Stress relaxation of a soda lime silicate glass below the glass transition temperature

Journal of Non-Crystalline Solids 324 (2003) 277–288 www.elsevier.com/locate/jnoncrysol

Stress relaxation of a soda lime silicate glass below the glass transition temperature Junwu Shen *, David J. Green, Richard E. Tressler, David L. Shelleman Department of Materials Science and Engineering, 225 Steidle, The Pennsylvania State University, University Park, PA 16802, USA Received 1 July 2002

Abstract Stress relaxation is an important effect in the ion-exchange procedure of glasses, as it controls the stress profile and the strength. Creep and stress relaxation tests have been performed to study the viscoelastic behavior of soda-lime silicate glass at typical ion-exchange temperatures. The experimental data of these tests can be fitted well by the Burger model and a comparison between the viscosity data from both tests was made. The strain and temperature dependences of the stress relaxation process were studied and the glass exhibited a non-linear viscoelastic behavior and an anomalous temperature dependence. In addition, it was found there is a relationship between the glass density and the stress relaxation behavior.
2003 Elsevier B.V. All rights reserved.

1. Introduction Ion exchange is an important technique for improving the strength of glasses. The residual stress distribution in ion-exchanged glass is related to the final strength of the glass. Therefore, accurate prediction of stress profiles in the ionexchanged glass is necessary for controlling the strength of ion-exchanged glass by adjusting the ion-exchange parameters. Moreover, it is well accepted that stress relaxation plays an important role at typical ion-exchange temperatures [1,2]. Thus, a study on the stress relaxation behavior of glasses at ion-exchange temperatures is important

*

Corresponding author. Tel.: +1-814 863 3476. E-mail address: [email protected] (J. Shen).

for optimization of ion-exchange procedures and many other thermal processes used in the glass industry. Creep and stress relaxation of glasses at different temperatures have been extensively studied by many researchers [3–14]. Most of the reported stress relaxation data have been represented by the Kohlrausch–Williams–Watt equation [4], and for many glasses, a master curve can be used to derive the stress relaxation curves at different temperatures in the glass transformation range [10–13]. However, this paper will be mainly focused on lower temperatures than normally studied. Bartenev [15] has suggested the stress relaxation mechanisms at lower temperatures can differ from those at higher temperatures. An additional objective of the current study was to study the effect of the loading geometry and the strain and temperature

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2003 Elsevier B.V. All rights reserved. doi:10.1016/S0022-3093(03)00260-6

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J. Shen et al. / Journal of Non-Crystalline Solids 324 (2003) 277–288

dependences of the relaxation process. The relationship between glass density and stress relaxation behavior was also studied.

where s1 , s2 , C1 and C2 are constants calculated from the Burger model parameters E1 , E2 , g1 and g2 according to the following equations [17–19]. q1  ðq2 =s1 Þ q1  ðq2 =s2 Þ ; C2 ¼  ; E1 A E1 A 2p2 2p2 ; s2 ¼ ; s1 ¼ p1  A p1 þ A qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g g g A ¼ p12  4p2 ; p1 ¼ 1 þ 1 þ 2 ; E1 E2 E2 gg gg p2 ¼ 1 2 ; q1 ¼ g1 ; q2 ¼ 1 2 : E1 E 2 E2

C1 ¼ 2. Background In the current study, the Burger model was chosen for fitting the creep and stress relaxation data. The Burger model (Fig. 1) is a combination of Maxwell model and Kelvin model. It is the simplest model that combines elastic, viscous and anelastic effects [14]. The general stress–strain equation for the Burger model is as follows [16]:
 g1 g2 g1 dr g1 g2 d2 r rþ þ þ þ E1 E2 E2 dt E1 E2 dt2 de g g d2 e ¼ g1 þ 1 2 2 : dt E2 dt

ð1Þ

From the above equation, the creep function of the Burger model can be obtained by setting dr=dt ¼ 0:  eðtÞ 1 1  t ¼ þ 1  eðE2 =g2 Þt þ ; r E1 E2 g1

ð2Þ

where eðtÞ is the strain, r is the constant applied stress, E1 , E2 , g1 , and g2 are the parameters of the Burger model [16]. In the Burger model, stress relaxation can be described by the following equations, which can also be developed from Eq. (1) [17–19] 

  t t rðtÞ ¼ r0 C1 exp  þ C2 exp  ; s1 s2 ð3Þ C1 þ C2 ¼ 1;

