- redactor@odeku.edu.ua
- +38 0482 326 758
References
- O.I. Gerasymov, N.N. Khudyntsev Voronoi tessellations: The classical theory of moments for structure description of granular materials
- Gerasymov O.I., Spivak A.Ya. Transmission (propagation) of perturbation in 1D granular chain under the confinement
- Gerasymov O., Spivak A. Modeling of impulse propagation in 1D inhomogeneous granular lattices: the influence of boundary conditions
- Gerasymov O.I., Somov M.M. Entropic lattice gas model for description of equilibrium vertical density profile in granular materials subjected to gravity field
- Gerasymov O.I., Somov M.M. Stereological analysis of local structure of granular materials (Voronoi method)
- Gerasymov O.I. A new class of rigorous solutions of the functional differential equation of motion for the mechanical impulses in the 1D nonhomogeneus granular chain A new rigorous solution of the functional differential equation for signal propagation through the vertical granular
- Gerasymov O.I., Somov M.M. Local structure of granular materials
- Gerasymov, O., N. Khudyntsev, O. Klymenkov, and A. Spivak The kinetics of processes occurring in granular materials in the field of vibroaccelerations
Papers
The theoretical description of the local structure of granular materials has been performed by means of Voronoi method. The detailed investigation of structure transformations has been carried on with help of Voronoi tessellation supplemented by direct modeling of the relevant distribution function in terms of classical moments theory. Analytical expression for distribution function of Voronoi figures has been constructed with the help of Nevanlinna’s formula from theory of orthogonal polynomials .Proposed approach permit to avoid the problem of week argumentation of applicability the statistical mechanics methods for description of the structure and physical properties of granular materials. We show that generated ordering in local structure are escorted by appearing of particular symmetries in Voronoi diagrams. We perform a numerical simulations of structural configurations in 2D system of hard discs. Proposed algorithm allow us to prove theoretical predictions about existence of correlations between configurational ordering and symmetry breaking in Voronoy tessellations. We study these effects in the vicinity of jammed states. Obtained results shows that criticality in structurisation (formation of jammed states)connected with particular behavior of the first two moments of Voronoi figures distribution function. We show nonhomogeneous character of jammed states in which kinematic freedom degrees become frozen. Namely, coexisting ordered domains which has a different symmetries in grain configurations are observed. Therefore given analysis fulfill the basis of research in the area of granular physics which are mostly based on the concepts of probabilistic stereology and do not use methods from statistical mechanics which in the case of granular materials are not enough argumeneted.
- Jaeger H.M., Nagel S.R., Behringer R.P. Rev. Mod. Phys., 1996, no. 68, pp. 1259.
- Duran J. Sands, Powders and Grains. New York: Springer, 2000.
- Kadanoff L. Rev.Mod.Phys., 1999, no. 71, pp. 435.
- De Gennes P.G., Rev. Mod. Phys., 1999, no. 71, pp. 374.
- Bideau D., Gervois A., Oger L., Troadec J.P. J.Physique, 1986, no. 47, pp. 1697.
- Lumay J., Vandewalle N. Phys. Rev. Lett., 2005, no. 95, pp. 028002.
- Voronoi G., Reine Angew J. Math., 1908, no. 134, pp. 198.
- Quickenden T.I., Tan G.K. Journal of Collloid and Interface Science , 1974, no. 48, pp. 382.
- Aste T., Di Matteo T. Phys. Rev., 2008, no. E77, pp. 021309.
- Edwards S.F. Physica, 2005, no. A353 , pp. 114.
- Achiezer N.I., Classical problem of moments. Мoscow: Nauka, 1961. 600 p.
- Gerasymov O.I. Ukr.Journ.Phys., 2010, vol.55, no. 5, pp. 560.
We use Bateman algorithm [1] to set exact solution of the linearized equations of motion for perturbation in vertical granular chain under the confinement. It is shown the wave dynamics is strongly influenced by boundary conditions and constitutive relations.
