Table Of ContentAdvances in Continuum Mechanics
Pasquale Giovine
Paolo Maria Mariano
Giuseppe Mortara
Editors
Views on
Microstructures
in Granular
Materials
Advances in Mechanics and Mathematics
Advances in Continuum Mechanics
Volume 44
SeriesEditors
PaoloMariaMariano,UniversityofFlorence
RichardD.James,UniversityofMinnesota
ConstantineM.Dafermos,BrownUniversity
Moreinformationaboutthissubseriesathttp://www.springer.com/series/16097
Pasquale Giovine • Paolo Maria Mariano
Giuseppe Mortara
Editors
Views on Microstructures
in Granular Materials
Editors
PasqualeGiovine PaoloMariaMariano
DepartmentofCivil,Energy,Environment, DepartmentofCivilandEnvironmental
andMaterialsEngineering(DICEAM) Engineering
University‘Mediterranea’ofReggioCalabria UniversityofFlorence
ReggioCalabria,Italy Firenze,Italy
GiuseppeMortara
DepartmentofCivil,Energy,Environment,
andMaterialsEngineering(DICEAM)
UniversityofReggioCalabria
ReggioCalabria,Italy
ISSN1571-8689 ISSN1876-9896 (electronic)
AdvancesinMechanicsandMathematics
ISSN2524-4639 ISSN2524-4647 (electronic)
AdvancesinContinuumMechanics
ISBN978-3-030-49266-3 ISBN978-3-030-49267-0 (eBook)
https://doi.org/10.1007/978-3-030-49267-0
Mathematics Subject Classification: 74L10, 74M25, 74A20, 74A60, 74E20, 76T25, 74L50, 70F35,
70F99
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Views on Microstructures in Granular
Materials: Preface
Granularmaterialsaresetsofthousandstomillionsofindividualparticles—grains
withadiameter greater than one micron—each able tomove independently of the
neighboring particles, unless the environment forces them to a certain degree of
compactnessortoflowintoconstrainedregionssuchaschannels.
Viewed with the naked eye, grains are simple, at times even described merely
by indicating the placement in space of their mass center. However, grain clusters
behaveascomplexmedia.Theirbehaviordependsonseveralfactors,e.g.,thesize
distributionofthegrains,theroughnessofthegrainboundaries,andtheirpotential
chemical activity, determining partial cohesion, history, whether they are dry or
embeddedinagroundfluid,etc.
Such materials oscillate from fluid-like behavior to solid-like states in which
grainsarejammedtovariousdegrees.Underthelatterconditions,wealsorecognize
groups of grains at equilibrium but additionally rattles, i.e., mechanically unstable
grain sub-clusters. Essentially, we meet metastable jammed states: the energy
exhibits local minimizers rather than global ones. Variety in the ways in which
grainscanbejammeddoesnotallowustoidentifyspecificreferencerepresentative
volumeelementsfromwhichwemaystarthomogenizingatcontinuumscale.
Duetosuchdifficulty,statisticalapproachesmayofferappropriateviewsatleast
for jammed states. With this aim, in 1989, S.F. Edwards and R.B.S. Oakeshott
proposed adopting for mechanically stable configurations of grains in a given
domain a microcanonical view, in which volume plays the same role as energy so
thatconfigurationsoccupyingthesamevolumeareconsideredtobeequi-probable.
¯
TheprobabilityP thatagranularsub-clusterofvolumeV < V isinagivenstate
ς
¯
ς isP =exp(− V ).Thescalarfactorχ(V),dubbedtheso-calledcompactions,
ς χ(V)
is the inverse of a temperature and behaves as an order parameter, which may
characterizenon-equilibriumphases.1
1EdwardsE.F.,OakeshottR.B.S.(1989),Theoryofpowders,PhysicaA,157,1080–1090.
