×

zbMATH — the first resource for mathematics

A nonlocal multiscale discrete-continuum model for predicting mechanical behavior of granular materials. (English) Zbl 1352.74082
Summary: A three-dimensional nonlocal multiscale discrete-continuum model has been developed for modeling mechanical behavior of granular materials. In the proposed multiscale scheme, we establish an information-passing coupling between the discrete element method, which explicitly replicates granular motion of individual particles, and a finite element continuum model, which captures nonlocal overall responses of the granular assemblies. The resulting multiscale discrete-continuum coupling method retains the simplicity and efficiency of a continuum-based finite element model, while circumventing mesh pathology in the post-bifurcation regime by means of staggered nonlocal operator. We demonstrate that the multiscale coupling scheme is able to capture the plastic dilatancy and pressure-sensitive frictional responses commonly observed inside dilatant shear bands, without employing a phenomenological plasticity model at a macroscopic level. In addition, internal variables, such as plastic dilatancy and plastic flow direction, are now inferred directly from granular physics, without introducing unnecessary empirical relations and phenomenology. The simple shear and the biaxial compression tests are used to analyze the onset and evolution of shear bands in granular materials and sensitivity to mesh density. The robustness and the accuracy of the proposed multiscale model are verified in comparisons with single-scale benchmark discrete element method simulations.

MSC:
74E20 Granularity
76T25 Granular flows
Software:
DEMPack
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Borst, A generalisation of J2-flow theory for polar continua, Computer Methods in Applied Mechanics and Engineering 103 (3) pp 347– (1993) · Zbl 0777.73014 · doi:10.1016/0045-7825(93)90127-J
[2] Vermeer, Non-associated plasticity for soils, concrete and rock, Heron 29 (3) pp 1– (1984)
[3] Kingston, General yield conditions in plane deformations of granular media, Journal of the Mechanics and Physics of Solids 18 (3) pp 233– (1970) · doi:10.1016/0022-5096(70)90026-8
[4] Pestana, Formulation of a unified constitutive model for clays and sands, International Journal for Numerical and Analytical Methods in Geomechanics 23 (12) pp 1215– (1999) · Zbl 0971.74054 · doi:10.1002/(SICI)1096-9853(199910)23:12<1215::AID-NAG29>3.0.CO;2-F
[5] Pestana, Evaluation of a constitutive model for clays and sands: Part I-sand behaviour, International Journal for Numerical and Analytical Methods in Geomechanics 26 (11) pp 1097– (2002) · Zbl 1008.74521 · doi:10.1002/nag.237
[6] Assimaki, Model for dynamic shear modulus and damping for granular soils, Journal of Geotechnical and Geoenvironmental Engineering 126 (10) pp 859– (2000) · doi:10.1061/(ASCE)1090-0241(2000)126:10(859)
[7] Wang, Bounding surface hypoplasticity model for sand, Journal of engineering mechanics 116 (5) pp 983– (1990) · doi:10.1061/(ASCE)0733-9399(1990)116:5(983)
[8] Manzari, A critical state two-surface plasticity model for sands, Geotechnique 47 (2) pp 255– (1997) · doi:10.1680/geot.1997.47.2.255
[9] Lade, Elastoplastic stress-strain theory for cohesionless soil, Journal of the Geotechnical Engineering Division 101 (10) pp 1037– (1975)
[10] Drucker, Soil mechanics and plastic analysis or limit design, Quarterly of Applied Mathematics 10 (2) pp 157– (1952) · Zbl 0047.43202 · doi:10.1090/qam/48291
[11] Rudnicki, Conditions for the localization of deformation in pressure-sensitive dilatant materials, Journal of the Mechanics and Physics of Solids 23 (6) pp 371– (1975) · doi:10.1016/0022-5096(75)90001-0
