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Modeling of Thermo‐Viscoelastic Material Behavior of Glass Over A Received: 30 August 2019 | Revised: 8 November 2019 | Accepted: 1 December 2019 DOI: 10.1111/jace.16963 ORIGINAL ARTICLE Modeling of thermo-viscoelastic material behavior of glass over a wide temperature range in glass compression molding Anh Tuan Vu1 | Anh Ngoc Vu1 | Tim Grunwald1 | Thomas Bergs1,2 1Fraunhofer Institute for Production Technology IPT, Aachen, Germany Abstract 2Laboratory for Machine Tools and In glass compression molding, most current modeling approaches of temperature- Production Engineering (WZL) of RWTH dependent viscoelastic behavior of glass materials are restricted to thermo-rheolog- Aachen University, Aachen, Germany ically simple assumption. This research conducts a detailed study and demonstrates Correspondence that this assumption, however, is not adequate for glass molding simulations over a Anh Tuan Vu, Fine Machining & Optics, wide range of molding temperatures. In this paper, we introduce a new method that Fraunhofer Institute for Production eliminates the prerequisite of relaxation functions and shift factors for modeling of Technology IPT, Steinbachstraße 17, 52074 Aachen, Germany. the thermo-viscoelastic material behavior. More specifically, the temperature effect Email: [email protected] is directly incorporated into each parameter of the mechanical model. The mechanical model parameters are derived from creep displacements using uniaxial compression Funding information German Academic Exchange Service experiments. Validations of the proposed method are conducted for three different (DAAD) glass categories, including borosilicate, aluminosilicate, and chalcogenide glasses. Excellent agreement between the creep experiments and simulation results is found in all glasses over long pressing time up to 900 seconds and a large temperature range that corresponds to the glass viscosity of log (η) = 9.5 – 6.8 Pas. The method eventu- ally promises an enhancement of the glass molding simulation. KEYWORDS FEM simulation, glass molding, thermo-rheologically simple, thermo-viscoelastic modeling 1 | INTRODUCTION increasing geometrical complexity, higher precision, and pro- duction volume. Accordingly, conventional fabrication tech- Glass, recently redefined as a nonequilibrium, noncrystal- niques via grinding and polishing become no longer feasible. line condensed state of matter,1 contains a glass transition, Instead, replicative manufacturing or glass molding promises the region at which it strongly exhibits viscoelastic behav- a distinct satisfaction of such ever-increasing demands. This iors. Owing to enormously precious properties including technology enables the fabrication of precision glass optics, high hardness, thermal resistance, refractive index, light where the desirable shape is readily achieved after a very permeability, and stability to extreme environment changes, short molding cycle and requires no subsequent machining such as humidity and temperature, glass has been replacing steps, such as grinding and polishing. polymers in many optical applications. Today, optical com- In the glass compression molding, a hot glass blank is ponents are challenging for glass optic manufacturers toward pressed in between a precision mold pair and then is cooled This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. © 2019 The Authors. Journal of the American Ceramic Society published by Wiley Periodicals, Inc. on behalf of American Ceramic Society (ACERS) J Am Ceram Soc. 2020;103:2791–2807. wileyonlinelibrary.com/journal/jace | 2791 2792 | VU ET AL. down to a solid state. Depending on the temperature differ- implementation where the coupling temperature-mechanical ence between the glass and mold during pressing, the glass model can be simplified as a solely mechanical model by incor- molding process is distinguished by isothermal glass molding porating the so-called reduced time, defined by a shift function. or precision glass molding (PGM) and nonisothermal glass It is emphasized that this implementation is valid if and only molding (NGM).2 Regardless, the glass undergoes the heating if the shape of stress relaxation functions at every tempera- stage to a liquid-like state at nearly softening temperature (Sp) ture, when plotted against a logarithmic time, is fully identical. in PGM or even above Sp in NGM and the following pressing However, Scherer stresses that, albeit a good approximation and cooling stages to a solid-like state, typically below tran- near the glass transition, it fails at high temperature where the 21 sition temperature (Tg), where its desirable shape is mostly glass exhibits liquid-like viscoelasticity. In particular, sensi- achieved. tivity studies of the TRS