming from the storedstress tensor, and all the terms depending on the plastic response ina plastic force term, gives for the current time t1 dte

2heff= × _etþdtetot þ Kv×eff=tr

�_ɛtþdtetot

�þ f tþdteext þ f telastic þ f tþdte

plastic ¼ 0

(14)

The force terms are updated at each time increment. However, loadingis such that plastic strains are not supposed to varymuch between onestep and another, and at a given time t1 dte, f t1dte

plastic is replaced withthe known quantity f tplastic. Because the plastic force term is explicit,different magnitudes of the loading increments have to be tested toverify that the solution does not depend on it. Pure elastic or elas-toplastic behaviors can be addressed in the limit by setting a very highvalue to the viscosity h, which becomes a numerical parameter. Thisway, heff can be defined, but almost no relaxation occurs along thetime steps.

To describe the solid-to-fluid transition process, the right-hand sideofEq. (11) takes into account, before the transition, one elastic and onenumerical viscous term. After the transition, that is to say if the geo-material fulfills the conditions for the flow (d2Wn # ɛ and J2s . s0),the formulation of the code considers an additional term for the totalviscous strain rate, and Eq. (11) is modified in the following way:

_etot 2 _epl ¼ _ee þ ð _evÞnum þ ð _evÞphy¼ ~s

2mþ s

2hnumþ J2s 2 s0

2hphy× sJ2s

(15)

In this equation, the expression of ð _evÞphy corresponds to thephysical behavior of the geomaterial flow, and it is expressedaccording to Eq. (10). This new term induces an additional forceterm in the formulation of the problem [Eq. (14)] equal to f t

52ðheff=hphyÞs0= × ðs=J2sÞt, and a modified expression of the ef-fective viscosity, such as

heff 5 hnumdte

dte þ T þ dte�hnum=hphy

�

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The different behaviors being accounted for in Ellipsis, a continuummodel of transition can be simulated.

Description of the ModelAunit square representative elementary volume of soil is consideredwith the plane strain conditions of the code. The mechanical pa-rameters, gathered in Table 1, are chosen to be representative of aclayey silt before failure (low elastic parameters, quite high fric-tion angle and cohesion). The friction and the dilatancy angles incompression are the same as the friction and the dilatancy angles inextension. A broad scattering for rheological parameters is found inthe literature for flow slides (Jeyapalan et al. 1983; Pastor et al. 2008;Soga 2011) or for artificial mud mixtures (Coussot and Boyer 1995;Coussot et al. 1996), with viscosity ranging from 50 to 1,000 Pa×s,and the stress threshold ranging from 100 to 5,000 Pa, depending onthe soil characteristics. In this preliminary analysis, low values of s0and h are arbitrarily chosen (100 Pa and 50 Pa×s, respectively). Thematerial has an initial pressure (i.e., a mean stress) of 30 kPa.Compression strain in the vertical direction z and extension strain inthe horizontal direction x are both increasingly applied to the squareto keep a constant volume ( _ɛzz 52 _ɛxx 5 0:6 s21). The strain rateconditions are considered according to the Ellipsis viscous for-mulation, although time has no physical meaning until the transi-tion. Even if the hydraulic phase is not accounted for in the presentmodel, this isochoric loading has the same effect on the sample asundrained conditions on a saturated material because of the quasi-incompressibility of water and solid grains. d2Wn is calculated foreach material point and at each step, according to Eq. (3), butconsidering only the current loading direction of stress and strainincrements. These are thus given by the difference between thetensors for the current and the previous time steps.

Results and Analysis

The second stress invariant J2s is plotted as a function of pressureand physical time (Figs. 7 and 8, respectively), because the pressureaxis is better suited to studying the solid-like response before failure,and the time axis is better adapted to describing thefluid-like response.

Once the elastic limit is reached (point Q in Figs. 7 and 8), thepressure consistently decreases (points Q–R) because the soil isinitially contractant at this stage (the mobilized dilatancy angle isco 5c2 ðwf 2woÞ5220� according to the Plasol model), and theisochoric conditions prevent the sample from contracting.

Concerning the transition, the expression of the second-orderwork for such an undrained test is d2Wn 5 dq × dɛzz with dq5 dszz 2 dsxx. For a linear increase of ɛzz, failure is thus reached atthe peak of q, which corresponds to the peak of J2s (Fig. 7, point R).At this step, J2s suddenly decreases, and the stress state turns back

into the elastic domain (points R–S). Accordingly, the computationshows that no plastic strain occurs, and the pressure remains constantafter failure (Fig. 7), since no viscous volumetric strains can takeplace because the Bingham relation is purely deviatoric and elasticvolumetric strains are null because the loading is isochoric.

