Discrete Element Modelling in Granular Mechanics WUHAN seminars, China, 28th august-2nd september 2017

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1 Discrete Element Modelling in Granular Mechanics WUHAN seminars, China, 28th august-2nd september 2017 F. DARVE, L. SIBILLE, F. DUFOUR Laboratoire Sols, Solides, Structures, Risques, INPG-UGA-CNRS, Grenoble, France ALERT, INDURA F. NICOT, IRSTEA Grenoble, ALERT, INDURA R. WAN, J. DURIEZ, Calgary University, Canada L. SCHOLTES, Nancy, Université de Lorraine I Discrete element modelling for partially saturated granular media 1 The proposed methodology for partially saturated conditions 2 Critical revisit of the expression of the total stress tensor II Discrete element modelling of failure and post-failure in granular media 3 The principles of modelling 4 A direct numerical analysis of failure phenomenon by a discrete model

2 Capillarity in partially saturated granular media ( Madeo A., Dell Isola F., Darve F., J.M.P.S., 2013) Gaz (air Solid grain Liquid (water) hygroscopic pendular funicular capillary At low water content levels interfacial phenomena lead to intergranular water menisci Sr 2

3 Capillarity in partially saturated granular media Capillary theory (Laplace): C = f y x Δu = u gas u liquid ; y(x) is the interface profile = C F capillary = 2π y 0 +πδu 2 V = π y meniscus xdx. y 2 0 3

4 DEM Modelling (L. Scholtes et al., IJES, 2009) Evolution of capillary force at constant suction Local Hysteresis F cap D 4

5 DEM Modelling using YADE - open DEM ( Resultant forces and moments Force-Displacement Law applied to each interaction Capillary forces F cap Law of Motion (Newton's 2 nd Law) applied to each particle k n k t cont Positions and contacts update (finite difference scheme) 5 5

6 DEM Simulations (L. Scholtes et al., IJES, 2009) DEM Sample : 1, 1 σ 2 σ spherical particles randomly positionned into a cubic box A unique homogeneous value of succion in the sample (thermodynamic equilibrium) compacted through radius expansion to ensure the initial isotropy of the packing rigid frictionless boundary walls guarantee the homogeneity of the loading 6

7 DEM Simulation: macroscale results Water retention hysteresis: Sr = N meniscus m= 1 V V meniscus void Local hysteresis «ink bottle effect» 7

8 DEM Simulation: macroscale results Dry Sample 8

9 DEM Simulation: macroscale results Wet Sample For several capillary pressures u in the pendular regime (0 < Sr < 12%) u a u w 9

10 DEM Simulation: macroscale results Plastic limit surfaces in the (q,p) plane No significant change of the internal friction angle Cohesion vs saturation degree Compares well with experiments done in Montpellier [Richefeu and al., 2007] for <D> = mm c DEM = 5 kpa 10

11 Effective Stress (L. Scholtes et al., IJNAMG, 2009) The «effective» intergranular mechanical behaviour of granular materials should not be affected by the presence of water Effective stress tensor in saturated granular materials [Terzaghi 1936 ]: σ : total stress u : pore pressure σ' ij = σ ij u ij 11

12 Generalized Effective Stress Partial saturation : [Bishop and Blight, Géotechnique, 1963] σ ij '= σ ij u a + χ u a u w δ ij? 2 questions : 1 are the stresses additive? 2 - is the capillary stress isotropic? 12

13 Generalized Effective Stress Plastic limit surfaces in the (p',q') plane : χ = Sr χ = u u a a u u w w 0.55 [Khalili and Khabbaz, Géotechnique, 1998] None of the common definitions results in a unique plastic limit surface... 13

14 Generalized Effective Stress Micromechanical interpretation: On each particle of the assembly : F = F + cont F cap Cauchy stress tensor by homogenisation : [Love, 1927] σ ij = 1 V N contacts c= 1 F cont i l j 1 + V N menisci m= 1 F cap i l j σ = σcontact + σ capillary 14