v
vi ViewsonMicrostructuresinGranularMaterials:Preface
Arelatedcrucialproblemistodecidewhatkindofcontinuumisthetargetofour
homogenization procedures.2 Is it Cauchy type, Cosserat type, or does it refer to
morecomplexmedia?Thesimplestmodelsrefertothestandardformatofcontin-
uummechanics,forwhichthesolekinematicdescriptoristheplacementofgrains,
its variations given just by the point velocity field, and can be considered in short
as schemes of frictional plasticity, friction described by Mohr–Coulomb’s critical
statecriterionorrelatedrefinements.3 Suchschemesdonotincludeintrinsiclength
scales, which appear when we refine the representation of kinematic modes, e.g.,
accounting for independent grain rotations, with the construction of the pertinent
Cosserat scheme, possibly coupled with the local grain strain, in a micromorphic-
typemodel(specificexamplesaresomeindependentworksbyJ.D.GoddardandP.
Giovine,butnotonlythese).4
Granular flows display even more intricate behavior and present a number of
delicate problems in modeling. First, a reference configuration from which we
may start homogenizing is hard (if not impossible) to find; thus, the description
at continuum scale is naturally Eulerian. In 1998, M.-L. Tan and I. Goldhirsch
have even questioned whether hydrodynamic representation of granular flow is
appropriate.5 Their argument rests on the consideration that hydrodynamic-like
equations are“intimatelyrelatedtothenotionofscaleseparation”andtheyfound
“that such a scale separation is nonexistent” in granular systems “except when
the system is very nearly elastic.” Such a viewpoint emerges from the analysis
of a simply sheared stationary monodisperse granular system composed of disks
(spheres) in two (three) dimensions, with a fixed normal restitution coefficient.
Two key aspects characterize it: the product of mean free time τ by shear rate
0
γ (the inverse of which is the macroscopic time scale, whereas the other is a
microscopic time scale) is not small unless the system is nearly elastic. The mean
path shows spatial variations. The pertinent conclusion already mentioned was
criticized1yearlaterbyJ.W.DuftyandJ.J.Brey.6First,theyanalyzeanelastically
colliding flow of grains undergoing uniform shear with nonsmall γτ and show
0
that the hydrodynamic description applies for t (cid:2) τ . Then, they find that the
0
spatial variations of mean path, used by Tan and Goldhirsch, are the same for
both elastic and inelastic collisions; they conclude that spatial variations are “due
2SeeremarksinKamrinK.(2017),Ahierarchyofgranularcontinuummodels:Whyflowinggrains
arebothsimpleandcomplex,EPJWebofConferences,140,art.01007.
3See, e.g., Mortara G. (2015), A constitutive framework for the elastoplastic modelling of
geomaterials,Int.J.SolidsStruct.,63,139–152,and(2019),Anewyieldcriterionforsoilswith
embeddedtensioncut-off,Meccanica,54,683–696.
4ExamplesareGoddardJ.D.(2014),Continuummodelingofgranularmedia,Appl.Mech.Rev.,
66,art.050801;GiovineP.(2017),“Extendedgranularmicromechanics”,EPJWebofConferences,
140,art.11009.
5TanM.-L.,GoldhirschI.(1998),Rapidgranularflowsasmesoscopicsystems,Phys.Rev.Lett.,
81,3022–3025.
6DuftyJ.W.,BreyJ.J.(1999),Commenton“Rapidgranularflowsasmesoscopicsystems”,Phys.
Rev.Lett.,82,4566.
ViewsonMicrostructuresinGranularMaterials:Preface vii
tothehydrodynamicsitself,nottheunderlyingkinetics.”Inanycase,oneproblem
remains:whatkindofhydrodynamicdescriptionandhowtogetit?
If we refer to flow with low grain density, we could think of starting from the
Boltzmann equation, possibly accounting for inelastic collisions, and we would
find the problem of rigorously obtaining the continuum limit. We could also
consider expansion of the distribution function, taking 13 moments as in Grad’s
proposalorenlargingtheviewinthespiritofI.Müller’sandT.Ruggeri’sextended
thermodynamics.7 In this setting, when we account for the widespread inelastic
collisions, Boltzmann’s distribution—the probability of a state is proportional to
theexponentialofitsenergy—doesnotapply.Atvariance,inthepresenceofelastic
collisionsand increasinggraindensity,wecould directlyapproach thedescription
at continuum scale through Hamiltonian terms, but reality introduces friction or
other non-Hamiltonian effects, although we could think of mimicking them by
using Hamiltonian structures involving pseudo-potentials, at the cost of losing the
possibilityofdescribingoscillatoryinstabilities,whichfalloutsidetheHamiltonian
realm.
