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The performance of laminated glass, which consists of two or more glass plies bonded together by polymeric interlayers, depends upon shear coupling between the plies through the polymer. This is commonly considered by defining the effective thickness, i.e., the thickness of a monolithic beam with equivalent bending properties in terms of stress and deflection. General expressions have been proposed on the basis of simplified models by Newmark and Wölfel-Bennison, but they are either difficult to apply or inaccurate. Here, a variational approach to the problem is presented. By choosing appropriate shape functions for the laminated-beam deformation, minimization of the strain energy functional gives new expressions for the effective thickness under any constraint-and load-conditions, embracing the classical formulations as particular cases. Comparisons with numerical experiments confirm the better accuracy of the proposed approach with respect to the previous ones.Keywords: Structural glass, laminated glass, composite structures, bending strength, effective thickness, variational approach. IntroductionIn order to reduce the risk of catastrophic collapse of structures made of glass, the brittle material par excellence, an effective technique is to bond two or more glass plies with 1 2 L. Galuppi & G. Royer-Carfagni thermoplastic polymeric interlayers with a treatment in autoclave at high pressure and temperature. This bond is quite strong because it is chemical in type, being due to the union between hydroxyl groups along the polymer and silanol groups on the glass surface. The resulting laminated glass is a safety glass because, after breakage, the fragments remain attached to the interlayer: risk of injuries is reduced and the element maintains a certain consistency that prevents detachment from fixings. But the iterlayer affects also the pre glass-breakage response because it allows the transfer of shear stresses among glass plies, at the price of a relative sliding due to the deformation of the polymer. The assessment of the degree of connection offered by the polymer is crucial for the design of glass structures in the serviceability limit state and this is why a great number of studies, including this one, have considered the response of the composite laminated package before first cracking occurs.Indeed, the polymeric interlayers are too soft to present flexural stiffness per se, but they can provide shear stresses that play an important role for the glass-layer interaction [7]. In general, the degree of coupling of two glass layers depends upon the shear stiffness of the polymeric interlayer, as first mentioned by Hooper in 1973 [3] while studying the bending of simply supported laminated-glass beams. Since then, the problem has been considered by many authors [15], one of the most recent contribution being the careful finite element analysis of [14], which includes an updated list of the most relevant literature.In the pre glass-breakage modeling no distinction has to be made for what the type of ...

Recently, Francfort and Marigo (J. Mech. Phys. Solids 46, 1319-1342 have proposed a novel approach to fracture mechanics based upon the global minimization of a Griffith-like functional, composed of a bulk and a surface energy term. Later on the same authors, together with Bourdin, introduced (in J. Mech. Phys. Solids 48, 797-826, 2000) a variational approximation (in the sense of -convergence) of such functional, essentially for computational purposes.Here, we utilize this new variational approach to show how it might be altered to incorporate the idea of less brittle, "deviatoric-type fracture" and apply to materials such as confined stone. To do so, we modify the original formulation of Francfort and Marigo, in particular its approximation of Bourdin, Francfort and Marigo, to only allow for discontinuities in the deviatoric part of the strain. We apply such modified model to gain insight on the deterioration and cracking in the ashlar masonry work of the French Panthéon, which are so repetitious and particular to be a distinguishable symptom of ongoing damage. Numerical experiments are performed and the results compared to those obtained using the original Francfort-Marigo model and to actual crack patterns from the Panthéon.The modified formulation allows one to reproduce fracture paths surprisingly similar to that observed in situ, to sort out the possible causes of damage, and to confirm, with a quantitative analysis, the main structural deficiencies in the French monument. This practical example enhances the importance of this promising new theory based in the mathematical sciences.

a b s t r a c tWe analytically solve the time-dependent problem of a simply-supported laminated beam, composed of two elastic layers connected by a viscoelastic interlayer, whose response is modeled by a Prony's series of Maxwell elements. This case applies in particular to laminated glass, a composite made of glass plies bonded together by polymeric films. A practical way to calculate the response of such a package is to consider also the interlayer to be linear elastic, assuming its equivalent elastic moduli to be the relaxed moduli under constant strain, after a time equal to the duration of the design action. The obtained results, that are confirmed by a full 3-D viscoelastic finite-element numerical analysis, emphasize that there is a noteworthy difference between the state of strain and stress calculated in the full-viscoelastic case or in the aforementioned ''equivalent'' elastic problem.

Soft and conductive nanomaterials like carbon nanotubes, graphene, and nanowire scaffolds have expanded the family of ultraflexible microelectrodes that can bend and flex with the natural movement of the brain, reduce the inflammatory response, and improve the stability of long-term neural recordings. However, current methods to implant these highly flexible electrodes rely on temporary stiffening agents that temporarily increase the electrode size and stiffness thus aggravating neural damage during implantation, which can lead to cell loss and glial activation that persists even after the stiffening agents are removed or dissolve. A method to deliver thin, ultraflexible electrodes deep into neural tissue without increasing the stiffness or size of the electrodes will enable minimally invasive electrical recordings from within the brain. Here we show that specially designed microfluidic devices can apply a tension force to ultraflexible electrodes that prevents buckling without increasing the thickness or stiffness of the electrode during implantation. Additionally, these "fluidic microdrives" allow us to precisely actuate the electrode position with micron-scale accuracy. To demonstrate the efficacy of our fluidic microdrives, we used them to actuate highly flexible carbon nanotube fiber (CNTf) microelectrodes for electrophysiology. We used this approach in three proof-of-concept experiments. First, we recorded compound action potentials in a soft model organism, the small cnidarian Hydra. Second, we targeted electrodes precisely to the thalamic reticular nucleus in brain slices and recorded spontaneous and optogenetically evoked extracellular action potentials. Finally, we inserted electrodes more than 4 mm deep into the brain of rats and detected spontaneous individual unit activity in both cortical and subcortical regions. Compared to syringe injection, fluidic microdrives do not penetrate the brain and prevent changes in intracranial pressure by diverting fluid away from the implantation site during insertion and actuation. Overall, the fluidic microdrive technology provides a robust new method to implant and actuate ultraflexible neural electrodes.

The flexural performance of laminated glass, a composite of two or more glass plies bonded together by polymeric interlayers, depends upon shear coupling between the glass components through the polymer. This effect is usually taken into account, in the design practice, through the definition of the effective thickness, i.e., the thickness of a monolith with equivalent bending properties in terms of stress and deflection. The traditional formulas à la Bennison-Wölfel are accurate only when the deformed bending shape of the plate is cylindrical and the plate response is similar to that of a beam under uniformly distributed load. Here, assuming approximating shape function for the deformation of laminated plates variously constrained at the edges, minimization of the corresponding strain energy furnishes new simple expressions for the effective thickness, which can be readily used in the design. Comparisons with accurate numerical simulations confirm the accuracy of the proposed simple method for laminated plates.

Due to deformability of the polymeric interlayer, stiffness and strength of laminated glass are usually less than those corresponding to a monolith with same total thickness. A practical design tool consists in the definition of the "effective thickness", i.e., the thickness of an equivalent monolithic glass that would correspond to the same deflection and peak stress of the laminated glass, under the same constraint and load conditions. Very recently, a new model has been proposed for the evaluation of the effective thickness. Here, a comparison is made with the classical approach by Wölfel-Bennison and the new method is specialized to the most common cases of the design practice, providing synthetic tables for ease of reference and immediate applicability.

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