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A constitutive model with microstructure evolution for flow of rate-independent granular materials

Published online by Cambridge University Press:  15 July 2011

JIN SUN  and
SANKARAN SUNDARESAN
JIN SUN*
Affiliation:
School of Engineering, University of Edinburgh, Edinburgh EH9 3JL, UK
SANKARAN SUNDARESAN
Affiliation:
Department of Chemical and Biological Engineering, Princeton University, Princeton, NJ 08544, USA
*
†Email address for correspondence: [email protected]
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Abstract

A constitutive model is developed for the complex rheology of rate-independent granular materials. The closures for the pressure and the macroscopic friction coefficient are linked to microstructure through evolution equations for coordination number and fabric. The material constants in the model are functions of particle-level properties and are calibrated using data generated through simulations of steady and unsteady simple shear using the discrete element method (DEM). This model is verified against DEM simulations at complex loading conditions.

JFM classification

Granular media: Granular media Non-Newtonian Flows: Rheology
Type
Papers
Information
Journal of Fluid Mechanics , Volume 682 , 10 September 2011 , pp. 590 - 616
Copyright
Copyright © Cambridge University Press 2011

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References

REFERENCES

Aarons, L. R., Sun, J. & Sundaresan, S. 2009 Unsteady shear of dense assemblies of cohesive granular materials under constant volume conditions. Ind. Engng Chem. Res. 49 (11), 5153–5165.CrossRefGoogle Scholar
Anand, L. & Gu, C. 2000 Granular materials: constitutive equations and strain localization. J. Mech. Phys. Solids 48 (8), 1701–1733.CrossRefGoogle Scholar
Azéma, E., Radjai, F. & Saussine, G. 2009 Quasistatic rheology, force transmission and fabric properties of a packing of irregular polyhedral particles. Mech. Mater. 41 (6), 729–741.CrossRefGoogle Scholar
Barnes, H. A., Hutton, J. F. & Walters, K. 1989 An Introduction to Rheology. Elsevier.Google Scholar
Bathurst, R. J. & Rothenburg, L. 1990 Observations on stress-force-fabric relationships in idealized granular materials. Mech. Mater. 9 (1), 65–80.CrossRefGoogle Scholar
Bocquet, L., Losert, W., Schalk, D., Lubensky, T. C. & Gollub, J. P. 2001 Granular shear flow dynamics and forces: Experiment and continuum theory. Phys. Rev. E 65 (1), 011307.Google ScholarPubMed
Briscoe, C., Song, C., Wang, P. & Makse, H. A. 2010 Jamming III: Characterizing randomness via the entropy of jammed matter. Physica A 389 (19), 3978–3999.CrossRefGoogle Scholar
Campbell, C. S. 2002 Granular shear flows at the elastic limit. J. Fluid Mech. 465, 261–291.CrossRefGoogle Scholar
Chaudhuri, P., Berthier, L. & Sastry, S. 2010 Jamming transitions in amorphous packings of frictionless spheres occur over a continuous range of volume fractions. Phys. Rev. Lett. 104 (16), 165701.CrossRefGoogle Scholar
da Cruz, F., Emam, S., Prochnow, M., Roux, J-N. & Chevoir, F. 2005 Rheophysics of dense granular materials: Discrete simulation of plane shear flows. Phys. Rev. E 72 (2), 021309.CrossRefGoogle ScholarPubMed
Cundall, P. A. & Strack, D. L. 1979 A discrete numerical model for granular assemblies. Géotechnique 29, 47–65.CrossRefGoogle Scholar
