Lattice with long-range interaction of power-law type for fractional non-local elasticity

Abstract Lattice models with long-range interactions of power-law type are suggested as a new type of microscopic model for fractional non-local elasticity. Using the transform operation, we map the lattice equations into continuum equation with Riesz derivatives of non-integer orders. The continuum equations that are obtained from the lattice model describe fractional generalization of non-local elasticity models. Particular solutions and correspondent asymptotic of the fractional differential equations for displacement fields are suggested for the static case.

[1]  Shaofan Li,et al.  Handbook of Micromechanics and Nanomechanics , 2013 .

[2]  V. E. Tarasov.  Universal electromagnetic waves in dielectric , 2008, 0907.2163.

[3]  S. Narendar,et al.  Wave Propagation in Nanostructures: Nonlocal Continuum Mechanics Formulations , 2013 .

[4]  A. Carpinteri,et al.  A fractional calculus approach to nonlocal elasticity , 2011 .

[5]  Martin Ostoja-Starzewski,et al.  Lattice models in micromechanics , 2002 .

[6]  E. Lieb,et al.  Phase transitions and reflection positivity. I. General theory and long range lattice models , 1978 .

[7]  V. E. Tarasov.  Map of discrete system into continuous , 2006, 0711.2612.

[8]  M. Zingales,et al.  FRACTIONAL DIFFERENTIAL CALCULUS FOR 3D MECHANICALLY BASED NON-LOCAL ELASTICITY , 2011 .

[9]  Vasily E. Tarasov,et al.  Lattice Model of Fractional Gradient and Integral Elasticity: Long-Range Interaction of Grunwald-Letnikov-Riesz Type , 2014, 1502.06268.

[10]  A. K. Jonscher,et al.  Low-loss dielectrics , 1999 .

[11]  R. Hilfer.  Applications Of Fractional Calculus In Physics , 2000 .

[12]  Freeman J. Dyson,et al.  An Ising ferromagnet with discontinuous long-range order , 1971 .

[13]  Nakano,et al.  Magnetic properties of quantum Heisenberg ferromagnets with long-range interactions. , 1995, Physical review. B, Condensed matter.

[14]  O. Agrawal,et al.  Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering , 2007 .

[15]  K. A. Lazopoulos.  Non-local continuum mechanics and fractional calculus , 2006 .

[16]  M. Di Paola,et al.  A generalized model of elastic foundation based on long-range interactions: Integral and fractional model , 2009 .

[17]  V. E. Tarasov.  Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media , 2011 .

[18]  S. Timoshenko,et al.  Theory of elasticity , 1975 .

[19]  T. Atanacković,et al.  Generalized wave equation in nonlocal elasticity , 2009 .

[20]  F. Mainardi,et al.  Fractals and fractional calculus in continuum mechanics , 1997 .

[21]  A. Cleland.  Foundations of nanomechanics , 2002 .

[22]  G. Zaslavsky,et al.  Fractional dynamics of systems with long-range interaction , 2006, 1107.5436.

[23]  CNRS,et al.  Statistical mechanics and dynamics of solvable models with long-range interactions , 2009, 0907.0323.

[24]  Harold S. Park,et al.  Nano Mechanics and Materials: Theory, Multiscale Methods and Applications , 2006 .

[25]  Vasily E. Tarasov,et al.  Fractional Gradient Elasticity from Spatial Dispersion Law , 2013, 1306.2572.

[26]  Mario Di Paola,et al.  The mechanically-based approach to 3D non-local linear elasticity theory: Long-range central interactions , 2010 .

[27]  V. E. Tarasov.  Review of Some Promising Fractional Physical Models , 2013, 1502.07681.

[28]  Yu. A. Brychkov,et al.  Integrals and series , 1992 .

[29]  V. E. Tarasov.  Continuous limit of discrete systems with long-range interaction , 2006, 0711.0826.

[30]  Vasily E. Tarasov,et al.  Lattice model with power-law spatial dispersion for fractional elasticity , 2013, 1501.01201.

[31]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[32]  Jacqueline Krim,et al.  Foundations of Nanomechanics: From Solid-State Theory to Device Applications , 2004 .

[33]  Pietro Cornetti,et al.  Static‐kinematic fractional operators for fractal and non‐local solids , 2009 .

[34]  Albert C. J. Luo,et al.  Long-range interactions, stochasticity and fractional dynamics , 2010 .

[35]  E. Aifantis,et al.  Gradient elasticity in statics and dynamics: An overview of formulations, length scale identification procedures, finite element implementations and new results , 2011 .

[36]  M. Di Paola,et al.  Fractional mechanical model for the dynamics of non-local continuum , 2009 .

[37]  A. K. Jonscher,et al.  The ‘universal’ dielectric response , 1977, Nature.

[38]  N. Challamel,et al.  On the fractional generalization of Eringenʼs nonlocal elasticity for wave propagation , 2013 .

[39]  Massimiliano Zingales,et al.  Wave propagation in 1D elastic solids in presence of long-range central interactions , 2011 .

[40]  V. E. Tarasov.  General Lattice Model of Gradient Elasticity , 2015, 1501.01435.

[41]  Mario Di Paola,et al.  Long-range cohesive interactions of non-local continuum faced by fractional calculus , 2008 .

[42]  S. Hsieh,et al.  Nanomechanics of Materials and Structures , 2006 .

[43]  E. Kröner,et al.  Elasticity theory of materials with long range cohesive forces , 1967 .

[44]  Alberto Carpinteri,et al.  Wave propagation in nonlocal elastic continua modelled by a fractional calculus approach , 2013, Commun. Nonlinear Sci. Numer. Simul..

[45]  Ralf Metzler,et al.  Fractional dynamics : recent advances , 2011 .

[46]  F. Mainardi.  Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models , 2010 .

[47]  Pietro Cornetti,et al.  Fractional calculus in solid mechanics: local versus non-local approach , 2009 .

[48]  G. Failla,et al.  Mechanically Based Nonlocal Euler-Bernoulli Beam Model , 2014 .

[49]  Mario Di Paola,et al.  Elastic waves propagation in 1D fractional non-local continuum , 2009 .