For many problems in science and engineering, it is necessary to describe the collective behavior of a very large number of grains. Complexity inherent in granular materials, whether due the variability of grain interactions or grain-scale morphological factors, requires modeling approaches that are both representative and tractable. In these cases, continuum modeling remains the most feasible approach; however, for such models to be representative, they must properly account for the granular nature of the material. The granular micromechanics approach has been shown to offer a way forward for linking the grain-scale behavior to the collective behavior of millions and billions of grains while keeping within the continuum framework. In this paper, an extended granular micromechanics approach is developed that leads to a micromorphic theory of degree n. This extended form aims at capturing the detailed grain-scale kinematics in disordered (mechanically or morphologically) granular media. To this end, additional continuum kinematic measures are introduced and related to the grain-pair relative motions. The need for enriched descriptions is justified through experimental measurements as well as results from simulations using discrete models. Stresses conjugate to the kinematic measures are then defined and related, through equivalence of deformation energy density, to forces conjugate to the measures of grain-pair relative motions. The kinetic energy density description for a continuum material point is also correspondingly enriched, and a variational approach is used to derive the governing equations of motion. By specifying a particular choice for degree n, abridged models of degrees 2 and 1 are derived, which are shown to further simplify to micro-polar or Cosserat-type and second-gradient models of granular materials.
MisraAPoorsolhjouyP.Grain- and macro-scale kinematics for granular micromechanics based small deformation micromorphic continuum model. Mech Res Commun2017; 81: 1–6.
2.
PoorsolhjouyPMisraA.Granular micromechanics based continuum model for grain rotations and grain rotation waves. J Mech Phys Solid2019; 129: 244–260.
RadjaïFDuboisF. Discrete-element modeling of granular materials. London: Wiley-Iste, 2011.
10.
TurcoEdell’IsolaFMisraA.A nonlinear Lagrangian particle model for grains assemblies including grain relative rotations. Int J Numer Anal Meth Geomech2019; 43: 1051–1079.
11.
AndradeJE, et al., Multiscale modeling and characterization of granular matter: from grain kinematics to continuum mechanics. J Mech Phys Solid2011; 59: 237–250.
12.
GuoNZhaoJ.A coupled FEM/DEM approach for hierarchical multiscale modelling of granular media. Int J Numer Meth Eng2014; 99: 789-818.
13.
ZhaoJShanT.Coupled CFD–DEM simulation of fluid–particle interaction in geomechanics. Powder Technol2013; 239: 248–258.
14.
ChangCSMisraA.Theoretical and experimental-study of regular packings of granules. J Eng Mech ASCE1989; 115: 704-720.
15.
ChangCSMisraA.Packing structure and mechanical-properties of granulates. J Eng Mech ASCE1990; 116: 1077–1093.
16.
DuffyJMindlinRD.Stress-strain relations and vibrations of a granular medium. J Appl Mech1957; 24: 585–593.
17.
JenkinsJ, et al. Fluctuations and the effective moduli of an isotropic, random aggregate of identical, frictionless spheres. J Mech Phys Solid2005; 53: 197–225.
18.
JenkinsJT. Volume change in small strain axisymmetric deformations of a granular material. In: SatakeMJenkinsJT (eds) Micromechanics of granular materials. Amsterdam: Elsevier Science Publishers, 1988, 245–252.
19.
JenkinsJT. Inelastic Behavior of Random Arrays of Identical Spheres. In: FleckNACocksACF (eds) IUTAM Symposium on Mechanics of Granular and Porous Materials. Solid Mechanics and its Applications. Dordrecht: Springer, 1997.
20.
KruytNRothenburgL.Kinematic and static assumptions for homogenization in micromechanics of granular materials. Mech Mater2004; 36: 1157–1173.
21.
WaltonK.The effective elastic moduli of a random packing of spheres. J Mech Phys Solid1987; 35: 213–226.
22.
