  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.
