Alena Zemanová

Numerical model of elastic laminated glass beams under finite strain

Abstract Laminated glass structures are formed by stiff layers of glass connected with a compliant plastic interlayer. Due to their slenderness and heterogeneity, they exhibit a complex mechanical response that is difficult to capture by single-layer models even in the elastic range. The purpose of this paper is to introduce an efficient and reliable finite element approach to the simulation of the immediate response of laminated glass beams. It proceeds from a refined plate theory due to Mau (1973), as we treat each layer independently and enforce the compatibility by the Lagrange multipliers. At the layer level, we adopt the finite-strain shear deformable formulation of Reissner (1972) and the numerical framework by Ibrahimbegovic and Frey (1993). The resulting system is solved by the Newton method with consistent linearization. By comparing the model predictions against available experimental data, analytical methods and two-dimensional finite element simulations, we demonstrate that the proposed formulation is reliable and provides accuracy comparable to the detailed two-dimensional finite element analyzes. As such, it offers a convenient basis to incorporate more refined constitutive description of the interlayer.

Simple Numerical Model of Laminated Glass Beams

This contribution presents a simple Finite Element model aimed at efficient simulation of layered glass units. The adopted approach is based on considering independent kinematics of each layer, tied together via Lagrange multipliers. Validation and verification of the resulting model against independent data demonstrate its accuracy, showing its potential for generalization towards more complex problems.

FINITE ELEMENT MODEL BASED ON REFINED PLATE THEORIES FOR LAMINATED GLASS UNITS

LAMINATED GLASS UNITS EXHIBIT COMPLEX RESPONSE AS A RESULT OF DIFFERENT MECHANICAL BEHAVIOR AND PROPERTIES OF GLASS AND POLYMER FOIL. WE AIM TO DEVELOP A FINITE ELEMENT MODEL FOR ELASTIC LAMINATED GLASS PLATES BASED ON THE REFINED PLATE THEORY BY MAU. FOR A GEOMETRICALLY NONLINEAR DESCRIPTION OF THE BEHAVIOR OF UNITS, EACH LAYER BEHAVES ACCORDING TO THE REISSNER-MINDLIN KINEMATICS, COMPLEMENTED WITH MEMBRANE EFFECTS AND THE VON KAiRMAiN ASSUMPTIONS. NODAL LAGRANGE MULTIPLIERS ENFORCE THE COMPATIBILITY OF INDEPENDENT LAYERS IN THIS APPROACH. WE HAVE DERIVED THE DISCRETIZED MODEL BY THE ENERGY-MINIMIZATION ARGUMENTS, ASSUMING THAT THE UNKNOWN FIELDS ARE APPROXIMATED BY BI-LINEAR FUNCTIONS AT THE ELEMENT LEVEL, AND SOLVED THE RESULTING SYSTEM BY THE NEWTON METHOD WITH CONSISTENT LINEARIZATION. WE HAVE DEMONSTRATED THROUGH VERIFICATION AND VALIDATION EXAMPLES THAT THE PROPOSED FORMULATION IS RELIABLE AND ACCURATELY REPRODUCES THE BEHAVIOR OF LAMINATED GLASS UNITS. THIS STUDY REPRESENTS A FIRST STEP TO THE DEVELOPMENT OF A COMPREHENSIVE, MECHANICS-BASED MODEL FOR LAMINATED GLASS SYSTEMS THAT IS SUITABLE FOR IMPLEMENTATION IN COMMON ENGINEERING FINITE ELEMENT SOLVERS.

Comparison of viscoelastic finite element models for laminated glass beams

Abstract Laminated glass elements, which consist of stiff elastic glass layers connected with a compliant viscoelastic polymer foil, exhibit geometrically non-linear and time/temperature-sensitive behavior. In computational modeling, the viscoelastic effects are often neglected or a detailed continuum formulation typically based on the volumetric-deviatoric elastic-viscoelastic split is used for the interlayer. Four layerwise beam theories are introduced in this paper, which differ in the non-linear beam formulation at the layer level (von Karman/Reissner) and in constitutive assumptions for the interlayer (a viscoelastic solid with the time-independent bulk modulus/Poisson ratio). We perform detailed verification and validation studies at different temperatures and compare the accuracy of the selected formulation with simplified elastic solutions used in practice. We show that all the four formulations predict very similar responses. Therefore, our suggestion is to use the most straightforward formulation that combines the von Karman model with the assumption of time-independent Poisson ratio. The simplified elastic model mostly provides response in satisfactory agreement with full viscoelastic solutions. However, it can lead to unsafe or inaccurate predictions for rapid changes of loading. These findings provide a suitable basis for extensions towards laminated plates and glass layer fracture, owing to the modular format of layerwise theories.

Numerical and Experimental Modal Analysis of Laminated Glass Beams

This paper presents a numerical and experimental modal analysis of laminated glass beams, i.e. a multilayer composite structure made of glass panes bonded to an interlayer foil. These polymer foils provide shear coupling of glass layers, damping of vibrations, and play a key role in post-breakage performance. In this contribution, three-layer beams with ethylene-vinyl acetate interlayer are investigated. Using a finite element discretization and the Newton method, we solve numerically a complex eigenvalue problem which is nonlinear due to the frequency/temperature-sensitive viscoelastic behavior of the interlayer. In our experimental investigations, a roving hammer test was carried out to identify the mode shapes, natural frequencies, and modal damping. The validation shows that there is a good agreement between the numerical predictions and experimental data in natural frequencies. However, the errors in loss factors can be high, because these values are very sensitive to the material properties of polymer, frequency, temperature, and boundary conditions. These effects are discussed in the concluding part of our study.

VARIATIONALLY-BASED EFFECTIVE DYNAMIC THICKNESS FOR LAMINATED GLASS BEAMS

Laminated glass, consisting of glass layers connected with transparent foils, has found its applications in civil, automotive, or marine engineering. Due to a high contrast in layer properties, mechanical response of laminated glass structures cannot be predicted using classical laminate theories. On the other hand, engineering applications demand easy-to-use formulas of acceptable accuracy. This contribution addresses such simplified models for free vibrations of laminated glass beams, with the goal to determine their natural frequencies and modal damping properties. Our strategy is to approximate the complex behavior of a laminated structure with that of an equivalent monolithic beam. Its effective thickness is determined by the variational method proposed by Galuppi and Royer-Carfagni for static problems, which we extended for modal analysis. We show that this new approach overcomes inaccuracies of the currently used dynamic effective thickness model by Lopez-Aenlle and Pelayo.

Numerical Modeling of Laminated Glass Units

Laminated glass has been developed to improve the impact resistance of brittle glass sheets and to prevent injuries and collapse of glass members. The goal of this contribution is to briefly introduce a finite element model based on the refined plate theory by Mau that can describe the response of laminated glass plates without the need for fully resolved three-dimensional simulations. Each layer is considered to behave according to the Reissner-Mindlin kinematics, complemented with membrane effects and the von Karman assumptions. The compatibility of independent layers is enforced by nodal Lagrange multipliers. Predictions of the finite element model, obtained with a MATLAB-based program LaPla (Laminated Plates) developed by the authors, are compared with simplified monolithic and layered limits and a semi-analytical solution.