A multiscale description of failure in granular materials

Transcription

1 A multisale desription of failure in granular materials Nejib Hadda, François Niot, Lu Sibille, Farhang Radjai, Antoinette Tordesillas et al. Citation: AIP Conf. Pro. 154, 585 (013); doi: / View online: View Table of Contents: Published by the AIP Publishing LLC. Additional information on AIP Conf. Pro. Journal Homepage: Journal Information: Top downloads: Information for Authors:

2 A Multisale Desription of Failure in Granular Materials Nejib Hadda 1, François Niot 1, Lu Sibille, Farhang Radjai 3, Antoinette Tordesillas 4, and Félix Darve 5 1 IRSTEA, Center of Grenoble, Frane LUNAM university, GeM institute, CNRS, Nantes, Frane 3 LMGC laboratory, University of Montpellier, CNRS, Frane 4 Department of Mathematis and Statistis, University of Melbourne, Vitoria 3010, Australia 5 UJF-INPG-CNRS, 3S-R Laboratory, Grenoble, Frane Abstrat. This paper presents onditions of initiation and development of failure in granular materials through a twodimensional disrete element model. General ondition for the effetive development of failure and its physial harateristis are realled. Then relation between failure and the seond order work expressed in terms of mirosopi variables is disussed. Eventually, orrespondene between a loalized mode of failure marked with shear band patterns and spae distribution of negative values of mirosopi seond-order work is investigated. Keywords: failure, strain loalization, seond-order work, miro-mehanis, disrete element method. PACS: 6.0.-x; 6.0.M-; i INTRODUCTION Failure in granular materials is lassially assoiated to the notion of plasti limit ondition represented in the stress spae by a Mohr-Coulombtype failure surfae. The objetive of this paper is to reonsider the question of failure from a more general point of view. Hene we reall first: what is the physial manifestation of failure, the relation between its effetive ourrene and the mode of ontrol of the mehanial state, and finally the way to detet with the seond-order work riterion the possible ourrene of failure. Then, in the framework of this general desription of failure, we fous on the desription of the mode of failure, either loalized or diffused aording to the development or the lak of loalization of deformations. In partiular, we investigate if a loal expression of the seond-order work (built from mirosopi variables), in diret relation with the marosopi expression of the seond order work and thus with failure at the sale of the representative elementary volume, ould also be linked to the mode of failure. Disussions and analyses are based on numerial simulations performed with the disrete element based software YADE [1]. The granular assemblies are made of elasti irular partiles (in D), or spherial ones (in 3D), and inter-partile ontats are purely fritional. Besides, we will make referene to the marosopi expression of the seond order work W, basially defined for a homogeneous volume V o in equilibrium at a given time t under a presribed external loading as: ( i ) W = ij dvo (1) X V uj o involving both inremental Piola-Kirhoff stress ( ui ) and strain F, with F ij = experiened by the X j system during a time inrement δt. In what follows, the lagrangian stress will be onfounded with the usual Cauhy stress, for the sake of simpliity. FAILURE DESCRIPTION Failure is generally assoiated to the notion of limit stress state. Suh limit stress states are easily observable in homogeneous laboratory tests where some loading paths lead to stress states that annot be exeeded. The drained triaxial ompression on a dense granular assembly onstitutes a lassial example. Fig. 1a shows the simulated response with the disrete element model to a two-dimensionnal drained biaxial ompression under a onfining pressure σ = 300 kpa. The axial stress σ 1 grows until reahing a maximum. This maximum onstitutes a limit stress state. This point an be verified by swithing the ontrol and response parameters. For the simulation we have just presented, the loading path was fixed by the loading parameter dσ = 0 (defining a straight line in the stress plane) and the mehanial state of the granular assembly along Powders and Grains 013 AIP Conf. Pro. 154, (013); doi: / AIP Publishing LLC /$

