Viscosity Measurement by Cylindrical Compression for NumericalModeling of Precision Lens Molding Process

Anurag Jain, Gregory C. Firestone, and Allen Y. Yiw

Department of Industrial, Welding and Systems Engineering, The Ohio State University, Columbus, Ohio 43210

Cylindrical compression tests were conducted on two differentoptical glass grades for determining the high-temperature vis-cosity and elastic parameters. Numerical simulations of thecompression tests and the precision lens molding process wereperformed by incorporating the data obtained from the com-pression tests using a commercial finite-element method pro-gram. Excellent agreement between the viscosity data from thecompression test and the beam bending test was obtained, and agood comparison between the measured and predicted deforma-tion load results was also observed. Further issues that haveemerged from this research that would be relevant to the ongo-ing research on the numerical modeling of the precision aspher-ical lens molding process include determination of high-temperature elastic properties of glass (i.e. elastic and shearmodulus) and friction characterization at the glass–mold inter-face.

I. Introduction

COMPRESSION molding of aspherical glass lenses is an emerg-ing technique for fabricating precision glass optics. For

over three decades, work has now been performed to make high-quality lenses by the hot compression method. Although it hasbeen shown that the molding process can produce lenses havingform accuracy, surface finish, and optical performance compa-rable with the lenses manufactured using conventional opticalfabrication techniques (grinding, polishing, and lapping), thereare some technical challenges associated with the process such asthermal expansion of the molds, mold life, lens curve shift aftercooling to room temperature, and residual stresses inside theformed lens that still need to be overcome.

With the recent advances in numerical simulation capabilitiesand computing technology, finite-element methods (FEMs) canbe applied to resolve some of the issues mentioned earlier andalso predict process performance.1,2 A reliable and accurate nu-merical model of the lens-forming process should incorporatethe viscoelastic nature of glass (i.e., stress and structural relax-ation) in addition to accurate temperature-dependent mechan-ical and thermal properties. Most of the past work carried out inthe area of numerical simulation of general glass forming hasneglected the viscoelastic response of glass at forming temper-atures.3–6

In this research, simple compression tests were performed ontwo different optical grades of glass in the forming temperaturerange (up to 1501C above transition temperature) followed byFEM simulations. From the standpoint of lens-forming processunderstanding, and its numerical simulation, it was consideredimportant to start with simple glass and mold geometries in or-der to establish a model, before proceeding with complicatedlens shape. Additionally, compression tests also enabled viscos-

ity determination of the two common grades of glass used in theoptics industry for fabricating lenses.

II. Research Objective

The overall objective of the work was to develop a fundamentalunderstanding of the lens molding process by performing cylin-der compression experiments and numerical simulations. Thespecific tasks were as follows:

(a) To perform cylinder compression test on two differentoptical glasses for a range of temperatures above the transfor-mation temperature of the respective glass and use analyticalrheology-based models available in Meinhard et al.,7 to anal-yze the measured data and determine the elastic and viscousparameters.

(b) To determine the viscosity of glass at the test tempera-ture by using the parallel plate viscometer methodology (ex-plained in Section III).

(c) To perform FEM simulations by using the viscosity dataand elastic coefficient values obtained from compression testsand compare the measured and predicted load curves.

(d) To perform an FEM simulation of a real lens moldingprocess by using the viscous and elastic coefficients determinedfrom the compression tests.

III. Theory

The methodology of a cylinder compression test is based upon aparallel plate viscometer, which is a test method that enables thedirect viscosity measurement of glass in the softening tempera-ture range. The compression test consists of a specimen of glassin the form of a solid right cylinder that is deformed betweentwo parallel rigid plates in uniaxial compression. The basicmeasurement is the determination of the change in height ofthe specimen with temperature and time. The relation that canthen be used to calculate glass viscosity (Z) is given by the fol-lowing equation:7–9

Z ¼ 2pFl5ðtÞ3V _l 2pl3ðtÞ þ Vð Þ

(1)

where F is the loading force, _l is the axial deformation rate, V isthe volume of the sample, and l(t) is the instantaneous length ofthe sample. During cylinder compression, the volume of thesample is assumed to remain constant and therefore:

V ¼ A0l0 ¼ pr2 tð Þl tð Þ (2)

whereA0 is the initial area of cross-section, l0 is the initial samplelength, and r(t) and l(t) are the instantaneous radius and lengthof the sample, respectively. A high exponential dependence ofthe test on the instantaneous sample length makes measurementaccuracy very critical. In the compression test, Eq. (1) becomesuseful only when pure viscous deformation has been achieved,

2409

Journal

J. Am. Ceram. Soc., 88 [9] 2409–2414 (2005)

DOI: 10.1111/j.1551-2916.2005.00477.x

r 2005 The American Ceramic Society

R. Condrate—contributing editor

wAuthor to whom correspondence should be addressed. e-mail: yi.71@osu.eduManuscript No. 20293. Received October 6, 2004; approved March 11, 2005.

i.e. when the application of load results in instantaneous dissi-pation of stresses.