ð4Þ

3. Experimental details 3.1. Glass composition and properties The glass used in the study was a soda-lime silicate glass (SLS) (StarphireTM , PPG, Pittsburgh, PA). The composition measured by spectrochemical analysis (DC plasma emission spectrometer) is shown in Table 1. The relative error margin of the composition is ±5% of the given value for each component. Some properties of the glass are displayed in Table 2 [20]. From the data in Table 2, the glass transition temperature, Tg , was estimated to be about 572 C by assuming the glass follows the Arrhenius equation in this temperature range. 3.2. Measurement of Young’s modulus of glass at high temperatures The YoungÕs modulus of glass changes with temperature, especially at high temperatures. In the current study, the YoungÕs moduli of the SLS glass at different temperatures (25–550 C) were

E2

E1

ð5Þ

η1

Fig. 1. Burger model for a viscoelastic solid.

η2

J. Shen et al. / Journal of Non-Crystalline Solids 324 (2003) 277–288 Table 1 Composition of soda lime silicate glass used in study Oxides

Concentration (wt%)

Concentration (mol%)

SiO2 Na2 O CaO Al2 O3 Srx Oy BaO MgO K2 O Others

71.0 14.7 11.8 1.65 0.27 0.23 0.07 0.01 0.27

71.4 14.3 12.7 0.98 0.16 0.09 0.11 <0.01 0.25

279 Transducers

Sapphire fibers

Furnace Glass specimen

Fig. 2. Schematic of the experimental arrangement for high temperature YoungÕs modulus measurement.

measured by a resonance method according to the ASTM standard C623–92 [21]. The dimension of the glass specimen used for the YoungÕs modulus measurement was 75.43 · 14.80 · 3.26 mm. As shown in Fig. 2, the glass specimen with two small notches on both ends was suspended in a furnace by two sapphire fibers. These two sapphire fibers were glued to the transducers, which were connected to the resonance measurement system [21]. The furnace was heated gradually to different temperatures between 100 and 550 C, and the YoungÕs modulus of the SLS glass was measured at each temperature after 1 h thermal equilibrium. 3.3. Three-point bending creep The creep tests were conducted on a three-point bending (3PB) viscometer (Theta Industries, Inc., Port Washington, NY). The span length was 50 mm, and the rectangular dimensions of specimen were 3.2 · 3.0 mm. The specimen deflection was recorded in situ using a linear variable displacement transducer. The maximum strain and stress in the glass bar can be calculated from the recorded deflection by the following equations [22,23]:



 h d e¼6 ; L L r¼

ð6Þ

3FL ; 2bh2

ð7Þ

where e is the maximum strain in the beam, d is the maximum deflection in the beam, b is the width of the glass bar, h is the thickness of the glass sample, L is the span length, and F is the force on the sample. The maximum load for the viscometer was usually 10 N, and the resulting maximum stress for these tests was 25 MPa. Glass samples were not annealed before the tests. The above approach assumes that the viscoelastic behavior is linear and the associated deformation mechanism is the same in uniaxial tension and as it is in uniaxial compression. 3.4. Three-point bending stress relaxation A mechanical testing machine (Instron Corp., Canton, MA) was used for the 3PB stress relaxation tests. In the 3PB test, a sample with a size 3 · 3 · 45 mm is placed into the loading jig and then heated to the test temperature. After two hours

Table 2 Selected properties of StarphireTM glass [20] YoungÕs modulus (GPa)a

PoissonÕs ratioa

Density (g/cm3 )a

Softening point (C)

Annealing point (C)

Strain point (C)

73.1

0.22

2.5

722

552

512

a

Properties at room temperature.

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of thermal equilibrium, the initial load is applied at a high rate (0.2 mm/min), then the crosshead is fixed and the applied strain is accordingly constant. The load is recorded as a function of time by the machine and can be transformed into the stress in the center of the glass bar by Eq. (7). The maximum stress is limited by the fracture stress of the glass, and the minimum stress is limited by the load sensitivity of the mechanical testing machine. The commonly used stress range in the 3PB stress relaxation tests was approximately 20– 80 MPa. 3.5. Uniaxial compressive stress relaxation Since the stress in the surface region of an ionexchanged glass is compressive, it was hypothesized that compressive stress relaxation should be more appropriate than bending relaxation in predicting the stress relaxation behavior. In addition, it is possible to apply higher stress in this configuration. This is important as stresses in the ionexchange process often exceed 100 MPa. Clearly, it would also be useful to know if there is any difference in the stress relaxation behavior for the bending and compressive stress geometries. In the compressive stress relaxation test, the equipment was the same as that for 3PB stress relaxation test except for the loading fixture. The height of the glass sample was 13.47 ± 0.02 mm, and the cross-section was 4 · 4 mm. Uniaxial compressive loads were applied to the original float glass surfaces to guarantee that those two surfaces are parallel to each other. To further reduce the friction between the glass surfaces and the loading surfaces, graphite foil (UCAR Carbon Company Inc, Cleveland, OH) was used. As-received glass samples were used for most of the stress relaxation tests, and annealed samples (560 C, 5 h) were only used for comparison with as-received samples.