[12] Borst, Fundamental issues in finite element analyses of localization of deformation, Engineering Computations 10 (2) pp 99– (1993) · doi:10.1108/eb023897
[13] Rechenmacher, Evolution of force chains in shear bands in sands, Geotechnique 60 (5) pp 343– (2010) · doi:10.1680/geot.2010.60.5.343
[14] Cundall PA A computer model for simulating progressive large scale movements in blocky rock systems Proc. Symp. Rock Fracture (ISRM) Nancy 2013 2 8
[15] Cundall, A discrete numerical model for granular assemblies, Geotechnique 9 (1) pp 47– (1979) · doi:10.1680/geot.1979.29.1.47
[16] Kouznetsova, Multi-scale constitutive modelling of heterogeneous materials with a gradient-enhanced computational homogenization scheme, International Journal for Numerical Methods in Engineering 54 (8) pp 1235– (2002) · Zbl 1058.74070 · doi:10.1002/nme.541
[17] Geers, Multi-scale computational homogenization: trends and challenges, Journal of computational and applied mathematics 234 (7) pp 2175– (2010) · Zbl 1402.74107 · doi:10.1016/j.cam.2009.08.077
[18] Wellmann, Homogenization of granular material modeled by a three-dimensional discrete element method, Computers and Geotechnics 35 (3) pp 394– (2008) · doi:10.1016/j.compgeo.2007.06.010
[19] Bardet, A numerical investigation of the structure of persistent shear bands in granular media, Geotechnique 41 (4) pp 599– (1991) · doi:10.1680/geot.1991.41.4.599
[20] Wellmann, A two-scale model of granular materials, Computer Methods in Applied Mechanics and Engineering 205 pp 46– (2012) · Zbl 1239.74083 · doi:10.1016/j.cma.2010.12.023
[21] Li, A bridging scale method for granular materials with discrete particle assembly-Cosserat continuum modeling, Computers and Geotechnics 38 (8) pp 1052– (2011) · doi:10.1016/j.compgeo.2011.07.001
[22] Regueiro, Bifurcations, Instabilities and Degradations in Geomaterials pp 251– (2011) · Zbl 1239.74018 · doi:10.1007/978-3-642-18284-6_14
[23] Miehe, A framework for micro-macro transitions in periodic particle aggregates of granular materials, Computer Methods in Applied Mechanics and Engineering 193 (3) pp 225– (2004) · Zbl 1075.76541 · doi:10.1016/j.cma.2003.10.004
[24] Miehe, Homogenization and two-scale simulations of granular materials for different microstructural constraints, International Journal for Numerical Methods in Engineering 83 (8 - 9) pp 1206– (2010) · Zbl 1197.74084 · doi:10.1002/nme.2875
[25] Stránský, Open Source FEM-DEM Coupling, Proc. 18th Int. Conf. Engineering Mechanics pp 1237– (2012)
[26] Nguyen, FEM\(\times\)DEM modelling of cohesive granular materials: numerical homogenisation and multi-scale simulations, Acta Geophysica 62 (5) pp 1109– (2014) · doi:10.2478/s11600-014-0228-3
[27] Guo, A coupled FEM/DEM approach for hierarchical multiscale modelling of granular media, International Journal for Numerical Methods in Engineering 99 (11) pp 789– (2014) · Zbl 1352.74359 · doi:10.1002/nme.4702
[28] Andrade, Multiscale framework for behavior prediction in granular media, Mechanics of Materials 41 (6) pp 652– (2009) · doi:10.1016/j.mechmat.2008.12.005
[29] Sun WC Kuhn MR Rudnicki JW A Micromechanical analysis on permeability evolutions of a dilatant shear band Proceedings of the 48th American Rock Mechanics Association Symposium 2014
[30] Fish, Practical Multiscaling (2013)
[31] Chen, A generalized space-time mathematical homogenization theory for bridging atomistic and continuum scales, International Journal for Numerical Methods in Engineering 67 (2) pp 253– (2006) · Zbl 1110.74814 · doi:10.1002/nme.1630
[32] Chen, A mathematical homogenization perspective of virial stress, International journal for numerical methods in engineering 67 (2) pp 189– (2006) · Zbl 1110.74813 · doi:10.1002/nme.1622
