[Paper Review] Paradoxes of differential nonlocal cantilever beams: Reasons and a novel solution

Mohamed Shaat, S. Faroughi+1 more|arXiv (Cornell University)|Dec 2, 2017

Nonlocal and gradient elasticity in micro/nano structures42 references4 citations

TL;DR

This paper resolves paradoxes in differential nonlocal cantilever beam models by introducing the iterative nonlocal residual approach, which applies local boundary conditions through iteratively formed nonlocal residuals. The method produces results identical to integral nonlocal models, effectively eliminating inconsistencies from nonlocal boundary condition formulation in static bending and free vibration analyses.

ABSTRACT

The paradoxes of differential nonlocal models have been attributed to the inconsistency in forming the nonlocal boundary conditions. To overcome these paradoxes, nonlocal boundary conditions should be correctly formed. However, still forming these boundary conditions for differential nonlocal models is a challenging task. To resolve the trouble, in this study, the iterative nonlocal residual approach is employed. In the context of this approach, the field equation is solved in the local field with an imposed nonlocal residuals. This nonlocal residual is iteratively formed (not determined) depending on a pre-determined local field. The iterative nonlocal residual approach permits applying the local boundary conditions. Thus, the paradoxes of differential nonlocal models due to the nonlocal boundary conditions are effectively solved. The paradoxes of differential nonlocal models and reasons behind them are discussed. Moreover, a nonlocal beam model based on the iterative nonlocal residual approach is proposed. This nonlocal beam model is employed to investigate the static bending and free vibration of differential nonlocal cantilever beams. Comparisons between the results of the iterative nonlocal residual approach and results of integral nonlocal beam models are carried out. It is demonstrated that the iterative nonlocal residual approach gives the same results as integral nonlocal models.

Motivation & Objective

• To identify and resolve the paradoxes arising from inconsistent nonlocal boundary conditions in differential nonlocal beam models.
• To develop a robust method for formulating nonlocal boundary conditions that avoids the inconsistencies plaguing existing differential nonlocal formulations.
• To propose a nonlocal beam model based on the iterative nonlocal residual approach that maintains compatibility with local boundary conditions.
• To validate the proposed approach by comparing its results with those of established integral nonlocal beam models.
• To demonstrate the effectiveness of the method in static bending and free vibration problems of cantilever beams.

Proposed method

• The iterative nonlocal residual approach solves the field equation in the local domain while imposing nonlocal residuals derived iteratively from a pre-determined local field.
• Nonlocal residuals are not directly prescribed but are formed through successive iterations based on the local field response.
• Local boundary conditions are applied directly, avoiding the need for ad hoc nonlocal boundary condition formulations.
• The method ensures consistency in nonlocal effects by iteratively refining the residual terms until convergence is achieved.
• The approach is implemented in a nonlocal beam model for static bending and free vibration analysis of cantilever beams.
• Results are compared with integral nonlocal beam models to validate accuracy and consistency.

Experimental results

Research questions

• RQ1 Why do differential nonlocal models produce paradoxical results in cantilever beam problems?
• RQ2 What causes the inconsistency in nonlocal boundary condition formulation within differential nonlocal models?
• RQ3 Can a nonlocal beam model be developed that avoids paradoxes while preserving the simplicity of differential formulations?
• RQ4 Does the iterative nonlocal residual approach yield results equivalent to integral nonlocal models?
• RQ5 How does the proposed method perform in static bending and free vibration analyses of nonlocal cantilever beams?

Key findings

• The iterative nonlocal residual approach successfully eliminates paradoxes in differential nonlocal cantilever beams by correctly formulating nonlocal effects through iterative residuals.
• The method allows the direct application of local boundary conditions, resolving the long-standing issue of inconsistent nonlocal boundary conditions.
• The results of the iterative nonlocal residual approach are numerically identical to those obtained from integral nonlocal beam models.
• The proposed nonlocal beam model accurately predicts static deflection and natural frequencies of cantilever beams under various loading and boundary conditions.
• The approach demonstrates robustness and consistency in both static bending and free vibration analyses, validating its reliability.
• The study confirms that the root cause of paradoxes lies in improper nonlocal boundary condition formulation, not in the nonlocal integral operator itself.