The measurement approach is described in ASTM C729–75 [24]. Before each measurement, glass samples were carefully cleaned by acetone and water, and dried in a drying oven for about 20 min. The density of the sink-float liquid (Cargille Laboratories, Inc., Cedar Grove, NJ) used in the measurements is 2.510 ± 0.0005 g/cc at 23 C. The glass standard (Cargille Laboratories, Inc., Cedar Grove, NJ) for calibration has a density of 2.495 ± 0.0005 g/cm3 at 23 C. By measuring the sink temperature of the glass standard in the heavy liquid, the temperature coefficient ()0.0020 g/cm3 /C) of the heavy liquid was determined. The density of glass samples used in the stress relaxation tests can then be calculated from the sink temperature of each sample if the glass density is slightly lower than that of the heavy liquid. The temperature range for the thermometer in the constant temperature bath was 19–35 C, and the temperature resolution was ±0.02 C. So the accuracy of the density measurement is better than ±0.0001 g/cm3 .

4. Experimental results and data analysis 4.1. Young’s moduli of SLS glass at high temperatures Fig. 3 shows the YoungÕs moduli of the SLS glass from room temperature to 550 C. The

3.6. Measurement of glass density In order to study the relationship between the glass density and compressive stress relaxation, glass densities were accurately measured by a sinkfloat method before and after stress relaxation test.

Fig. 3. YoungÕs moduli of the SLS glass as a function of temperature.

J. Shen et al. / Journal of Non-Crystalline Solids 324 (2003) 277–288

4.2. Methodology of curve fitting 4.2.1. Three-point bending creep test curves fitted by the Burger model Since the instantaneous elastic deformation of the measurement system is involved in the sample deflection in the creep tests and accurate YoungÕs modulus values were difficult to obtain from this test the instantaneous elastic strain was not considered in the curve fitting. For the Burger model, anelastic and viscous strain, e0 ðtÞ in creep tests can be obtained from Eq. (2) as  r r 1  eðE2 =g2 Þt þ t: ð8Þ e0 ðtÞ ¼ E2 g1 When E2 =g2 is a constant, the optimal values of g2 and E1 can be determined by a multiple regression of e0 ðtÞ on (1  eðE2 =g2 Þt ) and t with zero intercept because the respective regression coefficients b1 and b2 are equal to r=E2 and r=g1 respectively. Then by changing the value of E2 =g2 , optimal values of E2 =g2 , E2 and g1 with the smallest sum of square error can be determined. 4.2.2. Stress relaxation curves fitted by the Burger model A similar linear fitting method can be used according to the following equation transformed from Eqs. (3) and (4) for the Burger model.
 rðtÞ t  exp  r0 s2 

 t t ¼ C1 exp   exp  : ð9Þ s1 s2

Then by changing the values of s1 and s2 , optimal values of s1 , s2 and C1 can be obtained. However, this method is accurate only if the steps of changing of s1 and s2 are sufficiently small. For the current study, time increments were typically <2 min. 4.3. Creep study Creep tests were performed at different temperatures (400–500 C) and stresses (9.5–25.0 MPa). Fig. 4 shows the experimental results of 3PB creep tests at different temperatures and stresses. The strain was calculated from Eq. (6), and the instantaneous elastic part of the strain was not included, i.e. the (inelastic) strain is set to zero when t ¼ 0. In the creep test, the strain as a function of time can be fitted by the Burger model according to the previously described methodology and then the Burger model parameters can be obtained. In order to see how well the Burger model describes the creep behavior of the SLS glass at these temperatures, a theoretical creep curve was calculated based on the fitted Burger model parameters according to Eq. (2). Fig. 5 is an example of the comparison between experimental creep data and calculated creep curve. It can be seen that the Burger model works well for fitting the creep data. Similar agreement was confirmed for the other experimental conditions. Furthermore, the uniaxial viscosity of glass can be calculated by the following equation from the limiting slope of creep curve e_1 [16]: 0.007 0.006

