Abstract
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.
Highlights
The theory of derivatives and integrals of noninteger orders [1,2,3] allows us to investigate the behavior of materials and media that are characterized by nonlocality of power-law type
This paper focuses on the fractional generalization of gradient elasticity which describes a weak nonlocality of power type
Gradient elasticity is considered as a phenomenological theory representing continuum limit of lattice dynamics, where the length scales are much larger than interatomic distances
Summary
The theory of derivatives and integrals of noninteger orders [1,2,3] allows us to investigate the behavior of materials and media that are characterized by nonlocality of power-law type. 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 interest in the theory of elasticity As it was shown in [23, 24] (see [25,26,27]), the equations with fractional derivatives can be directly connected to lattice models with long-range interactions. We make the transformation to the continuous limit and derive the fractional equation, which describes the dynamics of the nonlocal elastic materials. We show how the continuous limit for the lattice with fractional weak spatial dispersion gives the corresponding continuum equation of the fractional gradient elasticity. The continuum equations of fractional elasticity are solved for some special cases: subgradient elasticity and supergradient elasticity
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