Surface Elasticity and Residual Stress Effects on the Dynamic Stability of Axially Excited Nanobeams Using Floquet–Lyapunov Theory

This work presents a unified, size-aware study of the dynamic stability of submicrometer beams embedded in an elastic medium and subjected to combined static and harmonic axial loading. Timoshenko and Euler–Bernoulli beam formulations are considered while explicitly accounting for surface elasticity and residual surface stress to capture size-dependent mechanical behavior at the nano- to micrometer scales. The governing partial differential equations are projected onto trigonometric admissible functions via the Galerkin method and cast into extended Mathieu–Hill form, and instability boundaries are determined using Floquet–Lyapunov theory. Numerical results, obtained by integrating the resulting ordinary differential systems with a Runge–Kutta scheme (Gill coefficients), systematically quantify how beam model choice, anodic-alumina crystallographic orientation ([100] versus [111]), elastic foundation stiffness, and excitation frequency alter parametric-instability regions, providing direct guidance for the design and stability assessment of nanoscale structural elements.