Variable time steps in the numerical implementation of viscoelastic fractional models for laminated glass

Abstract

We elaborate numerical approaches to calculate the rheological response of laminated glass beams, whose viscoelastic interlayer is modelled via fractional calculus. This mathematical description is very effective when the relaxation function of the polymer can be expressed by continuously connected branches of power-laws, as is the case for most materials used to laminate glass. The classical approach uses the Grünwald-Letnikov approximation of fractional derivatives, but it requires constant time steps, which would become very large to reasonably cover the entire observation time, thus losing accuracy. We propose to use the L1 algorithm with increasing time steps, which is well suited to the power law character of the relaxation function. This allows to follow the long-term creep response, providing a better approximation when needed. The method is implemented for beams laminated with viscolastic interlayers whose relaxation is described by four branches of power laws, to cover most practical cases. Numerical experiments shows its advantages over the Grünwald-Letnikov approach for characterizing the long-term structural response.