Analytical and Numerical Investigation of a Nonlinear Nanobeam Model

PurposeThis article investigates the mechanical behavior of a nonlinear nanobeam subjected to static and dynamic loading conditions using a nonlinear model that incorporates analytical and finite-element methods, and the non-local strain gradient theory.MethodsThe study derives the equation of motion for the nanobeam using Hamilton's principle and establishes dimensionless parameters. The deflection of the nanobeam under static loading conditions is analyzed using the Galerkin method, while the time-dependent nonlinear equation is obtained using the same method under initial conditions. The natural frequency and nonlinear frequency of oscillation are determined using the method of multiple scales. The forced vibrations of the nanobeam are analyzed using the multiple scale method, and the amplitude and phase of oscillation are determined.ResultsThe study examines the convergence of the finite-element method, and a comparison is made between the outcomes of two analytical and numerical methods. It investigates the role of non-local and length scale parameters on static and vibration behavior, and compares classical, non-local, and non-local strain gradient theories. ConclusionThe results reveal that the non-local strain gradient theory demonstrates a combined softening and stiffening behavior, dependent on the dimensions of the structures, with the boundary between the two regions displayed in the results.

Automated summary

This article investigates the mechanical behavior of a nonlinear nanobeam under static and dynamic loading conditions using a nonlinear model that combines analytical and finite-element methods, as well as the non-local strain gradient theory. The equation of motion for the nanobeam is derived using Hamilton's principle and dimensionless parameters are established. The study analyzes the deflection under static loading conditions using the Galerkin method and obtains the time-dependent nonlinear equation under initial conditions using the same method. The natural frequency, nonlinear frequency, and forced vibrations of the nanobeam are determined using the method of multiple scales.