- Pinney E. Ordinary Difference-differential equations. – University of California Press, Berkley and Los Angeles, 1958. – 250p.; Bateman H. Some simple differential difference equations and the related functions. //Bulletin of the American Mathematical Society.-1943.-Vol.49, N7.-С. 494-512.
- Fermi E., Pasta J., Ulam S. Studies of non linear problems. //Los Alamos Sc.Lab.Rep. -1955.- Vol.LA-1940, P.978-988.
- Hascoёt E., Herrmann H.J., Loreto V. Shock propagation in a granular chain. //Phys. Rev. E -1999.-Vol.59, P.3202–3206.
- Sarmiento A., Reigada R., Romero A.H., Lindenberg K. Enhanced pulse propagation in nonlinear arrays of oscillators. //Phys. Rev. E – 1999.-Vol.60, P.5317–5326.
- Rosas A., Lindenberg K. Pulse dynamics in a chain of granules with friction. //Phys. Rev. E – 2003.-Vol.68, P.041304.
- Герасимов О.І., Співак А.Я., Худинцев М.М., Клименков О.А. Транспорт енергії (імпульсу) в модельних низьковимірних дисипативних системах. //Вісник ОДЕКУ.-2008.-№6.-С.225-233.
- Герасимов О.И. О новом классе точных решений дифференциально- разностного уравнения движения для механических возбуждений в одномерной неоднородной гранулированной цепочке. //Вісник ОДЕКУ.-2011.-№11.-С.198-202.
- Герасимов О.І., Вандевалле Н. Щодо точних розв’язків задачі про перенесення імпульсу у неоднорідному гранульованому ланцюжку. //Доповіді НАН України.-2012.-№8.-С.67-72.
- Gerasymov O., Spivak A. Performing of the momentum dynamics in 1D models of granular systems. //Book of extended abstracts of I-st Belgian-Ukrainian Mini-Symposium Selected Topics of Soft Matter Physics: Granular Materials. Edited by O.Gerasymov (Odessa, 2012), P.34-40.
- Герасимов О.І., Співак А.Я. Моделювання руху механічних збуджень у одновимірних неоднорідних гранульованих ланцюжках: вплив граничних умов. //Вісник ОДЕКУ.-2012.-№14.-С.217-223.
- Lumay G., Dorbolo S., Gerasymov O., Vandewalle N. Experimental study of a vertical column of grains submitted to a series of impulses. //Eur. Phys. J. E – 2013.- Vol.36, N2, P.1-6.
- Корн Г., Корн Т. Справочник по математике для научных работников и инженеров. – М.: Наука, 1984.-835с.
- Sen S., Hong J., Bang J., Avalos E., Doney R. Solitary waves in the granular chain. //Physics Reports-2008.-Vol.462, N2.-P.21-66.
- Porter M.A., Daraio C., Szelengowicz I., Herbold E.B., Kevrekidis P.G. Highly nonlinear solitary waves in heterogeneous periodic granular media. //Physica D: Nonlinear Phenomena.-2009.-Vol. 238, N6.-P.666-676.
- Gerasymov O.I., Vandewalle N., Spivak A.Ya. Towards the problem of momentum propagation along inhomogeneous granular chain. //Phil. Mag.-2014
We perform the linearized equations governing the impulse propagation along a 1D weakly nonlinear inhomogeneous granular chain of identical beads. It has been sown that the rigorous solution which satisfy considered problem in every point (including boundaries) can be choose in form of linear combination of farmer basis which consist on Bessel function an integer order and valid in the internal points only. Obtained results thought to be useful practically for the more adequate parameterization of impulse transport in low dimensional granular chains.
- Bhatnagar P.L. Nonlinear waves in one dimensional dispersive systems.-Oxford: Clarendon, 1979.-250p.
- Jackson E.A. Perspectives of nonlinear dynamics.-Cambridge, 1990.-350p.
