Discrete instabilities in granular materials

Transcription

1 Discrete instabilities in granular materials Félix Darve, Guillaume Servant Laboratoire Sols, Solides, Structures INPG, UJF, CNRS RNVO ALERT Geomaterials - Grenoble, France Felix.Darve@hmg.inpg.fr and Farid Laouafa Institut National de l Environnement Industriel et des Risques Verneuil en Halatt ABSTRACT In the framework of continuum mechanics, it has been shown that Hill's condition of Stability can detect diffuse modes of failure as, for example, the ones exhibited by loose sands subjected to undrained triaxial loading, axially stress controlled. The purpose of this paper is to consider a discrete form of Hill's condition of Stability and to apply it to discrete element computations. The bidimensional discrete material is constituted of wooden piled cylinders and different boundary conditions simulating slope stability problems have been considered. The numerical results show that the kinetic energy is correlated in space and in time with local and global peaks of negative values of the second order work. It is concluded that grain avalanches can be analysed by Hill's condition of Stability, which might be considered as a precursor to certain modes of failure. INTRODUCTION In previous papers (Darve and Laouafa [2000], Laouafa and Darve [2002]), the application of Hill s Condition of Stability to granular materials viewed as a continuous medium has been extensively considered. A summary of the main results is given in section 2 of this paper. The purpose of the present paper is to investigate the interest to consider a discrete form of the second order work (as envisioned by Mandel [1964]) and to apply it to granular materials viewed as discrete media. When a slope is constituted by granular materials and if a localised mode of failure is not constrained by the loading, the failure is generally characterised by granular avalanches. These avalanches can take a local or a global character (from a spatial view point) and they have been extensively studied by physicists of granular media (they are a paradigm of self organised criticality ). The basic objective of this paper is thus to discuss if granular avalanches can be analysed as material instabilities in Hill s sense [1958].

2 The granular material is a 2D Schneebeli medium constituted by wood piled cylinders. The particles are big enough to be followed individually by an automatic image acquisition and processing technique. The discrete element method, which has been used to simulate the experiments, is based on the Contact Dynamics method as initially proposed by Moreau [1994]. The grains are assumed to be rigid and in contact to each other by Coulombian friction. The granular medium and the discrete model are respectively presented in sections 3 and 4. The most interesting results, given in section 5, basically show comparisons between local and global kinetic energy (defined respectively for one single particle or summed for all the particles) and local or global second order work. Spatial and temporal correlations are exhibited between the bursts of kinetic energy and the peaks of negative values of second order work. HILL S CONDITION OF STABILITY IN CONTINUUM MECHANICS Hill's condition of Stability [Hill, 1958] is linked to the sign of the second order work : d ² W = dσ : dε After a given stress-strain history, a stress-strain state is considered as stable if, for any values of dσ and dε linked by the constitutive relation, the second order work is strictly positive. From a mechanical point of view, a negative value of second order work in a certain stress direction means that, for a dead load acting in this direction, the deformation will continue without any input of energy from outside. According to Hill, the material can be considered as ''unstable'' under this definition. The links (or more interestingly the equivalence) between Hill's condition and Lyapunov's definition of stability [Lyapunov, 1907] have been discussed by Koïter [1969] in the framework of elasticity. Nova [1994] has established such a link between Hill's condition and the so-called ''non-controllability'' of the loading in the framework of incrementally linear constitutive relations and for classical elastoplastic laws. The incrementally linear relation for rate-independent materials is : dσ = Mdε (1) The second order work can be written successively : t t t s d ² W = dσ : dε= dε dσ= dε M dε= dε M dε (2) where t ( ) signifies the transposed matrix and ( ) s the symmetric part of the matrix. With eigen values of M s varying in a continuous manner from positive values at the virgin state : s d ² W > 0 det M > 0 (3) Let us consider now the possible existence of ''failure'' states, defined as limit stress points (possibly path dependent) in the six-dimensional stress space. This means that we assume non-vanishing strain modes under constant stresses : dσ = 0 and dε 0 (4) 2