Drucker, D. C., Gibson, R. E. & Henkel, D. J. 1957 Soil mechanics and work hardening theories of plasticity. Trans. Am. Soc. Civil Engng 122, 338–346.CrossRefGoogle Scholar
Evans, D. J. & Morriss, G. P. 1990 Statistical Mechanics of Nonequilibrium Liquids. Theoretical chemistry 1. Academic Press.Google Scholar
Gadala-Maria, F. & Acrivos, A. 1980 Shear-induced structure in a concentrated suspension of solid spheres. J. Rheol. 24 (6), 799–814.CrossRefGoogle Scholar
Gera, D., Syamlal, M. & O'Brien, T. J. 2004 Transport equation for modeling particle contacts. In 11th Intl Conf. on Fluidization (ed. Arena, U., Chirone, R., Miccio, M. & Salatino, P.). Engineering Conferences International.Google Scholar
Goddard, J. 1986 Dissipative materials as constitutive models for granular media. Acta Mechanica 63 (1), 3–13.CrossRefGoogle Scholar
Goddard, J. 2010 Parametric hypoplasticity as continuum model for granular media: from Stokesium to Mohr–Coulombium and beyond. Granular Matter 12 (2), 145–150.CrossRefGoogle Scholar
Goddard, J. D. 1984 Dissipative materials as models of thixotropy and plasticity. J. Non-Newtonian Fluid Mech. 14, 141–160.CrossRefGoogle Scholar
Goddard, J. D. 1998 Continuum modeling of granular assemblies. In Physics of Dry Granular Media (ed. Herrmann, H. J.), chap. 1, pp. 1–24. Kluwer.Google Scholar
Goddard, J. D. 2006 A dissipative anisotropic fluid model for non-colloidal particle dispersions. J. Fluid Mech. 568 (1), 1–17.CrossRefGoogle Scholar
Harris, D. 2006 Some properties of a new model for slow flow of granular materials. Meccanica 41 (3), 351–362.CrossRefGoogle Scholar
Hébraud, P. & Lequeux, F. 1998 Mode-coupling theory for the pasty rheology of soft glassy materials. Phys. Rev. Lett. 81 (14), 2934–2937.CrossRefGoogle Scholar
Hill, R. 1950 The Mathematical Theory of Plasticity. Oxford University Press.Google Scholar
Jackson, R. 1986 Some features of the flow of granular materials and aerated granular materials. J. Rheol. 30 (5), 907–930.CrossRefGoogle Scholar
Ketterhagen, W. R., Curtis, J. S., Wassgren, C. R. & Hancock, B. C. 2009 Predicting the flow mode from hoppers using the discrete element method. Powder Technol. 195 (1), 1–10.CrossRefGoogle Scholar
Keys, A. S., Abate, A. R., Glotzer, S. C. & Durian, D. J. 2007 Measurement of growing dynamical length scales and prediction of the jamming transition in a granular material. Nat. Phys. 3 (4), 260–264.CrossRefGoogle Scholar
KheiripourLangroudi, M. Langroudi, M., Sun, J., Sundaresan, S. & Tardos, G. I. 2010 Transmission of stresses in static and sheared granular beds: The influence of particle size, shearing rate, layer thickness and sensor size. Powder Technol. 203 (1), 23–32.CrossRefGoogle Scholar
Lade, P. V. 1977 Elasto-plastic stress-strain theory for cohesionless soil with curved yield surfaces. Intl J. Solids Struct. 13 (11), 1019–1035.CrossRefGoogle Scholar
Lees, A. W. & Edwards, S. F. 1972 The computer study of transport processes under extreme conditions. J. Phys. C 5 (15), 1921–1928.CrossRefGoogle Scholar
Lu, K., Brodsky, E. E. & Kavehpour, H. P. 2007 Shear-weakening of the transitional regime for granular flow. J. Fluid Mech. 587 (1), 347–372.CrossRefGoogle Scholar
Luding, S. 2004 Micro–macro transition for anisotropic, frictional granular packings. Intl J. Solids Struct. 41 (21), 5821–5836.CrossRefGoogle Scholar
