Correction of local elasticity for nonlocal residuals: application to Euler–Bernoulli beams

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 iterativenonlocal 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's nonlocal theory in accounting for the impacts of the material's Poisson's 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 beam's equation of motion.

Iterative Nonlocal Residual Elasticity

“…The boundary value problem of the homogeneous-isotropic-linear elastic material formed based on the proposed nonlocal elasticity constitutes the following system of equations: The iterative procedure implemented in this study is pioneered from an iterative-finite element-based model of nonlocal elasticity proposed in [16] and implemented in [17,18] → Recall 𝜀 𝑖𝑗 (𝑘−1) (𝒙), the determined strain field of the previous iteration 𝑘 − 1.…”

Nonlocal Timoshenko shear beam model for multilayer curved graphene nano-switches

“…The internal length and the characteristic of material microstructures have recently been accounted for with nonlocal strain gradient models and other combined models [29]. The nonlocal elasticity has been applied to a variety of problems such as wave propagation, crack growth, buckling, and coupled thermoelasticity on geometries such as beams [23,30,31], rods [32,33], plates [34][35][36][37] and NEMS [38][39][40]. A nonlocal elasticity theory-based analysis of the thermal buckling behavior of circular bilayer graphene sheets embedded in an elastic medium is presented in [41].…”

Some New Approximate Solutions in Closed-Form to Problems of Nanobars

“…In addition to those landmark studies cited in up to this point, the interested readers are kindly referred to [21,22] for relatively recent applications of Eringen's two-phase model, [23][24][25][26][27] for different approaches to the modelling of nanobars, and a review paper [28] for a better insight on classification, limitations, and mathematical aspects of nonlocal continuum models.…”