Limit behaviour of Eringen's two-phase elastic beams

TL;DR

This paper investigates the limit behavior of Eringen's two-phase local/nonlocal elasticity theory for small-scale beams, revealing that asymptotic solutions do not satisfy equilibrium or boundary conditions, challenging previous assumptions.

Contribution

It provides an analytical study of the limit behavior of Eringen's two-phase beam theory, showing that certain asymptotic fields are not valid solutions of the purely nonlocal model.

Findings

Limit solutions do not fulfill equilibrium requirements.

Asymptotic fields cannot be solutions of purely nonlocal theory.

Purely nonlocal beam problems admit no solutions.

Abstract

In this paper, the bending behaviour of small-scale Bernoulli-Euler beams is investigated by Eringen's two-phase local/nonlocal theory of elasticity. Bending moments are expressed in terms of elastic curvatures by a convex combination of local and nonlocal contributions, that is a combination with non-negative scalar coefficients summing to unity. The nonlocal contribution is the convolution integral of the elastic curvature field with a suitable averaging kernel characterized by a scale parameter. The relevant structural problem, well-posed for non-vanishing local phases, is preliminarily formulated and exact elastic solutions of some simple beam problems are recalled. Limit behaviours of the obtained elastic solutions, analytically evaluated, studied and diagrammed, do not fulfill equilibrium requirements and kinematic boundary conditions. Accordingly, unlike alleged claims in literature, such asymptotic fields cannot be assumed as solutions of the purely nonlocal theory of beam elasticity. This conclusion agrees with the known result which the elastic equilibrium problem of beams of engineering interest formulated by Eringen's purely nonlocal theory admits no solution.