[33] Fish, Generalized mathematical homogenization of atomistic media at finite temperatures in three dimensions, Computer Methods in Applied Mechanics and Engineering 196 (4) pp 908– (2007) · Zbl 1120.74716 · doi:10.1016/j.cma.2006.08.001
[34] Borja, Computational modeling of deformation bands in granular media. I. Geological and mathematical framework, Computer Methods in Applied Mechanics and Engineering 193 (27) pp 2667– (2004) · Zbl 1067.74570 · doi:10.1016/j.cma.2003.09.019
[35] Fish, A staggered nonlocal multiscale model for a heterogeneous medium, International Journal for Numerical Methods in Engineering 91 (2) pp 142– (2012) · Zbl 1246.74011 · doi:10.1002/nme.4259
[36] Liu, A regularized phenomenological multiscale damage model, International Journal for Numerical Methods in Engineering 99 (12) pp 867– (2014) · Zbl 1352.74024 · doi:10.1002/nme.4705
[37] Christoffersen, A micromechanical description of granular material behavior, Journal of Applied Mechanics 48 (2) pp 339– (1981) · Zbl 0471.73096 · doi:10.1115/1.3157619
[38] Rothenburg L Selvadurai A A Micromechanical Definition of the Cauchy Stress Tensor for Particulate Media 1981
[39] Bagi, Stress and strain in granular assemblies, Mechanics of materials 22 (3) pp 165– (1996) · doi:10.1016/0167-6636(95)00044-5
[40] Newland, Volume changes in drained taixial tests on granular materials, Geotechnique 7 (1) pp 17– (1957) · doi:10.1680/geot.1957.7.1.17
[41] Oda, Study on couple stress and shear band development in granular media based on numerical simulation analyses, International Journal of Engineering Science 38 (15) pp 1713– (2000) · doi:10.1016/S0020-7225(99)00132-9
[42] Mühlhaus, The thickness of shear bands in granular materials, Geotechnique 37 (3) pp 271– (1987) · doi:10.1680/geot.1987.37.3.271
[43] Brown, On the application of couple-stress theories to granular media, Geotechnique 22 (2) pp 356– (1972) · doi:10.1680/geot.1972.22.2.356
[44] Fish, Micro-inertia effects in nonlinear heterogeneous media, International Journal for Numerical Methods in Engineering 91 (13) pp 1406– (2012) · doi:10.1002/nme.4322
[45] Mühlhaus, Dispersion and wave propagation in discrete and continuous models for granular materials, International Journal of Solids and Structures 33 (19) pp 2841– (1996) · Zbl 0926.74052 · doi:10.1016/0020-7683(95)00178-6
[46] Belytschko, Nonlinear Finite Elements for Continua and Structures (2013)
[47] Bardet, Adaptative dynamic relaxation for statics of granular materials, Computers & Structures 39 (3) pp 221– (1991) · doi:10.1016/0045-7949(91)90020-M
[48] Sun, A multiscale DEM-LBM analysis on permeability evolutions inside a dilatant shear band, Acta Geotechnica 8 (5) pp 465– (2013) · doi:10.1007/s11440-013-0210-2
[49] Suzuki, Uniqueness of discrete element simulations in monotonic biaxial shear tests, International Journal of Geomechanics 14 (5) (2013)
[50] Tu, Criteria for static equilibrium in particulate mechanics computations, International Journal for Numerical Methods in Engineering 75 (13) pp 1581– (2008) · Zbl 1158.74339 · doi:10.1002/nme.2322
[51] Padbidri, Acceleration of DEM algorithm for quasistatic processes, International Journal for Numerical Methods in Engineering 86 (7) pp 816– (2011) · Zbl 1235.74367 · doi:10.1002/nme.3076
[52] Ng, Input parameters of discrete element methods, Journal of Engineering Mechanics 132 (7) pp 723– (2006) · doi:10.1061/(ASCE)0733-9399(2006)132:7(723)
[53] Filonova, Corotational formulation of reduced order homogenization, CMC: Computers, Materials & Continua 34 (3) pp 177– (2013)
[54] Yuan, Nonlinear multiphysics finite element code architecture in object oriented Fortran environment, Finite Elements in Analysis and Design 99 pp 1– (2015) · doi:10.1016/j.finel.2015.01.008
[55] Kuhn MR OVAL and OVALPLOT: Programs for analyzing dense particle assemblies with the discrete element method 2006
