We use a 3D Lattice Element Method, based on the discretization of the particles and binding matrix on a regular lattice, to investigate the particle-scale origins of the strength and failure of porous granular aggregates under tensile loading. Damage growth is analyzed by considering the evolution of stress probability density and the number of broken bonds in the particle phase. We show that the stress probability density functions are increasingly broader for a decreasing matrix volume fraction, the stresses being more
and more concentrated in the interparticle contact zones with an exponential distribution as in cohesionless granular media [4]. We carried out a detailed parametric study in order to evaluate the combined influence of the matrix volume fraction and particlematrix adherence. Our findings are in agreement with 2D results previously reported in the literature [6]. Three regimes of crack propagation are evidenced, corresponding to
no particle damage, particle abrasion and particle fragmentation, respectively. The crack morphology (tortuosity...) is another important feature that we investigate for different distributions of the particles and pores within porous granular aggregates

Affes, R.

Delenne, J-Y.

Monerie, Y.

Radja, F.

Topin, V.

English

UPCommons. Portal del coneixement obert de la UPC

II International Conference on Particle-based Methods - Fundamentals and ApplicationsPARTICLES 2011E. Oñate and D.R.J. Owen (Eds)FAILURE IN POROUS GRANULAR AGGREGATESR. Affes∗+, V. Topin++, J-Y. Delenne+, Y. Monerie++ AND F. Radja+++LMGC CNRS-Universit Montpellier II, Place Eugne Bataillon, 34095 Montpellier Cedex++IRSN, DPAM, CE Cadarache, Bat. 702, BP3-13115 St. Paul-lez-Durance Cedex+,++ Laboratoire MIST, IRSN-CNRS-Universit Montpellier II, France∗ rafik.affes@univ-montp2.frKey words: Porous granular aggregates, Lattice model, Fracture, Crack path, TortuosityAbstract. We use a 3D Lattice Element Method, based on the discretization of the parti-cles and binding matrix on a regular lattice, to investigate the particle-scale origins of thestrength and failure of porous granular aggregates under tensile loading. Damage growthis analyzed by considering the evolution of stress probability density and the number ofbroken bonds in the particle phase. We show that the stress probability density functionsare increasingly broader for a decreasing matrix volume fraction, the stresses being moreand more concentrated in the interparticle contact zones with an exponential distribu-tion as in cohesionless granular media [4]. We carried out a detailed parametric studyin order to evaluate the combined influence of the matrix volume fraction and particle-matrix adherence. Our findings are in agreement with 2D results previously reported inthe literature [6]. Three regimes of crack propagation are evidenced, corresponding tono particle damage, particle abrasion and particle fragmentation, respectively. The crackmorphology (tortuosity...) is another important feature that we investigate for differentdistributions of the particles and pores within porous granular aggregates.1 IntroductionDense granular materials are characterized either in terms of the network of solidparticles or by the properties of the pore space which can be fully or partially filled bya solid binding matrix or a liquid. At high particle volume fractions ρp (typically, forρp > 0.57), the stress transmission is basically guided by a percolating network of inter-particle contacts [5]. This role of the contact network in force transmission and rheologicalbehavior has been mainly investigated in granular materials in the absence of a bindingmatrix and under compressive confining stresses [2, 4].1295First A. Author, Second B. Author and Third CoauthorThe issue of stress concentration and the role of particles are much less evident in thepresence of a binding matrix and under tensile loading. Such porous granular aggregateshave been recently studied in some detail in a 2D geometry by means of numerical simula-tions [1, 6]. The overall stiffness and tensile strength of these materials are dependent onthe matrix volume fraction ρm, particle volume fraction ρp and particle-matrix adhesionσpm.In this paper, we introduce the lattice element method (LEM) in a 3D geometry forthe simulation of porous granular aggregates of spherical particles with a solid matrix.Based on a lattice discretization of both phases and their interface as well as an efficientquasi-static time-stepping scheme, the LEM algorithm allows us to analyze the fractureof cohesive aggregates as a function of phase volume fractions and local binding strength.2 Lattice Element MethodThe lattice element method (LEM) has been recently employed as an alternative to thefinite element method for the investigation of the fracture properties of granular materialsmixed with a binding matrix [1, 6]. Such materials, to which we refer in this paper asporous