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Complications exist when solving the field equation in the nonlocal field.This has been attributed to the complexity of deriving explicit forms of thenonlocal boundary conditions. Thus, the paradoxes in the existing solutions ofthe nonlocal field equation have been revealed in recent studies. In thepresent study, a new methodology is proposed to easily determine the elasticnonlocal fields from their local counterparts without solving the fieldequation. This methodology depends on the iterative-nonlocal residual approachin 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 thenonlocal residuals in materials are presented. These corrections are formedbased on the general nonlocal theory. In the context of the general nonlocaltheory, two distinct nonlocal parameters are introduced to form theconstitutive equations of isotropic-linear elastic continua. In this study, itis demonstrated that the general nonlocal theory outperforms Eringen-nonlocaltheory in accounting for the impacts of the material Poisson ratio on itsmechanics. To demonstrate the effectiveness of the proposed approach, thecorrections of the local static bending, vibration, and bucklingcharacteristics of Euler-Bernoulli beams are derived. Via these corrections,bending, vibration, and buckling behaviors of simple-supported nonlocalEuler-Bernoulli beams are determined without solving the beams equation ofmotion.

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