Top

Hint

Swipe to navigate through the chapters of this book

Published in:

Read chapter Read first chapter

2021 | OriginalPaper | Chapter

2. A 2D Lattice with Dense Packing of the Particles

Authors : Vladimir I. Erofeev, Igor S. Pavlov

Published in: Structural Modeling of Metamaterials

Publisher: Springer International Publishing

Abstract

Mechanical properties of a granular consolidated medium depend on the geometry of the microparticles, their location, and the forces of interaction between them. One of the main goals of mathematical modeling of such media is obtaining equations of motion and equations of state, which are capable to describe a discrete nature of a medium.

To get access to this content you need the following product:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt 90 Tage mit der neuen Mini-Lizenz testen!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt 90 Tage mit der neuen Mini-Lizenz testen!

inform now

previous chapter Theoretical Basis of the Structural Modeling Method

next chapter A Two-Dimensional Lattice with Non-dense Packing of Particles

Hereinafter, we will use the terms “grains” and “granules” as synonyms for the word “particles.” However, these terms do not have such a meaning, as in materials science.

Belyaeva, I.Y., Zaitsev, V.Y., Ostrovsky, L.A.: Nonlinear acoustical properties of granular media. Acoust. Phys. 39, 11–16 (1993)

Bykov, V.G.: Solitary shear waves in a granular medium. Acoust. Phys. 45(2), 138–142 (1999)

Goldstein, R.V., Gorodtsov, V.A., Lisovenko, D.S.: Young’s moduli and Poisson’s ratio of curvilinear anisotropic hexagonal and rhombohedral nanotubes. Nanotubes-auxetics. Dokl. Phys. 58(9), 400–404 (2013)

Chang, C.S., Gao, J.: Wave propagation in granular rod using high-gradient theory. J. Engn. Mech.-ASCE 1, 52–59 (1997)

Chang, C.S., Ma, L.: A micromechanical-based micropolar theory for deformation of granular solids. Int. J. Solids Struct. 28(1), 67–87 (1994)

Christoffersen, J., Mehrabadi, M.M., Nemat-Nasser, S.A.: A micromechanical description of granular material behavior. Trans. ASME J. Appl. Mech. 48(2), 339–344 (1981) CrossRef

Lisina, S.A., Potapov, A.I., Nesterenko, V.F.: Nonlinear granular medium with rotations of the particles one-dimensional model. Phys. Acoust. 47(5), 666–674 (2001) CrossRef

Pavlov, I.S., Potapov, A.I., Maugin, G.A.: A 2D Granular Medium With Rotating particles. Int. J. Solids Struct. 43(20), 6194–6207 (2006) CrossRef

Sadovskaya, O., Sadovskii, V.: Mathematical modeling in mechanics of granular materials. Springer, Heidelberg, New York, Dordrecht, London, 390 p (2012)

10.

Krivtsov, A.M., Podol’skaya E.A.: Modeling of elastic properties of crystals with hexagonal close-packed lattice. Mech. Solids. 45(3), 370–378 (2010)

11.

Krivtsov, A.M.: Deformation and destruction of microstructured solids. Fizmatlit Publ., Moscow, 304 p (2007). (in Russian)

12.

Askar, A.: Lattice Dynamics Foundation of Continuum Theory. World-Scientific, Singapore (1985)

13.

Berglund K.: Structural models of micropolar media. In: Mechanics of Micropolar Media. Brulin, O., Hsieh, R.K.T. (eds.) World Scientific, Singapore, pp. 35–86 (1982)

14.

Ostoja-Starzewski, M., Sheng, P.Y., Alzebdeh, K.: Spring network models in elasticity and fracture of composites and polycrystals. Comput. Mater. Sci. 7, 82–93 (1996) CrossRef

15.

Pavlov, I.S.: Acoustic Identification of the anisotropic nanocrystalline medium with non-dense packing of particles. Acoust. Phys. 56(6), 924–934 (2010) CrossRef

16.

Pavlov, I.S., Vasiliev, A.A., Porubov, A.V.: Dispersion properties of the phononic crystal consisting of ellipse-shaped particles. J. Sound Vibr. 384, 163–176 (2016) CrossRef

17.