granular aggregates or cemented granular materials can be found in very differentforms in nature and industry. Well-known examples are conglomerates and concrete.In LEM, the space is discretized as a regular or disordered grid of points (nodes) in-terconnected by one-dimensional elements (bonds). Each bond can transfer normal force,shear force and bending moment up to a threshold in force or energy. Various rheologicalbehaviors can be carried by these material lattice bonds, in contrast to the finite elementapproach where the local behavior is carried by volume elements. When several phasesare present as in a porous granular aggregates, each phase and its boundaries are materi-alized by lattice elements sharing the same properties and belonging to the same portionof space. We use linear elastic-brittle elements, each element characterized by a Hookeconstant and a breaking force threshold. The bonds transmit only normal forces betweenthe lattice nodes and thus the strength of the lattice in shear and distortion is ensuredonly by the high connectivity of the nodes. This simple kinematics allows to investigatehigh sampling statistical approch. A sample is defined by its contour and the configu-ration of the phases in space. The samples are deformed by imposing displacements orforces to nodes belonging to the contour. The initial state is the reference (unstressed)configuration. The total elastic energy of the system is a convex function of node dis-placements and thus finding the unique equilibrium configuration of the nodes amounts toa minimization problem (implemented here by means of the conjugate gradient method).Performing this minimization for stepwise loading corresponds to subjecting the systemto a quasi-static deformation process. The overloaded elements (exceeding a threshold)are removed according to a breaking rule. This corresponds to irreversible micro-crackingof the lattice. The released elastic energy between two successive equilibrium states isthus fully dissipated by micro-cracking. In the fast implementation used in the presentwork, all overloaded elements occurring within the same step are removed, as well as those2296First A. Author, Second B. Author and Third CoauthorParticleMatrixParticlematrixInterfaceFigure 1: Example of a discretized aggregate involving particles, matrix, voids and particle-particle andparticle-matrix interface zones.appearing recursively after energy minimization (within the same step). This correspondsphysically to unstable growth of the micro-cracks compared to the imposed strain rate.The 3D LEM has the advantage to be cheap in computational effort, making it pos-sible to simulate systems with an large number of nodes for reasonable computing time.It should be remarked that due to the simple additivity of the potential energy, the com-putation time depends only linearly on the number of nodes. It is also obvious that theLEM is a convenient model of brittle fracture in which the generation and propagation ofcracks are “naturally” taken into account.3 Application to granular aggregatesIn a granular aggregate, there are three bulk phases: particles, matrix and voids.There are also two interface phases: particle-particle and particle-matrix; see Fig. 1. Toconstruct the samples, we first generate a large dense packing of rigid spherical particlescompressed isotropically by means of the contact dynamics method. A cubic portionof this three-dimensional packing is overlaid on a disordered tetrahedral lattice. Theparticle properties are attributed to the bonds falling in the bulk of the particle phase.The binding matrix is then added in the form of bridges of variable width connectingneighboring particles within a prescribed gap between particles. The bonds belonging tothese bridges are given the properties of the matrix. In the same way, the bonds fallingbetween a particle and the matrix or between two particles are given the properties of thecorresponding interface. The width of solid bridges between particles is proportional tothe total volume of the binding material. At higher levels of the matrix volume fraction,the bridges overlap and the porosity declines to zero. The particles are polydisperse with3297First A. Author, Second B. Author and Third Coauthorρρε0 0.05 0.1 0.15 0.2 0.250.0050.010.0150.02σσFigure 2: Normalized vertical stress as a function of vertical strain in tension for two values of the matrixvolume fraction ρm (in %).diameters varying nearly uniformly is size in a range [0.8d, d]. The total particle volumefraction is about 0.6 corresponding to a dense close packing. The samples consist of thebulk phases: 1) particles, denoted p; 2) matrix, denoted m; and 3) void space or pores,denoted v, as well as the interface phases: 1) particle-particle interface, denoted pp, and2) particle-matrix interface, denoted pm. The elements belonging to each phase φ (bulkor interface) are given a Hooke constant kφand a