Discontinuity, Nonlinearity, and Complexity
Lattice Model with NearestNeighbor and NextNearestNeighbor Interactions for Gradient Elasticity
Discontinuity, Nonlinearity, and Complexity 4(1) (2016) 1123  DOI:10.5890/DNC.2016.03.002
Vasily E. Tarasov
Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow 119991, Russia
Download Full Text PDF
Abstract
Lattice models for the secondorder straingradient models of elasticity theory are discussed. To combine the advantageous properties of two classes of secondgradient models, we suggest a new lattice model that can be considered as a discrete microstructural basis for gradient continuum models. It was proved that two classes of the secondgradient models (with positive and negative sign in front the gradient) can have a general lattice model as a microstructural basis. To obtain the secondgradient continuum models we consider a lattice model with the nearestneighbor and nextnearestneighbor interactions with two different coupling constants. The suggested lattice model gives unified description of the secondgradient models with positive and negative signs of the strain gradient terms. The sign in front the gradient is determined by the relation of the coupling constants of the nearestneighbor and nextnearestneighbor interactions.
References

[1]  Born, M. and Huang, K. (1954), Dynamical Theory of Crystal Lattices, Oxford University Press, Oxford. 

[2]  Kosevich, A.M. (2005), The Crystal Lattice. Phonons, Solitons, Dislocations, Superlattices, Second Edition, Wiley VCH, Berlin, New York. 

[3]  Hahn, H.G., Elastizita Theorie Grundlagen der Linearen Theorie und Anwendungen auf undimensionale, ebene und zaumliche Probleme, B.G. Teubner, Stuttgart. (in German) 

[4]  Landau, L.D. and Lifshitz, E.M. (1986), Theory of Elasticity, Pergamon Press, Oxford, New York. 

[5]  Tarasov, V.E. (2011), Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, New York. 

[6]  Eringen, A.C. (1972), Nonlocal polar elastic continua, International Journal of Engineering Science, 10, 116. 

[7]  Eringen, A.C. (2002), Nonlocal Continuum Field Theories, Springer, New York. 

[8]  Rogula, D. (1983), Nonlocal Theory of Material Media, SpringerVerlag, New York. 

[9]  Mindlin, R.D. (1964),Microstructure in linear elasticity, Archive for Rational Mechanics and Analysis, 16, 5178. 

[10]  Mindlin, R.D. (1965), Second gradient of strain and surfacetension in linear elasticity, International Journal of Solids and Structures, 1, 417438. 

[11]  Mindlin, R.D. (1968), Theories of elastic continua and crystal lattice theories, In: E. Kroner,Mechanics of Generalized Continua, SpringerVerlag, Berlin, 312320. 

[12]  Eringen, A.C. (1983), On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of Applied Physics, 54, 47034710. 

[13]  Aifantis, E.C. (1992), On the role of gradients in the localization of deformation and fracture, International Journal of Engineering Science, 30, 12791299. 

[14]  Tarasov, V.E. (2006), Map of discrete system into continuous, Journal of Mathematical Physics, 47, 092901. (arXiv:0711.2612) 

[15]  Tarasov, V.E. (2006), Continuous limit of discrete systems with longrange interaction, Journal of Physics A, 39, 1489514910. (arXiv:0711.0826) 

[16]  Tarasov, V.E. and Zaslavsky, G.M. (2006), Fractional dynamics of coupled oscillators with longrange interaction, Chaos, 16, 023110. (13 pages) (arXiv:nlin/0512013) 

[17]  Tarasov, V.E. and Zaslavsky, G.M. (2006), §Fractional dynamics of systems with longrange interaction§, Communications in Nonlinear Science and Numerical Simulation, 11, 885898. (arXiv:1107.5436) 

[18]  Braides, A. and Gelli, M.S. (2002), Continuum limits of discrete systems without convexity hypotheses, Mathematics and Mechanics of Solids, 7, 4166. 

[19]  Lakes, R.S., Lee, T., Bersie, A., and Wang, Y.C. (2001), Extreme damping in composite materials with negativestiffness inclusions, Letters to Nature. Nature, 410, 565567. 

[20]  Bukreeva, K.A., Babicheva, R.I., Dmitriev, S.V., Zhou, K., and Mulyukov, R.R. (2013), Negative stiffness of the FeAl intermetallic nanofilm, Physics of the Solid State, 55, 19631967. 

[21]  Bukreeva, K.A., Babicheva, R.I., Dmitriev, S.V., Zhou, K., and Mulyukov, R.R. (2013) Inhomogeneous elastic deformation of nanofilms and nanowires of NiAl and FeAl alloys, Journal of Experimental and Theoretical Physics Letters, 98, 9195. 

[22]  Savin, A.V., Kikot, I.P., Mazo, M.A., and Onufriev, A.V. (2013), Twophase stretching of molecular chains, Proceedings of the National Academy of Sciences USA, 110, 28162821. 

[23]  Wang, Y.C., Swadener, J.G., and Lakes, R.S. (2007), Anomalies in stiffness and damping of a 2D discrete viscoelastic system due to negative stiffness components, Thin Solid Films, 515, 31713178. 

[24]  Drugan, W.J. (2007), Elastic composite materials having a negative stiffness phase can be stable, Physical Review Letters, 98, 055502. 