As stated earlier, Eq. (1) can be used to carry out viscositydetermination only when the material flow is purely viscous, asthe equation does not consider the elastic parts of deformation.Therefore, in order to analyze the stress–time data obtainedfrom the compression tests analytical, rheology-based model hasto be used to take into account the elastic and viscous parts ofdeformation. For the loading part of the curve (the focus of thisstudy), the measured stress data can be represented using thefollowing equation:7

s tð Þ ¼ _ecZ 1� exp �EM

Zt

� �� �þ EH _ect (3)

where s(t) is the instantaneous stress, e is the strain, EM and EH

are the elastic parameters, and _ec is the constant strain rate usedin the test.

In the class of viscoelastic materials such as glass, the appli-cation of a constant load is followed by a deformation, whichcan be made up of instantaneous deformation (elastic effect)followed by continual deformation with time (viscous effect),which results in the decay of the applied load and is termed asrelaxation. Stress relaxation in high-temperature glass has beenstudied extensively by glass researchers.10–12

Viscoelastic stress relaxation can best be illustrated with thehelp of mechanical models—a Maxwell model consisting of asingle spring and the dashpot in series or a generalized Maxwellmodel consisting of a series of springs and dashpots placed inparallel.12 The spring represents the elastic behavior and dash-pot represents the viscous behavior of glass at high tempera-tures. A simple Maxwell model, shown in Fig. 1, was used tomodel the stress relaxation behavior of glass at the test temper-atures. In the figure, G represents the elastic shear modulus ofthe spring, Z represents the viscosity of the dashpot, and e12represents the total applied shear strain. The stress relaxation isthen given by Eq. (4), where s12(t) is the stress at time t and ts isthe stress relaxation time given by Z/G. In glass, the viscosity is asteep function of temperature and therefore as the temperaturerises beyond the transition temperature, the viscosity of glassfalls sharply, resulting in very low relaxation times (from a fewseconds to a few milliseconds for the temperature range con-sidered in this study):

s12ðtÞ ¼ 2Ge12 0ð Þe�t=ts (4)

IV. Experiments

Compression experiments were performed on two different cer-tified grades of Schott optical glass obtained from Glass FabInc. (Rochester, NY): (1) BK7 glass (Tg5 5571C) and (2) SK5glass (Tg5 6601C). Both the diameter and the height of the testsamples was 6.3570.1 mm.

A custom home-built hot forming machine designed for per-forming lens molding tests was used to perform compressionexperiments. The details of the machine design and its perform-ance have been included in Firestone et al.13 Figure 2 shows theschematic of the machine setup. The machine basically consistedof a 2 kW furnace having a nickel–chromium resistive heatingelement and could go up to a maximum temperature of 10101C.A proportional-integral-derivative controller utilizing two k-type thermocouples was used to control the furnace at the de-sired temperature. For the drive system, a servo motor was se-lected with a continuous thrust rating of 1400 N. The motordrives an actuator, that uses a ball screw for high positionalcontrol. Positional feedback was accomplished through an en-coder that can resolve up to 0.6 mm. The motor also had a ded-icated drive that constantly monitored the applied torque, motortemperature, position, velocity, and acceleration.

The stiffness of the machine was determined by measuring thetotal y deflection of the machine, which was estimated to be 10mm for approximately 500 N of load. For the test conditionsused in this study, the maximum deformation load was below 40N. Therefore, the machine deflection for the loading conditionswas lower than 0.5% of the measured glass sample deformation,and was considered negligible in viscosity calculations.