breaking force fφ. We have f v = 0 andthe choice of the value of kv is immaterial. The interface phases pm and pp are transitionzones of finite width. But for large systems, the volume fractions of these transition zonesare negligible compared to those of the particles and matrix. The interface phases affectthe global behavior through their specific surface and their strengths represented by theHooke constants kpp and kpm and the corresponding tensile force thresholds f pp and f pm.In our simulations, we model the interface phases by a one bond-thick layer linking twoparticles or a particle to the matrix. The volume fractions of the interface phases are thusassumed to be zero (ρpp = ρpm = 0) and the volume fractions ρp, ρm and ρv are attributedonly to the three bulk phases, with ρp + ρm + ρv = 1. It is dimensionally convenient toexpress the bond characteristics in stress units. We thus define the bond breaking (ordebonding) stresses σφ = fφa2and the moduli Eφ = kφawhere a is the length of the latticevector. These bond moduli Eφ of the lattice should be carefully distinguished from theequivalent phase moduli which depend both on the bond moduli and the geometry ofthe lattice. We will use below square brackets to represent the phase moduli: E[p], E[m],E[pp] and E[pm]. It can be shown that the overall Young modulus and Poisson ratio of andisordered isotropic tetrahedral lattice are Eeff = 54√2Eφ and νeff = 0.25. We performeda serie of simple tension tests over samples composed of 516 particles. The particle volumefraction was kept constant ρp = 0.6, and ρm was varied from ρp/10 to 4ρp. Each sample4298First A. Author, Second B. Author and Third Coauthor    ρρρσ /σFigure 3: Probability densities of normalized vertical node stresses σizz for three values of the matrixvolume fraction ρm (in %).was discretized over a lattice containing about 1.5 × 106 elements. The results presentedbelow were obtained for hard particules Ep = 3Em, σp = σm and σpp = 0. The cubicsamples were subjected to uniaxial tension with free lateral sides. The nodes belonging tothe base were constrained to be immobile. Upward step-wise displacements were appliedto the nodes belonging to the upper surface.Fig. 2 shows the stress-strain plot under for ρm = 28% and ρm = 13%. We observea brittle behavior with a well-defined initial stiffness Eeff and a tensile strength σeff atthe stress peak. The post-peak behavior is characterized by nonlinear propagation of themain crack (initiated at the stress peak) in the form of a sequence of loading-unloadingevents. The stiffness declines due to progressive damage of the aggregate. The overalltensile strength is higher at larger ρm as a result of a weaker concentration of stresses.The probability distribution functions of vertical node stresses σzz are shown in Fig. 3.From the shapes of the pdf’s, we distinguish large stresses falling off exponentially asobserved for large contact forces in granular media [2, 4]. The weak stresses have nonzeroprobability (increasing as σzz → 0) reflecting the arching effect whereas intermediatestresses are centered on the mean and define a nearly Gaussian distribution. The largestresses mostly concentrate at the contact zones and they form well-defined chains thatcross the particles.The tensile strength and crack propagation are controlled by both ρm and σpm. For aquantitative evaluation of this effect, we consider here the proportion nb of broken bondsinside the particles with respect to the total number of broken bonds. Fig. 4 shows a mapof nb in the parameter space (ρm, σpm) following failure. We see that below a well-definedfrontier, no particle damage occurs (nb ≃ 0). For this range of parameter values, the crackspropagate either in the matrix or at the particle-matrix interface. Above this “particle-5299First A. Author, Second B. Author and Third Coauthor   ρ  σ      / σFigure 4: Grey level map of the fraction of broken bonds in the particle phase for different values ofmatrix volume fraction and particle-matrix adhesion.damage” limit, the isovalue lines become nearly parallel to the limit line with an increasinglevel of nb. This suggests three distinct regimes of crack propagation: 1) below theparticle-damage limit, the cracks bypass the particles and propagate through the matrix,the pores or along the particle-matrix interface; 2) above this limit and for ρm < 20, thecracks penetrate into the particles from solid bridges that strongly concentrate stresses;3) Above this limit, the cracks propagate inside the matrix as well as across the particles,causing the fragmentation of the particles. These results are qualitatively similar in 2Dcohesive granular aggregates [6].4 TORTUOSITYIn order to understand the role of disorder in granular aggregates on the overall crackpath, we investigate the effects of size polydispersity and particle volume fraction on thecrack tortuosity. A commonly-used estimate of tortuosity can be