{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,15]],"date-time":"2026-09-15T15:51:58Z","timestamp":1789487518882,"version":"build-2803163510"},"reference-count":56,"publisher":"MDPI AG","issue":"18","license":[{"start":{"date-parts":[[2026,9,15]],"date-time":"2026-09-15T00:00:00Z","timestamp":1789430400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Materials"],"abstract":"<jats:p>The \u03bc(I) framework for dense granular flows treats the macroscopic friction (\u03bc), defined as the ratio of shear to normal stress transmitted across the sheared assembly, and the solid volume fraction (\u03c6) as single-valued functions of the inertial number (I). To test if these responses evolve together across shear rates, we used 3D discrete-element simulations of ring shear on a monodisperse assembly of spherical glass beads (diameter d = 1 mm), confined between two parallel toothed platens at 100 kPa normal stress. Shear velocity was varied over six orders of magnitude (10\u22124 to 10 m\/s), spanning a nominal inertial number I \u2248 10\u22126 to 10\u22121. Throughout, I denotes a bulk value defined from the imposed shear rate v\/H and the applied normal stress; the local inertial number varies with height and is reported separately. We found that steady-state friction closely follows the \u03bc(I) law, rising monotonically with rate. In contrast, bulk solid fraction \u03c6 is non-monotonic: it falls as rate increases, reaches a minimum near I \u2248 10\u22123, and recovers at higher rates, making a single \u03c6(I) curve a poor fit. Height-resolved fields explain this decoupling. At higher rates, shear concentrates near the driving boundary, and strong vertical agitation drives cross-layer mixing that re-compacts the lower bed. Consequently, bulk \u03c6 averages a dilating top with a re-densifying base, losing its one-to-one link to I. Instead, over the conditions tested, the contact density (nc) decreases monotonically\u2014by the same factor whether it is normalised per unit height, per unit volume, or per particle\u2014and so remains in step with the friction where the bulk solid fraction does not. The contact-force distribution broadens, and the fabric anisotropy grows as the flow becomes inertial, so that load is carried through fewer, more strongly oriented chains. These results show that friction and packing can respond to shear rate on different terms, and that bulk \u03c6 should be used with care as a state variable in rate-dependent dense flows.<\/jats:p>","DOI":"10.3390\/ma19183920","type":"journal-article","created":{"date-parts":[[2026,9,15]],"date-time":"2026-09-15T13:33:42Z","timestamp":1789479222000},"page":"3920","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Decoupling of Friction and Solid Fraction in Dense Granular Flow Across Six Orders of Magnitude of Shear Rate"],"prefix":"10.3390","volume":"19","author":[{"given":"Yi","family":"Zhang","sequence":"first","affiliation":[{"name":"College of Intelligent Construction and Manufacturing, Sichuan Technology and Business University, Chengdu 611745, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yangshuai","family":"Zheng","sequence":"additional","affiliation":[{"name":"College of Intelligent Construction and Manufacturing, Sichuan Technology and Business University, Chengdu 611745, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0007-1359-832X","authenticated-orcid":false,"given":"Hui","family":"Luo","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Geo-Hazard Prevention and Geo-Environment Protection, Chengdu University of Technology, Chengdu 610059, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xin","family":"Gao","sequence":"additional","affiliation":[{"name":"The 2nd Geological Brigade of Sichuan, Chengdu 610041, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,9,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1259","DOI":"10.1103\/RevModPhys.68.1259","article-title":"Granular solids, liquids, and gases","volume":"68","author":"Jaeger","year":"1996","journal-title":"Rev. Mod. Phys."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Andreotti, B., Forterre, Y., and Pouliquen, O. (2013). Granular Media: Between Fluid and Solid, Cambridge University Press.","DOI":"10.1017\/CBO9781139541008"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1146\/annurev.fluid.40.111406.102142","article-title":"Flows of dense granular media","volume":"40","author":"Forterre","year":"2008","journal-title":"Annu. Rev. Fluid Mech."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"101157","DOI":"10.1016\/j.pmatsci.2023.101157","article-title":"A state-of-the-art review of experimental and computational studies of granular materials","volume":"138","author":"Tahmasebi","year":"2023","journal-title":"Prog. Mater. Sci."