The upper and the lower molds were prepared from 17-4PHgrade of stainless steel because of its high stability at high tem-perature. The surface of the molds was ground and polished to afinish Rao0.05 mm and was coated with chromium (a few mi-crons thick) in order to prevent sticking of glass to the molds athigh temperatures. Two thermocouples having a junction madeof chromel–alumel wires were placed below the surface of themolds (one each in the upper and lower molds) by means ofholes drilled on the sides of the molds. The thermocouples had ameasurement accuracy of 721C.

A load cell capable of accurately measuring less than 1/3 N ofcompressive force with a full-scale reading of 2 kN was mountedunder the lower mold assembly. High-temperature insulationsheets and a finned aluminum heat sink were placed between thelower mold and the load cell to prevent it from malfunctioningat high test temperatures. During the compression tests, tem-perature, load, and mold position were monitored and control-led simultaneously. The signal from each of the hardwareelements was fed to a 16-bit National Instrument DAQ card,which was then processed through the Labview software forfurther analysis.

The experiments were performed in a temperature range of6451–7001C for BK7 glass and 7001–7751C for SK5 glass. Ther-mal considerations of the molding machine, process time, and

Fig. 1. Maxwell model.

Fig. 2. Schematic illustration of the experimental setup for glass com-pression tests.

2410 Journal of the American Ceramic Society—Jain et al. Vol. 88, No. 9

small stress relaxation times resulting in negligible formingstresses in glass makes this a recommended range for moldinglenses.1,2 Depending on the test temperature, the strain rate usedin the experiments was varied from 3� 10�4 to 6� 10�3 s�1. Itwas desired to deform the cylindrical glass samples without anycrack formation. For a given temperature, there exists a criticalvalue of strain rate that increases with increasing temperature,which, if exceeded, causes the glass to crack.7,8 As no prior in-formation of this critical strain rate value was available for thetwo glasses under study, a very low range of strain rate (men-tioned above) was selected, in order to make sure that no crackinitiation occurred during the tests. To calculate viscosity andforming stress using Eqs. (1) and (3), respectively, it was impor-tant to maintain the value of strain rate constant, in a given test.The strain rate _ec is given by v/l, where v is the deformation ve-locity and l is the instantaneous length of the sample. In theexperiments, the velocity of the drive was maintained constantand as the deformation of the sample during each test was min-imal, it was assumed that the strain rate remained constant. Thiswas verified by calculating the strain rate from the measureddrive position and time information available from each test.

V. Numerical Model

Figure 3 shows the 2D axisymmetric simulation model of a cy-lindrical glass compression test. In the figure, d0 represents thediameter and l0 represents the length of the cylindrical glasssamples used in the compression tests. The simulations wereperformed using a commercial general-purpose, non-linearFEM program MARC, which is suitable for conducting visco-elastic stress analysis. The upper and lower molds were modeledas rigid bodies, while glass was modeled as a viscoelastic mate-rial. The glass object was meshed with 1000, four-node, quad-rilateral, axisymmetric elements. The viscosity of glass at eachtemperature to be input to the simulation was obtained from thecompression test results (using Eq. (1)). The elastic parameterneeded for the simulation model was obtained by curve fittingthe experimental stress–time curve with Eq. (3).

As the compression tests were performed under isothermalconditions, no heat transfer was modeled at the glass–mold in-terface. A true stick–slip friction model available in MARC wasused to represent the glass–mold interface friction. In this proc-ess, based on nodal forces, a node is assumed to completely stickto the surface until the tangential force reaches a critical valuedepending on the user input friction coefficient value. Because ofthe lack of any available data, an arbitrary value of the frictioncoefficient, m5 0.5, was used in all the numerical simulations.The deformation velocity used in the compression tests was ap-plied as a velocity boundary condition to the rigid upper mold.

VI. Results

(1) Experimental and Simulation Results from CylinderCompression

Figure 4 shows the measured stress–strain curves at differenttemperatures for two grades of glass studied in this research, i.e.,SK5 glass and BK7 glass, respectively. As already mentioneddepending on the temperature, the strain rate used in the exper-iments was varied within a certain range but for a given testtemperature a constant strain rate was maintained during theentire duration of the test.

The deformation stress initially rises as the glass sample isdeformed because of both elastic and viscoelastic mechanismsoperating inside glass. Once the elastic spring of the Maxwellmodel (see Fig. 1) is completely loaded, a pure viscous flow isattained, resulting in a constant stress for any further increase instrain (i.e. ds/de5 0). After the viscous flow is reached, the vis-cosity of the glass material at that temperature can be deter-mined using Eq. (1).