Instatics,suchinstabilitiesplayanontrivialroleassourcesoflarge-scaleshear-
banding and collapse mechanisms.8 In any case, even if we did feel satisfied by
Hamiltonianorpseudo-Hamiltonianstructures,weshouldconstructexplicitlytheir
elementsintermsofaverage,atleastinaportionofspace.Ifso,theprocedurewould
alsoopenthepathtowardcontinuumschemesthatarenotnecessarilyHamiltonian.
Also,inanalyzingawindowinspacewiththeaimofmakingappropriateaverages,
we realize that the microcanonical assumption mentioned above does not apply.
Thatwindowhastobeconsideredasagrandcanonicalensemble:thegrainclusterit
containsatacertaininstantmayloseelementsandreceivesenergyfrom(ortransfers
itto)theenvironmentthroughitsboundary,owingtocollisions.Aquestionishowto
considerfluctuationsbeyondtheaveragevelocityv,whichwecandefinepresuming
toknowthepertinentdistributionfunction.
Actually, to compute the average velocity explicitly, we need a specific form
of that distribution. In a rather traditional approach the fluctuation v¯ := w − v
is considered, with w the velocity of a grain at a point in the window and a
¯
certain instant as the ingredient for defining a Reynolds tensor H by averaging
v¯ ⊗ v¯ over the window considered in terms of the same distribution function
used to define v. Such a tensor essentially plays the role that we can call a
(tensor)granulartemperature.Startingfrom2003,G.Caprizproposedchangingthis
7MüllerI.,RuggeriT.(1998),Rationalextendedthermodynamics(2nded.),SpringerVerlag,New
York.
8ChattorajJ.,GendelmanO.,PicaCiamarraM.,ProcacciaI.(2019), Oscillatoryinstabilitiesin
frictional granular matter, Phys. Rev. Lett., 123, art. 098003 (1–5); di Prisco C., Matiotti R.,
NovaR.(1995),Theoreticalinvestigationoftheundrainedstabilityofshallowsubmergedslopes,
Geotechnique,45,479–496;MuirWoodD.(2002),Someobservationsofvolumetricinstabilities
insoils,Int.J.SolidsStruct.,39,3429–3449.
viii ViewsonMicrostructuresinGranularMaterials:Preface
view,9 a proposal further refined in independent collaborations with M. Brocato,
P. Giovine, and P.M. Mariano.10 The new approach foresees a decomposition of
v¯ into a homogeneous component and spurious fluctuations. The former accounts
for fluctuating expansion and contraction of the grain cluster within the window
(indeed, a loculus in space) at a given instant. Through it we can evaluate mass
gainorlossbeyondtheaveragedmotion.Also,wecanassociatewithsucha(low-
spatial-scale) mechanism true interactions power-conjugated with it, which satisfy
anappropriatebalance.Inthisway,wesetthedescriptionof(aboveallfast)granular
flows within the mechanics of complex materials, enriched by an energy balance
associatedwiththespuriousfluctuations.Whenweconsideragrainclustermoving
with and into a ground fluid, the previous view allows us to capture a refined
description of anisotropies that we may recognize in nature—falling hail is an
obviousexample.11
The size of grains larger than one micron, defining granular materials, avoids
thermal fluctuations. However, even injammed states wemay induce thermal-like
agitations by applying tapping or sharing protocols repeatedly.12 After the time
interval pertinent to the external excitation, they decay. A question is then how to
interpret thermodynamics for granular materials.13 If we think of jammed states
and the possible externally induced dynamics by repeated shaking, we can create,
at least in principle, different jammed states, and may factorize the state space by
the total stress pertaining to the cluster in a given window considered now as a
canonical ensemble for it exchanging energy with the environment. Frequency by
whichagivenstateisaccessibleallowsustodefineadensityofstates,depending
onthetotalstress(leavingapartfluctuationcomponentsduetoshaking),sothatwe
can obtain a temperature-like tensor as the derivative of a state density logarithm
with respect to the stress. For granular flow such a structure requires refinements,
eventobeabandoned,owingtothenontrivialroleofstressfluctuationsand(above
all) the lack of validity for the canonical ensemble assumption. In the way of
considering fluctuations as sketched above, the spurious ones, or at least some
of them, play a thermal-like role here. The pertinent Reynolds-type tensor could
itself be considered a (tensor) granular temperature, as its evolution equation is a
9CaprizG.(2003),Elementarypreambletoatheoryofgranulargases,Rend.Mat.Univ.Padova,
110,179–198.