15 Generalized Effective Stress Micromechanical interpretation: verification through DEM simulation σ boundary ij = σ cont ij + σ cap ij boundary σ ij cont σ ij cap σ ij cont cap σij σij 15

16 Generalized Effective Stress Micromechanical definition: Definition of the effective stress : σ contact ij 1 = V N contacts c= 1 The «effective» plastic limit surface in the (p contact, q contact ) plane is now unique : F contact i l j 15

17 Generalized Effective Stress Micromechanical definition: Bishop's formalism fails to describe the anisotropy of the fluid contribution σ contact ij = σ ij σ capillary ij σ ij '= σ ij + χ u a u w δ ij k κ = 2 κ = 0 if σ σ cap 1 cap 1 σ cap σ +σ cap 2 cap 2 = χ u a u w 16

18 Generalized Effective Stress (L. Scholtes et al., IJNAMG, 2009 Duriez J., Eghbalian M., Wan R., Darve F., J.M.P.S., 2017) Anisotropy of the water/air contribution Contacts and menisci orientation distributions 1 17

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22 Discrete analysis of bifurcation from D.E.M. computations (Sibille et al.,ijnamg, Acta Geotech., 2008 Tordesillas et al., Phil. Mag., 2012 Sibille et al., JMPS, 2015) The Discrete Element Model :. SDEC / YADE software (Donzé & Magnier 1997): molecular dynamics approach such as Cundall s one (1979).. Contact interaction defined by 3 mechanical parameters: kn / Dsphere 356MPa / 0.42 k t k n 35 cont k n k t cont. Cubic form of the specimen; spheres; continuous size distribution; dense specimen: 2D 9mm sphere For p = 100 kpa: n All paths in principal stress ( 1, 2, 3 ) or strain ( 1, 2, 3 ) spaces can be simulated.

23 Discrete analysis of stability from D.E.M. computations (Sibille et al., 2007) Stress probes : The initial axi-symmetric stress states: 1. isotropic compression (100, 200 or 300 kpa), 2. triaxial drained compression ( 2 = 3 = cst.) characterised by: q p The loading program: defined in the Rendulic plane of stress increments, d d1 2d 3 cte 1kPa 0 a d < 360 response vectors d defined in dual plane ( 2 d, d). 2 2 d W d W ad 2 2 ( ) 3 1

24 Discrete analysis of stability from D.E.M. computations (Sibille et al., 2007) Unstable directions : diagram of 2 f ( ad ) d Wnorm. c with: 2 dwnorm. d d d d c such as: f ( a ) 0 d 3 = 100 kpa 3 = 200 kpa Cones of instability" stress directions observed with the D.E.M.

25 Discrete analysis of stability from D.E.M. computations Cones of unstable stress directions : D.E.M. and Incremental Non-Linear model Discrete Element Model d²w > 0 Cones of unstable directions (L. Sibille) Macroscopic phenomenological relation (I.N.L.2 model) calibrated on the drained triaxial responses of D.E.M. (L.Scholtes).

26 From the bifurcation point to failure ( Sibille et al., IJNAMG,2007) ( I ) INFLUENCE OF THE CONTROL/LOADING PARAMETERS The control parameters can be linear combinations of stresses or strains (e.g. volumetric strains). Loose specimen, h = 0.46, a = deg (q = cst ) Stress probe is fully stress controlled no failure observed. Can we choose others control parameters? d²w = d 1 d d 3 d 3 = dq d 1 + d 3 d v Can we control the loading programme defined by: dq = 0 and d v = % (imposed dilatancy rate)? a = f d W c 2 ( a) norm.