Figures 5(a) and (b) compare the viscosity data obtained fromthe compression test (using Eq. (1)) with the measured viscosityvalues obtained from the standard beam bending test for SK5and BK7 glass, respectively. It should be noted that the viscosityvalues from the beam bending tests for the two glasses wereobtained from the manufacturer.

Fig. 3. 2D axisymmetric simulation model of a cylindrical glass com-pression test.

Fig. 4. Experimental stress versus strain curves of cylinder compressiontests at different temperatures for (a) SK5 glass and (b) BK7 glass; Tindicates the test temperature and _e indicates the strain rate used in thetest.

September 2005 Viscosity Measurement by Cylindrical Compression 2411

A very good agreement is observed between the viscosity val-ues measured using the two techniques. Greater deviation be-tween the results is obtained at higher test temperaturesprimarily because of the higher uncertainty in the experimentalmeasurement at those temperatures. Glass evidently becomesvery soft and it became difficult to measure any change in de-formation load at a higher test temperature.

Figures 5(a) and (b) also show the viscosity values for the twoglasses at different test temperatures obtained by curve fittingEq. (3) to the experimental stress–time data using the least-square regression procedure. The stress–time data, obtainedfrom different test cases fitted very well using Eq. (3), with anadjusted R2 value of approximately 0.995, and the value of theviscosity parameter within 70.5%. Glass–mold interface fric-tion, which was not accounted for by the rheology-based model,and the strain rate sensitivity of the equation may be some of thereasons for the higher viscosity values obtained by curve fittingEq. (3).

The viscosity values of SK5 and BK7 glass obtained from thecompression tests were also fitted using the famous Vogel, Fulc-her, and Tammnann (VFT) equation14,15 given by:

log Zð Þ ¼ Aþ B

T � T0(5)

where Z is the viscosity at temperature T, and A, B, and T0 areconstants. The corresponding curves for SK5 and BK7 glasseshave been shown by the solid line in Figs. 5(a) and (b), respec-tively. In the studied temperature range, the parameters of theVFT equation were found to be A5 0.5, B5 25221C, andT05 453.31C for SK5 glass and A5 6.495, B5 212.51C, andT05 584.51C for BK7 glass. Because of the limited data overwhich the VFT equation was fit, all the other parameters varied715% except parameter B for BK7 glass, which showed greatervariation. Overall, a reasonably good agreement is observed be-tween different experimental and analysis methods. Within ex-perimental uncertainty, both compression tests and rheology-based models can be used to carry out viscosity determination.

The elastic parameter EM was similarly calculated by curvefitting the measured stress–time data at different temperatures

with Eq. (3). The range of this parameter was approximately 100MPa for BK7 and 200 MPa for SK5 in the temperature rangeinvestigated. The temperature dependence of the elastic param-eter was observed from the results, wherein the parameter de-creases with increasing temperature. Similar trends of the elasticparameter determined using the cylindrical compression testshave been reported in Meinhard et al.16 for a different glass,around the transition temperature range. In the same way, thevalue of the other elastic parameter EH was estimated to be ap-proximately equal to zero for all the measurements, which sug-gests that in the temperature range investigated, the compressionof cylindrical glass samples can be modeled by a simple Maxwellmodel, with a good approximation.

Figure 6 shows the comparison of the measured stress–timecurve and the predicted curve during the cylinder compressiontest for SK5 glass at 7401C and BK7 glass at 6451C. The ex-perimentally determined viscosity and elastic parameter wasused in the simulations to model the viscous behavior of glass.

As far as the magnitude of the deformation load is concerned,a very good comparison between the measured load and thepredicted load is obtained. However, the shapes of the measuredand predicted loading curves are quite different for both theglasses. This difference is possibly because of a single Maxwellelement that was used to model the relaxation mechanism ofglass at higher temperatures. With one element, a single springgets completely loaded during cylinder deformation before theviscous flow is initiated. A generalized Maxwell element hasmultiple springs with different elastic constants and dashpotswith different viscosities that cause a different loading and re-laxation behavior, which may provide a better comparison be-tween the predicted and the measured results. Similar trendswere observed between the measured and predicted results atother temperatures for both the glasses. Using a generalizedMaxwell model to model the relaxation mechanism would be apart of future research.