Luding, S. 2005 Anisotropy in cohesive, frictional granular media. J. Phys. Condens. Matter 17 (24), 2623–2640.CrossRefGoogle Scholar
Majmudar, T. S. & Behringer, R. P. 2005 Contact force measurements and stress-induced anisotropy in granular materials. Nature 435 (7045), 1079–1082.CrossRefGoogle ScholarPubMed
Majmudar, T. S., Sperl, M., Luding, S. & Behringer, R. P. 2007 Jamming transition in granular systems. Phys. Rev. Lett. 98 (5), 058001.CrossRefGoogle ScholarPubMed
Makse, H., Johnson, D. L. & Schwartz, L. M. 2000 Packing of compressible granular materials. Phys. Rev. Lett. 84 (18), 4160–4163.CrossRefGoogle ScholarPubMed
Mehrabadi, M. M. & Cowin, S. C. 1978 Initial planar deformation of dilatant granular materials. J. Mech. Phys. Solids 26 (4), 269–284.CrossRefGoogle Scholar
MiDi, G. D. R. 2004 On dense granular flows. Eur. Phys. J. E 14 (4), 341–305.CrossRefGoogle Scholar
Mohan, L. S., Rao, K. K. & Nott, P. R. 2002 A frictional Cosserat model for the slow shearing of granular materials. J. Fluid Mech. 457 (1), 377–409.CrossRefGoogle Scholar
Mueth, D. M., Debregeas, G. F., Karczmar, G. S., Eng, P. J., Nagel, S. R. & Jaeger, H. M. 2000 Signatures of granular microstructure in dense shear flows. Nature 406 (6794), 385–389.CrossRefGoogle ScholarPubMed
Nedderman, R. M. 1992 Statics and Kinematics of Granular Materials. Cambridge University Press.CrossRefGoogle Scholar
Nemat-Nasser, S. 2000 A micromechanically-based constitutive model for frictional deformation of granular materials. J. Mech. Phys. Solids 48 (6–7), 1541–1563.CrossRefGoogle Scholar
Oda, M., Konishi, J. & Nemat-Nasser, S. 1980 Some experimentally based fundamental results on the mechanical behaviour of granular materials. Géotechnique 30 (4), 479–495.CrossRefGoogle Scholar
O'Hern, C. S., Silbert, L. E., Liu, A. J. & Nagel, S. R. 2003 Jamming at zero temperature and zero applied stress: The epitome of disorder. Phys. Rev. E 68 (1), 011306.CrossRefGoogle ScholarPubMed
Okada, N. & Nemat-Nasser, S. 1994 Energy dissipation in inelastic flow of saturated cohesionless granular media. Géotechnique 44, 1–19.CrossRefGoogle Scholar
Peyneau, P.-E. & Roux, J.-N. 2008 Frictionless bead packs have macroscopic friction, but no dilatancy. Phys. Rev. E 78 (1), 011307.CrossRefGoogle ScholarPubMed
Plimpton, S. 1995 Fast parallel algorithms for short-range molecular dynamics. J. Comput. Phys. 117 (1), 1–19.CrossRefGoogle Scholar
Radjai, F., Wolf, D. E., Jean, M. & Moreau, J.-J. 1998 Bimodal character of stress transmission in granular packings. Phys. Rev. Lett. 80 (1), 61–64.CrossRefGoogle Scholar
Reynolds, O. 1885 On the dilatancy of media composed of rigid particles, with experimental illustrations. Phil. Mag. 20 (127), 469–481.CrossRefGoogle Scholar
Rothenburg, L. & Kruyt, N. P. 2004 Critical state and evolution of coordination number in simulated granular materials. Intl J. Solids Struct. 41 (21), 5763–5774.CrossRefGoogle Scholar
Rycroft, C. H., Kamrin, K. & Bazant, M. Z. 2009 Assessing continuum postulates in simulations of granular flow. J. Mech. Phys. Solids 57 (5), 828–839.CrossRefGoogle Scholar
Savage, S. B. 1979 Gravity flow of cohesionless granular materials in chutes and channels. J. Fluid Mech. 92, 53–96.CrossRefGoogle Scholar
Savage, S. B. 1983 Granular flows down rough inclines – review and extension. In Mechanics of Granular Materials: New Models and Constitutive Relations (ed. Jenkins, J. T. & Satake, M.), pp. 261–282. Elsevier.CrossRefGoogle Scholar
