Abstract
We introduce a mathematical formalism towards the construction of a continuum-field theory for particulate fluids and solids. We briefly outline a research program aimed at unifying the fundamentals of the coarse-graining theory and the numerical method of Smoothed Particle Hydrodynamics (SPH). We show that the coarse-graining functions must satisfy well defined mathematical properties that comply with those of the SPH kernel integral representation of continuous fields. Given the appropriate dynamics for the macroscopic response, the present formalism is able to describe both the solid and fluid-like behaviour of granular materials.
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Notes
- 1.
For granular systems this approach seems doomed from the outset: because energy is lost through internal friction, and gained by a non-thermal source such as tapping or shearing, the dynamical equations do not leave the canonical or any other known ensemble invariant.
- 2.
Remark 1 In general, an \(n\)-rank tensor function \(G:\fancyscript{D}\rightarrow \mathbb {R}^4\otimes ^{[n]}\mathbb {R}^4\) (where \(\otimes ^{[n]}\) denotes the \(n\)-rank tensor product) is not an observable. Nevertheless, each component of the tensor is an observable. Typical examples are the components of the stress tensor (\(\sigma ^{\alpha \beta }\), where Greek indices denote Cartesian coordinates), or the components of the velocity field (\(v^{\alpha }\)).
References
de Gennes PG (1999) Granular matter. A tentative view. Rev Mod Phys 71:S374–S382
Gingold RA, Monaghan JJ (1977) Smoothed particle hydrodynamics–Theory and application to non-spherical stars. Monthly Not R Astron Soc 181:375–389
Glasser BJ, Goldhirsch I (2001) Scale dependence, correlations, and fluctuations of stresses in rapid granular flows. Phys Fluids 13:407–420
Goldhirsch I (2003) Rapid granular flows. Annu Rev Fluid Mech 35:267–293
Gray JP, Monaghan JJ, Swift RP (2001) SPH elastic dynamics. Comput Methods Appl Mech Eng 190:6641–6662
Babic M (1997) Average balance equations for granular materials. Int J Eng Sci 5:523–548
Irving JH, Kirkwood JG (1950) The statistical mechanical theory of transport processes. IV. The equations of hydrodynamics. J Chem Phys 18:817–829
Jaeger HM, Nagel SR, Behringer RP (1996) Granular solids, liquids, and gases. Rev Mod Phys 68:1259–1273
Kadanoff LP (1999) Built upon sand: theoretical ideas inspired by granular flows. Rev Mod Phys 71:435–444
Kirkwood JG (1946) The statistical mechanical theory of transport processes I. General theory. J Chem Phys 14:180–201
Li S, Liu WK (2007) Meshfree particle methods. Springer-Verlag, Berlin
Liu GR, Liu MB (2003) Smoothed particle hydrodynamics: a meshfree particle method. World Scientific, Singapore
Liu MB, Liu GR, Lam KY (2003) Constructing smoothing functions in smoothed particle hydrodynamics with applications. J Comp Appl Math 155:263–284
Liu MB, Liu GR (2006) Restoring particle consistency in smoothed particle hydrodynamics. Appl Num Math 56:19–36
Liu MB, Liu GR (2010) Smoothed particle hydrodynamics (SPH): an overview and recent developments. Arch Comput Methods Eng 17:25–76
Lucy LB (1977) A numerical approach to the testing of the fission hypothesis. Astro J 82:1013–1024
Monaghan JJ (1992) Smoothed particle hydrodynamics. Ann Rev Astron Astrophys 30:543–574
Murdoch AI, Kowalski SJ (1992) On fluid-fluid coupling within porous media: a mixture theoretic approach based upon molecular considerations. Transp Porous Media 8:47–70
Murdoch AI, Bedeaux D (1994) Continuum equations of balance via weighted averages of microscopic quantities. Proc R Soc A 445:157–179
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Marín, J.F., Petit, J.C., Sigalotti, L.D.G., Trujillo, L. (2014). Integral Representation for Continuous Matter Fields in Granular Dynamics. In: Sigalotti, L., Klapp, J., Sira, E. (eds) Computational and Experimental Fluid Mechanics with Applications to Physics, Engineering and the Environment. Environmental Science and Engineering(). Springer, Cham. https://doi.org/10.1007/978-3-319-00191-3_33
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DOI: https://doi.org/10.1007/978-3-319-00191-3_33
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