Abstract
We discuss a multi-scale continuum representation of bodies made of several mass particles flowing independently each other. From an invariance procedure and a nonstandard balance of inertial actions, we derive the balance equations introduced in earlier work directly in pointwise form, essentially on the basis of physical plausibility. In this way, we analyze their foundations. Then, we propose a Boltzmann-type equation for the distribution of kinetic energies within control volumes in space and indicate how such a distribution allows us to propose a definition of (granular) temperature along processes far from equilibrium.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bacci, M., Mariano, P.M.: Protein dynamics: an approach based on the Cauchy–Born rule. Physica E 61, 69–82 (2014)
Brocato, M., Capriz, G.: Clockwork, ephemeral and hybrid continua. Phys. Mesomech. 14, 124–144 (2011)
Capriz, G.: Elementary preamble to a theory of granular gases. Rend. Mat. Univ. Padova 110, 179–198 (2003)
Capriz, G.: Pseudofluids. In: Capriz, G., Mariano, P.M. (eds.) Material Substructures in Complex Bodies: From Atomic Level to Continuum, pp. 238–261. Elsevier, Amsterdam (2006)
Capriz, G.: A quest for an ‘extended’ continuum mechanics. Note di Matematica 27, 27–41 (2007)
Capriz, G.: On ephemeral continua. Phys. Mesomech. 11, 285–298 (2008)
Capriz, G., Giovine, P.: Hypocontinua. In: Albers, B., Kuczma, M. (eds.) Continuum Media with Microstructure 2, pp. 23–43. Springer, Berlin (2016)
Capriz, G., Giovine, P.: Classes of ephemeral continua. Math. Methods Appl. (2017) https://doi.org/10.1002/mma.4658
Capriz, G., Mariano, P.M.: Objective fluxes in a multiscale continuum description of sparse medium dynamics. Physica A 415, 354–365 (2014)
Capriz, G., Mazzini, G.: Invariance and balance in continuum mechanics. Nonlinear Analysis and Continuum Mechanics, pp. 27–35. Springer, New York (1998)
Cohen, H., Muncaster, R.G.: The Theory of Pseudo-rigid Bodies. Springer, New York (1988)
de Groot, S.R., Mazur, P.: Non-equilibrium Thermodynamics. Dover, New York (1984)
Giovine, P.: Nonclassical thermomechanics of granular materials. Math. Phys. Anal. Geom. 2, 179–196 (1999)
Giovine, P.: An extended continuum theory for granular media. Mathematical Models of Granular Matter, Lecture Notes in Mathematics pp. 167–192. Springer, Berlin (2008)
Landau, L.D., Lifshitz, E.M.: Course of Theoretical Physics. Vol. 5: Statistical Physics, second revised and enlarged edition. Pergamon Press, Oxford-Edinburgh-New York (1968)
Landau, L.D., Lifshitz, E.M.: Course of Theoretical Physics. Vol. 3: Quantum Mechanics: Non-relativistic Theory. Pergamon Press Ltd., London-Paris; for U.S.A. and Canada: Addison-Wesley Publishing Co., Inc., Reading, MA (1958)
Lieb, E.H., Yngvason, J.: The physics and mathematics of the second law of thermodynamics. Phys. Rep. 310, 1–96 (1999)
Lumley, J.L.: Toward a turbulent constitutive relation. J. Fluid Mech. 41, 413–434 (1970)
Lumley, J.L.: Turbulence modeling. J. Appl. Mech. 50, 1097–1103 (1983)
Mariano, P.M.: Multifield theories in mechanics of solids. Adv. Appl. Mech. 38, 1–93 (2002)
Mariano, P.M.: Mechanics of material mutations. Adv. Appl. Mech. 47, 1–91 (2014)
Müller, I., Ruggeri, T.: Rational Extended Thermodynamics, 2nd edn. With supplementary chapters by H. Struchtrup and Wolf Weiss, Springer Tracts in Natural Philosophy, vol. 37. Springer, New York (1998)
Noll, W.: Die Herleitung der Grundgleichungen der Thermomechanik der Kontinua aus der statistichen Mechanik. J. Ration. Mech. Anal. 4, 627–646 (1955)
Sagaut, P., Deck, S., Terracol, M.: Multiscale and Multiresolution Approaches in Turbulence. Imperial College Press, London (2006)
Segev, R.: Fluxes and flux-conjugated stresses. In: Capriz, G., Mariano, P.M. (eds.) Advances in Multifield Theories of Continua with Substructure, pp. 149–165. Birkäuser, Basel (2004)
Šilhavý, M.: Cauchy’s stress theorem and tensor fields with divergences in L\(^{p}\). Arch. Ration. Mech. Anal. 116, 223–255 (1991)
Truesdell, C.A.: A First Course in Rational Continuum Mechanics. Academic Press, New York (1977)
Truesdell, C.A.: Hypoelasticity. J. Ration. Mech. Anal. 4, 83–133 (1955)
Truesdell, C.A.: Rational Thermodynamics. A Course of Lectures on Selected Topics. McGraw-Hill Book Co., New York (1969)
Truesdell, C.A., Muncaster, R.G.: Fundamentals of Maxwell’s Kinetic Theory of a Simple Monatomic Gas. Treated as a Branch of Rational Mechanics. Academic Press, Inc., New York-London (1980)
Ván, P.: Weakly nonlocal irreversible thermodynamics. Annalen der Physik (Leipzig) 12, 142–169 (2003)
Vilar, J.M.G., Rubi, J.M.: Thermodynamics “beyond” local equilibrium. Proc. Nat. Acad. Sci. USA 98, 11081–11084 (2001)
Acknowledgements
This work has been developed within the programs of the research group in ‘Theoretical Mechanics’ of the ‘Centro di Ricerca Matematica Ennio De Giorgi’ of the Scuola Normale Superiore in Pisa. The support of GNFM-INDAM is acknowledged.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Andreas Öchsner.
Rights and permissions
About this article
Cite this article
Capriz, G., Mariano, P.M. Multi-scale kinetic description of granular clusters: invariance, balance, and temperature. Continuum Mech. Thermodyn. 30, 1323–1342 (2018). https://doi.org/10.1007/s00161-017-0613-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00161-017-0613-7