The time evolution of J2s for such a viscoelastic material sub-jected to an axial strain rate is the solution of a partial differentialequation, and it is expressed as a function of physical time t

J2s ¼ ðJ2so2 2h _ɛzz 2 soÞexp�2m

ht

�þ 2h _ɛzz þ so (16)

where J2so 5 second stress invariant at failure. Along the time axis(Fig. 8), J2s after failure coincides with this exponential analyticalsolution. When time tends to infinity, J2s is defined by a pureBingham relation, with one fraction of the stress induced by theconstant strain rate loading and the other because of the yield stresss0, which cannot be relaxed.

This computation describes a sudden change of the materialbehavior at failurewith a relaxation of the deviatoric part of the stresstensor, which cannot be sustained by fluids (unless J2s is smallerthan so). Fig. 7 shows that the stress path obtained differs from thepresented undrained elastoplastic path (EP reference curve). It mustbe noted in the present results that, although the stress state beforethe transition does correspond to the intergranular stresses withinthe soil, the stress state after the transition corresponds to the stresseswithin a homogenized material (the granular suspension), whichstands for the water-grain mixture. This is a consequence of the

Table 1. Constitutive Parameters Considered

Parameter Value

r (kg=m3) 0E (MPa) 15n 0.29w0=wf (degrees) 3=28c (degrees) 15co=cf (kPa) 1=10Bw 0.01Bc 0.02n 20:229h (Pa×s) 50s0 (Pa) 100

Fig. 7. Stress path with regard to the elastoplastic (EP) case with notransition; for the transition curve: PQ 5 elastic response; QR 5 elas-toplastic response; RS 5 viscoelastoplastic response

Fig. 8. J2s over physical time and comparison with the analyticalsolution; for the transition curve: PQ 5 elastic response; QR 5 elas-toplastic response; RS 5 viscoelastoplastic response

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absence of coupling between two phases. It involves that, after thetransition, the reference curve of Fig. 7 and the calculated stress pathcannot be compared in a rigorousmanner. Besides, itmust be noticedthat, for such an undrained test driven by a strain loading, the failurecannot develop because of the kinematic constraints, which is not thecase if the test is stress controlled. The disorganization of thematerialthus appears to be progressive during the loading. Because of theabsence of coupling in this approach, the transition has to be in-stantaneous, whatever the loading conditions. Nonetheless, thetriggering of mudflows in the field is generally linked to mixedloading conditions (stress and displacement ones) with an importantrole of the stress conditions (e.g., rising of the water table level). Thetotal constraint of the kinematic, which prevents the effective failuredevelopment, is thus unlikely in the field.

Parametric Analysis

Two parametric studies are conducted to evaluate the performanceof the transitional model through the large realistic range of visco-sity and Bingham’s threshold found in the literature for mudflows.First, for a fixed s0 5 100 Pa, five values of h are chosen: 50, 100,300, 600, and 1,000 Pa×s. The time evolution of the second stressinvariant J2s is presented in Fig. 9(a). As predicted by the analyti-cal solution [Eq. (16)], the increase of viscosity leads to a higher

final stress and a slower relaxation of stresses (less decreasingexponential).

Then, for h5 50 Pa×s, four tests are performed with s0 5 100,300, and 2,400 Pa and s0 5 4,800 Pa (i.e., defined as the secondstress invariant at failure J2sf ). Results are presented in Fig. 9(b). Inthe last case, viscosity does not develop because the Binghammodelcannot induce a viscous strain rate if there is no difference betweenthe second stress invariant and the stress threshold [Eq. (10)]. Itwould be the same for any s0 larger than J2sf . The other cases are alsoconsistent with Eq. (16) because the higher s0, the higher the hor-izontal asymptote of J2sðtÞ.

Drained Triaxial Path

In this second test, the mechanical parameters are the same as theundrained test. A stress path is followed with an initial confinementof po 5 100 kPa (state marked as A in all figures), an increase of theaxial deviatoric stress q5szz 2 po up to the failure (marked as B),a constant stress state to let the viscous strains develop (fromB toC),and finally a decrease of q up to 0 (point D) to highlight the jammingof the flow. The evolution of q is plotted along computational time[Fig. 10(a)] and pressure [Fig. 10(b)].

