Non-Standard Extensions of Gradient Elasticity: Fractional Non-Locality, Memory and Fractality
arXiv:1404.5241 · doi:10.1016/j.cnsns.2014.10.002
Abstract
Derivatives and integrals of non-integer order may have a wide application in describing complex properties of materials including long-term memory, non-locality of power-law type and fractality. In this paper we consider extensions of elasticity theory that allow us to describe elasticity of materials with fractional non-locality, memory and fractality. The basis of our consideration is an extension of the usual variational principle for fractional non-locality and fractality. For materials with power-law non-locality described by Riesz derivatives of non-integer order, we suggest a fractional variational equation. Equations for fractal materials are derived by a generalization of the variational principle for fractal media. We demonstrate the suggested approaches to derive corresponding generalizations of the Euler-Bernoulli beam and the Timoshenko beam equations for the considered fractional non-local and fractal models. Various equations for materials with fractional non-locality, fractality and fractional acceleration are considered.
LaTeX, 53 pages
References in corpus (9)
- Fractional Hydrodynamic Equations for Fractal Media
- Review of Some Promising Fractional Physical Models
- Calculus of variations with fractional derivatives and fractional integrals
- Vector Calculus in Non-Integer Dimensional Space and its Applications to Fractal Media
- Anisotropic Fractal Media by Vector Calculus in Non-Integer Dimensional Space
- Lattice Model of Fractional Gradient and Integral Elasticity: Long-Range Interaction of Grunwald-Letnikov-Riesz Type
- Lattice with Long-Range Interaction of Power-Law Type for Fractional Non-Local Elasticity
- Dispersion Relations and Wave Operators in Self-Similar Quasi-Continuous Linear Chains
- Chains with Fractal Dispersion Law
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