Fractional viscoelastic modeling of laminated glass beams in the pre-crack state under explosive loads

Abstract

A mathematical approach, alternative to the classical formulation of linear viscoelasticity, defines the constitutive laws in terms of integro-differential equations involving fractional, rather than ordinary, derivatives. Here, we discuss the possible advantages of this formulation in the modeling of laminated glass under loads of very short duration that change by sign, such as those from an explosion, with reference to the model problem of a beam composed of two glass plies coupled by a viscoelastic polymeric interlayer in the pre-glass-breakage stage, amenable of a zig-zag warping of the cross section. The Galerkin solution of the fractional integro-differential equations is obtained with a mixed analytical/numerical approach, by separating the discontinuous part of the blast pressure history and proceeding to a preliminary integration, which optimizes convergence. This also allows to recognize the specific role of the compression and suction phases of the blast pressure, with respect to the memory effect of viscoelasticity. Only two parameters are needed to define the viscoelastic properties of the commercial polymers used as interlayers, thus facilitating the model calibration on the base of simple relaxation tests on laminated glass specimens. A parametric analysis defines the polymer response as intermediate between purely elastic and a purely viscous, depending on its viscoelastic parameters. The comparison with the classical method relying on an expansion in Prony series of the relaxation law, highlights the simplifications obtainable with the fractional approach.