The small length scale effect for a non-local cantilever beam: a paradox solved

Area of Science:

• Continuum Mechanics
• Nanotechnology
• Materials Science

Background:

• Non-local continuum mechanics accounts for small length scale effects crucial in micro/nanostructures.
• Classical local elasticity assumes stress depends only on strain at a single point.
• Existing integral-based non-local elastic beam models sometimes fail to show small scale effects, yielding classical solutions.

Purpose of the Study:

• To present simplified non-local elastic beam models for bending analysis of small scale rods.
• To resolve the paradox of non-local models producing classical solutions by developing a new model.
• To demonstrate the effectiveness of the proposed model in capturing small scale effects.

Main Methods:

• Development of simplified non-local elastic beam models, including integral-type and gradient models.
• Investigation of bending analyses for small scale rods using these models.
• Formulation of a novel integral non-local elastic model combining local and non-local curvatures.

Main Results:

• Identified a paradox where some integral non-local beam solutions match classical local solutions, negating small scale effects.
• Demonstrated that a gradient elastic model can overcome this paradox.
• Showed that a combined integral non-local elastic model, incorporating local and non-local curvatures, successfully introduces small length scale terms into solutions.

Conclusions:

• The proposed combined integral non-local elastic model effectively resolves the paradox of vanishing small scale effects.
• This novel model, encompassing classical gradient and Eringen's integral models, provides accurate bending solutions for small scale beams.
• The findings are significant for applications in microelectromechanical systems and nanoelectromechanical systems where small scale effects are critical.