Viscoelastic modeling via fractional calculus of the cold bending of laminated glass

Abstract

A viscoelastic description via fractional calculus is used to theoretically determine the time-varying stress state in single-curvature cold-bent laminated glass. This approach is proven effective when the relaxation function of the polymeric interlayer can be approximated by branches of power laws, as in most commercial materials. Solutions are obtained numerically by approximating the fractional time derivatives with the L1 formula. This conveniently allows to use a variable time step for a phenomenon characterized by two time-scales, corresponding to the loading process and the long-term relaxation. A parametric analysis shows the effects of polymer type, interlayer thickness, deformation history and operating temperature. A comparison is made with the results from the quasi-elastic approximation, which neglects the memory effect of viscoelasticity, showing that, since the interlayer strain is kept constant in the long term, it provides accurate results in term of peak and asymptotic stresses in glass, although the early stages of the deformation history give rise to differences when the structure is far from the layered or monolithic limits. This study can be useful for the optimization of the cold-bending process.