3 At such failure states, equation (1) gives : det M = 0, (5) which represents the plastic limit condition and : Mdε = 0, (6) which is the flow rule of the material at failure. A limit stress point, as defined by (4), is always unstable according to Lyapunov [1907], since, if we add at this stress state a ''small'' additional stress load, a sudden and brutal failure (a response which is very different from the strain controlled response, which remains ''small'') is obtained. In the framework of associated elasto-plasticity, the constitutive matrix M is symmetric. Thus Hill's condition (3) coincides with the plastic limit condition (5), and nothing new is exhibited by Hill's condition. On the contrary, for non-associated materials (like granular materials and, more generally, geomaterials), M is no longer symmetric and Hill's condition (3) will always be satisfied ''before'' (according to a given loading parameter) the plastic limit condition (5) [Nova, 1994, Bigoni and Hueckel, 1991]. We can thus conjecture the existence of a whole domain of instabilities, bifurcation, losses of uniqueness (i.e. failures) strictly inside the plastic limit condition. As an example of classical failure state, we can consider the drained triaxial loading path defined by a constant lateral pressure σ 3. In such axisymmetric conditions, the constitutive relation for a rate-independent incrementally linear behaviour is given by : dσ 1 dε1 = M (7) 2dσ 3 2dε 3 As σ 3is constant along this path and at the limit stress point in the axial direction, we have : dε1 0 det M = 0 and M =, 2 (8) dε 3 0 which corresponds respectively to the plastic limit condition and to the flow rule at failure. The second order work in such axisymmetric conditions is equal to the expression below : d ² W = t dσ dε = dσ 1dε1 + 2dσ 3dε3 = dσ 1dε1, (9) a quantity which vanishes at the peak value of σ 1 and takes negative values along the softening descending stress branch. As an example of a failure state inside the plastic limit condition, we can now consider the undrained triaxial loading path on a loose sand. The undrained condition corresponds to the isochoric relation given by : dε v = dε + dε 0 (10) = q = σ 1 σ 3 and ε 1 versus respectively ε v and σ 3 (the effective radial stress) are conjugated variables, since : 3

4 W = σ ε + σ ε = qε + σ ε. (11) v Thus the incrementally linear constitutive relation can be written as follows : dq dε1 = N (12) dε v dσ 3 The experiments show that, for a loose sand, q passes through a maximum located strictly inside the plastic limit condition. Along the whole path : dε v = 0 Moreover, at q peak : dq = 0 Thus the bifurcation criterion is obtained : det N = 0 (13) and the failure rule : dε1 0 N = (14) dσ 3 0 The second order work is equal to : d ² W = t dσ dε = dσ1dε1 + 2dσ 3dε3 = dqdε1, (9) which means that it vanishes at the peak of q and takes negative values along the q descending branch. Let us assume that this path is axially force controlled. As : q = F S, where F is the axial force and S the cross sectional area of the sample, at q peak, a sudden brutal failure of the whole sample with chaotic motions of the grains is observed. Moreover there is no strain localisation pattern. This diffuse mode of failure which occurs strictly inside the plastic limit condition can be detected by the negative sign of the second order work. Thus we have investigated in a systematic manner [Darve and Laouafa, 1999, 2000, 2001] the sign of the second order work for a loose and dense sand, in axisymmetric conditions, in plane strain conditions and in more general 3D conditions. For this, we have used our piece-wise incrementally linear model [Darve and Labanieh, 1982] and our thoroughly incrementally non-linear model [Darve et al., 1995]. 4

5 Plastic limit condition Instability d i 2 Plastic limit condition Instability d i 2 Fig. 1 Domains of instability in Rendulic Plane and axisymmetric conditions for the loose sand at the top, for the dense sand at the bottom and for 2 different constitutive relations (symbols o and + for the incrementally non-linear and the octolinear models respectively). 5

6 The unstable stress domain for axisymmetric, plane strain and 3D conditions are presented in Fig. 1, 2 and 3 respectively. These figures show that, as conjectured by the non-associated elasto-plasticity theory, there is a whole domain of instabilities, bifurcations, losses of uniqueness, (i.e. in fine : failure states) strictly inside the plastic limit condition. ϕ Fig. 2 Lowest unstable stress levels in plane strain conditions for the loose and dense sand, for 2 different constitutive relations. 6