27 Stresses: 1 3 (kpa) Strains: 1 3 (%) Kinetic energy (J) Comparison of the response of the numerical specimen controlled by: dq = 0 and d 3 < 0 or by dq = 0 and d v < 0 Loss of controllability for dq = 0 and d v < 0, the specimen totally collapses. dq = 0, d 3 < 0 Simulation time (s) dq = 0, d 3 < 0 dq = 0, d 3 < 0 dq = 0, d 3 < 0 Simulation time (s) Simulation time (s)

28 From the bifurcation point to the failure ( Sibille et al., IJNAMG,2007) ) ( II ) EXISTENCE OF INSTABILITY CONES 2 d W d1 d1 2d 3d3 2 d3 d3 d W d1d1 d1 2Rd3 R R with R = cst. for a given loading path. Can we control the loading programme defined by: d 1 - d 3 / R =0 and d 1 + 2R d 3 = %? with: R = 1.94 (a = 200, d²w > 0) R = 1.22 (a = 210, d²w < 0) R = 1.00 (a = 215.3, d²w < 0) R = (a = 220, d²w < 0) R = (a = 230, d²w < 0) R = (a = 240, d²w > 0) f d W c 2 ( a) norm.

29 From the bifurcation point to failure The loading programme is controllable for R = 1.94 and (d²w > 0). Loss of controllability for R = 1.00; 0.843; (d²w < 0). For R = 1.22 (d²w < 0 but stress direction close to the border of the cone) : the loss of controllability is not total. Simulation time (s) Simulation time (s)

30 From the bifurcation point to the failure ( Sibille et al., IJNAMG,2007) For R = 1.94 and (d²w > 0) a new stable state close to the initial one is reached. Total collapse for R = 1.00; 0.843; (d²w < 0). For R = 1.22 (d²w < 0 but stress direction close to the border of the cone ) the collapse is partial. Simulation time (s) Simulation time (s) Loading parameters exist such as the specimen collapses from a bifurcation point detected by the sign of d²w.

31 From the bifurcation point to the failure ( III ) INFLUENCE OF A PERTURBATION (Notion of loss of sustainability, Nicot et al., IJSS, 2009). Control of the mechanical state by: d 1 - d 3 / R =0 and d 1 + 2R d 3 = 0 with: R = 4.01 (a = 190, d²w > 0) R = 1.94 (a = 200, d²w > 0) R = 1.22 (a = 210, d²w < 0) R = 1.00 (a = 215.3, d²w < 0) R = (a = 220, d²w < 0) R = (a = 230, d²w < 0) R = (a = 240, d²w > 0) R = (a = 250, d²w > 0) 2 f ( a) d Wnorm. c Small perturbation of the numerical specimen External input of kinetic energy ( J) to the specimen by excitements of some floating grains (simulation without gravity).

32 Kinetic energy (J) From the bifurcation point to the failure For R = 4.01; 1.94; 0.408; (d²w > 0) a new stable state is reached. Total collapse for R = 1.00; 0.843; (d²w < 0). For R = 1.22 (d²w < 0 but stress direction close to the border of the cone ) the collapse is partial. Simulation time (s) Simulation time (s)

33 CONCLUSIONS 1 partially saturated granular media : From a methodological point of view, DEM is able to describe properly the hydromechanical coupling through the capillary forces. Analytical homogenisation techniques and numerical DEM have given fully consistent results showing ( in particular) the tensorial nature of the capillary stress, which is here firmly established for the first time. 2 failure and post-failure of granular media : From again a methodological point of view, DEM is also able to describe the intricate features of failure behaviour of granular media as plastic non-associate materials. The single plastic limit surface has to be replaced by a bifurcation domain with instability cones. Effective failure is governed by 3 necessary and sufficient conditions: bifurcation criterion (det Ms < 0), second order work criterion ( d2w < 0) and controllability criterion. ( see a list of reference papers on Google Scholar «Darve F.»)

34 DEM simulation of failure for an undrained compression Triggering of failure after the peak of q (Nicot et al., Gran.Matter,2011 IJP, 2012) Relation between D2W and kinetic energy verified until t 0.08 s after the failure triggering.