10The list includes Brocato M., Capriz G. (2011), Clockwork, ephemeral and hybrid continua,
Phys. Mesomech., 14, 124–144; Capriz G., Giovine P. (2017), Classes of ephemeral continua,
Math. Meth. Appl. Sci., 43, 1175–1196; Capriz G., Mariano P.M. (2014), Objective fluxes in a
multiscalecontinuumdescriptionofsparsemediumdynamics,Phys.A,415,354–365.
11MarianoP.M.(2020),Flowsofsuspendedgrains:mixingasparsephasewithsimpleandcomplex
continua,Int.J.SolidsStruct.,187,141–161.
12See, for example, Becker V., Kassner K. (2015), Protocol-independent granular temperature
supportedbynumericalsimulations,Phys.Rev.E.Stat.Nonlin.SoftMatterPhys.,92,art.052201
(1–15).
13See,forexample,GoldhirschI.(2008),Introductiontogranulartemperature,PowderTech.,182,
130–136.
ViewsonMicrostructuresinGranularMaterials:Preface ix
low-scaleenergybalance.Butwecanalsodefineascalargranulartemperatureby
usingthedistributionofspuriousfluctuation(scalar)kineticenergy.14
Clearly,thedescriptionofgranularmatterbehavioroffersanumberofnontrivial
challenges,whichcanbetackledfromdifferentviewpoints,usingrefinedmathemat-
icaltheoriesaswellasadvancednumericalandexperimentalmethods.Inparticular,
experimentsinsoilmechanicshavebeenenrichedinlastdecadesbytheintroduction
of new techniques, which enable the observation of granules at the contact scale.
Also, the introduction of numerical tools such as the Discrete Element Method
revealedtheroleofthemicrostructureincharacterizingthebehaviorofsoils.These
new tools help in understanding the complex behavior of granular materials at the
macroscale. To take into account all the complex phenomena that occur at the
microscale,interactionamongdifferentareasofexpertise,engineering,physics,and
mathematics,aswellaschemistry,isthusoffundamentalimportanceforacomplete
and rational description of a granular material, even if it is concretely arduous to
account critically in a unique volume for the variety of existing approaches and
relatedresults.
Forthisreason,inthepresentbookwefocusattentiononlyonspecificissuesand
viewpoints, trying at least to present a wide range of scenarios on the matter. Our
choices of the material presented here include experimental questions connected
with X-ray tomography techniques for analyzing sand, the theoretical analysis
of dense inhomogeneous granular shearing, the evaluation of effective stress in
unsaturatedsoils,themodelingofsolidswithnanoporesasaschemeforcompacted
soils, hyper-plasticity, wave propagation in granular systems, yield criteria, the
scheme of partly saturated porous media, some of the computational problems
emerging in the analysis of granular matter, multi-scale phenomena in flows of
grains with internal structure, and related continuum modeling. All the chapters
that constitute the present book offer an overview intended for addressing further
researchwork.
ReggioCalabria,Italy PasqualeGiovine
Firenze,Italy PaoloMariaMariano
ReggioCalabria,Italy GiuseppeMortara
14CaprizG.,Mariano(2019)P.M.,Interactionsinamulti-scalerepresentationofsparsemedia:
frommechanicstothermodynamics,J.Elast.,135,91–115.
