Abstract
Micromechanical analysis has the potential to resolve many of the deficiencies of constitutive equations of granular continua by incorporating information obtained from particle-scale measurements. The outstanding problem in applying micromechanics to granular media is the projection scheme to relate continuum variables to particle-scale variables. Within the confines of a projection scheme that assumes affine motion, contact laws based on binary interactions do not fully capture important instabilities. Specifically, these contact laws do not consider mesoscale mechanics related to particle group behaviour such as force chains commonly seen in granular media. The implications of this are discussed in this paper by comparison of two micromechanical constitutive models to particle data observed in computer simulations using the discrete element method (DEM). The first model, in which relative deformations between isolated particle pairs are projected from continuum strain, fails to deliver the observed behaviour. The second model accounts for the contact mechanics at the mesoscale (i.e. particle group behaviour) and, accordingly, involves a nonaffine projection scheme. In contrast with the first, the second model is shown to display strain softening behaviour related to dilatancy and produce realistic shear bands in finite element simulations of a biaxial test. Importantly, the evolution of microscale variables is correctly replicated.
Similar content being viewed by others
References
Aifantis E.C. (1984). Remarks on media with microstructure. Int. J. Eng. Sci. 22: 961–968
Bagi K. (1996). Stress and strain in granular assemblies. Mech. Mater. 22: 165–177
Bathurst R.J., Rothenburg L. (1990). Observations on stress–force–fabric relationships in idealized granular materials. Mech. Mater. 9: 65–80
de Borst R., Mühlhaus H.-B. (1992). Gradient-dependent plasticity: formulation and algorithmic aspects. Int. J. Numer. Methods Eng. 35: 521–539
de Borst R., Sluys L.J. (1991). Localisation in a Cosserat continuum under static and dynamic loading conditions. Comput. Methods Appl. Mech. Eng. 90: 805–827
Calvetti F., Combe G., Lanier J. (1997). Experimental micromechanical analysis of a 2D granular material: relation between structure evolution and loading path. Mech. Cohesive-Fric. Mater. 2: 121–163
Chang C.S., Liao C.L. (1990). Constitutive relation for a particulate medium with the effect of particle rotation. Int. J. Solids Struct. 26(4): 437–453
Collins I.F. (2005). The concept of stored plastic work or frozen elastic energy in soil mechanics. Geotechnique 55(5): 373–382
Collins I.F., Houlsby G.T. (1997). Applications of thermomechanical principles to the modeling of geotechnical materials. Proc. Roy. Soc. London A 453: 1975–2001
Cook, B.K., Jensen, R.P. (eds.): Discrete Element Methods: Numerical Modeling of Discontinua. ASCE Geotechnical Special Publication No. 117. ASCE, Reston, VA (2002)
Edwards S.F., Grinev D.V. (2003). Statistical mechanics of granular materials: stress propagation and distribution of contact forces. Granular Matter 4: 147–153
Eringen A.C. (1968). Theory of micropolar elasticity. In: Liebowitz, H. (eds) Fracture—An Advanced Treatise, vol. II, pp 621–693. Academic Press, New York
Gardiner B.S., Tordesillas A. (2003). Micromechanical constitutive modelling of granular media: evolution and loss of contacts in particle clusters. J. Eng. Math. 52(1): 93–106
Goldenberg C., Goldhirsch I. (2002). Force chains, microelasticity and macroelasticity. Phys. Rev. Lett. 89(8): 084302
Houlsby G.T., Puzrin A.M. (2000). A thermomechanical framework for constitutive models for rate-independent dissipative materials. Int. J. Plasticity 16: 1017–1047
Jaeger H.M., Nagel S.R., Behringer R.P. (1996). Granular solids, liquids and gases. Rev. Mod. Phys. 64(4): 1259–1273
Liao C.-L., Chang T.-P., Young D.-H., Chang C.S. (1997). Stress–strain relation for granular materials based on hypothesis of best fit. Int. J. Solids Struct. 64(31–32): 4087–4100
Majmudar T.S., Behringer R.P. (2005). Contact force measurements and stress-induced anisotropy in granular materials. Nature 435: 1079–1082
