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A non-local constitutive model for slow granular flow that incorporates dilatancy

Published online by Cambridge University Press:  17 February 2020

Peter Varun Dsouza  and
Prabhu R. Nott Open the ORCID record for Prabhu R. Nott [Opens in a new window]
Peter Varun Dsouza
Affiliation:
Department of Chemical Engineering, Indian Institute of Science, Bangalore560 012, India
Prabhu R. Nott*
Affiliation:
Department of Chemical Engineering, Indian Institute of Science, Bangalore560 012, India
*
†Email address for correspondence: prnott@iisc.ac.in
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Abstract

Over the past two decades several attempts have been made to formulate constitutive models for slow granular flow to remedy the deficiencies of classical plasticity. All the proposed models assume the medium to be incompressible, though it is well known that density change accompanies deformation in granular materials. A particularly important aspect of density change that is distinctive of granular materials is dilatancy, or volume deformation caused by shear deformation. No constitutive model for sustained flow has thus far captured dilatancy. Here we present a non-local constitutive model wherein the deformation rate and density at a point depend on the state of stress in a mesoscopic region around it. Apart from incorporating dilatancy, our model has a physical origin that is distinct from that of the previously proposed non-local models. We test our model on simple shear flow in the absence and presence of gravity, and find its predictions to be in good agreement with particle dynamics simulations.

JFM classification

Granular media: Granular media

Information

Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 888 , 10 April 2020 , R3
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

