  This study presents a generalized multiscale nonlocal elasticity theory that
leverages distributed order fractional calculus to accurately capture
coexisting multiscale and nonlocal effects within a macroscopic continuum. The
nonlocal multiscale behavior is captured via distributed order fractional
constitutive relations derived from a nonlocal thermodynamic formulation. The
governing equations of the inhomogeneous continuum are obtained via the
Hamilton principle. As a generalization of the constant order fractional
continuum theory, the distributed order theory can model complex media
characterized by inhomogeneous nonlocality and multiscale effects. In order to
understand the correspondence between microscopic effects and the properties of
the continuum, an equivalent mass-spring lattice model is also developed by
direct discretization of the distributed order elastic continuum. Detailed
theoretical arguments are provided to show the equivalence between the discrete
and the continuum distributed order models in terms of internal nonlocal
forces, potential energy distribution, and boundary conditions. These
theoretical arguments facilitate the physical interpretation of the role played
by the distributed order framework within nonlocal elasticity theories. They
also highlight the outstanding potential and opportunities offered by this
methodology to account for multiscale nonlocal effects. The capabilities of the
methodology are also illustrated via a numerical study that highlights the
excellent agreement between the displacement profiles and the total potential
energy predicted by the two models under various order distributions.
Remarkably, multiscale effects such as displacement distortion, material
softening, and energy concentration are well captured at continuum level by the
distributed order theory.
