Towards a unified approach to nonlocal elasticity via fractional-order mechanics

This study presents a fractional-order continuum mechanics approach that allows combining selected characteristics of nonlocal elasticity, typical of classical integral and gradient formulations, under a single frame-invariant framework. The resulting generalized theory is capable of capturing both stiffening and softening effects and it is not subject to the inconsistencies often observed under selected external loads and boundary conditions. The governing equations of a 1D continuum are derived by continualization of the Lagrangian of a 1D lattice subject to long-range interactions. This approach is particularly well suited to highlight the connection between the fractional-order operators and the microscopic properties of the medium. The approach is also extended to derive, by means of variational principles, the governing equations of a 3D continuum in strong form. The positive definite potential energy, characteristic of our fractional formulation, always ensures well-posed governing equations. This aspect, combined with the differ-integral nature of fractional-order operators, guarantees both stability and the ability to capture dispersion without requiring additional inertia gradient terms. The proposed formulation is applied to the static and free vibration analyses of either Timoshenko beams or Mindlin plates. Numerical results, obtained by a fractional-order finite element method, show that the fractional-order formulation is able to model both stiffening and softening response in these slender structures. The numerical results provide the foundation to critically analyze the physical significance of the different fractional model parameters as well as their effect on the response of the structural elements.