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Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com


Lattice Model with Nearest-Neighbor and Next-Nearest-Neighbor Interactions for Gradient Elasticity

Discontinuity, Nonlinearity, and Complexity 4(1) (2016) 11--23 | DOI:10.5890/DNC.2016.03.002

Vasily E. Tarasov

Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow 119991, Russia

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Abstract

Lattice models for the second-order strain-gradient models of elasticity theory are discussed. To combine the advantageous properties of two classes of second-gradient 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 second-gradient models (with positive and negative sign in front the gradient) can have a general lattice model as a microstructural basis. To obtain the second-gradient continuum models we consider a lattice model with the nearest-neighbor and next-nearestneighbor interactions with two different coupling constants. The suggested lattice model gives unified description of the second-gradient 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 nearest-neighbor and next-nearest-neighbor interactions.

References

  1. [1]  Born, M. and Huang, K. (1954), Dynamical Theory of Crystal Lattices, Oxford University Press, Oxford.
  2. [2]  Kosevich, A.M. (2005), The Crystal Lattice. Phonons, Solitons, Dislocations, Superlattices, Second Edition, Wiley- VCH, Berlin, New York.
  3. [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. [4]  Landau, L.D. and Lifshitz, E.M. (1986), Theory of Elasticity, Pergamon Press, Oxford, New York.
  5. [5]  Tarasov, V.E. (2011), Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, New York.
  6. [6]  Eringen, A.C. (1972), Nonlocal polar elastic continua, International Journal of Engineering Science, 10, 1-16.
  7. [7]  Eringen, A.C. (2002), Nonlocal Continuum Field Theories, Springer, New York.
  8. [8]  Rogula, D. (1983), Nonlocal Theory of Material Media, Springer-Verlag, New York.
  9. [9]  Mindlin, R.D. (1964),Micro-structure in linear elasticity, Archive for Rational Mechanics and Analysis, 16, 51-78.
  10. [10]  Mindlin, R.D. (1965), Second gradient of strain and surface-tension in linear elasticity, International Journal of Solids and Structures, 1, 417-438.
  11. [11]  Mindlin, R.D. (1968), Theories of elastic continua and crystal lattice theories, In: E. Kroner,Mechanics of Generalized Continua, Springer-Verlag, Berlin, 312-320.
  12. [12]  Eringen, A.C. (1983), On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of Applied Physics, 54, 4703-4710.
  13. [13]  Aifantis, E.C. (1992), On the role of gradients in the localization of deformation and fracture, International Journal of Engineering Science, 30, 1279-1299.
  14. [14]  Tarasov, V.E. (2006), Map of discrete system into continuous, Journal of Mathematical Physics, 47, 092901. (arXiv:0711.2612)
  15. [15]  Tarasov, V.E. (2006), Continuous limit of discrete systems with long-range interaction, Journal of Physics A, 39, 14895-14910. (arXiv:0711.0826)
  16. [16]  Tarasov, V.E. and Zaslavsky, G.M. (2006), Fractional dynamics of coupled oscillators with long-range interaction, Chaos, 16, 023110. (13 pages) (arXiv:nlin/0512013)
  17. [17]  Tarasov, V.E. and Zaslavsky, G.M. (2006), §Fractional dynamics of systems with long-range interaction§, Communications in Nonlinear Science and Numerical Simulation, 11, 885-898. (arXiv:1107.5436)
  18. [18]  Braides, A. and Gelli, M.S. (2002), Continuum limits of discrete systems without convexity hypotheses, Mathematics and Mechanics of Solids, 7, 41-66.
  19. [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, 565-567.
  20. [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, 1963-1967.
  21. [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, 91-95.
  22. [22]  Savin, A.V., Kikot, I.P., Mazo, M.A., and Onufriev, A.V. (2013), Two-phase stretching of molecular chains, Proceedings of the National Academy of Sciences USA, 110, 2816-2821.
  23. [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, 3171-3178.
  24. [24]  Drugan, W.J. (2007), Elastic composite materials having a negative stiffness phase can be stable, Physical Review Letters, 98, 055502.