MisraAPlacidiLTurcoE. Variational methods for continuum models of granular materials. In: AltenbachHOchsnerA (eds) Encyclopedia of continuum mechanics. Berlin: Springer GmbH, 2019.
23.
MisraAPoorsolhjouyP.Granular micromechanics based micromorphic model predicts frequency band gaps. Continuum Mech Thermodynam2016; 28: 215–234.
24.
AbaliBEMüllerWHdell’IsolaF.Theory and computation of higher gradient elasticity theories based on action principles. Arch Appl Mech2017; 87: 1495–1510.
25.
NejadsadeghiN, et al. Frequency band gaps in dielectric granular metamaterials modulated by electric field. Mech Res Commun2019; 95: 96–103.
26.
MisraANejadsadeghiN.Longitudinal and transverse elastic waves in 1D granular materials modeled as micromorphic continua. Wave Motion2019; 90: 175–195.
27.
NejadsadeghiNMisraA.Axially moving materials with granular microstructure. Int J Mech Sci2019; 161–162: 105042.
28.
JirásekM.Nonlocal theories in continuum mechanics. Acta Polytechnica2004; 44: 16–34.
29.
CosseratECosseratF. Theory of deformable bodies. Paris: Scientific Library A. Hermann and Sons, 1909.
30.
PlacidiL, et al. Variational methods in continuum damage and fracture mechanics. In: AltenbachHOchsnerA (eds) Encyclopedia of continuum mechanics. Berlin: Springer, 2017.
31.
EugsterSRdell’IsolaF. Exegesis of the Introduction and Sect. I from “Fundamentals of the Mechanics of Continua”** by HellingerE.ZAMM J Appl Math Mech/Zeitschrift für Angewandte Mathematik und Mechanik2017; 97: 477–506.
32.
EugsterSRdell’IsolaF. Exegesis of Sect. III. B from “Fundamentals of the Mechanics of Continua” by HellingerE.ZAMM J Appl Math Mech/Zeitschrift für Angewandte Mathematik und Mechanik2018; 98: 69–105.
33.
EugsterSRDell’IsolaF. Exegesis of Sect. II and III. A from “Fundamentals of the Mechanics of Continua” by HellingerE.ZAMM J Appl Math Mech/Zeitschrift für Angewandte Mathematik und Mechanik2018; 98: 31–68.
34.
MisraAJiangH.Measured kinematic fields in the biaxial shear of granular materials. Comput Geotechn1997; 20: 267–285.
35.
ShuklaASaddMHMeiH.Experimental and computational modeling of wave propagation in granular materials. Exp Mech1990; 30: 377–381.
36.
MisraAPoorsolhjouyP.Elastic behavior of 2D grain packing modeled as micromorphic media based on granular micromechanics. J Eng Mech2016; 143: C4016005.
37.
MühlhausHBOkaF.Dispersion and wave propagation in discrete and continuous models for granular materials. Int J Solid Struct1996; 33: 2841–2858.
38.
DeresiewiczH.Stress-Strain relations for a simple model of a granular medium. J Appl Mech1958; 25: 402–406.
NejadsadeghiN, et al. Parametric experimentation on pantographic unit cells reveals local extremum configuration. Exp Mech2019; 59: 927–939.
41.
KanitT, et al. Determination of the size of the representative volume element for random composites: statistical and numerical approach. Int J Solid Struct2003; 40: 3647–3679.
42.
MatheronG. The theory of regionalized variables and its applications. Vol. 5. Paris: Ecole nationale supérieure des mines de Paris, 1971.
43.
MatheronG.Random sets and integral geometry. New York: John Wiley Sons, 1975.
44.
MatheronG.Estimating and choosing. Berlin: Springer, 1989.
45.
StroevenMAskesHSluysLJ.Numerical determination of representative volumes for granular materials. Comput Meth Appl Mech Eng2004; 193: 3221–3238.
46.
WellmannCLillieCWriggersP.Homogenization of granular material modeled by a three-dimensional discrete element method. Comput Geotechn2008; 35: 394–405.
47.