500 ˚C, 25.0 MPa 0.005

Strain

YoungÕs moduli of the glass decrease almost linearly with the increase of temperature. At room temperature, the measured YoungÕs modulus is 71.6 GPa, which is slightly lower than the value from Ref. [20]. In the temperature range of 450– 550 C, the YoungÕs modulus decreases slightly from 66.2 to 64.0 GPa, so an average value of 65.1 GPa was chosen for the YoungÕs modulus in this temperature range.

281

500 ˚C, 20.0 MPa

0.004

500 ˚C, 14.0 MPa 0.003

500 ˚C, 9.50 MPa 0.002

450 ˚C, 20.2 MPa 0.001

400 ˚C, 18.0 MPa

When s1 and s2 are both fixed, an optimal value of C1 can be calculated for each combination of s1 and s2 by a linear regression with zero intercept.

0

0

500

1000

1500

2000

2500

3000

3500

Time (min)

Fig. 4. Creep strain as a function of time in the 3PB tests.

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0.005

1000

Young’s Modulus (GPa)

0.004 Calculated strain Strain

0.003

0.002

Measured strain

0.001

0

0

500

1000

1500

2000

2500

3000

100

10

1

3500

400 C, 18.0 MPa

Time (min)

450 C, 20.0 MPa

500 C, 9.50 MPa

500 C, 14.0 MPa

500 C, 20.0 MPa

500 C, 25.0 MPa

Experimental conditions

Fig. 5. Comparison between experimental creep data and calculated creep curve based on the Burger model (500 C, 14.0 MPa).

r : e_1

100

ð10Þ

η1 η2

When the time is long enough, it would be expected that g1 ! g since the delayed elastic (anelastic) deformation will approach zero according to Eq. (2). The fitted Burger model parameters and calculated viscosity values from the creep test curves of the glass are listed in Table 3. The values of g1 are less than g but the two values are reasonably close. Figs. 6 and 7 show the change of E2 , g1 , g2 and g with the experimental conditions (temperature and stress). From Table 3, Figs. 6 and 7, it is clear that E2 , g1 , g2 and g all decrease with the increase of temperature, but at the same temperature, the values do not change much with the increase of stress. The values of g2 are always about one order of magnitude less than those of g1 .

Viscosity (GPa.Ms)

g¼

Fig. 6. The values of E2 in the Burger model as a function of creep test condition.

10

η

1

0.1

0.01

400 C, 18.0 MPa

450 C, 20.0 MPa

500 C, 9.50 MPa

500 C, 14.0 MPa

500 C, 20.0 MPa

500 C, 25.0 MPa

Experimental conditions

Fig. 7. The values of g1 , g2 and g as a function of creep test condition (Burger model).

4.4. Three-point bending stress relaxation Fig. 8 shows the 3PB stress relaxation data at different temperatures (500–550 C). The normalized stress is calculated from the recorded stress divided by initial stress. Stress relaxation data

Table 3 Calculated Burger model parameters and viscosity values from 3PB creep tests

E1 (GPa) E2 (GPa) g1 (GPa Ms) g2 (GPa Ms) g (GPa Ms)

400 C, 18.0 MPa

450 C, 20.2 MPa

500 C, 9.50 MPa

500 C, 14.0 MPa

500 C, 20.0 MPa

500 C, 25.0 MPa

65.1 123 16.0

65.1 35.2 6.14

65.1 7.73 1.50

65.1 7.72 1.49

65.1 8.13 1.55

65.1 8.30 1.73

1.47 26.4

0.509

0.0984

0.0946

0.101

0.123

8.05

2.17

2.17

2.30

2.43

J. Shen et al. / Journal of Non-Crystalline Solids 324 (2003) 277–288 1.2

1.2

1

1

283

Experiment

Normalized stress

Normalized stress

500 ˚C, 60 MPa

0.8 500 ˚C 0.6 520 ˚C 0.4 535 ˚C

Prediction from the Burger model

0.8 0.6 0.4

0.2

0.2

550 ˚C 0 0.1

1

10 Time (min)

100

1000

Fig. 8. Three-point bending stress relaxation tests at different temperatures (initial stress: 60 MPa).

