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arXiv · 1307.6999

Towards Fractional Gradient Elasticity

Vasily E. Tarasov·Elias C. Aifantis
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Abstract

An extension of gradient elasticity through the inclusion of spatial derivatives of fractional order to describe power-law type of non-locality is discussed. Two phenomenological possibilities are explored. The first is based on the Caputo fractional derivatives in one-dimension. The second involves the Riesz fractional derivative in three-dimensions. Explicit solutions of the corresponding fractional differential equations are obtained in both cases. In the first case it is shown that stress equilibrium in a Caputo elastic bar requires the existence of a non-zero internal body force to equilibrate it. In the second case, it is shown that in a Riesz type gradient elastic continuum under the action of a point load, the displacement may or may not be singular depending on the order of the fractional derivative assumed.

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Vasily E. Tarasov, Elias C. Aifantis. 2013-07-26. Towards Fractional Gradient Elasticity. https://doi.org/10.1515/jmbm-2014-0006

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