Stochastic instability via nonlocal continuum mechanics

The dynamic stability problem is solved for one-dimensional structures subjected to time-dependent deterministic or stochastic axial forces. The stability analysis of structures under time-dependent forces strongly depends on dissipation energy. The paper is concerned with the stochastic parametric vibrations of micro- and nano-rods based on Eringen’s nonlocal elasticity theory and Euler–Bernoulli beam theory. The asymptotic instability, and almost sure asymptotic instability criteria involving a damping coefficient, structure and loading parameters are derived using Liapunov’s direct method. Using the appropriate energy-like Liapunov functional sufficient conditions for the asymptotic instability, and the almost sure asymptotic instability of undeflected form of beam are derived. The nonlocal Euler–Bernoulli beam accounts for the scale effect, which becomes significant when dealing with short micro- and nano- rods. From obtained analytical formulas it is clearly seen that the small scale effect increases the dynamic instability region. Instability regions are functions of the axial force variance, the constant component of axial force and the damping coefficient.