During the first loading stage (A–B), the total axial strainɛzz increases in a classical elastoplastic way (Fig. 11), whereas the

Fig. 9. Influence of (a) viscosity; (b) yield stress

Fig. 10. Stress loading path along (a) time; (b) pressure

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viscous component of ɛzz remains null [Fig. 12(c)]. The second-order work first presents a slight discontinuity at the elastic limit,then goes through a maximum value before sharply decreasing as itapproaches the plastic limit criterion, reaching the limit value of1026 (B) [Figs. 12(a and b)]. In this case, it thus coincides with theplastic limit criterion. Fig. 12(c) shows that at failure, ɛzz evolvescontinuously over time, with a continuous derivative (no singularpoints). From this point, the viscous axial strain is activated ac-cording to the transition model, and because the axial stress is keptconstant, ɛzz linearly increases [B–C in Fig. 12(c)].

During the last stage (C–D), the application of a linear decrease ofq up to 0 induces a linear decrease of _ɛzz. The flow finally stops[Fig. 13(b)] for a value of the second stress invariant, which is notzero but 100 Pa (i.e., equal to s0) [Fig. 13(a)]. The model thusdescribes the Bingham jamming well.

Conclusions and Perspectives

The 3D transitional constitutive model presented here has beenformulated to describe, within a unified framework, both theelastoplastic behavior of an in situ soil and its transition towarda fluid-like behavior, which occurs, for example, during mudflow

triggering. Themodel is based on a transition criterion defined as thesecond-order work failure criterion; therefore, it is able to detect alltypes of divergence instabilities and is independent of the elasto-plastic constitutive relation used. It also links adaptable solid andfluid constitutive models. In the second step, if a viscous relationwith a yield stress is chosen, the arrest of the flow (i.e., the fluid-to-solid transition) is also described. A Plasol elastoplastic model anda Bingham viscous model are first considered, with an inverted ex-pression of the 3D Bingham relation with respect to usual for-mulations. The two tests performed with this behavior validate theconsistency of the model for both drained and undrained conditions,not only at the material point scale, but also implemented in a con-tinuum FE code. The drained triaxial test more specifically shows theconsistency of the failure detection and jamming of the flow, and ithighlights the continuity of the total strains at the solid–fluid transi-tion. The undrained testmakes it possible to verify the postfailureflowaccording to the viscoelastic analytical response. In the latter case, themodel describes a drastic transition from an elastoplastic behaviortoward a viscous one, although elastoplasticity still coexists in themodel during the flow. This could describe the sudden collapse ofsoils observed during such a test, when it is axially force controlled.

The parametric analysis shows that the Bingham threshold needsto be lower than the second stress invariant at failure to enable thetransition. In the field, it is consistent to consider that the yield stressin mudflows is lower than the in situ soil stress invariant at failure.This condition can be viewed as a reduction of the mechanicalstrength of a deeply remolded soil.

In conclusion, because of the transitional behavior it describes,thismodel, now correctly implemented into a numerical code,wouldbe very efficient in simulating gravitational flows, such as landslidesof the flow type (mudflows, debris flows). The next step will thus beto apply it, with Ellipsis, to a heuristic unstable slope evolving intoflow at failure. Further developments are already underway to ac-count for hydromechanical coupling to model partially saturatedgeomaterials.

References

Balmforth, N. J., and Craster, R. V. (1999). “A consistent thin-layer theoryfor Bingham plastics.” J. Non-Newt. Fluid Mech., 84(1), 65–81.

Barnichon, J.-D. (1998). “Finite element modeling in structural and pe-troleum geology.” Ph.D. thesis, Univ. de Liège, Liège, Belgium.

Fig. 11. Deviatoric stress q versus axial strain ɛzz

Fig. 12. d2Wn over (a) time; (b) zoom; (c) onset of the solid–fluidtransition

Fig. 13. (a) Reaching of the yield stress by the second stress invariant;(b) induced arrest of the flow

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ite D

e L

iege

on

03/0

6/14

. Cop

yrig

ht A

SCE

. For

per

sona

l use

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Bigoni, D., and Hueckel, T. (1991). “Uniqueness and localization_I. As-sociative and non-associative elastoplasticity.” Int. J. Solids Struct.,28(2), 197–213.

Bigoni, D., and Noselli, G. (2011). “Experimental evidence of flutter anddivergence instabilities induced by dry friction.” J. Mech. Phys. Solids,59(10), 2208–2226.