obtained as the ratioof the length of the crack path to the distance between its end points. This measure isa dimensionless number greater than 1 and it can be applied to a 2D or 3D crack pathFig.5.6300First A. Author, Second B. Author and Third CoauthorFigure 5: Measures of curve tortuosityWe first conducted a parametric study in 2 dimensions on square samples under tensileloading simulated by LEM. The particle volume fraction ρp varies from 0.35 to 0.55 andthe polydispersity varies with the size ratio r = dmax/dmin between the largest and small-est particle sizes dmax and dmin for each configuration of ρp, from 1 (monodisperse) to 7.In order to get sufficient statistics, five different samples for each set of parameters aregenerated using a deal-leave process ensuring the control of the particule volume fractionand size polydispersity.The main elastic parameters of the matrix and the aggregate are those of ordinaryconcrete. The interface between aggregate and mortar is known to have a remarkableeffect on the tensile properties of concrete [3] since it constitutes a preferential locus ofthe failure. To preserve this property, the stiffness of the aggregate/matrix and the ag-gregate/aggregate interfaces are assumed to be two times less than that of the matrixThe LEM simulation provides the evolution of fracture as a sequence of 3D images. Adedicated image processing algorithm was developed to extract the crack path from theseimages and to analyze its morphological properties Fig.6.7301First A. Author, Second B. Author and Third CoauthorFigure 6: Image processing: (a) extracting the crack from sample, (b) binarized crack, (c) skeletonizedcrack, (d) extraction of largest shortest path (in red)The evolution of the tortuosity for a given particle volume fraction and three values ofsize ratio r is displayed in Fig. 7. The mean tortuosity increases with polydispersity inthe samples: since the crack is locally reoriented by the boundaries of the aggregates, theapparent tortuosity obtained in a mono-disperse configuration is reproduced at variousscales when the polydispersity increases.8302First A. Author, Second B. Author and Third Coauthor0 1 2 3 4 5 6r = dmin/dmax1.21.251.31.351.4TortuosityAverage valuesFigure 7: Evolution of tortuosity for three values of size ratio rFig 8 displays two snapshots after the appearance of the cracks for the monodispersesample (a) and sample with r=5 (b).In Fig 8(a) the crack passes through the samplebetween the particles without being deflected from its trajectory. On the other handin Fig.8(b), the crack is clearly deviated from its trajectory to reach the large particles,increasing thus the tortuosity.9303First A. Author, Second B. Author and Third Coauthor Figure 8: Two snapshots after the appearance of the cracks for the monodisperse sample (a) and samplewith r=5 (b)In a forthcoming work, this result will be confirmed in the three dimensional case, andthe connection between microstructure and the overall permeability will be investigated.5 ConclusionIn this paper, a lattice-based discretization approach (lattice element method) was in-troduced and illustrated by application to the brittle failure of porous granular aggregates.In contrast to dilute particle-reinforced composites, such materials involve a high level ofparticle volume fraction and thus a jammed skeleton of solid particles interconnected viaa binding matrix. The overall behavior depends on the bulk phase volume fractions andthe properties of the particle-particle and particle-matrix interface zones. We found thatthe presence of the particle skeleton controls stress concentration and thus the strengthproperties of these materials. It was also shown that for a range of the values of theparticle-matrix adhesion and matrix volume fraction, no particle damage occurs. Thetrends are very similar to those previously established for 2D aggregates by the samemodel.REFERENCES[1] J. G. M. Van Mier, M. R. A. Van Vliet, and T. K. Wang. Fracture mechanisms inparticle composites: statistical aspects in lattice type analysis. Mechanics of Materials,34(11):705–724, 2002.10304First A. Author, Second B. Author and Third Coauthor[2] D. M. Mueth, H. M. Jaeger, and S. R. Nagel. Force distribution in a granular medium.Phys. Rev. E, 57:3164, 1998.[3] A.M. Neville. Properties of concrete. London, 1981.[4] F. Radjäı, M. Jean, J.-J. Moreau, and S. Roux. Force distribution in dense two-dimensional granular systems. Phys. Rev. Lett., 77(2):274, 1996.[5] M. Satake and J. T. Jenkins. Micromechanics of granular materials. Elsevier, Ams-terdam, 1988.[6] V. Topin, J.-Y. Delenne, F. Radjäı, L. Brendel, and F. Mabille. Strength and fractureof cemented granular matter. The European Physical Journal E, 23:413–429, 2007.11305

Failure in porous granular aggregates

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Failure in porous granular aggregates

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