  25. [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, 2230-2254.
  26. [26]  Lee, C.-M. and Goverdovskiy, V.N. (2012), A multi-stage high-speed railroad vibration isolation system with ※negative§ stiffness, Journal of Sound and Vibration, 331, 914-921.
  27. [27]  Dyskin, A.V. and Pasternak, E. (2012), Elastic composite with negative stiffness inclusions in antiplane strain, International Journal of Engineering Science, 58, 45-56.
  28. [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, 167-183.
  29. [29]  Kilbas, A.A., Srivastava H.M., and Trujillo, J.J. (2006), Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam.
  30. [30]  Tarasov, V.E. (2013), Lattice model with power-law spatial dispersion for fractional elasticity, Central European Journal of Physics, 11, 1580-1588.
  31. [31]  Tarasov, V.E. (2014), Lattice model of fractional gradient and integral elasticity: Long-range interaction of Grünwald- Letnikov-Riesz type, Mechanics of Materials, 70, 106-114.
  32. [32]  Tarasov, V.E. (2014), Fractional gradient elasticity from spatial dispersion law, ISRN Condensed Matter Physics, 2014, 794097.
  33. [33]  Tarasov, V.E. (2014), Lattice with long-range interaction of power-law type for fractional non-local elasticity, International Journal of Solids and Structures, 51, 2900-2907.
  34. [34]  Tarasov, V.E. and Aifantis, E.C. (2014), Towards fractional gradient elasticity, Journal of the Mechanical Behavior of Materials, 23, 41-46.
  35. [35]  Tarasov, V.E. (2008), Chains with fractal dispersion law, Journal of Physics A, 41, 035101. (arXiv:0804.0607)
  36. [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 self-similar quasicontinuous linear chains, Physical Review E, 80, 011135. (arXiv:0904.0780)
  37. [37]  Michelitsch, T.M., Maugin, G.A., Nicolleau, F.C.G.A., Nowakowski, A.F., and Derogar, S. (2011), Wave propagation in quasi-continuous linear chains with self-similar harmonic interactions: Towards a fractal mechanics, Mechanics of Generalized Continua: Advanced Structured Materials, 7, 231-244.
  38. [38]  Tarasov, V.E. and Aifantis, E.C. (2015), Non-standard extensions of gradient elasticity: fractional non-locality,memory and fractality, Communications in Nonlinear Science and Numerical Simulation, 22 (1-3) 197-227(arXiv:1404.5241).
  39. [39]  Metrikine, A.V. and Askes H. (2002), One-dimensional dynamically consistent gradient elasticity models derived from a discrete microstructure Part 1: Generic formulation, European Journal of Mechanics A/Solids, 21, 555-572.
  40. [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, 1962-1990.
  41. [41]  Tarasov, V.E. (2015), Vector calculus in non-integer dimensional space and its applications to fractal media, Communications in Nonlinear Science and Numerical Simulation, 20, 360-374.
  42. [42]  Ostoja-Starzewski, M., Li, J., Joumaa, H., and Demmie, P.N. (2014), From fractal media to continuum mechanics, Journal of Applied Mathematics and Mechanics, 94, 373-401.
  43. [43]  Druzhinin, O.A. and Ostrovskii, L.A. (1991), Solitons in discrete lattices, Physics Letters A, 160, 357-362
  44. [44]  Kartashov, Y.V., Malomed, B.A., and Torner, L. (2011), Solitons in nonlinear lattices Reviews of Modern Physics, 83, 247-306. (arXiv:1010.2254)
  45. [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, 306-311.
  46. [46]  Joarder, M.A., Minato, A., Ozawa, S., and Hiki, Y. (2001), Computer experiments on solitons in three-dimensional crystals with nearest-neighbor and next-nearest-neighbor atomic interactions, Japanese Journal of Applied Physics, Part 1, 40, 3501-3504.
  47. [47]  Kevrekidis, P.G., Malomed, B.A., Saxena, A., Bishop, A.R., and Frantzeskakis, D.J. (2003), Higher-order lattice diffraction: solitons in the discrete NLS equation with next-nearest-neighbor interactions, Physica D: Nonlinear Phenomena, 183, 87-101.
  48. [48]  Zhang, Y. (2007), Soliton excitations in pernigraniline-base polymer: Effects of next-nearest-neighbor hopping, Solid State Communications, 143, 304-307.
  49. [49]  Luo, A.C.J. (2010), Nonlinear Deformable-body Dynamics (2010) Higher Education Press and Springer-Verlag, Beijing, Berlin.