- Nesterenko V. Dynamics of Heterogeneous materials.-New York: Springer, 2001.-350p.
- Coste C., Falcon E., Fauve S. Stationary states in vertical cradle excited from the bottom // Phys. Rev. E-1997.-Vol.56.- P.6104.
- Sen S., Hong J., Bang J., Avalos E., Doney R. Soliton-like waves in low-dimensional granular systems //Phys. Rep.-2008.-N462, P.21-66.
- E.Somfai, J-N.Roux, T.Snoeijer, M.van Hecke, W.van Saarlos. Dispersive waves in 1D granular chain //Phys. Rev. E-2005.-Vol.72.-P.021301.
- Герасимов О.І., Вандевалле Н., Співак А.Я., Худинцев М.М., Люмє Г., Дорболло С., Клименков О.А. Стаціонарні стани у 1D системі непружних частинок //Укр.фіз.журн. – 2008.- Т.53, № 11. – С.1129-1137.
- Герасимов О.І. Щодо двох точних розв’язків задачі про транспорт імпульсу в неоднорідному гранульованому ланцюжку //Доповіді НАН України. – 2012 (прийнято до друку).
- Landau L.D. Theory of elasticity, 3rd ed.- Heinemann: Butterworth, 1986.-200p.
- Pinney E. Ordinary Difference-differential equations.-Berkley and Los Angeles: University of California Press, 1955.-500p.
Entropy of lattice gas model used for description of equilibrium vertical density profile in granular materials subjected to gravity field. We perform the parameter (like Lindeman parameter in the theory of melting of the quantum crystals) which describe the structural deformations in considered systems. We observe good qualitative agreement between the data of theory and experimental observations, which became quantitative in the closest neighborhoods of the point of maximum compactisation.
- Gerasymov O.I. Structure and Dynamics of Granular Materials Perturbed by External Fields. //Ukr. Journ. Phys.-2010.-Vol.55, N 5.-P.586-592.
- Vandewalle N., Lumay G., Gerasymov O., Ludewig F. The influence of grain shape, friction and cohesion on granular compaction dynamics. //Eur.Phys.Journ.E.-2007.-22.- P.241-248.
- Герасимов О.І. Структура та динаміка гранульованих матеріалів. //Доповіді НАН України.-2010.-№11.-С.59-65.
- А. Исихара. Статистическая физика.-М.:МИР,1973.-443с.
- Holly Kokstein, Paul V. Quinn Sr. Fermi statistics applied to a weakly excited column of granular particles in a vibrating bed.//Physica A: Statistical Mechanics and its Applications.-2007.-Vol.374, N 1.-P.66-76.
- Киттель Ч. Введение в физику твёрдого тела М.: Наука, 1978.-600с.
- Abramovitz M., Stegun I.A. Handbook of mathematical functions with formulas, graphs, and mathematical tables.-Washington:US GPO, 10th, 1972.-1046p.
The method of Voronoi diagrams has been used for the local structure analysis of 2D system consist on hard disk. We propose the analytical model for the parameterization of the Voronoi squares distribution. Within the given approach the particular behavior of the parameters (moments) of the deterministic distribution influenced by the structure defects has been outlined.
- Jaeger H.M., Nagel S.R., Behringer R.P. The physics of granular materials. // Rev.Mod.Phys.-1996.-Vol.68.- P.1259-1273.
- Duran J. Sands, Powders and Grains.-New York: Springer Verlag, 2000.- 200p.
- Kadanoff L. Built upon sand: theoretical ideas inspired by granular flows. //Rev.Mod.Phys.- 1999.-Vol.71.-P.435-447.
- De Gennes P.G. Granular matter: a tentative view //Rev.Mod.Phys.-1999.-71,S.- P.374-385.
- Mehta A. Granular Physics .- Harvard, Massachusetts, 2009.-318p.
- Н.Н. Медведев, Метод Вороного-Делоне в исследовании структуры некристаллических систем .-Новосибирск: СО РАН, 2000.-214с.