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1140\/epje\/i2003-10153-0","article-title":"On dense granular flows","volume":"14","author":"Midi","year":"2004","journal-title":"Eur. Phys. J. E"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"021309","DOI":"10.1103\/PhysRevE.72.021309","article-title":"Rheophysics of dense granular materials: Discrete simulations of plane shear flows","volume":"72","author":"Emam","year":"2005","journal-title":"Phys. Rev. E"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"208","DOI":"10.1016\/j.powtec.2005.12.008","article-title":"Granular material flows\u2014An overview","volume":"162","author":"Campbell","year":"2006","journal-title":"Powder Technol."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"2937","DOI":"10.1002\/nag.2169","article-title":"From solid to granular gases: The steady state for granular materials","volume":"37","author":"Vescovi","year":"2013","journal-title":"Int. J. Numer. Anal. Methods Geomech."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"80","DOI":"10.1002\/nag.2391","article-title":"A visco-elasto-plastic model for granular materials under simple shear conditions","volume":"40","author":"Redaelli","year":"2016","journal-title":"Int. J. Numer. Anal. Methods Geomech."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"727","DOI":"10.1038\/nature04801","article-title":"A constitutive law for dense granular flows","volume":"441","author":"Jop","year":"2006","journal-title":"Nature"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1146\/annurev-fluid-121021-022045","article-title":"Advances in modeling dense granular media","volume":"56","author":"Kamrin","year":"2024","journal-title":"Annu. Rev. Fluid Mech."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"103737","DOI":"10.1016\/j.compgeo.2020.103737","article-title":"Particle size effects on small-scale avalanches and a \u03bc(I) rheology-based simulation","volume":"126","author":"Fei","year":"2020","journal-title":"Comput. Geotech."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"021306","DOI":"10.1103\/PhysRevE.79.021306","article-title":"Annular shear of cohesionless granular materials: From the inertial to quasistatic regime","volume":"79","author":"Koval","year":"2009","journal-title":"Phys. Rev. E"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"178301","DOI":"10.1103\/PhysRevLett.108.178301","article-title":"Nonlocal constitutive relation for steady granular flow","volume":"108","author":"Kamrin","year":"2012","journal-title":"Phys. Rev. Lett."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"238301","DOI":"10.1103\/PhysRevLett.111.238301","article-title":"Nonlocal rheology of granular flows across yield conditions","volume":"111","author":"Bouzid","year":"2013","journal-title":"Phys. Rev. Lett."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"6730","DOI":"10.1073\/pnas.1219153110","article-title":"A predictive, size-dependent continuum model for dense granular flows","volume":"110","author":"Henann","year":"2013","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"385","DOI":"10.1038\/35019032","article-title":"Signatures of granular microstructure in dense shear flows","volume":"406","author":"Mueth","year":"2000","journal-title":"Nature"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1428","DOI":"10.1103\/PhysRevLett.85.1428","article-title":"Particle dynamics in sheared granular matter","volume":"85","author":"Losert","year":"2000","journal-title":"Phys. Rev. Lett."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"094301","DOI":"10.1103\/PhysRevLett.92.094301","article-title":"Universal and wide shear zones in granular bulk flow","volume":"92","author":"Fenistein","year":"2004","journal-title":"Phys. Rev. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1918","DOI":"10.1002\/nag.988","article-title":"Fabric evolution within shear bands of granular materials and its relation to critical state theory","volume":"35","author":"Fu","year":"2011","journal-title":"Int. J. Numer. Anal. Methods Geomech."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"928","DOI":"10.1038\/nature01394","article-title":"Logarithmic rate dependence of force networks in sheared granular materials","volume":"421","author":"Hartley","year":"2003","journal-title":"Nature"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"3999","DOI":"10.1007\/s11440-023-02115-9","article-title":"Effect of overlying pressure and applied shear velocity on the nonlinear behavior of confined shear flow","volume":"19","author":"Ding","year":"2024","journal-title":"Acta Geotech."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1525","DOI":"10.1002\/nag.3355","article-title":"General friction law for velocity-stress dependent phase transition in granular flow","volume":"46","author":"Wang","year":"2022","journal-title":"Int. J. Numer. Anal. Methods Geomech."