Pavlov, I.S., Potapov, A.I.: Structural Models in Mechanics of Nanocrystalline Media. Dokl. Phys. 53(7), 408–412 (2008) CrossRef

18.

Potapov, A.I., Pavlov, I.S., Lisina, S.A.: Acoustic identification of nanocrystalline media. J. Sound Vib. 322(3), 564–580 (2009) CrossRef

19.

Suiker, A.S.J., Metrikine, A.V., de Borst, R.: Comparison of wave propagation characteristics of the Cosserat continuum model and corresponding discrete lattice models. Int. J. Solids Struct. 38, 1563–1583 (2001)

20.

Suiker, A.S.J., Metrikine, A.V., de Borst, R.: Dynamic behaviour of a layer of discrete particles. Part 1: analysis of body waves and eigenmodes. J Sound Vib. 240(1), 1–18 (2001)

21.

Vasiliev, A.A., Pavlov, I.S.: Auxetic properties of hiral hexagonal cosserat lattices composed of finite-sized particles. Phys. Status Solidi B 3(257), 1900389 (2020). https://doi.org/10.1002/pssb.201900389 CrossRef

22.

Vasiliev, A.A., Pavlov, I.S.: Models and some properties of Cosserat triangular lattices with chiral microstructure. Lett. Mater. 9(1), 45–50 (2019) www.lettersonmaterials.com, https://doi.org/10.22226/2410-3535-2019-1-45-50

23.

Erofeev, V.I., Pavlov, I.S., Vasiliev, A.A., Porubov, A.V.: Dispersion properties of a closed-packed lattice consisting of round particles. In: Altenbach, H., et al. (eds.) Generalized Models and Non-classical Approaches in Complex Materials 2, Advanced Structured Materials 90. © Springer International Publishing AG, part of Springer Nature 2018, pp. 101–117

24.

Sedov, L.I.: Models of continuous media with internal degrees of freedom. J. Appl. Math. Mech. 32(5), 803–819 (1968) CrossRef

25.

Sedov, L.I.: Mechanics of continuous medium. World Scientific Publ, Singapore, vol. 1, (1997)

26.

Potapov, A.I., Pavlov, I.S., Maugin, G.A.: Nonlinear wave interactions in 1D crystals with complex lattice. Wave Motion 29, 297–312 (1999)

27.

Bogomolov, V.N., Parfen’eva, L.S., Smirnov, I.A., Misiorek, H., Jzowski, A.: Phonon propagation through photonic crystals—media with spatially modulated acoustic properties. Phys. Solis State 44, 181–185 (2002)

28.

Vetrov, S.Y., Timofeev, I.V., Rudakova, N.V.: Band structure of a two-dimensional resonant photonic crystal. Phys. Solid State 52, 527–532 (2010)

29.

Yablonovitch, E., Gmitter, T.J., Leung, K.M.: Photonic band structure: The face-centered cubic case employing nonspherical atoms. Phys. Rev. Lett. 67, 2295 (1991)

30.

Fujii, M., Kanzaea, Y., Hayashi, S., Yamamoto, K.: Raman scattering from acoustic phonons confined in Si nanocrystals. Phys Rev. B 54, R8373 (1996) CrossRef

31.

Kushwaha, M.S., Halevi, P., Martinez, G., Dobrzynski, L., Djafari-Rouhani, B.: Theory of band structure of periodic elastic composites. Phys. Rev. B 49, 2313 (1994)

32.

Samusev, K.B., Rybin, M.V., Limonov, M.F., Yushin, G.N.: Structural parameters of synthetic opals: statistical analysis of electron microscopy images. Phys. Solid State 50(7), 1280–1286 (2008) CrossRef

33.

Stroscio, M.A., Dutta, M.: Phonons in nanostructures. Cambridge University Press, 274 p (2001)

34.

Pierce, J.R., Almost All about Waves, Dover Publications (2006)

35.

Normal waves. In: Prokhorov, A.M., (ed.) The physical encyclopedia in 5 volumes. The Big Russian encyclopedia, Moscow, vol. 3. p. 360 (1992). (in Russian)

36.

