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2013 | 11 | 11 | 1580-1588
Article title

Lattice model with power-law spatial dispersion for fractional elasticity

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Abstracts
EN
A lattice model with a spatial dispersion corresponding to a power-law type is suggested. This model serves as a microscopic model for elastic continuum with power-law non-locality. We prove that the continuous limit maps of the equations for the lattice with the power-law spatial dispersion into the continuum equations with fractional generalizations of the Laplacian operators. The suggested continuum equations, which are obtained from the lattice model, are fractional generalizations of the integral and gradient elasticity models. These equations of fractional elasticity are solved for two special static cases: fractional integral elasticity and fractional gradient elasticity.
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Journal
Year
Volume
11
Issue
11
Pages
1580-1588
Physical description
Dates
published
1 - 11 - 2013
online
10 - 12 - 2013
Contributors
References
  • [1] B. Ross, Lect. Notes Math. 457, 1 (1975) http://dx.doi.org/10.1007/BFb0067096[Crossref]
  • [2] J. T. Machado, V. Kiryakova, F. Mainardi, Commun. Nonlinear Sci. Num. Simul. 16, 1140 (2011) http://dx.doi.org/10.1016/j.cnsns.2010.05.027[Crossref]
  • [3] S. G. Samko, A. A. Kilbas, O. I. Marichev, Integrals and Derivatives of Fractional Order and Applications (Nauka i Tehnika, Minsk, 1987)
  • [4] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives Theory and Applications (Gordon and Breach, New York, 1993)
  • [5] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations (Elsevier, Amsterdam, 2006)
  • [6] A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics (Springer, New York, 1997)
  • [7] R. Hilfer (Ed.), Applications of Fractional Calculus in Physics (World Scientific, Singapore, 2000)
  • [8] R. Metzler, J. Klafter, Phys. Rep. 339, 1 (2000) http://dx.doi.org/10.1016/S0370-1573(00)00070-3[Crossref]
  • [9] J. Sabatier, O. P. Agrawal, J. A. Tenreiro Machado (Eds.), Advances in Fractional Calculus, Theoretical Developments and Applications in Physics and Engineering (Springer, Dordrecht, 2007)
  • [10] A. C. J. Luo, V. S. Afraimovich (Eds.), Long-range Interaction, Stochasticity and Fractional Dynamics (Springer, Berlin, 2010)
  • [11] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models (World Scientific, Singapore, 2010) http://dx.doi.org/10.1142/9781848163300[Crossref]
  • [12] J. Klafter, S. C. Lim, R. Metzler (Eds.), Fractional Dynamics, Recent Advances (World Scientific, Singapore, 2011)
  • [13] V. E. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media (Springer, New York, 2011)
  • [14] J. A. Tenreiro Machado, Commun. Nonlinear Sci. Num. Simul. 16, 4596 (2011) http://dx.doi.org/10.1016/j.cnsns.2011.01.019[Crossref]
  • [15] V. E. Tarasov, Int. J. Mod. Phys. B. 27, 1330005 (2013) http://dx.doi.org/10.1142/S0217979213300053[Crossref]
  • [16] K. A. Lazopoulos, Mech. Res. Commun. 33, 753 (2006) http://dx.doi.org/10.1016/j.mechrescom.2006.05.001[Crossref]
  • [17] A. Carpinteri, P. Cornetti, A. Sapora, App. Math. Mech. 89, 207 (2009)
  • [18] A. Carpinteri, P. Cornetti, A. Sapora, Eur. Phys. J. Special Topics 193, 193 (2011) http://dx.doi.org/10.1140/epjst/e2011-01391-5[Crossref]
  • [19] A. Sapora, P. Cornetti, A. Carpinteri, Commun. Nonlinear Sci. Num. Simul. 18, 63 (2013) http://dx.doi.org/10.1016/j.cnsns.2012.06.017[Crossref]
  • [20] G. Cottone, M. Di Paola, M. Zingales, Physica E. 42, 95 (2009) http://dx.doi.org/10.1016/j.physe.2009.09.006[Crossref]
