Toward fractional gradient elasticity - Tarasov Vasily E.; Aifantis Elias C. (Journal of the Mechanical Behavior of Materials, 2014)
The use of an extension of gradient elasticity through the inclusion of spatial derivatives of fractional order to describe the 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, stress equilibrium in a Caputo elastic bar requires the existence of a nonzero internal body force to equilibrate it. In the second case, 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.