Nonlocal Elasticity Theory as a Continuous Limit of 3D Networks of Pointwise Interacting Masses
DOI:
https://doi.org/10.15407/mag15.02.203Ключові слова:
нелокальна еластичність, усереднення, інтегральна модель, модель Ерінгена.Анотація
Розглядаються малі коливання пружної системи точкових мас (часток) з нелокальною взаємодією. Вивчається асимптотичне поводження системи, коли кількість часток прямує до нескінченності, а відстані між ними та сили взаємодії прямують до нуля. Перший член асимптотики описується усередненою системою рівнянь, що є нелокальною моделлю коливань пружного середовища.
Mathematics Subject Classification: 35Q70, 35Q74, 35B27.
Посилання
E.C. Aifantis, Gradient effects at macro, micro and nano scales, J. Mech. Behav. of Mater. 5 (1994), 355–375. https://doi.org/10.1515/JMBM.1994.5.3.355
M. Berezhnyy and L. Berlyand, Continium limit for three-dimensional mass-spring network and disret Korn’s inequality, J. Mech. Phys. Solids 54 (2006), 635–669. https://doi.org/10.1016/j.jmps.2005.09.006
M. Born and K. Huang, Dynamical Theory of Crystal Lattices, Oxford University Press, London, 1954.
M. Di Paola, G. Failla, and M. Zingales, Physically-based approach to the mechanics of strong non-local elasticity theory, J. Elasticity 97 (2) (2008), 103–130. https://doi.org/10.1007/s10659-009-9211-7
M. Di Paola, A. Pirrotta, and M. Zingales, Mechanically-based approach to nonlocal elasticity: variational principles, Int. J. Solids Struct. 47 (5) (2010), 539–548. https://doi.org/10.1016/j.ijsolstr.2009.09.029
M. Di Paola, G. Failla, and M. Zingales, The mechanically-based approach to 3D non-local linear elasticity theory: long-range central interactions, Int. J. Solids Struct. 47 (2010), 2347–2358. https://doi.org/10.1016/j.ijsolstr.2010.02.022
A.C. Eringen, Non-Local Polar Field Models, Academic Press, New York, 1996.
A.C. Eringen, Non-local polar elastic continua, Int. J. Eng. Sci. 10 (1972), 1–16. https://doi.org/10.1016/0020-7225(72)90050-X
A.C. Eringen and D.G.B. Edelen, On non-local elasticity, Int. J. Eng. Sci. 10 (1972), 233–248. https://doi.org/10.1016/0020-7225(72)90050-X
S. Gopalakrishnan and S. Narendar, Wave Propagation in Nanostructures. Nonlocal Continuum Mechanics Formulations. NanoScience and Technology, Springer, Cham, 2013. https://doi.org/10.1007/978-3-319-01032-8
E. Kröner, Elasticity theory of materials with long range cohesive forces, Int. J. Solids Struct. 3 (1967), 731–742. https://doi.org/10.1016/0020-7683(67)90049-2
E. Kröner, The problem of non-locality in the mechanics of solids: review on present status, Fundamental Aspects of Dislocation Theory. Conference Proceedings, Nat. Bur. Stand. (U.S.), Spec. Publ. 317, II, 1970, 729–736.
J.A. Krumhansl, Generalized continuum field representation for lattice vibrations, Lattice Dynamics, Proceedings of International Conference. Pergamon Press, London, 1963.
I.A. Kunin, Elastic Media with Microstructure, I, Springer-Verlag, Berlin, 1982. https://doi.org/10.1007/978-3-642-81748-9
K.M. Liew, Y. Zhang, and L.W. Zhang, Nonlocal elasticity theory for graphene modeling and simulation: prospects and challenges, J. Model. Mech. and Mater. 1 (2017), 2328–2355. https://doi.org/10.1515/jmmm-2016-0159
A.I. Markushevich, The Theory of Analytic Functions: a Brief Course, Moscow, Mir, 1983 (Russian).
V.A. Marchenko and E.Ya. Khruslov, Homogenization of Partial Differential Equations, Birkhöuser, Boston–Basel–Berlin, 2006. https://doi.org/10.1007/978-0-8176-4468-0
R.D. Mindlin, Theories of elastic continua and crystal lattice theories, Mechanics of Generalized Continua. IUTAM Symposia (International Union of Theoretical and Applied Mechanics), Springer, Berlin–Heidelberg, 1968, 312–320. https://doi.org/10.1007/978-3-662-30257-6_38
O.A. Oleı̆nik , A.S. Shamaev, and G.A. Yosifian, Mathematical Problems in Elasticity and Homogenization, Studies in Mathematics and its Applications, 26, NorthHolland Publishing Co., Amsterdam, 1992.
A.L. Piatnitski, G.A. Chechkin, and A.S. Shamaev, Homogenization. Methods and Applications, Novosibirsk, 2007 (Russian): Engl. transl: Translations of Mathematical Monographs, 234, Amer. Math. Soc., Providence, RI, 2007.
C. Polizzotto, Non-local elasticity and related variational principles, Int. J. Solids Struct. 38 (2001), 7359–7380. https://doi.org/10.1016/S0020-7683(01)00039-7
E. Sanchez-Palencia, Non-Homogeneous Media and Vibration Theory, SpringerVerlag, Berlin–Heidelberg, 1980.
S.A. Silling, Reformulation of elasticity theory for discontinuities and long-range forces, J. Mech. Phys. Solids 48 (2000), 175–209. https://doi.org/10.1016/S0022-5096(99)00029-0
V.E. Tarasov, General lattice model of gradient elasticity, Mod. Phys. Lett. B 28 (2014), No. 7, 1450054. https://doi.org/10.1142/S0217984914500547
V.E. Tarasov, Three-dimensional lattice models with long-range interactions of Grünwald–Letnikov type for fractional generalization of gradient elasticity, Meccanica 51 (2016), 125–138. https://doi.org/10.1007/s11012-015-0190-4