- John C. Russ, Robert T. Dehoff, Practical stereology.- New York: Plenum Press, 1999.- 312p.
- Z. M. Jakˇsi´c, S. B. Vrhovac, B. M. Pani´c and B. M. Jelenkovi´. Upward penetration of grains through a granular medium //Eur.Phys.Journ.E-2006.-Vol.27.-P.345-356.
A new rigorous solution of the functional differential equation for signal propagation through the vertical granular chain with a nonlinear contacts has been found under the approximation of the weak nonlinearity in form of the Bessel function of first order. The linear combinations of the solutions we found in principle can satisfy an appropriate initial conditions. This new solution also is a significant supplement to the dispersive wave modes which has been discovered for such a system earlier and could be considered as an envelope function for them. The relevant experiments directed to the experimental studying of the discovered modes are under the current proceeding.
- Zabusky N.Z., Kruskal M.D. Inverse problem of scattering. //Phys. Rev. Lett.-1965-Vol.15.-P.240-245.
- Zakharov V.T., Shabat A.V. Nonlinear waves. //Zh.Trsper.Teor.Fiz.(JETP) -1971.-619.-P.118-124.
- Bhatnagar P.L. Nonlinear waves in one dimensional dispersive systems.-Oxford: Clarendon 1979.-250p.
- Jackson E.A. Perspectives of nonlinear dynamics.-Cambridge, 1990.-350p.
- Ford J. Solitary waves. //Phys. Rep.-1992.-N213.-P.271-296.
- Nesterenko V.F. Tapered granular chains sound propagation. //J. Appl. Mech. Tech. Phys.-1984.-N5. P.733-738.
- Nesterenko V.F. Solitary waves in 1D system of inelastic balls. //J. Phys. (France)-1994.-Vol.IV, N55.-P.729-736.
- Nesterenko V. Dynamics of Heterogeneous materials.-New York: Springer, 2001.-350p.
- Coste C., Falcon E., Fauve S. Stationary states in vertical cradle excited from the bottom. //Phys. Rev. E-1997.-Vol.56.- P.6104.
- Sen S., Hong J., Bang J., Avalos E., Doney R. Soliton-like waves in low-dimensional granular systems.//Phys. Rep.-2008.-N462, P.21-66.
- E.Somfai, J-N.Roux, T.Snoeijer, M.van Hecke, W.Van Saarlos, Dispersive waves in 1D granular chain.//Phys. Rev. E-2005.-Vol.72.-P.021301.
- Герасимов О.І., Вандевалле Н., Співак А.Я., Худинцев М.М., Люмє Г., Дорболло С., Клименков О.А. Стаціонарні стани у 1D системі непружних частинок //Укр.фіз.журн. -2008.- Т.53, № 11. – С.1129-1137.
- Falcon E., Laroche C., Fauve S., Coste C. Bouncing balls under the vibrating substrate. //Euro. Phys. J. B-1998.-Vol.5.-P.111-117.
- E.J.Hinch, S.Saint-Jean, Signal transmission through the cradle. //Proc.Roy.Soc.London A-1999.-N455.-P.3201-3210.
- E.Avalos, T.Krishna Mohan, S.Sen, Propagation of energy throughout granular chains. //Phys. Rev. E-2003. – Vol.67.-P.060301(R)
- Job S., Santibanez F., Tapia F., Melo F. Mechanical impulse passing the inhomogeneous systems.//Phys. Rev. E-2009.-Vol.80.-P.025602(R)
- U.Harbola, A.Rosas, A.Romero, M.Esposito, K.Lindenberg, Wave transport in granular chains. //Phys. Rev. E-2009.-Vol.80.-P.051302.
- J.Lima Dias Pinto, A.Rosas, A.Romero, K.Lindenberg, Rigorous solutions for a equations governing the energy distribution in discrete nonlinear systems. //Phys. Rev. E-2010.-Vol.82.-P.031308.