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"2525","DOI":"10.1002\/nag.2985","article-title":"Three-dimensional steady-state locus for dry monodisperse granular materials: DEM numerical results and theoretical modelling","volume":"43","author":"Redaelli","year":"2019","journal-title":"Int. J. Numer. Anal. Methods Geomech."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1016\/j.powtec.2005.09.002","article-title":"A comparison of discrete element simulations and experiments for \u2019sandpiles\u2019 composed of spherical particles","volume":"160","author":"Li","year":"2005","journal-title":"Powder Technol."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1017\/S0022112007009329","article-title":"Dense flows of cohesive granular materials","volume":"596","author":"Rognon","year":"2008","journal-title":"J. Fluid Mech."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1115\/1.4010702","article-title":"Elastic spheres in contact under varying oblique forces","volume":"20","author":"Mindlin","year":"1953","journal-title":"J. Appl. Mech."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1680\/geot.1979.29.1.47","article-title":"A discrete numerical model for granular assemblies","volume":"29","author":"Cundall","year":"1979","journal-title":"G\u00e9otechnique"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1016\/0032-5910(92)88030-L","article-title":"Lagrangian numerical simulation of plug flow of cohesionless particles in a horizontal pipe","volume":"71","author":"Tsuji","year":"1992","journal-title":"Powder Technol."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1016\/j.powtec.2010.09.030","article-title":"Assessment of rolling resistance models in discrete element simulations","volume":"206","author":"Ai","year":"2011","journal-title":"Powder Technol."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"409","DOI":"10.1016\/j.powtec.2011.10.057","article-title":"Rolling friction as a technique for modelling particle shape in DEM","volume":"217","author":"Wensrich","year":"2012","journal-title":"Powder Technol."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1016\/j.compgeo.2015.02.017","article-title":"Modeling the direct shear test of a coarse sand using the 3D discrete element method with a rolling friction model","volume":"67","author":"Salazar","year":"2015","journal-title":"Comput. Geotech."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"785","DOI":"10.1080\/19648189.2008.9693050","article-title":"Introduction to discrete element methods","volume":"12","author":"Luding","year":"2008","journal-title":"Eur. J. Environ. Civ. Eng."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"527","DOI":"10.1007\/s10035-010-0211-x","article-title":"Comprehensive study of the effects of rolling resistance on the stress\u2013strain and strain localization behavior of granular materials","volume":"12","author":"Mohamed","year":"2010","journal-title":"Granul. Matter"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"032913","DOI":"10.1103\/PhysRevE.96.032913","article-title":"Granular temperature measured experimentally in a shear flow by acoustic energy","volume":"96","author":"Taylor","year":"2017","journal-title":"Phys. Rev. E"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"130","DOI":"10.1016\/j.powtec.2007.12.002","article-title":"Introduction to granular temperature","volume":"182","author":"Goldhirsch","year":"2008","journal-title":"Powder Technol."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1103\/PhysRevLett.80.61","article-title":"Bimodal character of stress transmission in granular packings","volume":"80","author":"Radjai","year":"1998","journal-title":"Phys. Rev. Lett."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"041307","DOI":"10.1103\/PhysRevE.72.041307","article-title":"Characterization of force chains in granular material","volume":"72","author":"Peters","year":"2005","journal-title":"Phys. Rev. E"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"2345","DOI":"10.1007\/s00707-014-1120-6","article-title":"Fabric, force and strength anisotropies in granular materials: A micromechanical insight","volume":"225","author":"Li","year":"2014","journal-title":"Acta Mech."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"e2023GL104410","DOI":"10.1029\/2023GL104410","article-title":"Velocity profile geometries and granular temperature distributions in very dense granular flows","volume":"51","author":"Li","year":"2024","journal-title":"Geophys. Res. Lett."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"108190","DOI":"10.1016\/j.enggeo.2025.108190","article-title":"Evolution of velocity and temperature profiles in very dense granular flows: Insights from ring-shear experiments","volume":"348","author":"Ge","year":"2025","journal-title":"Eng. Geol."