Abstract
In the present work we treat granular materials as mixtures composed of a solid and a surrounding void continuum, proposing then a continuum thermodynamic theory for it. In contrast to the common mass-weighted balance equations of mass, momentum, energy and entropy for mixtures, the volume-weighted balance equations and the associated jump conditions of the corresponding physical quantities are derived in terms of volume-weighted field quantities here. The evolution equations of volume fractions, volume-weighted velocity, energy, and entropy are presented and explained in detail. By virtue of the second law of thermodynamics, three dissipative mechanisms are considered which are specialized for a simple set of linear constitutive equations. The derived theory is applied to the analysis of reversible and irreversible compaction of cohesionless granular particles when a vertical oscillation is exerted on the system. In this analysis, a hypothesis for the existence of a characteristic depth within the granular material in its closely compacted state is proposed to model the reversible compaction.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Oda M. and Iwashita K. (1999). Mechanics of Granular Materials. Balkema Publishers, Netherlands
Duran J. (2000). Sand, Power and Grains. Springer, Berlin
Hinrichsen H. and Wolf D.E. (2004). The Physics of Granular Media. WILEY-VCH, Weinheim
Hutter K., Svendsen B. and Rickenmann D. (1996). Debris flow modeling: a review. Contin. Mech. Thermodyn. 8: 1–35
Ottino J.M. and Khakhar D.V. (2000). Mixing and segregation of granular materials. Annu. Rev. Fluid Mech. 32: 55–91
Wang Y. and Hutter K. (2001). Granular material theories revisited. In: Balmforth, N.J. and Provenzale, A. (eds) Geomorphological Fluid Mechanics., pp 79–107. Springer, Berlin
Chang C.S. and Liao C.L. (1994). Estimate of elastic modulus for media of randomly packed granules. Appl. Mech. Rev. 47(1): 197–206
Mullins W.W. (1972). Stochastic theory of particle flow under gravity. J. Appl. Phys. 43: 665–678
Caram H.S. and Hong D.C. (1992). Diffusing void model for granular flow. Mod. Phys. Lett. B 6: 761–771
Fried E. and Roy B.C. (2003). Gravity-induced segregation of cohensionless granular mixtures. In: Hutter, K. and Kirchner, N. (eds) Dynamic Response of Granular and Porous Materials under Large and Catastrophic Deformation. Lecture Notes in Applied and Computational Mechanics 11, pp 393–421. Springer, Berlin
Fried E., Gurtin M.E. and Hutter K. (2004). A void-based description of compaction and segregation in flowing granular materials. Contin. Mech. Thermodyn. 16: 199–219
Möbius M.E., Cheng X., Karczmar G.S., Nagel S.R. and Jaeger H.M. (2004). Intruders in the dust: air-driven granular size separation. Phys. Rev. Lett. 93: 198001
Yan X., Shi Q., Hou M., Lu K. and Chan C.K. (2003). Effects of air on the segregation of particles in a shaken granular bed. Phys. Rev. Lett. 91: 014302
Möbius M.E., Cheng X., Eshuis P., Karczmar G.S., Nagel S.R. and Jaeger H.M. (2005). Effects of air on granular size separation in a vibrated granular bed. Phys. Rev. E. 72: 011304
Naylor M.A., Swift M.R. and King P.J. (2003). Air-driven Brazil nut effect. Phys. Rev. E. 68: 012301
Truesdell C.A. and Toupin R.A. (1957). The classical field theories. In: S., Flügge (eds) Encyclopedia of Physics, Vol. III/1, pp 226–793. Springer, Berlin
Bowen R.M. (1976). Theory of mixtures. In: Eringen, A.C. (eds) Continuum Physics III, pp 1–127. Academic, New York
Goodman M.A. and Cowin S.C. (1972). A continuum theory for granular materials. Arch. Rational Mech. Anal. 44: 249–266
Vardoulakis I. and Aifantis E.C. (1991). A gradient flow theory of plasticity for granular materials. Acta Mech. 87: 197–217
Fang C., Wang Y. and Hutter K. (2006). A thermo-mechanical continuum theory with internal length for cohesionless granular materials Part I. a class of constitutive models. Contin. Mech. Thermodyn. 17: 545–576
Ahuja A.S. (1972). Formulation of wave equation for calculating velocity of sound in suspensions. J. Acoust. Soc. Am. 51: 916–919
Ahuja A.S. (1973). Wave equation and propagation parameters for sound propagation in suspensions. J. Appl. Phys. 44: 4863–4868
Wilmański K. (1998). A thermodynamic model of compressible porous materials with the balance equation of porosity. Trans. Porous Media 32: 21–47
Wilmański K. (1998). Thermomechanics of Continua. Springer, Berlin
Hutter K. and Jöhnk K. (2004). Continuum Methods of Physical Modeling. Springer, Berlin
Müller I. (1971). The coldness, a universal function in thermoelastic bodies. Arch. Rational Mech. Anal. 41: 319–332
Liu I.-S. (1972). Method of Lagrange multipliers for exploitation of the entropy principle. Arch. Rational Mech. Anal. 46: 131–148
Nowak E.R., Knight J.B., Povinelli M.L., Jaeger H.M. and Nagel S.R. (1997). Reversibility and irreversibility in the packing of vibrated granular material. Powder Technol. 94: 79–83
Chuang, W.F., Tai, Y.C., Chen, K.C.: On the reversibility in the packing of vertically vibrated granular materials. In: The 30th Conference of Theoretical and Applied Mechanics, Changhua, Taiwan, ROC, December, 2006
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by S.L. Gavrilyuk
Rights and permissions
About this article
Cite this article
Chen, KC., Tai, YC. Volume-weighted mixture theory for granular materials. Continuum Mech. Thermodyn. 19, 457–474 (2008). https://doi.org/10.1007/s00161-007-0064-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00161-007-0064-7