BargmannS, et al. Generation of 3D representative volume elements for heterogeneous materials: a review. Progr Mater Sci2018; 96: 322–384.
48.
GitmanIMAskesHSluysLJ.Representative volume: existence and size determination. Eng Fract Mech2007; 74: 2518–2534.
49.
GermainP. Method of virtual power in continuum mechanics. 2. Microstructure. Siam J Appl Math1973; 25: 556–575.
50.
MindlinRD.Micro-structure in linear elasticity. Arch Ration Mech Anal1964; 16: 51-78.
51.
EringenA.Microcontinuum field theories I: foundations and solids. New York: Springer, 1999.
52.
dell’IsolaFSeppecherPDella CorteA.The postulations á la D’Alembert and á la Cauchy for higher gradient continuum theories are equivalent: a review of existing results. Proc R Soc A2015; 471: 20150415.
53.
dell’IsolaFSeppecherPMadeoAHow contact interactions may depend on the shape of Cauchy cuts in Nth gradient continua: approach “à la D’Alembert”. Zeitschrift für angewandte Mathematik und Physik2012; 63: 1119–1141.
54.
SeppecherPAlibertJ-Jdell’IsolaF. Linear elastic trusses leading to continua with exotic mechanical interactions. J Phys Conf Ser2011; 319: 012018.
55.
dell’IsolaFAndreausUPlacidiL. At the origins and in the vanguard of peri-dynamics, non-local and higher gradient continuum mechanics. An underestimated and still topical contribution of Gabrio Piola. Mech Math Solid2015; 20: 887-928.
56.
dell’IsolaF, et al. The complete works of Gabrio Piola: volume I commented English translation - English and Italian edition. Berlin: Springer Publishing Company, Incorporated, 2014, 813.
YangYMisraA.Micromechanics based second gradient continuum theory for shear band modeling in cohesive granular materials following damage elasticity. Int J Solid Struct2012; 49: 2500–2514.
59.
GrayWG, et al. Mathematical tools for changing scale in the analysis of physical systems. New York: CRC Press, 1993.
60.
Dell’IsolaF, et al. Pantographic metamaterials: an example of mathematically driven design and of its technological challenges. Continuum Mech Thermodynam2018; 31: 851-884.
61.
MisraA, et al. Pantographic metamaterials show atypical Poynting effect reversal. Mech Res Commun2018; 89: 6–10.
62.
dell’IsolaF, et al. Advances in pantographic structures: design, manufacturing, models, experiments and image analyses. Continuum Mech Thermodynam2019; 31: 1231–1282.
63.
TurcoE, et al. King post truss as a motif for internal structure of (meta) material with controlled elastic properties. R Soc Open Sci2017; 4: 171153.
64.
TurcoE, et al. Enhanced Piola–Hencky discrete models for pantographic sheets with pivots without deformation energy: numerics and experiments. Int J Solid Struct2018; 147: 94–109.
65.
GiorgioI, et al. Pattern formation in the three-dimensional deformations of fibered sheets. Mech Res Commun2015; 69: 164–171.
66.
GiorgioIRizziNTurcoE.Continuum modelling of pantographic sheets for out-of-plane bifurcation and vibrational analysis. Proc R Soc A Math Phys Eng Sci2017; 473: 20170636.
67.
PlacidiLBarchiesiE.Energy approach to brittle fracture in strain-gradient modelling. Proc R Soc A2018; 474: 20170878.
68.
PlacidiLBarchiesiEMisraA.A strain gradient variational approach to damage: a comparison with damage gradient models and numerical results. Math Mech Complex Syst2018; 6: 77–100.
69.
PlacidiLMisraABarchiesiE.Two-dimensional strain gradient damage modeling: a variational approach. Zeitschrift für angewandte Mathematik und Physik2018; 69: 56.
70.
MisraAPoorsolhjouyP.Identification of higher-order elastic constants for grain assemblies based upon granular micromechanics. Math Mech Complex Syst2015; 3: 285–308.