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"274","DOI":"10.1103\/PhysRevLett.77.274","article-title":"Force distributions in dense two-dimensional granular systems","volume":"77","author":"Radjai","year":"1996","journal-title":"Phys. Rev. Lett."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"061303","DOI":"10.1103\/PhysRevE.66.061303","article-title":"Statistics of the contact network in frictional and frictionless granular packings","volume":"66","author":"Silbert","year":"2002","journal-title":"Phys. Rev. E"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"4987","DOI":"10.1080\/14786430701594848","article-title":"Force chain buckling, unjamming transitions and shear banding in dense granular assemblies","volume":"87","author":"Tordesillas","year":"2007","journal-title":"Philos. Mag."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"1079","DOI":"10.1038\/nature03805","article-title":"Contact force measurements and stress-induced anisotropy in granular materials","volume":"435","author":"Majmudar","year":"2005","journal-title":"Nature"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"469","DOI":"10.1080\/14786448508627791","article-title":"On the dilatancy of media composed of rigid particles in contact","volume":"20","author":"Reynolds","year":"1885","journal-title":"Philos. Mag."},{"key":"ref_47","first-page":"500","article-title":"The stress-dilatancy relation for static equilibrium of an assembly of particles in contact","volume":"269","author":"Rowe","year":"1962","journal-title":"Proc. R. Soc. A"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"2191","DOI":"10.1007\/s00707-014-1125-1","article-title":"Extended kinetic theory applied to dense, granular, simple shear flows","volume":"225","author":"Berzi","year":"2014","journal-title":"Acta Mech."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"138001","DOI":"10.1103\/PhysRevLett.120.138001","article-title":"Controlled viscosity in dense granular materials","volume":"120","author":"Gnoli","year":"2018","journal-title":"Phys. Rev. Lett."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"118201","DOI":"10.1103\/PhysRevLett.130.118201","article-title":"Frictional weakening of vibrated granular flows","volume":"130","author":"Clark","year":"2023","journal-title":"Phys. Rev. Lett."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"L032010","DOI":"10.1103\/nzmt-b7bf","article-title":"Rheology of dense vibrated granular flows: Nonmonotonic response controlled by granular temperature","volume":"8","author":"Plati","year":"2026","journal-title":"Phys. Rev. Res."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.compgeo.2012.07.002","article-title":"The signature of shear-induced anisotropy in granular media","volume":"47","author":"Guo","year":"2013","journal-title":"Comput. Geotech."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"051302","DOI":"10.1103\/PhysRevE.65.051302","article-title":"Shear profiles and localization in simulations of granular materials","volume":"65","author":"Aharonov","year":"2002","journal-title":"Phys. Rev. E"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"591","DOI":"10.1061\/(ASCE)1090-0241(2006)132:5(591)","article-title":"Particle shape effects on packing density, stiffness, and strength: Natural and crushed sands","volume":"132","author":"Cho","year":"2006","journal-title":"J. Geotech. Geoenviron. Eng."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"e2020GL087763","DOI":"10.1029\/2020GL087763","article-title":"A weakening rheology of dry granular flows with extensive brittle grain damage in high-speed rotary shear experiments","volume":"47","author":"Hu","year":"2020","journal-title":"Geophys. Res. Lett."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"104476","DOI":"10.1016\/j.compgeo.2021.104476","article-title":"Effects of gravel content and shape on shear behavior of soil-rock mixture: Experiment and DEM modelling","volume":"141","author":"Yao","year":"2022","journal-title":"Comput. Geotech."}],"container-title":["Materials"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1996-1944\/19\/18\/3920\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,9,15]],"date-time":"2026-09-15T13:37:02Z","timestamp":1789479422000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1996-1944\/19\/18\/3920"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,9,15]]},"references-count":56,"journal-issue":{"issue":"18","published-online":{"date-parts":[[2026,9]]}},"alternative-id":["ma19183920"],"URL":"https:\/\/doi.org\/10.3390\/ma19183920","relation":{},"ISSN":["1996-1944"],"issn-type":[{"value":"1996-1944","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,9,15]]}}}