The Initial Boundary-Value Problem for a Mathematical Model for Granular Medium

Abstract:

In this work we consider a mathematical model for granular medium. Here we claim that Reduced Cosserat continuum is a suitable model to describe granular materials. Reduced Cosserat Continuum is an elastic medium, where all translations and rotations are independent. Moreover a force stress tensor is asymmetric and a couple stress tensor is equal to zero. Here we establish the variational (weak) form of an initial boundary-value problem for the reduced Cosserat continuum. We calculate the variation of corresponding Hamiltonian to obtain motion differential equation.

• Periodical: Applied Mechanics and Materials (Volumes 725-726)
• Pages: 863-868
• DOI: https://doi.org/10.4028/www.scientific.net/AMM.725-726.863
• Online since: January 2015
• Authors: Vladimir Lalin, Elizaveta Zdanchuk*
• Keywords: Granular Medium, Hamilton’s Principle, Reduced Cosserat Continuum, Variational Form of the Initial Boundary-Value Problem

References

[1] Daraio, C., Nesterenko, V.F., Herbold, E.B.,  Jin S.  Energy Trapping and Shock Disintegration in a Composite Granular Medium (1996) Physical Review Letters, 96 (5).

DOI: 10.1103/physrevlett.96.058002

[2] Hall, S.A., Wood, D.M., Ibraim, E., and Viggiani G. Localised deformation patterning in 2D granular materials revealed by digital image correlation (2010) Granular Matter 12, pp.1-14.

DOI: 10.1007/s10035-009-0155-1

[3] Majmudar, T.S., Behringer R.P. Contact force measurements and stress-induced anisotropy in granular materials (2005) Nature 435, pp.1079-1082.

DOI: 10.1038/nature03805

[4] Kvapil R. Gravity flow of granular materials in Hoppers and bins in mines (1965) International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, Volume 2, Issue 3, p.277–292.

DOI: 10.1016/0148-9062(65)90029-x

[5] Revuzhenko A.F. Mechanics of Granular Media (2006) Mechanics of Granular Media, 308 p.

[6] Springman, S., Laue, J., Seward L. Physical Modelling in Geotechnics (2010) Physical Modelling in Geotechnics, 1552 p.

DOI: 10.1201/b10554

[7] Peijun Guo, Xubin Su, Shear strength, interparticle locking, and dilatancy of granular materials (2007) Canadian Geotechnical Journal, 44(5), pp.579-591.

DOI: 10.1139/t07-010

[8] Kenney, T. C., Lau, D., Ofoegbu G. I. Permeability of compacted granular materials (1984) Canadian Geotechnical Journal, 21(4), pp.726-729.

DOI: 10.1139/t84-080

[9] Aki, K., Richards P.G. Quantitative seismology. Theory and methods (1980) Quantitative seismology. Theory and methods, 700p.

[10] Dhaar. Foundations of Community Medicine, 2/e (2008) Foundations of Community Medicine, 2/e, 820 p.

[11] Bártolo, P.J., Bidanda B. Bio-Materials and Prototyping Applications in Medicine (2007) Bio-Materials and Prototyping Applications in Medicine, 227 p.

DOI: 10.1007/978-0-387-47683-4

[12] Schwartz, L.M., Johnson, D.L., Feng S. Vibrational modes in granular materials (1984) Physical review letters, v. 52, №10, pp.831-834.

DOI: 10.1103/physrevlett.52.831

[13] Grekova, E.F., Kulesh, M.A., Herman G.C. Waves in linear elastic media with microrotations, part 2: Isotropic reduced Cosserat model (2009) Bulletin of the Seismological Society of America, 99 (2 B), pp.1423-1428.

DOI: 10.1785/0120080154

[14] Grekova E.F. Nonlinear isotropic elastic reduced Cosserat continuum as a possible model for geomedium and geomaterials. Spherical prestressed state in the semilinear material (2012) Journal of seismology, vol 16, issue 4, pp.695-707.

DOI: 10.1007/s10950-012-9299-2

[15] Lalin, V., Zdanchuk E. Reduced Cosserat continuum as a possible model for granular medium (2014) Proceedings of the International Conference Innovative Materials, Structures and Technologies, Riga, pp.90-93.

DOI: 10.7250/iscconstrs.2014.15

[16] Lalin, V., Zdanchuk E. Nonlinear thermodynamic model for granular medium (2014).

[17] Lalin, V., Zdanchuk E. Nonlinear thermodynamic model for reduced Cosserat continuum (2014) International Journal of Mathematical Models and Methods in Applied Sciences, V. 8, pp.208-213.

[18] Zdanchuk, E., Lalin V. The theory of continuous medium with free rotation without coupled stresses (2010) Proceedings of the XXXVIII Summer School- Conference Advanced problems in mechanics, pp.771-775.

[19] Lurie A.I. Nonlinear theory of elasticity (1990) Nonlinear theory of elasticity, 617 p.