Size-dependent nonlinear free vibration of magneto-electro-elastic nanobeams by incorporating modified couple stress and nonlocal elasticity theory

Magneto-Electro-Elastic (MEE) Composites, as an innovative functional material blend, are composed of multiple materials, boasting exceptional strength, rigidity, and an extraordinary magneto-electric interaction effect. This paper establishes a nonlocal modified couple stress (NL-MCS) magneto-electro-elastic nanobeam dynamic model. To accurately capture the intricate influences of scale effects on nanostructures, This model meticulously examines scale effects from two distinct perspectives: leveraging nonlocal elasticity theory to elucidate the softening phenomena in nanostructures stemming from long-range particle interactions, and employing modified couple stress theory to reveal the hardening effects attributed to the rotational behavior of particles within the structure. By incorporating Von Karman geometric nonlinearity, Reddy's third-order shear deformation theory and Maxwell's equations, the governing equations for the nonlinear free vibration of MEE nanobeams are derived using Hamilton's principle. Finally, a two-step perturbation method is employed to solve these equations. Two-step perturbation method disintegrates the solution process into two stages, iteratively approximating and refining the solution, thereby progressively unraveling the intricate details and enhancing the precision of the solution in a systematic manner. Finally, the nonlinear free vibration behavior of MEE nanobeams is explored under the coupled magnetic-electric-elastic fields, with a focus on the effects of various factors that including length scale parameters, nonlocal parameters, Winkler-Pasternak coefficients, span-to-thickness ratios, applied voltages and magnetic potentials.