Filippo Gerin

A main challenge in laminated glass design is determining the degree of shear coupling between the glass piles through the interlayer, which leads to behavior somewhere between the monolithic limit (perfect coupling) and the layered limit (free-sliding glass layers). The warping of cross-sections caused by significant transverse shear strains in soft interlayers makes traditional laminated plate theories based on the plane-section assumption unreliable. Many commercial software packages exist to assist with this, but they often have limitations, particularly with geometric nonlinearities. Some of these programs only incorporate second-order theories, which are insufficient for thoroughly analyzing curved structures. Here, we introduce an in-house developed FEM code, which employs a nonlinear theory based on solid-shell models to overcome these limitations.
Our code applies the quasi-elastic approximation, where polymers are linear-elastic materials with a secant shear modulus depending on temperature and time. The geometrically exact solid-shell finite element approach allows us to use one single element per layer (glass or polymer) in the thickness direction but can also accommodate the multiple sheets in multi-material interlayers, used to impart hybrid characteristics to the laminate. Visual programming tools can import any curved geometry into the model, which approximates elements using displacement nodal values on the top and bottom surfaces of each layer with inherent regularity. An alternative solid-shell model with fewer parameters is derived by enforcing equal finite rotation of the rigid layers at each surface point through a local rotation-free re-parametrization of nodal displacements and by imposing plane stress conditions. This approach, which facilitates coupling with a solid discretization for modelling connections, is based on a straightforward strain measure quadratic in the displacement unknowns, suitable for handling finite strains. Comprehensive numerical examples for laminated glass plates and curved shells with large deflections are included to illustrate the code potentiality.

Insulating Glass Units (IGUs) are widely used to enhance thermal and acoustic performance of windows and façades. In addition to the challenges associated with non-linear modelling of the possibly-laminated glass panes, delimiting the cavities filled with air or inert gas, there is the added complexity of accurately considering the load sharing between the panes through the gas. Indeed, when external loads are applied to one of the panes, its deflection alters the volume of the trapped gas, leading to a pressure variation that helps support the loaded pane while pressing the unloaded pane. Commercial software can consider this effect, but with limitations for what concerns the modelling of the gas and the possibility of analyzing curved geometries. From a constitutive point of view, the pressure variation depends nonlinearly on the volume variation which, in turn, is a highly nonlinear function of the displacements of the two surfaces that delimit the cavity. This leads, in general, to a sparse tangent stiffness matrix, which challenges the numerical solver. Many existing codes introduce linear approximations to simplify the problem, but these are not always suitable under large deformations. Anyway, without these simplifications the computation time can become excessively long.
We complemented our existing in-house FEM code, which applies a nonlinear solid-shell model for laminated glass, by adding a module to analyze the response of IGUs, including curved configurations, under general loading conditions. The model incorporates both geometric and constitutive nonlinearities and can readily consider variations in temperature and barometric pressure. Our numerical strategies account for the exact volumetric changes in the cavities, enabling precise determination of load-sharing effects. Results from multiple case studies are compared with those from commercial FEM software for structural analysis. While the agreement is excellent, our approach requires significantly less computational time.