Anomalous Nonlinear Dynamics Behavior of Fractional Viscoelastic Structures

Abstract

Fractional models and their parameters are sensitive to changes in the intrinsic micro-structures of anomalous materials. We investigate how such physics-informed models propagate the evolving anomalous rheology to the nonlinear dynamics of mechanical systems. In particular, we analyze the vibration of a fractional, geometrically nonlinear viscoelastic cantilever beam, under base excitation and free vibration, where the viscoelastic response is general through a distributed-order fractional model. We employ Hamilton's principle to obtain the corresponding equation of motion with the choice of specific material distribution functions that recover a fractional Kelvin-Voigt viscoelastic model of order α. Through spectral decomposition in space, the resulting time-fractional partial differential equation reduces to a nonlinear time-fractional ordinary differential equation, in which the linear counterpart is numerically integrated by employing a direct L1-difference scheme. We further develop a semi-analytical scheme to solve the nonlinear system through a method of multiple scales, which yields a cubic algebraic equation in terms of the frequency. Our numerical results suggest a set of α-dependent anomalous dynamic qualities, such as far-from-equilibrium power-law amplitude decay rates, super-sensitivity of amplitude response at free vibration, and bifurcation in steady-state amplitude at primary resonance.

Data-Driven Fractional Subgrid-scale Modeling for Scalar Turbulence: A Nonlocal LES Approach

“…Unlike the integer-order (standard) differential operators, fractional-order operators are fundamentally defined based on heavy-tailed stochastic processes; therefore, they are inherently nonlocal operators and are suitable to incorporate long-range interactions in a mathematical modeling [28,29]. Among the wide-range of applications employing the fractional-order operators, modeling of visco-elasto-plastic materials for structural analysis [30], their nonlinear vibration analysis [31], memory-dependent modeling of damage mechanics [32], and nonlocal elasticity modeling of solids [33] are listed among the outstanding works reported in the literature.…”