Abstract
A two-dimensional model of the crystalline (granular) medium is considered that represents a square lattice consisting of elastically interacting particles, which possess translational and rotational degrees of freedom. In the long-wavelength approximation the partial derivatives equations have been derived that describe propagation of longitudinal, transverse and rotational waves in such a medium. In the field of low frequencies, when the rotational degree of freedom of particles can be neglected, the obtained nonlinear three-mode system degenerates into a two-mode system. Analytical dependencies of the velocities of elastic waves and the nonlinearity coefficients on the sizes of particles and the parameters of interactions between them have been found for both nonlinear models. Due to these dependencies, numerical estimations of the nonlinearity coefficients are performed. The two-mode system is shown to be reduced by the multi-scale method to Kadomtsev–Petviashvili evolutionary equation for transverse deformation, which has a soliton solution. For some crystals with a cubic symmetry it is found out, whether soliton is steady and what kind of polarity it has.
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References
Potapov, A.I. (ed.): Introduction to Micro- and Nanomechanics: Mathematical Models and Methods. Nizhny Novgorod Technical State University (2010) (in Russian)
Miller, R.E., Shenoy, V.B.: Size-dependent elastic properties of nanosized structural elements. Nanotechnology 11(3), 139–147 (2000)
Lauke, B.: On the effect of particle size on fracture toughness of polymer composites. Compos. Sci. Technol. 68, 3365–3372 (2008)
Maksimov, E.G., Zinenko, V.I., Zamkova, N.G.: Ab initio calculations of the physical properties of ionic crystals. Phys. Usp. 47, 1075–1099 (2004)
Eringen, A.C.: Microcontinuum Field Theories-1: Foundation and Solids. Springer, New York (1999)
Lisina, S.A., Potapov, A.I.: Generalized continuum models in nanomechanics. Doklady Phys. 53(5), 275–277 (2008)
Cosserat, E., Cosserat, F.: Theorie des Corps Deformables. Librairie Scientifique A, Hermann et Fils, Paris (1909, Reprint, 2009)
Maugin, G.A., Metrikine, A.V. (eds.): Mechanics of Generalized Continua: One Hundred Years After the Cosserats. Springer, New York (2010)
Altenbach, H., Maugin, G.A., Erofeev, V.I. (eds.): Mechanics of Generalized Continua. Springer, Berlin, Heidelberg (2011)
Li, C., Chou, T.-W.: A structural mechanics approach for the analysis of carbon nanotubes. Int. J. Solids Struct. 40, 2487–2499 (2003)
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)
Pavlov, I.S., Potapov, A.I.: Structural models in mechanics of nanocrystalline media. Doklady Phys. 53(7), 408–412 (2008)
Erofeyev, V.I.: Wave Processes in Solids with Microstructure. World Scientific Publishing, New Jersey (2003)
Potapov, A.I., Pavlov, I.S., Lisina, S.A.: Acoustic identification of nanocrystalline media. J. Sound Vib. 322(3), 564–580 (2009)
Born, M., Kun, H.: Dynamical Theory of Crystal Lattices. Clarendon Press, Oxford (1954)
Gross, E., Korshunov, A.: Rotational oscillations of molecules in a crystal lattice of organic substances and scattering spectra. J. Exp. Theor. Phys. 16(1), 53–59 (1946) (in Russian)
Vardoulakis, I., Sulem, J.: Bifurcation Analysis in Geomechanics. Blackie Academic and Professional, London (1995)
Pelevin, A., Lauke, B., Heinrich, G., Svistkov, A., Adamov, A.A.: Algorithm of constant definition for a visco-elastic rubber model based cyclic experiments, stress relaxation and creep data. In: Heinrich, G., Kaliske, M., Lion, A., Reese, S. (eds.) Constitutive Models for Rubber, vol. 1. CRC Press, Boca Raton (2009)
Tucker, J.W., Rampton, V.W.: Microwave Ultrasonics in Solid State Physics. North-Holland Publishing Company, Amsterdam (1972)
Pavlov, I.S.: Acoustic identification of the anisotropic nanocrystalline medium with non-dense packing of particles. Acoust. Phys. 56(6), 924–934 (2010)
Frantsevich, I.N., Voronov, F.F., Bakuta, S.A.: Elastic Constants and Elasticity Moduli of Metals and Nonmetals. Reference Book Frantsevich, I.N. (ed.) Naukova Dumka, Kiev (1982) (in Russian)
Porubov, A.V.: Amplification of Nonlinear Strain Waves in Solids. World Scientific, Singapore (2003)
Pelinovsky, D.E., Stepanyants, YuA: Self-focusing instability of plane solitons and chains of two-dimensional solitons in positive-dispersion media. J. Exp. Theor. Phys. 77(4), 602–609 (1993)
Press, W.H., Teukolsky, S.L., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in C. The Art of Scientific Computing. Cambridge University Press, Cambridge (1992)
Acknowledgments
The research was carried out under the financial support of the RFBR (grants Nr. 12-08-90032-Bel-a, 10-08-01108-a).
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Erofeev, V.I., Kazhaev, V.V., Pavlov, I.S. (2013). Nonlinear Localized Strain Waves in a 2D Medium with Microstructure. In: Altenbach, H., Forest, S., Krivtsov, A. (eds) Generalized Continua as Models for Materials. Advanced Structured Materials, vol 22. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36394-8_6
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DOI: https://doi.org/10.1007/978-3-642-36394-8_6
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