Stochastic analysis of a nonlocal fractional viscoelastic bar forced by Gaussian white noise

Recently, a displacement-based non-local bar model has been developed. The model is based on the assumption that non-local forces can be modeled as viscoelastic long-range interactions mutually exerted by non-adjacent bar segments due to their relative motion; the classical local stress resultants are also present in the model. A finite element (FE) formulation with closed-form expressions of the elastic and viscoelastic matrices has also been obtained. Specifically, Caputo’s fractional derivative has been used in order to model viscoelastic long-range interaction. The static and quasistatic response has been already investigated. This work investigates the stochastic response of the non-local fractional viscoelastic bar introduced in previous papers, discretized with the FEM, forced by a Gaussian white noise. Since the bar is forced by a Gaussian white noise, dynamical effects cannot be neglected. The system of coupled fractional differential equations ruling the bar motion can be decoupled only by means of the fractional order state variable expansion. It is shown that following this approach Monte Carlo simulation can be performed very efficiently. For simplicity, here the work is limited to the axial response, but can be easily extended to transverse motion.