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

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Abstract

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.

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Tarasov, V.E. Three-dimensional lattice models with long-range interactions of Grünwald–Letnikov type for fractional generalization of gradient elasticity. Meccanica 51, 125–138 (2016). https://doi.org/10.1007/s11012-015-0190-4

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