, Volume 51, Issue 1, pp 125–138 | Cite as

Three-dimensional lattice models with long-range interactions of Grünwald–Letnikov type for fractional generalization of gradient elasticity

  • Vasily E. Tarasov


Models of three-dimensional lattices with long-range interactions of Grünwald–Letnikov type for fractional gradient elasticity of non-local continuum are suggested. The lattice long-range interactions are described by fractional-order difference operators. Continuous limit of suggested three-dimensional lattice equations gives continuum differential equations with the Grünwald–Letnikov derivatives of non-integer orders. The proposed lattice models give a new microstructural basis for elasticity of materials with power-law type of non-locality. Moreover these lattice models allow us to have a unified microscopic description for fractional and usual (non-fractional) gradient elasticity continuum.


Gradient elasticity Nonlocal continuum Fractional derivatives Lattice model Long-range interactions Fractional-order difference 


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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Skobeltsyn Institute of Nuclear PhysicsLomonosov Moscow State UniversityMoscowRussia

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