Abstract
A local constitutive model for anisotropic granular materials is introduced and applied to isobaric (homogeneous) axial-symmetric deformation. The simplified model (in the coordinate system of the bi-axial box) involves only scalar values for hydrostatic and shear stresses, for the volumetric and shear strains as well as for the new ingredient, the anisotropy modulus. The non-linear constitutive evolution equations that relate stress and anisotropy to strain are inspired by observations from discrete element method (DEM) simulations. For the sake of simplicity, parameters like the bulk and shear modulus are set to constants, while the shear stress ratio and the anisotropy evolve with different rates to their critical state limit values when shear deformations become large. When applied to isobaric deformation in the bi-axial geometry, the model shows ratcheting under cyclic loading. Fast and slow evolution of the anisotropy modulus with strain. Lead to dilatancy and contractancy, respectively. Furthermore, anisotropy acts such that it works “against” the strain/stress, e.g., a compressive strain builds up anisotropy that creates additional stress acting against further compression.
Article PDF
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
References
Alonso-Marroquin F., Herrmann H.J.: Ratcheting of granular materials. Phys. Rev. Lett 92(5), 054301 (2004)
Alonso-Marroquin F., Luding S., Herrmann H.J., Vardoulakis I.: Role of anisotropy in the elastoplastic response of a polygonal packing. Phys. Rev. E 71, 0513404 (2005)
Chen Y., Ishibashi I., Jenkins J.: Dynamic shear modulus and fabric: part I, depositional and induced anisotropy. Géotechnique 1(38), 25–32 (1988)
Chen Y., Ishibashi I., Jenkins J.: Dynamic shear modulus and fabric: part II, stress reversal. Géotechnique 1(38), 33–37 (1988)
Gennes P.G.: Granular matter: a tentative view. Rev. Mod. Phys. 71(2), 374–382 (1999)
Geng L., Reydellet G., Clément E., Behringer R.P.: Green’s function measurements in 2D granular materials. Physica D 182, 274–303 (2003)
Goddard, J.: Granular hypoplasticity with Cosserat effects. In: Goddard, J., Giovine, P., Jenkins, J.T. (eds.) IUTAM-ISIMM Symposium on Mathematical Modeling and Physical Instances of Granular Flows, AIP, 14–18 September 2009, pp. 323–332. Reggio Calabria (Italy) (2010)
Göncü F., Duran O., Luding S.: Constitutive relations for the isotropic deformation of frictionless packings of polydisperse spheres. C. R. Mecanique 338(10–11), 570–586 (2010)
Jaeger H.M., Nagel S.R., Behringer R.P.: Granular solids, liquids, and gases. Rev. Mod. Phys 68(4), 1259–1273 (1996)
Jenkins J., Cundall P., Ishibashi I.: Micromechanical modeling of granular materials with the assistance of experiments and numerical simulations. In: Biarez, J., Gourves, M. (eds) Powders and Grains, pp. 257–264. Balkema, Rotterdam (1989)
Jiang Y., Liu M.: Granular solid hydrodynamics. Granul. Matter 11(3), 139–156 (2009)
Khidas Y., Jia X.: Anisotropic nonlinear elasticity in a spherical-bead pack: influence of the fabric anisotropy. Phys. Rev. E 81, 21303 (2010)
Kolymbas D.: An outline of hypoplasticity. Arch. App. Mech. 61, 143–154 (1991)
Luding S.: Micro-macro models for anisotropic granular media. In: Vermeer, P.A., Ehlers, W., Herrmann, H.J., Ramm, E. (eds) Modelling of Cohesive-Frictional Materials, pp. 195–206. Balkema, Leiden, The Netherlands (2004)
Luding S.: Anisotropy in cohesive, frictional granular media. J. Phys.: Condens. Matter 17, S2623–S2640 (2005)
Luding S., Alonso-Marroquin F.: The critical-state yield stress (termination locus) of adhesive powders from a single numerical experimen. Granul. Matter 13(2), 109–119 (2011)
Luding, S., Perdahcioglu, S.: A local constitutive model with anisotropy for various homogeneous 2D biaxial deformation modes. CIT, doi:10.1002/cite.201000180 (2011)
Muhlhaus H., Moresi L., Gross L., Grotowski J.: The influence of non-coaxiality on shear banding in viscous-plastic materials. Granul. Matter 12, 229–238 (2010)
Sun, J., Sundaresan, S.: A plasticity model with microstructure evolution for quasi-static granular flows. In: Goddard, J., Giovine, P., Jenkins, J.T. (eds.) IUTAM-ISIMM Symposium on Mathematical Modeling and Physical Instances of Granular Flows, 14–18 September 2009, AIP, pp. 280–289. Reggio Calabria (Italy) (2010)
Sun, J., Sundaresan, S.: A plasticity model with microstructure evolution for quasi-static granular flows. J. Fluid Mech. (2010 submitted)
Tatsuoka F., Ishihara K.: Drained deformation of sand under cyclic stresses reversing direction. Soils. found. 3(14), 51–65 (1974)
Tejchman J., Bauer E., Wu W.: Effect of fabric anisotropy on shear localization in sand during plane strain compression. Acta Mech. 189, 23–51 (2007)
Vardoulakis I., Frantziskonis G.: Micro-structure in kinematic-hardening plasticity. Eur. J. Mech. A Solids 11(4), 467 (1992)
Vardoulakis I., Sulem J.: Bifurcation Analysis in Geomechanics. Chapman and Hall, London (1995)
Zhang J., Majmudar T.S., Tordesillas A., Behringer R.P.: Statistical properties of a 2D granular material subjected to cyclic shear. Granul. Matter 12, 159–172 (2010)
Acknowledgments
This work is dedicated to late Prof. I. Vardoulakis and his inspiring publications. Helpful discussions with J. Goddard, D. Krijgsman, M. Liu, S. McNamara, E. S. Perdahcıog̃lu, A. Singh, S. Srivastava, H. Steeb, J. Sun and S. Sundaresan are gratefully acknowledged. The work was financially supported by an NWO-STW VICI grant.
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Magnanimo, V., Luding, S. A local constitutive model with anisotropy for ratcheting under 2D axial-symmetric isobaric deformation. Granular Matter 13, 225–232 (2011). https://doi.org/10.1007/s10035-011-0266-3
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10035-011-0266-3