- Landau L.D. Theory of elasticity, 3rd ed.- Heinemann: Butterworth, 1986.-200p.
- Pinney E. Ordinary Difference-differential equations.-Berkley and Los Angeles: University of California Press, 1955.-500p.
- Vandewalle N., Lumay G., Dorbolo S. and Gerasymov O. Towards the problem of energy (impulse) transport in 1D inhomogeneous granular chains. //Phys.Rev.E.-2011. (in preparation)
Different methods of describing of the local structure of complex many-particle systems which based on conception of the translational and orientational order parameters are developed. The presence of anisotropic fluid phase which is the intermediate between states with short–range and long-range order is discovered. Inner-shell model for the description of the local structure of granular materials has been proposed.
- Jaeger H.M., Nagel S.R., Behringer R.P. The physics of granular materials. //Rev.Mod.Phys.-1996.-Vol.68.- P.1259-1273.
- Duran J. Sands, Powders and Grains._New York: Springer Verlag, 2000.- 200p.
- Kadanoff L. Built upon sand: theoretical ideas inspired by granular flows. //Rev.Mod.Phys.- 1999.-Vol.71.-P.435-447.
- De Gennes P.G. Granular matter: a tentative view //Rev.Mod.Phys.-1999.-71,S.- P.374-385.
- Mehta A. Granular Physics .- Harvard, Massachusetts, 2009.-318p.
- Patashinskii A.Z., Shumilo B.I. Theory of condensed matter based on the hypothesis of a local crystalline order //JETP.-1985.-Vol.62,N1.- P.177-184.
- Gerasymov O.I. Structure and Dynamics of Granular Materials Perturbed by External Fields. //Ukr. Journ. Phys.-2010.-Vol.55, N 5.-P.586-592.
- Reis P.M., Ingale R.A., Shattuck M.D. Crystallization of a quasi-two-dimensional granular fluid. //Phys.Rev.Lett.-2006.-Vol.96.-P.258001.
Theoretical modeling of granular compaction and segregation processes have been done. The relevant order parameter field relaxation is performed by means of solution of the appropriate equation of motion. The important character of obtained results for the general theory of phase transitions is emphasized.
- Duran J. Sands, Powders and Grains.-NY.: Springer-Verlag, 2000.-200p.
- Jaeger H.M., Nagel S.R., Behringer R.P. The physics of granular materials. // Rev.Mod.Phys. -1996.-68.- P.1259-1272.
- Kadanoff L. Built upon sand: theoretical ideas inspired by granular flows.. // Rev.Mod.Phys. -1999.-71.-P.435-447.
- De Gennes P.G. Granular matter: a tentative view. // Rev.Mod.Phys. -1999.-71,S.- P.374-385.
- Gerasimov O.I., Schram P.P.J.M. Differential equation of state of a model system with a singular measure: application to granular materials in steady states. // Physica A. -2002.-Vol.312.-P.172-180.
- Гиршфельдер Д.О. (и др.) Молекулярная теория жидкостей и газов. М.: Мир, 1973.-350с.
- Gerasymov O.I., Khudyntsev N.N., Klymenkov O.A., Spivak A.Ya. The kinetics of processes occurring in granular materials in the field of vibroaccelerations. // Ukr.Journ.Phys. -2005.-Vol.50, № 6.-P.624-632.
- Knight J.B., Fandrich C.G., Lau C.N., Jaeger H.M., Nagel S.R. Density relaxation in vibrated granular materials. // Phys.Rev.E. -1995.-Vol.51.-P.3957-3965.
- Vilarruel F. (and all) Compaction relaxation and ordering in granular materials. // Phys.Rev.E. -2000.-Vol.61.-P.-6914-6932.
- Gerasimov O.I., Schram P., Kitahara K. Kinetics of granular segregation. // Ukr.Journ.Phys. -2003.-Vol.48, № 8.-P.885-896.