  • [21] G. Cottone, M. Di Paola, M. Zingales, Adv. Num. Method. Lect. Notes El. Eng. 11, 389 (2009) http://dx.doi.org/10.1007/978-0-387-76483-2_33[Crossref]
  • [22] I. A. Kunin, Media with Microstructure, Vol. I (Springer-Verlag, Berlin, New York, 1982) http://dx.doi.org/10.1007/978-3-642-81748-9[Crossref]
  • [23] I. A. Kunin, Media with Microstructure, Vol.II (Springer-Verlag, Berlin, New York, 1983) http://dx.doi.org/10.1007/978-3-642-81960-5[Crossref]
  • [24] J. A. Krumhansl, In R. F. Wallis (Ed.), Generalized continuum field representation for lattice vibrations, Lattice Dynamics (Pergamon, London, 1965) 627–634
  • [25] A. C. Eringen, B. S. Kim, Crystal Latt. Def. 7, 51 (1977)
  • [26] M. Ostoja-Starzewski, App. Mech. Rev. 55, 35 (2002) http://dx.doi.org/10.1115/1.1432990[Crossref]
  • [27] H. Askes, A. Metrikine, Int. J. Soli. Struct. 42, 187 (2005) http://dx.doi.org/10.1016/j.ijsolstr.2004.04.005[Crossref]
  • [28] M. Charlotte, L. Truskinovsky, J. Mech. Phys. Solids. 60, 1508 (2012) http://dx.doi.org/10.1016/j.jmps.2012.03.004[Crossref]
  • [29] V. E. Tarasov, J. Phys. A. 39, 14895 (2006) http://dx.doi.org/10.1088/0305-4470/39/48/005[Crossref]
  • [30] V. E. Tarasov, J. Math. Phys. 47, 092901 (2006) http://dx.doi.org/10.1063/1.2337852[Crossref]
  • [31] V. E. Tarasov, G. M. Zaslavsky, Chaos. 16, 023110 (2006) http://dx.doi.org/10.1063/1.2197167[Crossref]
  • [32] N. Laskin, G. M. Zaslavsky, Physica A. 368, 38 (2006) http://dx.doi.org/10.1016/j.physa.2006.02.027[Crossref]
  • [33] V. E. Tarasov, G. M. Zaslavsky, Commun. Nonlinear Sci. Num. Simul. 11, 885 (2006) http://dx.doi.org/10.1016/j.cnsns.2006.03.005[Crossref]
  • [34] V. E. Tarasov, J. J. Trujillo, Ann. Phys. 334, 1 (2013) http://dx.doi.org/10.1016/j.aop.2013.03.014[Crossref]
  • [35] G. H. Hardy, J. London Math. Soc. 20, 45 (1945)
  • [36] M. M. Dzherbashyan, A. B. Nersesian, Izv. An. Armyanskoi SSR. Fiz.-Mat. 11, 85 (1958) (in Russian)
  • [37] M. M. Dzherbashyan, A. B. Nersesian, Dokl. Akad. Nauk. 121, 210 (1958) (in Russian)
  • [38] J. J. Trujillo, M. Rivero, B. Bonilla, J. Math. Anal. App. 231, 255 (1999) http://dx.doi.org/10.1006/jmaa.1998.6224[Crossref]
  • [39] Z. M. Odibat, N. T. Shawagfeh, App. Math. Comput. 186, 286 (2007) http://dx.doi.org/10.1016/j.amc.2006.07.102[Crossref]
  • [40] M. Di Paola, M. Zingales, Int. J. Solid. Struct. 45, 5642 (2008) http://dx.doi.org/10.1016/j.ijsolstr.2008.06.004[Crossref]
  • [41] H. Askes, E. C. Aifantis, Int. J. Solid. Struct. 48, 1962 (2011) http://dx.doi.org/10.1016/j.ijsolstr.2011.03.006[Crossref]
  • [42] L. D. Landau, E. M. Lifshitz, Theory of Elasticity, Third Edition (Pergamon Press, Oxford, 1986)
  • [43] H. Bateman, A. Erdelyi, Tables of integral transforms, Volume 1 (New York, McGraw-Hill, 1954) or (Moscow, Nauka, 1969) in Russian
  • [44] A. K. Jonscher, Nature 267, 673 (1977) http://dx.doi.org/10.1038/267673a0[Crossref]
  • [45] A. K. Jonscher, Universal Relaxation Law (Chelsea Dielectrics, London, 1996)
  • [46] A. K. Jonscher, J. Mat. Sci. 34, 3071 (1999) http://dx.doi.org/10.1023/A:1004640730525[Crossref]
  • [47] V. E. Tarasov, J. Phys. Condensed Matter. 20, 175223 (2008) http://dx.doi.org/10.1088/0953-8984/20/17/175223[Crossref]
  • [48] V. E. Tarasov, Jo. Phys. Condensed Matter. 20, 145212 (2008) http://dx.doi.org/10.1088/0953-8984/20/14/145212[Crossref]
  • [49] V. E. Tarasov, Cent. Eur. J. Phys. 10, 382 (2012) http://dx.doi.org/10.2478/s11534-012-0008-0[Crossref]
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bwmeta1.element.-psjd-doi-10_2478_s11534-013-0308-z
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