Buscarnera, G., and di Prisco, C. (2012). “Discussing the definition of thesecond-order work for unsaturated soils.” Int. J. Numer. Anal. MethodsGeomech., 36(1), 36–49.

Challamel, N., Nicot, F., Lerbet, J., and Darve, F. (2010). “Stability of non-conservative elastic structures under additional kinematics constraints.”Eng. Struct., 32(10), 3086–3092.

Colliat-Dangus, J. L. (1986). “Comportement des matériaux granulairessous fortes contraintes.” Ph.D. thesis, Université Scientifique et Médi-cale de Grenoble–Institut National Polytechnique de Grenoble (USMG–INPG), Grenoble, France.

Coussot, O., and Boyer, S. (1995). “Determination of yield stress fluidbehaviour from inclined plane test.” Rheol. Acta, 34(6), 534–543.

Coussot O., Nguyen Q. D., Huynh H. T., and Bonn D. (2002). “Avalanchebehavior in yield stress fluids.” Phys. Rev. Lett., 88(17), 175501.

Coussot, O., Proust, S., and Ancey, C. (1996). “Rheological interpretationof deposits of yield stress fluids.” J. Non-Newt. Fluid Mech., 66(1),55–70.

Coussot O., Roussel N., Jarny S., and Chanson H. (2005). “Continuous orcatastrophic solid-liquid transition in jammed systems.” Phys. Fluids,17(1), 011704.

Cuomo, S., Prime, N., Iannone, A., Dufour, F., Cascini, L., and Darve, F.(2013). “Large deformation FEMLIP drained analysis for a vertical cut.”Acta Geotech., 8(2), 125–136.

Daido, A. (1971). “On the occurrence of mud-debris flow.” Bull. Disas.Prev. Res. Inst., 21(2), 109–135.

Daouadji, A., et al. (2011). “Diffuse failure in geomaterials: Experiments,theory and modelling.” Int. J. Numer. Anal. Methods Geomech., 35(16),1731–1773.

Darve, F., and Laouafa, F. (2000). “Instabilities in granular material andapplication to landslides.” Mech. Cohes.-Frict. Mater., 5(8), 627–652.

Darve, F., Servant, G., Laouafa, F., and Khoa, H. D. V. (2004). “Failure ingeomaterials: Continuous and discrete analysis.” Comput. Methods Appl.Mech. Eng., 193(27–29), 3057–3085.

Desrues, J., andChambon, R. (2002). “Shear band analysis and shearmodulicalibration.” Int. J. Solids Struct., 39(13–14), 3757–3776.

Dufour, F., and Pijaudier-Cabot, G. (2005). “Numerical modeling of con-crete flow. Homogeneous approach.” Int. J. Numer. Anal. MethodsGeomech., 29(4), 395–416.

Duvaut, G., and Lions, J. L. (1972). “Transfert de chaleur dans un fluide deBingham dont la viscosité dépend de la température.” J. Func. Anal,11(1), 93–110.

Ellipsis 2012 [Computer software]. Grenoble, France, 3SR Laboratory.Hill, R. (1958). “A general theory of uniqueness and stability in elasto-

plastic solids.” J. Mech. Phys. Solids, 6(3), 236–249.Jaeger, H.M., andNagel, S. R. (1996). “Granular solids, liquids, and gases.”

Rev. Mod. Phys., 68(4), 1259–1273.Jeyapalan J. K., Duncan J. M., and Seed H. B. (1983). “Investigation of flow

failures of tailing dams.” J. Geotech. Engrg., 109(2), 172–189.Jop, P., Forterre, Y., and Pouliquen, O. (2006). “A constitutive law for dense

granular flows.” Nature, 441(7094), 727–730.Klubertanz, G., Laloui, L., and Vulliet, L. (2009). “Identification of mech-

anisms for landslide type initiation of debris flows.”Eng. Geol., 109(1–2),114–123.

Laouafa, F., and Darve, F. (2002). “Modelling of slope failure by materialinstability mechanism.” Comput. Geotech., 29(4), 301–325.

Lerbet, J., Aldowaji, M., Challamel, N., Nicot, F., Prunier, F., and Darve, F.(2012). “P-positive definite matrices and stability of nonconservativesystems.” Z. Angew. Math. Mech. ZAMM, 92(5), 409–422.

Lignon, S., Laouafa, F., Prunier, F., Khoa, H. D. V., and Darve, F. (2009).“Hydro-mechanical modelling of landslides with a material instabilitycriterion.” Geotechnique, 59(6), 513–524.

