Piecewise power law approximation of the interlayer relaxation curve for the long term viscoelastic fractional modeling of laminated glass

Laminated glass is composed of glass layers coupled by thin polymeric films which, being highly viscoelastic, influence the bending response over time. Here, an approximation of the polymer relaxation curve is proposed by means of continuously-connected power law branches. In the bi-logarithmic stress-time plane, this corresponds to a piecewise trilinear path, which well reproduces the experimental tests on most commercial polymers. A power law is naturally suited to be modeled via fractional calculus, as the Boltzmann convolution integral coincides with the definition of fractional derivative. However, the piecewise characterization through multiple power laws requires to properly extend the Grunwald-Letnikov integration scheme for fractional differential equations. For the paradigmatic case of a simply supported three-layer laminated beam, numerical results show an excellent correspondence with experiments under long duration loads. The comparison with the classic Prony series approach highlights, on the one hand, how calibration of material parameters is simplified; on the other hand, how the fractional approach is computationally more efficient and stable, even when the integration time step is wide enough to follow long term phenomena.