7 Fig. 3 Domain of instability in Π plane for the loose and dense sand, and for 2 different constitutive relations. It is obvious that the second order work, at² a given stress-strain state and after a given strain history, varies with the direction of the stress : this is essentially a directional quantity. Thus, in general, there are cones of unstable directions. In Fig. 4, these cones have been plotted in the cases of a loose and a dense sand. It is interesting to remark that some directions will never give rise to diffuse rupture by material instabilities as the drained triaxial compressions and the axisymmetric compressions or extensions at constant mean pressures for example. 2 7

8 2 Fig. 4 Cones of unstable directions in the cases of a loose sand (at the top) and of a dense sand (at the bottom) for an incrementally non-linear constitutive relation. Finally Hill s condition of stability can be introduced into a finite element code and the sign of the second order work can be checked at each stress point of the mesh for each loading increment. Fig.5 presents such an example of an excavated slope. The iso-value curves of second order work are plotted for the last excavated sand layer in front of the slope. Let us emphasise that the second order work is computed for the effective values of the local incremental stress and incremental strain (i.e. there is no directional search for negative values). Fig. 5 shows that there is an unstable superficial layer of sand along the slope. The corresponding failure mechanism is clearly different from a plastic circular deeper rupture with a circular shear band. Fig. 5 Finite element computation of an heuristic excavated slope from an initial rectangular domain. Negative values of the second order work are exhibited for the last excavated horizontal layer of sand in front of the slope. 8

9 We have briefly recalled above the results given by the application of Hill s condition of stability in the framework of continuum mechanics. The purpose of this paper is now to consider applications in the framework of discrete mechanics. DISCRETE MEDIA Granular materials exhibit a double nature : for some aspects (mechanical properties, engineering works, ) they can be considered as continuous media and very efficient models are now available, while for other aspects (segregation of grains, rockfalls, )they have to be simulated as discrete media and some specific modelling tools have been developed for this purpose. From now on, we will consider discrete aspects. The material, constituted of piled cylinders (i.e. the so-called Schneebeli material), is inserted in a shear apparatus 1γ2ε (see Fig. 6) which can perform one distortion and two deformations. For further details about the experimental device, refer to Joer [1992]. Thus, the sample is a bidimensional material whose individual grain displacements and rotations can be easily determined by photographic techniques. An example of a configuration of a 335 disk assembly is given in Fig. 6. Fig. 6 An example of configuration of the 335 disk assembly. An automatic image acquisition and processing technique is used in order to measure the position of the centre of particles as well as their rotation. Thus we can follow the motion of each disk during the loading programme. Four different configurations and types of loading have been considered. They are given in Fig. 7. Case 7-a corresponds to a slope which leads to failure in an active state since the downstream plate is pushed outside the granular assembly. In case 7-d we have a rotation of the whole body. In cases 7-b and 7-c, respectively a symmetric and an asymmetric slope have been simulated with passive failure states because the plate at the top of the slope is pushed against the granular medium. In this paper only case 7-a is presented and discussed. 9

10 7 a 7 b 7 c Fig. 7 The four configurations of disk assemblies and types of loading which have been simulated. 10

11 DISCRETE MODELLING The discrete element method which has been used is the Contact Dynamics method [Lanier and Jean, 1999]. This method is essentially different from the various Molecular Dynamics methods (as proposed initially by P. Cundall [1979]), because the intergranular forces are not directly taken into account through springs, slides and dashpots linking the grains to each other as in Molecular Dynamics. Contact Dynamics considers strictly rigid particles interacting with each other through Coulombian friction. The kinematic condition is thus constituted by the no-penetration condition (i.e. the so-called Signorini Condition ), while the static condition is the dry Coulombian friction (an intergranular sliding is allowed only if the tangential intergranular force R t is equal to the normal force R n multiplied by the friction coefficient µ). Both these conditions are illustrated in Fig. 8. Fig. 8 Illustration, on the left, of the Signorini condition (there is no intergranular penetration δ : δ 0 ) and, on the right, of the Coulomb friction (there is no intergranular tangential sliding U t, if the tangential force R t is strictly included between -µr n et +µr n ). Newton s law is applied and the problem is solved numerically by an implicit method, based on the convex analysis from a mathematical point of view [Moreau, 1994]. The characteristics of the computations are thus given by : - number of particles : velocity driven boundary condition (Fig. 7-a) : du/dt = m/s - number of loading steps : time step : t = s - value of the intergranular friction coefficient : µ 1 = value of the wall-particle friction coefficient : µ 2 = 0.25 it is important to notice that the initial numerical configuration of the disk assembly is taken as close as possible to the initial experimental configuration (defined through the initial photographs). A certain validation of the computations can be appreciated in Fig. 9, where the experimental discrete displacement field is compared to the numerical grain displacement field at given values of the translation of the downstream plate (see Fig. 7-a). The numerical model is able to simulate reasonably well both small as well as larger landslides. This good agreement is essentially due to the great care with which the initial experimental configuration 11