[25]  Kochmann, D.M. and Drugan, W.J. (2012), Analytical stability conditions for elastic composite materials with a nonpositive definite phase, Proceedings of the Royal Society A, 468, 22302254. 

[26]  Lee, C.M. and Goverdovskiy, V.N. (2012), A multistage highspeed railroad vibration isolation system with ※negative§ stiffness, Journal of Sound and Vibration, 331, 914921. 

[27]  Dyskin, A.V. and Pasternak, E. (2012), Elastic composite with negative stiffness inclusions in antiplane strain, International Journal of Engineering Science, 58, 4556. 

[28]  Yang, J., Xiong, Y.P., and Xing, J.T. (2013), Dynamics and power flow behaviour of a nonlinear vibration isolation system with a negative stiffness mechanism, Journal of Sound and Vibration, 332, 167183. 

[29]  Kilbas, A.A., Srivastava H.M., and Trujillo, J.J. (2006), Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam. 

[30]  Tarasov, V.E. (2013), Lattice model with powerlaw spatial dispersion for fractional elasticity, Central European Journal of Physics, 11, 15801588. 

[31]  Tarasov, V.E. (2014), Lattice model of fractional gradient and integral elasticity: Longrange interaction of Grünwald LetnikovRiesz type, Mechanics of Materials, 70, 106114. 

[32]  Tarasov, V.E. (2014), Fractional gradient elasticity from spatial dispersion law, ISRN Condensed Matter Physics, 2014, 794097. 

[33]  Tarasov, V.E. (2014), Lattice with longrange interaction of powerlaw type for fractional nonlocal elasticity, International Journal of Solids and Structures, 51, 29002907. 

[34]  Tarasov, V.E. and Aifantis, E.C. (2014), Towards fractional gradient elasticity, Journal of the Mechanical Behavior of Materials, 23, 4146. 

[35]  Tarasov, V.E. (2008), Chains with fractal dispersion law, Journal of Physics A, 41, 035101. (arXiv:0804.0607) 

[36]  Michelitsch, T.M., Maugin, G.A., Nicolleau, F.C.G.A., Nowakowski, A.F., and Derogar, S. (2009), Dispersion relations and wave operators in selfsimilar quasicontinuous linear chains, Physical Review E, 80, 011135. (arXiv:0904.0780) 

[37]  Michelitsch, T.M., Maugin, G.A., Nicolleau, F.C.G.A., Nowakowski, A.F., and Derogar, S. (2011), Wave propagation in quasicontinuous linear chains with selfsimilar harmonic interactions: Towards a fractal mechanics, Mechanics of Generalized Continua: Advanced Structured Materials, 7, 231244. 

[38]  Tarasov, V.E. and Aifantis, E.C. (2015), Nonstandard extensions of gradient elasticity: fractional nonlocality,memory and fractality, Communications in Nonlinear Science and Numerical Simulation, 22 (13) 197227(arXiv:1404.5241). 

[39]  Metrikine, A.V. and Askes H. (2002), Onedimensional dynamically consistent gradient elasticity models derived from a discrete microstructure Part 1: Generic formulation, European Journal of Mechanics A/Solids, 21, 555572. 

[40]  Askes, H. and Aifantis, E.C. (2011), Gradient elasticity in statics and dynamics: An overview of formulations, length scale identification procedures, finite element implementations and new results, International Journal of Solids and Structures, 48, 19621990. 

[41]  Tarasov, V.E. (2015), Vector calculus in noninteger dimensional space and its applications to fractal media, Communications in Nonlinear Science and Numerical Simulation, 20, 360374. 

[42]  OstojaStarzewski, M., Li, J., Joumaa, H., and Demmie, P.N. (2014), From fractal media to continuum mechanics, Journal of Applied Mathematics and Mechanics, 94, 373401. 

[43]  Druzhinin, O.A. and Ostrovskii, L.A. (1991), Solitons in discrete lattices, Physics Letters A, 160, 357362 

[44]  Kartashov, Y.V., Malomed, B.A., and Torner, L. (2011), Solitons in nonlinear lattices Reviews of Modern Physics, 83, 247306. (arXiv:1010.2254) 

[45]  Gorshkov, K.A., Ostrovskii, L.A., and Papko, V.V. (1976), Interactions and bound states of solitons as classical particles, Soviet Physics  JETP (Journal of Experimental and Theoretical Physics), 44, 306311. 

[46]  Joarder, M.A., Minato, A., Ozawa, S., and Hiki, Y. (2001), Computer experiments on solitons in threedimensional crystals with nearestneighbor and nextnearestneighbor atomic interactions, Japanese Journal of Applied Physics, Part 1, 40, 35013504. 

[47]  Kevrekidis, P.G., Malomed, B.A., Saxena, A., Bishop, A.R., and Frantzeskakis, D.J. (2003), Higherorder lattice diffraction: solitons in the discrete NLS equation with nextnearestneighbor interactions, Physica D: Nonlinear Phenomena, 183, 87101. 

[48]  Zhang, Y. (2007), Soliton excitations in pernigranilinebase polymer: Effects of nextnearestneighbor hopping, Solid State Communications, 143, 304307. 

[49]  Luo, A.C.J. (2010), Nonlinear Deformablebody Dynamics (2010) Higher Education Press and SpringerVerlag, Beijing, Berlin. 