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

Abstract

Models of three-dimensional lattices with long-range interactions of Grunwald-Letnikov type for fractional gradient elasticity of non-local con- tinuum are suggested. The lattice long-range interac- tions are described by fractional-order difference operators. Continuous limit of suggested three-dimen- sional lattice equations gives continuum differential equations with the Grunwald-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.