Equivalence between micromorphic, nonlocal gradient, and two-phase nonlocal beam theories

This paper explores the potential to unify gradient and nonlocal elastic beam theories using strain- or stress-driven nonlocal frameworks, focusing on nonlocal Euler–Bernoulli beam kinematics. It presents a two-length-scale gradient/nonlocal beam model that connects bending moments to curvature through a two-scale differential law, derivable via strain- or stress-based variational principles. The strain-driven micromorphic, two-phase strain-driven nonlocal, and nonlocal strain gradient beam theories share identical governing equations and higher-order boundary conditions for normalized nonlocal kernels on finite beams. However, the positive nonlocal potential energy constrains each theory’s validity, depending on the length-scale ratio. The nonlocal strain gradient theory can encompass the others to ensure positive potential energy across varying length scales. Differences between the finite-beam exponential kernel model and the infinite-beam model are clarified; though governed by the same differential equation but each has unique higher-order boundary conditions. The interest of the theory based on a normalized kernel along the finite beam is highlighted on a pure bending test which preserves the uniform curvature field. Additionally, the stress-driven micromorphic, two-phase stress-driven nonlocal, and nonlocal stress gradient theories share the same governing equations and boundary conditions for finite-beam kernels. The study concludes that these theories—micromorphic, nonlocal gradient, and two-phase nonlocal—can be unified within strain- and stress-driven frameworks for specific nonlocal kernels.