Ostrovsky, L.A., Papko, V.V., Pelinovsky, E.N.: Solitary electromagnetic waves in nonlinear lines. Radiophys. Quantum Electron. 15, 438–446 (1972) CrossRef

37.

Ostrovsky, L.A., Potapov, A.I.: Modulated waves: theory and applications. The Johns Hopkins University Press, Baltimore, MD (1999)

38.

Vinogradova, M.B., Rudenko, O.V., Sukhorukov, A.P.: Theory of Waves. Nauka, Moscow (1990). (in Russian)

39.

Kittel, C.: Introduction to Solid State Physics. 8th edn. Wiley, Inc., (2005)

40.

Kaganov, M.I.: Electrons, Phonons, Magnons, 268 pp. English Translation. Mir Publishers, Moscow (1981)

41.

Nikitenkova, S.P., Potapov, A.I.: Dispersion properties of two-dimensional phonon crystals with a hexagonal structure. Acoust. Phys. 56(6), 909–918 (2010) CrossRef

42.

Potapov, A.I., Pavlov, I.S., Nikitenkova, S.P., Shudyaev, A.A.: Structural models in nanoacoustics: control of dispersion properties of phonon crystals. Acoustics of inhomogeneous media. In: Proceedings of the Russian acoustic society, GEOS, Moscow, no 10, pp. 9–16 (2009). (in Russian)

43.

Reisland, J.A.: Physics of Phonons. Wiley, London, New York, Sydney, Toronto (1973)

44.

Porubov, A.V., Krivtsov, A.M., Osokina, A.E.: Two-dimensional waves in extended square lattice. Int. J. Non-Linear Mech. 99, 281–287 (2018) (бeз poтaциoнныx cтeпeнeй, yчeт нeлoкaльнocти, диcкpeтнaя мoдeль)

45.

Potapov, A.I., Pavlov, I.S., Lisina, S.A.: Identification of nanocrystalline media by acoustic spectroscopy methods. Acoust. Phys. 56(4), 588–596 (2010) CrossRef

46.

Berryman, J.G.: Long-wavelength propagation in composite elastic media I, II. J. Acoust. Soc. Am. 68(6), 1809–1831 (1980) CrossRef

47.

Goldshtein, R.V., Chentsov, A.V.: A discrete-continual model for a nanotube. Mech. Solids 4, 57–74 (2005)

48.

Porubov, A.V., Aero, E.L., Maugin, G.A.: Two approaches to study essentially nonlinear and dispersive properties of the internal structure of materials. Phys. Rev. E79. 046608 (2009)

49.

Vasiliev, A.A., Dmitriev, S.V., Miroshnichenko, A.E.: Multi-field approach in mechanics of structural solids. Int. J. Solids Struct. 47, 510–525 (2010)

50.

Vasiliev, A.A, Dmitriev, S.V., Miroshnichenko, A.E.: Multi-field continuum theory for medium with microscopic rotations. Int. J. Solids Struct. 42, pp. 6245–6260 (2005)

51.

Vasiliev, A.A., Miroshnichenko A.E., Ruzzene, M.: Multifield model for Cosserat media. J. Mech. Mater. Struct. 3(7), 1365–1382 (2008)

52.

Vasiliev, A.A., Miroshnichenko A.E., Dmitriev, S.V.: Multi-field modeling of a cosserat lattice: models, wave filtering, and boundary effects. Eur. J. Mech. A/Solids. 46, 96–105 (2014)

53.

Andrianov, I.V., Kholod, E.G., Weichert, D.: Application of quasi-continuum models for perturbation analysis of discrete kinks. Nonlinear Dyn. 68, 1–5 (2012) CrossRef

Title: A 2D Lattice with Dense Packing of the Particles
Authors: Vladimir I. Erofeev
Igor S. Pavlov
Publisher: Springer International Publishing
Book: Structural Modeling of Metamaterials
Print ISBN: 978-3-030-60329-8

Electronic ISBN: 978-3-030-60330-4

Copyright Year: 2021
DOI: https://doi.org/10.1007/978-3-030-60330-4_2