A mechanically based approach to non-local beam theories

A mechanically based non-local beam theory is proposed. The key idea is that the equilibrium of each beam volume element is attained due to contact forces and long-range body forces exerted, respectively, by adjacent and non-adjacent volume elements. The contact forces result in the classical Cauchy stress tensor while the long-range forces are modeled as depending on the product of the interacting volume elements, their relative displacement and a material-dependent distance-decaying function. To derive the beam equilibrium equations and the pertinent mechanical boundary conditions, the total elastic potential energy functional is used based on the Timoshenko beam theory. In this manner, the mechanical boundary conditions are found coincident with the corresponding mechanical boundary conditions of classical elasticity theory. Numerical applications are also reported.

Research highlights
► A mechanically-based non-local Timoshenko beam model is presented.
► Each beam volume element is acted upon by contact forces and long-range forces.
► The governing equations are derived by applying a variational approach.
► Non-local terms are not involved in the mechanical boundary conditions.
► Non-local effects are encountered for arbitrary loading conditions.