Correction of local-linear elasticity for nonlocal residuals: Application to Euler-Bernoulli beams

TL;DR

This paper introduces a new iterative residual approach to correct local-linear elastic fields for nonlocal residuals in materials, simplifying the analysis of Euler-Bernoulli beams without solving complex nonlocal boundary value problems.

Contribution

It proposes a novel methodology based on the iterative-nonlocal residual approach to efficiently incorporate nonlocal effects into local elastic solutions.

Findings

The method accurately predicts bending, vibration, and buckling of nonlocal Euler-Bernoulli beams.

General nonlocal theory with two parameters outperforms Eringen's theory in capturing Poisson ratio effects.

The approach avoids solving complex nonlocal boundary value problems.

Abstract

Complications exist when solving the field equation in the nonlocal field. This has been attributed to the complexity of deriving explicit forms of the nonlocal boundary conditions. Thus, the paradoxes in the existing solutions of the nonlocal field equation have been revealed in recent studies. In the present study, a new methodology is proposed to easily determine the elastic nonlocal fields from their local counterparts without solving the field equation. This methodology depends on the iterative-nonlocal residual approach in which the sum of the nonlocal fields is treaded as a residual field. Thus, in this study the corrections of the local-linear elastic fields for the nonlocal residuals in materials are presented. These corrections are formed based on the general nonlocal theory. In the context of the general nonlocal theory, two distinct nonlocal parameters are introduced to form the constitutive equations of isotropic-linear elastic continua. In this study, it is demonstrated that the general nonlocal theory outperforms Eringen-nonlocal theory in accounting for the impacts of the material Poisson ratio on its mechanics. To demonstrate the effectiveness of the proposed approach, the corrections of the local static bending, vibration, and buckling characteristics of Euler-Bernoulli beams are derived. Via these corrections, bending, vibration, and buckling behaviors of simple-supported nonlocal Euler-Bernoulli beams are determined without solving the beams equation of motion.