12 of the grain assembly has been simulated numerically. These initial geometric details of the granular assembly play a fundamental role on the spatial and temporal distributions of granular avalanches. 9 a Loading step : 200 Loading step : b Loading step : 300 Loading step : 300 Loading step : 560 Loading step : c Loading step : 600 Loading step : d Loading step : 900 Loading step : e Fig. 9 Discrete incremental displacement fields for the boundary value problem of Fig. 7-a : experimental results on the left, numerical results by the Contact Dynamics method on the right 12

13 Fig. 9 shows that the numerical model is able to simulate various types of granular avalanches ; the small ones (Fig. 9-c and 9-e) concern only a part of the granular body while the largest ones (Fig. 9-a, 9-b and 9-d) put in motion almost all the grains. Anyway, in all these cases, it is impossible to identify a localisation pattern with a shear banding phenomenon. A localised mode of failure would have implied rigid body movements for the granular assembly above the shear band. On the contrary, it is interesting to notice on Fig. 9-a,b,c,d,e that the grain motions have a chaotic character. It seems that granular avalanches correspond most often to diffuse modes of failure and not to localised ones. This conclusion, which is based only on a qualitative analysis, will be discussed in a quantitative manner in the next section. DISCRETE SECOND ORDER WORK AND KINETIC ENERGY For each loading increment it is possible to associate with every particle a value of its second order work by : d ² W = df. dl + dc. dω (16) where df is the increment of the total force applied to the particle, dl its increment of displacement, dc the increment of the total moment applied to the particle and dω the increment of rotation. The physical meaning of negative values of the discrete second order work for a particle is still an open question. We can only conjecture that these negative values might be linked to local or more global failure, what is called grain avalanches in the framework of discrete media. In order to analyse this possible link, we have introduced and computed the kinetic energy of a particle given by : E 2 c = mv + Iω, (17) 2 2 where v is the grain velocity, m the mass, I the moment of inertia and ω the rotational velocity. Bursts of kinetic energy correspond obviously to brutal motions of one or several grains. In Fig. 10, a comparison between the negative values of second order work of some grains and the bursts of kinetic energy of some particles is presented at given values of the loading parameter (which is also the physical time because the velocity of plate translation is constant, see Fig. 7-a). All these quantities have been compiled from the numerical data given by the discrete element method which has been presented in section 4. We can notice the good correlation in space between both these quantities. The second order work vanishes for certain modes of failure, according to the analyses recalled in the section 2 of this paper. If we make a spatial correlation between the negative values of the second order work and the area of grain avalanches (characterised by bursts of kinetic energy), so the second order work might be seen as an indicator of potentially unstable domains. We have also computed global values of d ² W an d E c by adding the values of each grain. Thus we have obtained global values of the second order work of the whole assembly of disks and global values of the kinetic energy of all the grains. These quantities can be plotted versus 13

14 Fig. 10 Comparison between negative values of the discrete second order work (on the left) and bursts of the kinetic energy of the grains (on the right). 14

15 the loading parameter time increments have been considered to describe the translation of the plate (Fig. 7-a). thus we have 1080 such values, represented by 1080 points in Fig. 11. As regards the variations of the kinetic energy, we note large variations. The most important peaks correspond to large grain avalanches, while the smallest peaks are linked to focused events involving few particles with limited motions. Plate Displacement (cm) *10 - max= J. min= Plate Displacement (cm) max= J. Fig. 11 Variations of the global second order work ( at the top) and of the global kinetic energy (at the bottom) versus plate displacement. The correlation between peaks of kinetic energy and the negative values of the second order work versus the loading parameter are to be noticed. As regards the variation of the global second order work, large variations from positive to negative values have to be noticed. Surprisingly many negative values are exhibited by Fig. 15

