Nonlocal Euler–Bernoulli Beam Theories: A Comparative Study

This book presents a comparative study on the static responses of the Euler-Bernoulli beam governed by nonlocal theories, including the Eringen’s stress-gradient beam theory, the Mindlin’s strain-gradient beam theory, the higher-order beam theory and the peridynamic beam theory. Benchmark examples are solved analytically and numerically using these nonlocal beam equations, including the simply-supported beam, the clamped-clamped beam and the cantilever beam. Results show that beam deformations governed by different nonlocal theories at different boundary conditions show complex behaviors. Specifically, the Eringen’s stress-gradient beam equation and the peridynamic beam equation yield a much softer beam deformation for simply-supported beam and clamped-clamped beam, while the beam governed by the Mindlin’s strain-gradient beam equation is much stiffer.
The cantilever beam exhibits a completely different behavior. The higher-order beam equation can be stiffer or softer depending on the values of the two nonlocal parameters. Moreover, the deformation fluctuation of the truncated order peridynamic beam equation is observed and explained from the singularity aspect of the solution expression. This research casts light on the fundamental explanation of nonlocal beam theories in nano-electromechanical systems.