Joe GODDARD Department of Mechanical and Aerospace ...

ColloqueTraffic and Granular Flows 07Paris-Sud University, OrsayJoe GoddardDepartment of Mechanical& Aerospace EngineeringUniversity of California,San Diego (La Jolla)From Granular NetworkstoGraded Continua

Outline• The multipolar continuum as a model for granular medawith “length-scale” effects• A recently proposed homogenization scheme based ongraph theoretical ideas and energy principles*• A toy model of particulate flow, with conclusions andopen questions• See “From Granular Matter to Generalized Continuum”, in Mathematical Models ofGranular Matter, P. Mariano et al. (eds), Springer Lecture Notes in Mathematics, toappear 2007. (Qualified research students are invited to attend a school“Structures of the mechanics of complex bodies” in the Centro di Giorgio,University of Pisa, Italy, October 1-7, 2007.)

Multipolar effects - Why should we care?• Granular media are susceptible to a numberof material instabilities. In quasi-staticmotions, these lead to “strain localization”,usually in the form of “shear bands” of theorder of 10 grain diameters thick, in bothexperiment and numerical simulations.• Experimental observations and computersimulations suggest that the particle rotationsin such regions are different from the globalrotation (or vorticity), sometimes interpretedas “Cosserat” effects.• Some type of “enriched continuum” model isnecessary to regularize simple continuum models,numerically if not physically. The associatedgradient effects may influence material instability(Aifantis, Vardoulakis and coworkers, 80’s +).• Same ideas may apply to continuum models oftraffic flow.

Mechanical Networks (& Delaunay Graphs)Examples:Materialrubberdry foamgranular mediumElementspolymer chainsPlateau bordersparticle contactsNodesentanglementsborder junctionsparticle centroids

From grains to branchesand simple continuummodelal_

Granular contact (Satake) graphs

From weightedgraph to continuum

From simple to multipolarcontinuum(Toupin, Mindlin, Green & Rivlin,“Graded Continuum”)(Cosserat, Green & Rivlin, Eringen“Micromorphic” Continuum)

Extrinsic granularmodesSubscripts “k ” referto contacts or edges (ij) .

Homogenization (extrinsic modes)

LSE in GAM** Proposed by M. Rosenblatt some 30 years ago for spline smoothing and later dubbed “GAM”(Wood, S. N. J. Am. Stat. Assoc. 99, 467, 2004).

Possible alternative

Toy “microfluidics” problem** Future calculation will involve MD data of J. Koplik for flow of Lennard -Jones fluid in a slot. Similar data for granular mechanics would be welcome.

Homogenization with latest LSE+GAMLSE fit to noisy quadraticu = Δv vs. y (/particle diam.)with 5th-deg. polynomialLSE+GAM fit• Note that GAM transfers energy from quadratic to linear terms

Conclusions and Questions• One can associate a graded multipolar continuum to theextrinsic degrees of freedom represented by granular centroids.• Although not discussed in detail above, this modifies the standarddefinition of (Cauchy) stress, since lower-order stresses areaffected by higher-order terms.• A proper energy-based homogenization still leaves some freedomof choice in least squares estimates (LSE) with generalized additivemodel (GAM) to avoid overfitting. This is a basic physical problemof distinguishing fluctuations from higher-gradient effects. Is therethere some physically preferred resolution?• Simulation data and new ideas welcome.Thanks for your attention!