[56] Onate, Combination of discrete element and finite element methods for dynamic analysis of geomechanics problems, Computer Methods in Applied Mechanics and Engineering 193 (27) pp 3087– (2004) · Zbl 1079.74646 · doi:10.1016/j.cma.2003.12.056
[57] Kuhn, Structured deformation in granular materials, Mechanics of Materials 31 (6) pp 407– (1999) · doi:10.1016/S0167-6636(99)00010-1
[58] Cundall, Distinct element models of rock and soil structure, Analytical and Computational Methods in Engineering Rock Mechanics 4 pp 129– (1987)
[59] Thornton, Numerical simulations of deviatoric shear deformation of granular media, Géotechnique 50 (1) pp 43– (2000) · doi:10.1680/geot.2000.50.1.43
[60] Lin, A three-dimensional discrete element model using arrays of ellipsoids, Geotechnique 47 (2) pp 319– (1997) · doi:10.1680/geot.1997.47.2.319
[61] Makse, Why effective medium theory fails in granular materials, Physical Review Letters 83 (24) pp 5070– (1999) · doi:10.1103/PhysRevLett.83.5070
[62] Hill, Elastic properties of reinforced solids: some theoretical principles, Journal of the Mechanics and Physics of Solids 11 (5) pp 357– (1963) · Zbl 0114.15804 · doi:10.1016/0022-5096(63)90036-X
[63] Ostoja-Starzewski, Material spatial randomness: from statistical to representative volume element, Probabilistic Engineering Mechanics 21 (2) pp 112– (2006) · Zbl 1204.74006 · doi:10.1016/j.probengmech.2005.07.007
[64] Ostoja-Starzewski, Comparisons of the size of the representative volume element in elastic, plastic, thermoelastic, and permeable random microstructures, International Journal for Multiscale Computational Engineering 5 (2) pp 73– (2007) · doi:10.1615/IntJMultCompEng.v5.i2.10
[65] Sun, Multiscale method for characterization of porous microstructures and their impact on macroscopic effective permeability, International Journal for Numerical Methods in Engineering 88 (12) pp 1260– (2011) · Zbl 1242.74165 · doi:10.1002/nme.3220
[66] Meier, Towards multiscale computation of confined granular media-Contact forces, stresses and tangent operators, Technische Mechanik 28 (1) pp 32– (2008)
[67] Belytschko, Multiscale aggregating discontinuities: a method for circumventing loss of material stability, International Journal for Numerical Methods in Engineering 73 (6) pp 869– (2008) · Zbl 1195.74008 · doi:10.1002/nme.2156
[68] Belytschko, Assumed strain stabilization of the eight-node hexahedral element, Computer Methods in Applied Mechanics and Engineering 105 (2) pp 225– (1993) · Zbl 0781.73061 · doi:10.1016/0045-7825(93)90124-G
[69] Sun, A stabilized assumed deformation gradient finite element formulation for strongly coupled poromechanical simulations at finite strain, International Journal for Numerical and Analytical Methods in Geomechanics 37 (16) pp 2755– (2013)
[70] Bao, Large-scale simulation of elastic wave propagation in heterogeneous media on parallel computers, Computer Methods in Applied Mechanics and Engineering 152 (1) pp 85– (1998) · Zbl 0961.74056 · doi:10.1016/S0045-7825(97)00183-7
[71] Zeghal, Site response and vertical seismic arrays, Progress in Structural Engineering and Materials 2 (1) pp 92– (2000) · doi:10.1002/(SICI)1528-2716(200001/03)2:1<92::AID-PSE11>3.0.CO;2-6
[72] Zeghal, Local system identification analyses of the dynamic response of soil systems, Soil Dynamics and Earthquake Engineering 22 (9) pp 985– (2002) · doi:10.1016/S0267-7261(02)00123-9
[73] Borja, Estimating inelastic sediment deformation from local site response simulations, Acta Geotechnica 2 (3) pp 183– (2007) · doi:10.1007/s11440-007-0044-x
[74] Borja, Coseismic sediment deformation during the 1989 Loma Prieta earthquake, Journal of Geophysical Research: Solid Earth (1978-2012) 113 (2008) · doi:10.1029/2007JB005265