Figure 7 shows the predicted von Mises stress distributioninside the BK7 glass cylinder during compression at a temper-

Fig. 5. Comparison of viscosity value obtained from the cylinder com-pression test and curve-fitting experimental data using Eq. (3) with themeasured viscosity from beam bending test for (a) SK5 glass and (b)BK7 glass. Fig. 6. Comparison of the measured stress versus time data with the

predictions from the numerical simulations for (a) SK5 glass at 7401Cand (b) BK7 glass at 6451C.

2412 Journal of the American Ceramic Society—Jain et al. Vol. 88, No. 9

ature of 6451C. As mentioned earlier, stick–slip friction approx-imation was used to model the friction at the glass–mold inter-face in FEM, which causes the cylindrical sample to bulgetoward the center. Because of the friction at the contact moldfaces, the material flow is relatively constrained at the ends, re-sulting in a higher stress buildup near that region. A zone ofhigh stress is also observed near the cylinder center because ofthe constrained material flow in that region as compared withthe surrounding regions where the material flow is relativelyeasier. A similar sample shape was also observed from the com-pression experiments performed on both the glass types.

(2) Results from the Simulation of Glass Lens Molding

The underlying objective of this research work was viscositymeasurement and development of a better understanding ofglass deformation mechanics at high temperatures so as to beable to model the hot compression molding of precision glasslens manufacturing process in FEM. The details of manufac-turing glass lenses using the molding process can be obtainedfrom Yi and Jain.1,2 The viscosity measurements made from thecompression tests and the elastic parameters obtained fromcurve fitting of the experimental stress data were used as theinput parameters to the lens molding simulations. It should benoted that the molding conditions were arbitrarily chosen inorder to test a methodology and use the property data obtainedfrom the cylindrical compression tests discussed earlier.

Figure 8 shows the predicted equivalent stress distributioninside the formed BK7 glass lens at a forming temperature of6711C. The axisymmetric cross-section of the lens has beenshown in the figure. The viscosity of glass is very low at the se-lected forming temperature, and therefore, very low stresses areobserved during most part of the forming processes. But as themolds begin to come closer together, the material flow becomesmore constrained, resulting in the stresses rising to a maximumvalue of approximately 1.5 MPa at a stage when the molds arecompletely closed. These stresses would begin to relax during thesubsequent annealing stage of the lens molding process, whichincluded within the scope of this research.

Figure 9 shows the corresponding predicted load curve for thecompression molding of the lens. Negligible load is requiredduring the initial part of the glass deformation. As the moldsclose in together, the material flow becomes constrained, whichleads to the gradual increase in load that reaches a maximumvalue at the point when the molds are in a completely closedposition.

VII. Conclusions

Cylindrical compression experiments were performed on twodifferent grades of optical glass to develop a better understand-ing of the molding process for glass lens manufacture. High-temperature viscosity and elastic properties determined from thecompression tests were used as inputs to the numerical simula-tion of cylindrical compression and the lens molding process.Some of the conclusions from the work are listed below:

A reasonably good comparison between the measured vis-cosity values using parallel plate viscometer analogy, curve-fitdata, and viscosity measurements performed using the standardbeam bending test for the two glasses was obtained, whichshows that both cylindrical compression and rheologicalcurve-fitting methods can be used to determine viscosity of glassat the desired temperature.

Predicted load results compared well with the experimentallymeasured loads, but the shape of the measured loading andpredicted curves was considerably different for cylindrical glasscompression.

Fig. 7. Predicted equivalent von Mises stress distribution in a BK7glass cylinder under compression at 6451C.

Fig. 8. Predicted equivalent von Mises stress distribution during com-pression molding of a glass lens at a forming temperature of 6711C (a)initial glass-blank and mold setup (b) compression of the glass blank,and (c) final lens shape.

0

200

400

600

800

1000

1200

1400

0 50 100 150 200 250Time [sec]

Def

orm

atio

n lo

ad [N

]

Fig. 9. Predicted load curve for compression molding of BK7 glass lensfor the selected molding conditions.

September 2005 Viscosity Measurement by Cylindrical Compression 2413

Stress distribution inside a formed lens and the deformationload curve was obtained from the numerical simulation of thelens molding process. The predicted results from the lens-form-ing simulation will have to be validated by performing experi-ments under similar conditions.

Other issues that have emerged from this research that de-serve future attention in order to numerically model and predictlens-forming process include modeling of viscoelastic stress re-laxation during lens forming by implementing a generalizedMaxwell model, more accurate information of high-tempera-ture elastic properties of glass such as elastic modulus, shearmodulus, and Poisson’s ratio using other methods like Brillouinspectroscopy, and knowledge of the friction behavior at theglass–mold interface.

