Abstract Summary
The question of predicting the long-term evolution of a mechanical structure in operation (e.g. fatigue) faces the associated issue of computational cost, particularly when the problem is intended to be numerically solved in the time domain (which can be required to account for the history of loads applied on the structure). When fast phenomena have to be described for a very long time interval, a scale separation assumption allows the use of time homogenization methods such as in (Guennouni & Aubry. Comptes rendus de l’Académie des sciences, Série II. 1986), (Oskay & Fish. International Journal for Numerical Methods in Engineering. 2004) or (Puel & Aubry, International Journal for Multiscale Computational Engineering, 2014), which are based on a quasi-periodicity assumption. When fast loads can be described by stochastic processes, a so-called stochastic time homogenization method can be defined and be seen as a transposition to time of the classical stochastic space homogenization method (Sab, European Journal of Mechanics: A. Solids, 1992). The time-homogenized equations are determined by averaging the different quantities allowing to separate slow-evolving phenomena from fast-time components, as in (Puel & Sab, VII European Congress on Computational Methods in Applied Sciences and Engineering, 2016). The main difficulty is then to substitute to the ensemble expectations a time averaging over a suitable time interval in order to derive the time-homogenized equations. The aim of the talk is to give theoretical as well as practical insights of this specific method. Examples will focus on structures with typical nonlinear material behaviors (e.g. viscoplasticity, damage), with loadings that can be described as the superposition of a slow deterministic component and a fast stochastic component.