[75] Bielak, On the effective seismic input for non-linear soil-structure interaction systems, Earthquake Engineering & Structural Dynamics 12 (1) pp 107– (1984) · doi:10.1002/eqe.4290120108
[76] Yoshimura, Domain reduction method for three-dimensional earthquake modeling in localized regions, part II: Verification and applications, Bulletin of the Seismological Society of America 93 (2) pp 825– (2003) · doi:10.1785/0120010252
[77] Prevost, A simple plasticity theory for frictional cohesionless soils, International Journal of Soil Dynamics and Earthquake Engineering 4 (1) pp 9– (1985) · doi:10.1016/0261-7277(85)90030-0
[78] Zamani, Analysis of wave propagation in dry granular soils using DEM simulations, Acta Geotechnica 6 (3) pp 167– (2011) · doi:10.1007/s11440-011-0142-7
[79] Sadd, DEM simulation of wave propagation in granular materials, Powder Technology 109 (1-3) pp 222– (2000) · doi:10.1016/S0032-5910(99)00238-7
[80] Sadd, Contact law effects on wave propagation in particulate materials using distinct element modeling, International Journal of Non-Linear Mechanics 28 (2) pp 251– (1993) · Zbl 0800.73054 · doi:10.1016/0020-7462(93)90061-O
[81] Zeghal, Discrete-element method investigation of the resilient behavior of granular materials, Journal of Transportation Engineering 130 (4) pp 503– (2004) · doi:10.1061/(ASCE)0733-947X(2004)130:4(503)
[82] Wang, Numerical analysis of the stability of heavily jointed rock slopes using PFC2D, International Journal of Rock Mechanics and Mining Sciences 40 (3) pp 415– (2003) · doi:10.1016/S1365-1609(03)00004-2
[83] Casagrande A Characteristics of cohensionless soils affecting the stability of slops and earth fills Boston Contributions to Soil Mechanics 1925-1940 1940 Boston Society of Civil Engineers 257 276
[84] Bazant, and Jirásek,M. Nonlocal integral formulations of plasticity and damage: survey of progress, Journal of Engineering Mechanics 128 (11) pp 1119– (2002) · doi:10.1061/(ASCE)0733-9399(2002)128:11(1119)
[85] Sun, A multiscale overlapped coupling formulation for large-deformation strain localization, Computational Mechanics pp 1– (2014)
[86] Fleck, A reformulation of strain gradient plasticity, Journal of the Mechanics and Physics of Solids 49 (10) pp 2245– (2001) · Zbl 1033.74006 · doi:10.1016/S0022-5096(01)00049-7
[87] Belytschko, A finite element with embedded localization zones, Computer Methods in Applied Mechanics and Engineering 70 (1) pp 59– (1988) · Zbl 0653.73032 · doi:10.1016/0045-7825(88)90180-6
[88] Borja, A finite element model for strain localization analysis of strongly discontinuous fields based on standard Galerkin approximation, Computer Methods in Applied Mechanics and Engineering 190 (11) pp 1529– (2000) · Zbl 1003.74074 · doi:10.1016/S0045-7825(00)00176-6
[89] Borja, Finite element simulation of strain localization with large deformation: capturing strong discontinuity using a Petrov-Galerkin multiscale formulation, Computer Methods in Applied Mechanics and Engineering 191 (27) pp 2949– (2002) · Zbl 1030.74049 · doi:10.1016/S0045-7825(02)00218-9
[90] Dolbow, A finite element method for crack growth without remeshing, International Journal for Numerical Methods in Engineering 46 (1) pp 131– (1999) · Zbl 0955.74066 · doi:10.1002/(SICI)1097-0207(19990910)46:1<131::AID-NME726>3.0.CO;2-J
[91] Yang, A class of variational strain-localization finite elements, International Journal for Numerical Methods in Engineering 62 (8) pp 1013– (2005) · Zbl 1081.74045 · doi:10.1002/nme.1199
[92] Holtzman, Mechanical properties of granular materials: a variational approach to grain-scale simulations, International Journal for Numerical and Analytical Methods in Geomechanics 33 (3) pp 391– (2009) · Zbl 1272.74109 · doi:10.1002/nag.725
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.