  This study presents the framework to perform a stability analysis of nonlocal
solids whose response is formulated according to the fractional-order continuum
theory. In this formulation, space fractional-order operators are used to
capture the nonlocal response of the medium by introducing nonlocal kinematic
relations. First, we use the geometrically nonlinear fractional-order kinematic
relations within an energy-based approach to establish the Lagrange-Dirichlet
stability criteria for fractional-order nonlocal structures. This energy-based
approach to nonlocal structural stability is possible due to a
positive-definite and thermodynamically consistent definition of deformation
energy enabled by the fractional-order kinematic formulation. Then, the
Rayleigh-Ritz coefficient for the critical load is derived for linear buckling
conditions. The fractional-order formulation is finally used to determine
critical buckling loads of slender nonlocal beams and plates using a dedicated
fractional-order finite element solver. Results establish that, in contrast to
existing studies, the effect of nonlocal interactions is observed on both the
material and the geometric stiffness, when using the fractional-order
kinematics approach. We support these observations quantitatively with the help
of case studies focusing on the critical buckling response of fractional-order
nonlocal slender structures, and qualitatively via direct comparison of the
fractional-order approach with the classical nonlocal approaches.