Acknowledgments

The authors would like to acknowledge the support and guidance provided byMary A. Hartzler (Laboratory supervisor, IWSE, the Ohio State University) dur-ing the machine building and experimental stage of this research. The authorswould also like to thank Ron Klimek (Manager, R&D Technical Services, SchottNorth America Inc.) for his technical support. The support of the technical team atMSC Software Corporation, USA is also acknowledged.

References

1A. Y. Yi and A. Jain, ‘‘Compression Molding of Aspherical Glass Lenses—ACombined Experimental and Numerical Analysis,’’ J. Am. Ceram. Soc., 88 [3] 579–86 (2005).

2A. Jain and A. Y. Yi, ‘‘Numerical Modeling of Viscoelastic Stress Relaxationduring Glass Lens Forming Process,’’ J. Am. Ceram. Soc., 88 [3] 530–5 (2005).

3G. Weidmann, K. Holtberg, and H. Eisermann, ‘‘Pressing of ‘‘Drinking-Glass’’Stems;’’ pp. 307–16 in Mathematical Simulation in Glass Technology, Edited by H.Loch and D. Krause. Springer-Verlag, Berlin, Germany, 2002.

4U. Lange, ‘‘The Use of Re-Meshing Methods in Pressing Simulations;’’ pp.317–26 in Mathematical Simulation in Glass Technology, Edited by H. Loch andD. Krause. Springer-Verlag, Berlin, Germany, 2002.

5A. Hoque, C. E. Fischer, and W. T. Wu, ‘‘Simulation of the Glass PressingProcess Using 3-Dimensional Large Deformation Finite Element Software;’’ Sci-entific Forming Technologies Corporation, Ohio, USA, Paper #353, paper ob-tained through internal communication (2001).

6D. Lochegnies, T. Vicart, and J. Oudin, ‘‘Finite Element Elasto-ViscoplasticModels for Optimization of Horizontal and Vertical Glass Bending of Sheets ofGlass,’’ Glass Technol., 36 [5] 166–70 (1995).

7H. Meinhard, W. Franzel, and P. Grau, ‘‘Viscosity of Glass Below the Trans-formation Temperature,’’ Glastech. Ber., Glass Sci. Technol., 74 [1] 6–16 (2001).

8R. Bruckner, Y. Yue, and A. Habeck, ‘‘Determination of the RheologicalProperties of High-Viscous Melts by the Cylinder Compression Method,’’ Glas-tech. Ber., Glass Sci. Technol., 67 [5] 114–29 (1994).

9A. N. Gent, ‘‘Theory of the Parallel Plate Viscometer,’’ Br. J. Appl. Phys., 11,85–7 (1960).

10C. R. Kurkjian, ‘‘Relaxation of Torsional Stress in the Transformation Rangeof a Soda–Lime–Silica Glass,’’ Phys. Chem. Glasses, 4 [4] 128–3 (1963).

11J. De Bast and P. Gilard, ‘‘Variation of the Viscosity of Glass and the Re-laxation of Stresses During Stabilisation,’’ Phys. Chem. Glasses, 4 [4] 117–28(1963).

12S. Rekhson, ‘‘Viscoelasticity of Glass;’’ pp. 1–115 in Glass Science and Tech-nology, Edited by D. R. Uhlmann and N. J. Kreidl. Academic Press Inc., Florida,USA, 1986.

13G. C. Firestone, A. Jain, and A. Y. Yi, ‘‘A Laboratory Apparatus for HighTemperature Compression Molding of Precision Glass Optics,’’ Rev. ScientificInstrum. J., (2004), in press.

14G. S. Fulcher, ‘‘Analysis of Recent Measurements of the Viscosity of Glasses,’’J. Am. Ceram. Soc., 8 [6] 339–55 (1925).

15G. S. Fulcher, ‘‘Analysis of Recent Measurements of the Viscosity of Glasses,’’J. Am. Ceram. Soc., 8 [12] 789–94 (1925).

16H. Meinhard, W. Franzel, and P. Grau, ‘‘Mechanical Properties of SheetGlass at High Pressure During Indentation Experiments,’’ Glastech. Ber., GlassSci. Technol., 74 [11–12] 333–40 (2001). &

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