A relationship between discrete and continuous fractional-order nonlocal elasticity theory is discussed. As a discrete system we consider three-dimensional lattice with long-range interactions that are described by fractional-order lattice operators.We prove that the continuous limit of suggested three-dimensional lattice equations gives continuum differential equations with the Riesz 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.
E. Tarasov, Vasily
"Three-Dimensional Lattice Approach to Fractional Generalization of Continuum Gradient Elasticity,"
Progress in Fractional Differentiation & Applications: Vol. 1
, Article 2.
Available at: https://dc.naturalspublishing.com/pfda/vol1/iss4/2