16 11. Now, if we compare the values of the loading parameter for which the bursts of kinetic energy appear with the values corresponding to the most negative second order works, a strong correlation can be shown. Thus, according to the loading parameter, there is a strong correlation with respect to the space but also with respect to the loading history. It can be concluded (subject to further confirmations, because these are the first results of this type) that, from a map of values of the second order work, we can determine the potentially unstable areas and that, from the variations of the global second order work versus a loading parameter, we can conjecture which are the critical values of this loading parameter. CONCLUSIONS In the first part of this paper we have seen, in the framework of continuum mechanics, that there is a whole stress domain of instabilities, bifurcations and losses of uniqueness. Uniqueness has been directly demonstrated by Chambon and Caillerie [1999] in the case of hypoplastic constitutive relations, when the second order work is positive at each point. These stress states, where instabilities, bifurcations and losses of uniqueness can occur, correspond in practice to the so-called phenomenon of failure at the limit stress points. The modes of failure can be of different types as, for example, the localized mode which corresponds to a plastic strain localization and to the formation of shear bands. Another mode has been identified as a diffuse mode of failure or an homogeneous bifurcation [Darve and Roguiez, 1998]. The theory shows that the localization criterion is not satisfied at these stress states and the experiments indicate a field of chaotic motions, limited by a discontinuity line or surface. It appears that Hill s condition of stability might predict these diffuse modes of failure. The field domain where the second order work is negative could correspond to the unstable body and the negative sign of the second order work integrated over the whole domain might be an indicator of failure. In practice various modes of failure can appear successively. So we can have first a diffuse mode of failure when the stress state reaches the boundary of the unstable domain (in so far as the stress directions are located inside the cone of unstable directions and a proper loading mode by dead loads is applied), followed by a localized mode of failure when the localization criterion is met. For example, in triaxial extension tests, necking formation generally precedes shear band formation. In the second part we considered the application of the same notion of stability in the case of discrete mechanics. From this viewpoint we used a discrete form of second order work and we showed that grain avalanches might be described by Hill s condition of stability by characterising at the same time the unstable mass of grains and the degree of loading at which failure might occur. Here also the negative sign of the global second order work for the whole assembly of grains might be an indicator of failure. It is obvious that grain avalanches is a mode of failure which is typical of discrete media. It might be relevant for rockfalls. The case simulated in this paper where 335 disks have been considered, can not be analysed in principle in the framework of continuum mechanics. For example the small local landslide events along the slope (see Fig. 9) are clearly due to the geometrical irregularities of the slope surface. If we increase the number of grains by a factor of 10 (for example), the experiments show that for a regular slope surface these small events tend to be less numerous and to be replaced by more important landslides, which are not however constituted by the classical sliding along a circular line (in 2D situations). The experiments, performed in rotating drums [Raschenbach, 1990] or with conical piles continuously supplied with sand at the top [Herrmann et al., 1998], show sudden and regular ruptures at an inclination varying from the dynamical repose angle to the static repose 16

17 angle. These differ from each other by about 3. This mode of failure, which has been characterised as large grain avalanches is not of localized type, because the experiments do not show any rigid or elastic body sliding along shear bands. It could also be an example of diffuse failure. In our experiments such a large grain avalanche has been illustrated by case d of Fig. 7, but it has not been presented in this paper. Case a of Fig. 7, which has been extensively analysed here, exhibits a succession of small or large grain avalanches (see on Fig. 11 the various amplitudes of the bursts of kinetic energy). What is exhibited by the displacement fields of Fig. 9 (measured as computed) is the absence of any localisation pattern with shear bands. In this case, grain avalanches seem to correspond to diffuse modes of failure. What has been shown by the second order work computations is the correlation between these successive failures and the negative values of the second order work. These conclusions will be assessed in further papers, where localised modes of failure will also be exhibited. ACKNOWLEDGEMENT This research has been supported by the ROMICO contract (network of technological research RGCU) with the French Ministries of Research and Engineering Works, and is presently being developed inside the European project DIGA (RNT ) inside 5 th PCRD. The support of these national and European organisations is gratefully acknowledged. 17

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