Finite-element formulation of a nonlocal hereditary fractional-order Timoshenko beam

A mechanically-based nonlocal Timoshenko beam model, recently proposed by the authors, hinges on the assumption that nonlocal effects can be modeled as elastic long-range volume forces and moments mutually exerted by nonadjacent beam segments, which contribute to the equilibrium of any beam segment along with the classical local stress resultants. Long-range volume forces/moments linearly depend on the product of the volumes of the interacting beam segments, and on pure deformation modes of the beam, through attenuation functions governing the space decay of nonlocal effects. This paper investigates the response of this nonlocal beam model when viscoelastic long-range interactions are included, modeled by Caputo's fractional derivatives. The finite-element method is used to discretize the pertinent fractional-order equations of motion. Closed-form solutions are obtained for creep tests by typical tools of fractional calculus. Numerical results are presented for various nonlocal parameters.