Savage, S. B. & Sayed, M. 1984 Stresses developed by dry cohesionless granular materials sheared in an annular shear cell. J. Fluid Mech. 142 (1), 391–430.CrossRefGoogle Scholar
Schaeffer, D. G. 1987 Instability in the evolution equations describing incompressible granular flow. J. Differ. Equ. 66, 19–50.CrossRefGoogle Scholar
Schofield, A. & Wroth, P. 1968 Critical State Soil Mechanics. McGraw-Hill.Google Scholar
Shäfer, J., Dippel, S. & Wolf, D. E. 1996 Force schemes in simulations of granular materials. J. Phys. I 6 (1), 5–20.Google Scholar
Shundyak, K., van Hecke, M. & van Saarloos, W. 2007 Force mobilization and generalized isostaticity in jammed packings of frictional grains. Phys. Rev. E 75 (1), 010301(R).CrossRefGoogle ScholarPubMed
Silbert, L. E., Ertas, D., Grest, G. S., Halsey, T. C., Levine, D. & Plimpton, S. J. 2001 Granular flow down an inclined plane: Bagnold scaling and rheology. Phys. Rev. E 64 (5), 051302.CrossRefGoogle ScholarPubMed
Silbert, L. E., Grest, G. S., Brewster, R. & Levine, A. J. 2007 Rheology and contact lifetimes in dense granular flows. Phys. Rev. Lett. 99 (6), 068002.CrossRefGoogle ScholarPubMed
Somfai, E., van Hecke, M., Ellenbroek, W. G., Shundyak, K. & van Saarloos, W. 2007 Critical and noncritical jamming of frictional grains. Phys. Rev. E 75 (2), 020301.CrossRefGoogle ScholarPubMed
Song, C., Wang, P. & Makse, H. A. 2008 A phase diagram for jammed matter. Nature 453 (7195), 629–632.CrossRefGoogle ScholarPubMed
Spencer, A. J. M. 1964 A theory of the kinematics of ideal soils under plane strain conditions. J. Mech. Phys. Solids 12 (5), 337–351.CrossRefGoogle Scholar
Subhash, G., Nemat-Nasser, S., Mehrabadi, M. M. & Shodj, H. M. 1991 Experimental investigation of fabric–stress relations in granular materials. Mech. Mater. 11 (2), 87–106.CrossRefGoogle Scholar
Suiker, A. S. J. & Fleck, N. A. 2004 Frictional collapse of granular assemblies. J. Appl. Mech. 71 (3), 350–358.CrossRefGoogle Scholar
Tardos, G. I., McNamara, S. & Talu, I. 2003 Slow and intermediate flow of a frictional bulk powder in the Couette geometry. Powder Technol. 131 (1), 23–39.CrossRefGoogle Scholar
Thornton, C. 2000 Numerical simulations of deviatoric shear deformation of granular media. Géotechnique 50 (1), 43–53.CrossRefGoogle Scholar
Toiya, M., Stambaugh, J. & Losert, W. 2004 Transient and oscillatory granular shear flow. Phys. Rev. Lett. 93 (8), 088001.CrossRefGoogle ScholarPubMed
Utter, B. & Behringer, R. P. 2004 Transients in sheared granular matter. Eur. Phys. J. E 14 (4), 373–380.CrossRefGoogle ScholarPubMed
Wood, D. M. & Budhu, M. 1980 The behaviour of Leighton Buzzard sand in cyclic simple shear tests. In Soilds under Cyclic and Transient Loading (ed. Pande, G. N. & Zienkiewicz, O. C.), p. 9. Balkema.Google Scholar
Youd, T. L. 1971 Maximum density of sand by repeated straining in simple shear. Highway Res. Rec. 374, 1–6.Google Scholar
Zhang, H. P. & Makse, H. A. 2005 Jamming transition in emulsions and granular materials. Phys. Rev. E 72 (1), 011301.CrossRefGoogle ScholarPubMed
Zhu, H., Mehrabadi, M. M. & Massoudi, M. 2006 Incorporating the effects of fabric in the dilatant double shearing model for planar deformation of granular materials. Intl J. Plast. 22 (4), 628–653.CrossRefGoogle Scholar