Lyapunov, A. M. (1907). “Problème général de la stabilité des mouve-ments.” Ann. Faculté Sci. Toulouse, 9, 203–274.

Malet, J.-P., Maquaire, O., Locat, J., and Remaître, A. (2004). “Assessingdebris flow hazards associated with slow moving landslides: Method-ology and numerical analyses.” Landslides, 1(1), 83–90.

Moresi, L., Dufour, F., and Mühlhaus, H.-B. (2003). “A Lagrangian in-tegration point finite element method for large deformation modelingof viscoelastic geomaterials.” J. Comput. Phys., 184(2), 476–497.

Mühlhaus H., Moresi L., Hobbs B., and Dufour F. (2002a). “Large am-plitude folding in finely layered viscoelastic rock structures.” Pure Appl.Geophys., 159(10), 2311–2333.

Mühlhaus, H.-B., Dufour, F.,Moresi, L., andHobbs, B. (2002b). “Adirectortheory for viscoelastic folding instabilities in multilayered rock.” Int. J.Solids Struct., 39(13–14), 3675–3691.

Nicot, F., Challamel, N., Lerbet, J., Prunier, F., and Darve, F. (2011a).“Bifurcation and generalizedmixed loading conditions in geomaterials.”Int. J. Numer. Anal. Methods Geomech., 35(13), 1409–1431.

Nicot, F., Daouadji, A., Laouafa, F., and Darve, F. (2011b). “Second-orderwork, kinetic energy and diffuse failure in granular materials.” Granul.Matter, 13(1), 19–28.

Nicot, F., and Darve, F. (2011). “Diffuse and localized failure modes, twocompetingmechanisms.” Int. J. Numer. Anal.MethodsGeomech., 35(5),586–601.

Nova, R. (1994). “Controllability of the incremental response of soilspecimens subjected to arbitrary loading programmes.” J.Mech. Behave.Mater., 5(2), 193–202.

O’Neill, C., Moresi, L., Muller, D., Albert, R., and Dufour, F. (2006).“Ellipsis 3D: A particle-in-cell finite-element hybrid code for modellingmantle convection and lithospheric deformation.” Comput. Geosci.,32(10), 1769–1779.

Pastor, M., et al. (2008). “Mathematical, constitutive and numerical mod-elling of catastrophic landslides and related phenomena.” Rock Mech.Rock Eng., 41(1), 85–132.

Pastor, M., Haddad, B., Sorbino, G., Cuomo, S., and Drempetic, V. (2009).“A depth-integrated, coupled SPH model for flow-like landslides andrelated phenomena.” Int. J. Numer. Anal. Methods Geomech., 33(2),143–172.

Perzyna, P. (1966). “Fundamental problems in viscoplasticity.” Adv. Appl.Mech., 9(2), 244–368.

Prunier, F., Nicot, F., Darve, F., Laouafa, F., and Lignon, S. (2009). “Three-dimensional multiscale bifurcation analysis of granular media.” J. Eng.Mech., 10.1061/(ASCE)EM.1943-7889.0000003, 493–509.

Rice, J. R. (1971). “Inelastic constitutive relations for solids: An internal-variable theory and its application to metal plasticity.” J. Mech. Phys.Solids, 19(6), 433–455.

Roussel, N., Geiker, M. R., Dufour, F., Trane, L. N., and Szabo, P. (2007).“Computational modeling of concrete flow: General overview.”CementConcr. Res., 37(9), 1298–1307.

Rudnicki, J. W., and Rice, J. (1975). “Conditions for the localization ofdeformation in pressure sensitive dilatant materials.” Int. J. SolidsStruct., 23, 371–394.

Servant, G., Darve, F., Desrues, J., andGeorgopoulos, I. O. (2005). “Diffusemodes of failure in geomaterials.” Deformation characteristics of geo-materials, H. Di Benedetto, T. Doanh, H. Geoffroy, andC. Sauzéat, eds.,Taylor & Francis, London, 181–200.

Soga K. (2011). “Failure observed in landslides.” Presentations of theALERT Workshop, Alliance of Laboratories in Europe for Research andTechnology, Univ. of Cambridge, Cambridge, U.K.

Taylor, D. (1948). Fundamentals of soil mechanics, Wiley, London.Van Eekelen, H. A. M. (1980). “Isotropic yield surface in three dimensions

for use in soil mechanics.” Int. J. Numer. Anal. Methods Geomech., 4(1),89–101.

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