3 this path were ontrolled through the axial strain ε 1 by imposing an axial ompression (dε 1 > 0). The axial stress σ 1 was then a response parameter. We an renew this experiment by ontrolling σ 1 and imposing a stress inreases (dσ 1 > 0), while ε 1 is the response parameter. The response to this loading program is superimposed to the previous one in Fig. 1a. Even though a onstant inrease of σ 1 is imposed, the limit stress state is not exeeded (atually it is here slightly exeeded due to inertial terms no more negligible at failure initiation). While it is approahed, deformations inrease strongly just as the strain rate and the kineti energy as displayed in Fig. 1b. Thus, it is shown that the peak of σ 1 is a limit stress state and that failure (haraterized by unlimited strains and a transition from a quasi-stati response to a dynami one []) effetively ours when the load apply to the granular assembly exeed this limit state. Limit stress states are lassially desribed in soils by failure riterion of Mohr-Coulomb type. They an also be identified with the seond-order work riterion stating that a limit state is reahed if W 0 [3] (for the biaxial loading path dσ = 0 is imposed and W vanishes together with dσ 1 ). This last riterion is more general than the Mohr-Coulomb riterion sine it offers the possibility to detet mehanial states stritly inluded within the Mohr-Coulomb riterion from whih failure may develop. This has been deeply disussed in previous papers (for instane [3] & [4]). Another important disussion in the desription of failure is about the loss of homogeneity of the strain field and the development of strain loalization patterns. For the drained ompression previously disussed, the sample is initially in a dense state and as shown in Fig., shear bands develop when the limit stress state is approahed, indifferently for an axial stress or an axial strain ontrol of the loading. Analytially, shear bands ourrene are deteted with the Rie's riterion orresponding to the vanishing of the determinant of the aousti tensor [5]. Consequently, the ourrene of failure haraterized by strain loalization pattern in shear bands will be deteted along a given loading path when both riteria (seond-order work for failure and Rie's riterion for shear band) are verified. Note that the effetive ourrene of failure along the onsidered loading path depends on the mode of ontrol of the loading (here for the drained ompression an axial stress ontrol is neessary). However, limit states and failure an also be assoiated with homogeneous strain fields as shown in Fig. 3, where inremental deviatori strain fields are displayed for a medium dense D partile assembly subjeted to an undrained (isohori) biaxial ompression. For suh a loading path the seond-order (a) (b) FIGURE 1. Responses of the disrete element model to a drained biaxial ompression axially strain ontrolled (a), or stress ontrolled (a & b). work writes W = V dq dε 1 and thus vanishes at the peak of the stress deviator q, orresponding to a limit state from whih failure an effetively develops if the stress deviator q is the ontrol parameter. However strain field stays rather homogeneous after the peak of q, and we an imagine that Rie's riterion never holds along this loading path for this granular assembly. LOCAL SECOND-ORDER WORK We onsider a homogeneous volume V o of granular material omprised of N grains. The shape of eah grain p is arbitrary. The total number of ontats at time t within the assembly is denoted N. The Lagrangian formulation given in Eq. (1) an be readily differentiated, then providing the following expression of the seond-order work (see [6] for more details): W = f l + f x () where ontating partiles, p p i i i i p V l is the branh vetor relating the entres of f is the ontat fore between 586