assumption highlight that non-negligi- In this temperature regime, viscoelastic properties dom- ble deviation occurs in the numerical predictions of the molded inate the deformation of the glass under thermo-mechanical lens shape22 or creep displacements.5,23 loads. Previous studies demonstrate essential contributions Finally, to characterize the thermo-viscoelastic material of viscoelastic properties to the final shape of molded behavior of the glass, the most common approach, saying lenses.3‒5 Numerical predictions of the final lens shape re- conventional approach, is to perform experiments, either quires a coupling thermo-mechanical constitutive model creep or stress relaxation, at several temperatures. The stress to comprehend thermo-viscoelastic responses of glass de- relaxation experiment is likely more favorable because the formation over the entire molding process. For mechanical model's parameters, for example, the generalized Maxwell modeling of the viscoelastic responses, a large number of model, can be directly obtained by fitting the stress relax- rheological models have been investigated in the field of ation data measured at one specific temperature, called the glass molding simulation over years. In the early days, be- reference temperature.24 The disadvantage of this approach is havior of glass at the molding temperature was commonly that the stress relaxation experiments are mostly conducted at considered the pure viscous flow, where flow stress is pro- temperatures in the vicinity of Tg, which are quite far from the 6‒8 portional to the strain rate. Other studies use either simple actual molding temperature, normally near Sp. In this context, Maxwell or Kelvin model to approximate the stress relax- the viscoelastic properties at the actual molding temperature ation and creep responses, respectively.9,10 Those simple are determined by using the shift function. In contrast, the models are chosen to qualitatively study the form deviation characterization of viscoelastic properties at the real mold- of molded glass shapes and stress characteristics. A later ing temperature is restricted to the creep experiment. At such study by Zhou et al reports that neither Maxwell nor Kelvin high temperature, in fact, the glass becomes liquid-like where model satisfactorily describes the experiment data in the its relaxation time is extremely fast (τ«seconds) so that the temperature regime of glass molding, but Burgers model—a data obtained from the stress relaxation experiment is not combination of both Maxwell and Kelvin model—does.11 sufficient and reliable for postprocessing. Based on the creep The Burgers model, in fact, can be represented by a two- approach, the mold displacements are acquired and used to term generalized Maxwell model with an identical math- compute creep compliances.25,26 The drawback of this ap- ematical description.12 The generalized Maxwell model, a proach is that the temperature-dependent viscoelasticity is combination of several Maxwell elements in parallel, has not accomplished. Instead, the modulus relaxations need to been widely utilized in many recent studies to characterize be derived from the creep compliance functions using inter- the viscoelastic property of glass above the transition tem- conversion theory. Relying on the creep setup, in addition, perature.13‒15 The sufficient number of Maxwell elements previous studies realize a non-negligible discrepancy be- is determined by fitting the relaxation data, either gained tween the experiment and numerical prediction of the creep by stress relaxation experiments or derived from creep ex- displacements.26,27 According to their thoughts, the discrep- periments. It is emphasized that the number of Maxwell ancy is attributed to friction at the glass–mold contact in- elements strongly depends on the temperature chosen for terface. Nevertheless, satisfactory verifications of the creep the experimental characterization. For example, a six-term displacement cannot be obtained by varying the friction coef- generalized Maxwell model is commonly found for investi- ficients. A recent study by Yu et al shows that if the frictional 16 gating temperatures in the vicinity of Tg, while three terms disturbance can be diminished from the creep data so that are used at higher characterizing temperatures.17 mere purity of the viscoelastic property is secured by creep In addition, the thermal-mechanical modeling of the vis- experiments, good agreement can be found.28 coelastic responses necessitates an incorporation of the tem- Three major contributions are aimed in this research. First, perature effect into the mechanical models. Most of the present a comprehensive study on the conventional characterization
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