Ananda, K. S., Moka, S. & Nott, P. R. 2008 Kinematics and statistics of dense, slow granular flow through vertical channels. J. Fluid Mech. 610, 69–97.Google Scholar
Barker, T., Schaeer, D. G., Bohorquez, P. & Gray, J. M. N. T. 2015 Well-posed and ill-posed behaviour of the 𝜇(i) rheology for granular flow. J. Fluid Mech. 779, 794–818.CrossRefGoogle Scholar
Barker, T., Schaeffer, D. G., Shearer, M. & Gray, J. M. N. T. 2017 Well-posed continuum equations for granular flow with compressibility and 𝜇(I)-rheology. Proc. R. Soc. Lond. A 473 (2201), 20160846.CrossRefGoogle ScholarPubMed
Bird, R. B., Armstrong, R. C. & Hassager, O. 1977 Dynamics of Polymeric Liquids, vol. 1. John Wiley.Google Scholar
Bocquet, L., Colin, A. & Ajdari, A. 2009 Kinetic theory of plastic flow in soft glassy materials. Phys. Rev. Lett. 103, 036001.CrossRefGoogle ScholarPubMed
Bouzid, M., Izzet, A., Trulsson, M., Clément, E., Claudin, P. & Andreotti, B. 2015 Nonlocal rheology in dense granular flows. Eur. J. Phys. E 38 (125), 238301.Google Scholar
Bouzid, M., Trulsson, M., Claudin, P., Clément, E. & Andreotti, B. 2013 Nonlocal rheology of granular flows across yield conditions. Phys. Rev. Lett. 111, 238301.CrossRefGoogle ScholarPubMed
Desrues, J., Chambon, R., Mokni, M. & Mazerolle, F. 1996 Void ratio evolution inside shear bands in triaxial sand specimens by computed tomography. Géotechnique 46 (3), 529–546.CrossRefGoogle Scholar
Eringen, A. C. 1983 Theories of nonlocal plasticity. Intl J. Engng Sci. 21 (7), 741–751.CrossRefGoogle Scholar
GDR MiDi 2004 On dense granular flows. Eur. Phys. J. E 14, 341–365.Google Scholar
Goddard, J. D. & Lee, J. 2017 On the stability of the 𝜇(I) rheology for granular flow. J. Fluid Mech. 833, 302–331.CrossRefGoogle Scholar
Gutam, K. J., Mehandia, V. & Nott, P. R. 2013 Rheometry of granular materials in cylindrical Couette cells: anomalous stress caused by gravity and shear. Phys. Fluids 25 (7), 070602.CrossRefGoogle Scholar
Henann, D. L. & Kamrin, K. 2013 A predictive, size-dependent continuum model for dense granular flows. Proc. Natl Acad. Sci. USA 110 (17), 6730–6735.CrossRefGoogle ScholarPubMed
Henann, D. L. & Kamrin, K. 2014 Continuum modeling of secondary rheology in dense granular materials. Phys. Rev. Lett. 113, 178001.CrossRefGoogle ScholarPubMed
Heyman, J., Delannay, R., Tabuteau, H. & Valance, A. 2017 Compressibility regularizes the 𝜇(I)-rheology for dense granular flows. J. Fluid Mech. 830, 553–568.CrossRefGoogle Scholar
Jackson, R. 1983 Some mathematical and physical aspects of continuum models for the motion of the granular materials. In Theory of Dispersed Multiphase Flow (ed. Meyer, R. E.), pp. 291–337. Academic Press.CrossRefGoogle Scholar
Jop, P., Forterre, Y. & Pouliquen, O. 2006 A constitutive law for dense granular flows. Nature 441, 727–730.CrossRefGoogle ScholarPubMed
Kabla, A. J. & Senden, T. J. 2009 Dilatancy in slow granular flows. Phys. Rev. Lett. 102, 228301.CrossRefGoogle ScholarPubMed
Krishnaraj, K. P. & Nott, P. R. 2016 A dilation-driven vortex flow in sheared granular materials explains a rheometric anomaly. Nature Commun. 7, 10630.CrossRefGoogle ScholarPubMed
Losert, W., Bocquet, L., Lubensky, T. C. & Gollub, J. P. 2000 Particle dynamics in sheared granular matter. Phys. Rev. Lett. 85, 1428–1431.CrossRefGoogle ScholarPubMed
Majmudar, T. S. & Behringer, R. P. 2005 Contact force measurements and stress-induced anisotropy in granular materials. Nature 435 (7045), 1079–1082.CrossRefGoogle ScholarPubMed
Mehandia, V., Gutam, K. J. & Nott, P. R. 2012 Anomalous stress profile in a sheared granular column. Phys. Rev. Lett. 109, 128002.CrossRefGoogle Scholar
Mohan, L. S., Nott, P. R. & Rao, K. K. 1999 A frictional Cosserat model for the flow of granular materials through a vertical channel. Acta Mechanica. 138, 75–96.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, 377–409.CrossRefGoogle Scholar
Moka, S. & Nott, P. R. 2005 Statistics of particle velocities in dense granular flows. Phys. Rev. Lett. 95, 068003.CrossRefGoogle ScholarPubMed
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, 385–389.CrossRefGoogle ScholarPubMed
Natarajan, V. V. R., Hunt, M. L. & Taylor, E. D. 1995 Local measurements of velocity fluctuations and diffusion coefficients for a granular material flow. J. Fluid Mech. 304, 1–25.CrossRefGoogle Scholar
Nedderman, R. M. & Laohakul, C. 1980 The thickness of shear zone of flowing granular materials. Powder Technol. 25, 91–100.CrossRefGoogle Scholar
Nott, P. & Jackson, R. 1992 Frictional-collisional equations of motion for granular materials and their application to flow in aerated chutes. J. Fluid Mech. 241, 125–144.CrossRefGoogle Scholar
Nott, P. R. 2009 Classical and Cosserat plasticity and viscoplasticity models for slow granular flow. Acta Mechanica 205, 151–160.CrossRefGoogle Scholar
Pouliquen, O. & Forterre, Y. 2009 A non-local rheology for dense granular flows. Phil. Trans. R. Soc. Lond. A 367, 5091–5107.Google ScholarPubMed
Prakash, J. R. & Rao, K. K. 1988 Steady compressible flow of granular materials through a wedge-shaped hopper: the smooth wall radial gravity problem. Chem. Engng Sci. 43, 479–494.CrossRefGoogle Scholar
Rao, K. K. & Nott, P. R. 2008 An Introduction to Granular Flow. Cambridge University Press.CrossRefGoogle Scholar
Reddy, K. A., Forterre, Y. & Pouliquen, O. 2011 Evidence of mechanically activated processes in slow granular flows. Phys. Rev. Lett. 106 (10), 108301.CrossRefGoogle ScholarPubMed
Reynolds, O. 1885 On the dilatancy of media composed of rigid particles in contact. With experimental illustrations. Phil. Mag. 20, 469–481.CrossRefGoogle Scholar
Schofield, A. N. & Wroth, C. P. 1968 Critical State Soil Mechanics. McGraw-Hill.Google Scholar
Srivastava, A. & Sundaresan, S. 2003 Analysis of a frictional-kinetic model for gas-particle flow. Powder Technol. 129, 72–85.CrossRefGoogle Scholar
Sun, J. & Sundaresan, S. 2011 A constitutive model with microstructure evolution for flow of rate-independent granular materials. J. Fluid Mech. 682, 590–616.CrossRefGoogle Scholar