4 FIGURE. Inremental deviatori strain fields omputed at states numbered from 1 to 3 in Fig. 1, for the axially strain ontrolled loading (top) and stress ontrolled one (bottom); olor represents the intensity of inremental deviatori strain. FIGURE 3. Stress response path for an undrained biaxial ompression (top), and inremental deviatori strain fields omputed at states numbered from 1 to 3 (bottom) ontating partiles, and f p denotes the resultant fore applied to the partile p of position x p. As speified in [6], the reation or the deletion of ontats is aounted for in this approah. The symbol denotes the summation over p and q varying over [ ] 1, N with q p, and refers to the ontating pair (p, q). When no ontat exists between partiles p and q, f is set to zero. It is worth noting that in the absene of inremental unbalaned fore and in quasi-stati regime, Eq. () simplifies into: i i (3) W = f l The validity of this relation has been numerially heked for 3D granular assemblies in axisymmetri onditions. After a first triaxial loading path under a onfining pressure of 100 kpa, different strain probes were simulated from the same loading point (orresponding to a deviatori ratio = 0.48 ). The strain probes have the same amplitude (10-4 ), and are haraterized by different orientations in the Renduli strain plane. The stress response is omputed for eah orientation and W is dedued from Eq. (1). Likewise, the quantity fi li an also be omputed, whih allows the validity of Eq. (3) to be assessed. As seen in Fig. 4, an exellent agreement between both expressions of the seond-order work with miro and maro variables is obtained. Equation (3) expresses the internal seond-order work from miromehanial variables, namely the ontat fores existing between ontating granules, and the branh vetors joining these granules. The attempt of suh a formulation is to go down to the mirosopi sale, to try to eluidate what are the basi mirostrutural origins giving rise to the vanishing of the internal seond-order work, and therefore what are the mirostrutural ontexts prone to instabilities. To progress along this line, we plotted for the drained biaxial ompressions presented in previous setion the spae distribution of inter-partile ontats - where the loal seond-order work ( fi l i ) is negative in Fig. 5, for both axially strain and stress ontrolled loading paths. These distributions an be ompared with the inremental deviatori strain fields in Fig.. The patterns of these distributions are very similar, - ontats are onentrated in the shear bands (as shown by the theory) where failure ours and are sparsely distributed outside the shear band where the 587

5 FIGURE 4. Marosopi and mirosopi expressions of the seond-order work. material is unloading. Inversely, - ontats for the undrained loading path (presented in Fig. 3) stay homogeneously distributed, even after the point of potential failure ourrene (q peak) as shown in Fig. 6. CONCLUSION Effetive failure an be desribed as the bifuration of the response of the granular assembly from a quasistati one, before failure initiation, to a dynami one during failure development. Along a given loading path, the ourrene of this bifuration (and thus of failure) depends on the mode of ontrol of the granular assembly. Possible points of bifuration are deteted with the seond-order work riterion. This later an be equivalently expressed in the framework of ontinuum mehanis or in the disrete miro-mehanis framework. Both expressions give at the sale of the granular assembly an idential information about the possibility of failure ourrene. Nevertheless the loal expression of the seond-order work riterion seems to give a riher information, sine the spatial distribution of inter-partile ontats satisfying this riterion is apparently diretly related to the mode of failure (loalized or diffused). However, additional statistial analysis need to be performed to improve the understanding of potential links between failure mode and loal seond-order work. ACKNOWLEDGMENTS The authors would like to express their sinere thanks to the Frenh Researh Network MeGe (Multisale and multi-physis ouplings in geoenvironmental mehanis GDR CNRS 3176, ) for having supported this work. FIGURE 5. Fields of - ontats for the drained biaxial ompression axially strain ontrolled (top) and stress ontrolled (bottom). FIGURE 6. Fields of - ontats for the undrained biaxial ompression presented in Fig. 3. REFERENCES 1. Yade: Open soure disrete element method. F. Niot, L. Sibille, F. Darve, Int. Journal of Plastiity 9, (01). 3. F. Darve, G. Servant, F. Laouafa, H. Khoa, Computer Meth. Appl. Meh. Eng. 85 (7-9), (004). 4. F. Darve, L. Sibille, A. Daouadji, F. Niot, Comptes Rendus Meanique 335, (007). 5. J.R. Rie, The loalization of plasti deformation, in Theoretial and Applied Mehanis, edited by W.T. Koiter, Delft: North-Holland Publishing Compagny, 1976, pp F. Niot, N. Hadda, F. Bourrier, L. Sibille, R. Wan, F. Darve, Int. J. of Solids and Strutures 49 (10), (01). 588