Mindlin R.D. (1965). Second gradient of strain and surface tension in linear elasticity. Int. J. Solids Struct. 1: 417–438
Oda M. (1993). Inherent and induced anisotropy in plasticity theory of granular soils. Mech. Mater. 16: 35–45
Oda, M., Iwashita, K. (eds.): Mechanics of Granular Materials: An Introduction. A.A. Balkema, Rotterdam (1999)
Oda M., Iwashita K. (2000). Study on couple stress and shear band development in granular media based on numerical simulation analyses. Int. J. Eng. Sci. 38: 1713–1740
Oda M., Kazama H. (1998). Microstructure of shear bands and its relation to the mechanisms of dilatancy and failure of dense granular soils. Geotechnique 48(4): 465–481
Oda M., Yoshida T. (1999). Recent laboratory study 1: shear band development. In: Oda, M., Iwashita, K. (eds) Mechanics of Granular Materials, pp 299–308. A.A. Balkema, Rotterdam
Peters J.F., Muthuswamy M., Wibowo J., Tordesillas A. (2005). Characterization of force chains in granular material. Phys. Rev. E 72: 041307
Puzrin A.M., Houlsby G.T. (2001). A thermomechanical framework for rate-independent dissipative materials with internal functions. Int. J. Plasticity 17: 1147–1165
Radjai F., Wolf D.E., Jean M., Moreau J. (1998). Bimodal character of stress transmission in granular packings. Phys. Rev. Lett. 80(1): 61–64
Roscoe K.H., Schofield A.N., Wroth C.P. (1958). On the yielding of soils. Geotechnique 8(1): 25–53
Roscoe, K.H., Burland, J.B.: On the Generalized Stress–Strain Behavior of ‘Wet’ Clay, Engineering Plasticity, pp. 539–609. Cambridge University Press, Cambridge (1968)
Rothenburg L., Bathurst R.J. (1989). Analytical study of induced anisotropy in idealized granular materials. Geotechnique 39(4): 601–614
Tejchman J., Herle I., Wehr J. (1999). FE-studies on the influence of initial void ratio, pressure level and mean grain diameter on shear localization. Int. J. Numer. Anal. Methods Geomech. 23: 2045–2074
Tordesillas A., Peters J.F., Gardiner B. (2004). Shear band evolution and accumulated microstructural development in Cosserat media. Int. J. Numer. Anal. Methods Geotech. Eng. 29: 981–1010
Tordesillas A., Walsh S.D.C. (2002). Incorporating rolling resistance and contact anisotropy in micromechanical models of granular media. Powder Technol. 124: 106–111
Valanis K.C. (1996). A gradient theory of internal variables. Acta Mech. 116: 1–14
Valanis K.C., Peters J.F. (1991). An endochronic plasticity theory with shear-volumetric coupling. Int. J. Numer. Anal. Methods Geomech. 15: 77–102
Valanis K.C., Peters J.F. (1996). Ill-posedness of the initial and boundary value problems in non-associative plasticity. Acta Mech. 114: 1–25
Vardoulakis I., Graf B. (1985). Calibration of constitutive models for granular materials using data from biaxial experiments. Geotechnique 35: 299–317
Tordesillas, A., Walsh, S.D.C.: Analysis of deformation and localization in thermomicromechanical Cosserat models of granular media. In: Garcia-Rojo, H.J., Herrmann, S., McNamara, R. (eds.) Proceedings of the Fifth International Conference on the Micromechanics of Granular Media Powders and Grains 2005. Powders and Grains, vol. 1, pp. 419–424. A.A. Balkema, Rotterdam (2005)
Walsh S.D.C., Tordesillas A. (2004). A thermomechanical approach to the development of micropolar constitutive models of granular media. Acta Mech. 167(3–4): 145–169
Tordesillas A., Walsh S.D.C., Gardiner B. (2004). Bridging the length scales: Micromechanics of granular media. BIT Numer. Maths. 44: 539–556
Tordesillas, A., Walsh, S.D.C., Muthuswamy, M.: Role of mesoscale kinematics and non-affine motion in the transition from particle to bulk mechanical properties. J. Engng. Mech. ASCE (2007) (in press)
Author information
Authors and Affiliations
Corresponding author
Additional information
This paper is dedicated to Professor Ching S. Chang on the occasion of his 60th birthday.
Rights and permissions
About this article
Cite this article
Walsh, S.D.C., Tordesillas, A. & Peters, J.F. Development of micromechanical models for granular media. Granular Matter 9, 337–352 (2007). https://doi.org/10.1007/s10035-007-0043-5
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10035-007-0043-5