%0 Journal Article
%T Fractional Gradient Elasticity from Spatial Dispersion Law
%A Vasily E. Tarasov
%J ISRN Condensed Matter Physics
%D 2014
%R 10.1155/2014/794097
%X Nonlocal elasticity models in continuum mechanics can be treated with two different approaches: the gradient elasticity models (weak nonlocality) and the integral nonlocal models (strong nonlocality). This paper focuses on the fractional generalization of gradient elasticity that allows us to describe a weak nonlocality of power-law type. We suggest a lattice model with spatial dispersion of power-law type as a microscopic model of fractional gradient elastic continuum. We demonstrate how the continuum limit transforms the equations for lattice with this spatial dispersion into the continuum equations with fractional Laplacians in Riesz's form. A weak nonlocality of power-law type in the nonlocal elasticity theory is derived from the fractional weak spatial dispersion in the lattice model. The continuum equations with derivatives of noninteger orders, which are obtained from the lattice model, can be considered as a fractional generalization of the gradient elasticity. These equations of fractional elasticity are solved for some special cases: subgradient elasticity and supergradient elasticity. 1. Introduction The theory of derivatives and integrals of noninteger orders [1¨C3] allows us to investigate the behavior of materials and media that are characterized by nonlocality of power-law type. Fractional calculus has a wide application in mechanics and physics (e.g., see [4¨C14]). Nonlocal elasticity theories in continuum mechanics can be treated with two different approaches [15]: the gradient elasticity theory (weak nonlocality) and the integral nonlocal theory (strong nonlocality). The fractional calculus allows us to formulate a fractional generalization of nonlocal elasticity models in two forms: the fractional gradient elasticity models (weak power-law nonlocality) and the fractional integral nonlocal models (strong power-law nonlocality). The idea to include some fractional integral term in the equations of the elasticity has been proposed by Lazopoulos in [16]. Fractional models of integral nonlocal elasticity are considered in different papers; see, for example, [16¨C22]. The microscopic models of fractional integral elasticity are also described. For this reason, the fractional integral elasticity models are not discussed here. This paper focuses on the fractional generalization of gradient elasticity which describes a weak nonlocality of power type. We suggest a lattice model with spatial dispersion of power-law type as a microscopic model of fractional gradient elastic continuum. Complex lattice dynamics has been the subject of continuing
%U http://www.hindawi.com/journals/isrn.cmp/2014/794097/