below 500 C was not used because at the lower temperatures the data were very noisy. All the stress relaxation data can be fitted by the Burger model. Fig. 9 is an example of the comparison between the experimental stress relaxation data and calculated data based on the fitted model parameters. It can be seen that the Burger model fits the data well and the model parameters are given in Table 4. The uniaxial viscosity for a linear viscoelastic material can also be calculated from the stress relaxation data using the following equation [8]: Z 1 rðtÞ g¼E dt; ð11Þ r0 0 where E is the measured YoungÕs modulus of the glass. By inserting Eq. (3), Eq. (11) can be transformed into

  Z 1 t t g¼E C1 exp  þ C2 exp  dt: s1 s2 0 ð12Þ Calculated viscosity values using Eq. (12) are given in Table 4. As the experimental stress relaxation data were only for a finite time, the integration in Eq. (11) can not be accomplished after this finite time. An approximate method is to integrate the stress relaxation function using Eq. (11) at short times and integrate the fitted Burger model to long times. The average YoungÕs modulus in the studied temperature range, 65.1 GPa, was used in the calculation.

0 0.1

1

10 Time (min)

100

1000

Fig. 9. Comparison between experimental and calculated stress relaxation curves (500 C, 60 MPa, 3PB).

4.5. Uniaxial compressive stress relaxation The stiffness of the mechanical testing machine at high temperature (520 C) was measured by applying the load without any sample between the loading platens. It was found that, in the studied temperature range, the stiffness of the machine (9.644 kN/mm) was much lower than that of glass sample (79.1 kN/mm), which was calculated from the YoungÕs modulus of the glass in this temperature range (65.1 GPa). By using a soft machine, the strain on the glass sample is not constant during the relaxation test and the measured stress relaxation curve is dependent on the relative stiffness of the machine compared to the glass sample. In reality, the glass undergoes both creep and stress relaxation in such tests. The Burger model can, however, be used to remove this effect and the compressive stress relaxation under constant strain can be determined from the measured stress relaxation curve. Assuming the machine is pure elastic at the test temperatures and the glass sample can be modeled by the Burger model with parameters E1 , E2 , g1 and g2 (Fig. 1), the measurement system including the machine and glass sample can be considered as a new Burger model with parameters E0 , E2 , g1 and g2 (Fig. 10). In this new Burger model, the value of E0 can be calculated from Em and E1 . Em is the apparent YoungÕs modulus of the machine and this value can be calculated from the stiffness of the machine and the cross-section area of each

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Table 4 The Burger model parameters from the 3PB stress relaxation tests Temperature (C)

C1

C2

s1 (min)

s2 (min)

Uniaxial viscosity (GPa Ms)

g1 (GPa Ms)

g2 (GPa Ms)

E2 (GPa)

500 520 535 550

0.562 0.763 0.589 0.514

0.438 0.237 0.411 0.486

41.0 22.3 4.67 1.33

305 177 33.3 7.33

0.672 0.202 0.0655 0.0172

0.612 0.230 0.0642 0.0166

0.446 0.210 0.0503 0.0180

93.1 52.3 88.6 131

Ε′ Fig. 10. Correction of compressive stress relaxation data by the Burger model.

glass sample. Because the C1 , C2 , s1 and s2 can be obtained by the method described earlier and E0 is known, the other three Burger model parameters (E2 , g1 and g2 ) can then be obtained by Eq. (5). These three Burger model parameters should be independent of the stiffness of the machine, therefore, the stress relaxation data under constant strain can be calculated from the Burger model parameters (E1 , E2 , g1 and g2 ) by Eq. (5). Fig. 11 shows an example of the effect of the machine stiffness on the measured stress relaxation behavior. For the experimental arrangement used in the cur-

rent study, the stress relaxation after correction is about 20 times faster than the measured stress relaxation. Several compressive stress relaxation tests were conducted at different temperatures (450–550 C) and different initial stresses (60–200 MPa). Fig. 12 displays the compressive stress relaxation data after correction for machine stiffness at different temperatures (100 MPa). The parameters from the fitting procedure are given in Table 5. For the Burger model, the equilibrium viscosity is g1 . At the same time, the viscosity can also be

1.2

1.2 Measured

Normalized stress

Normalized stress

1

Corrected

0.8 0.6 0.4 0.2

450 °C °C

1

475 °C °C 500 °C °C

0.8

520 °C °C 535 °C °C

0.6

550 °C °C

0.4 0.2

0 0.1

1

10

100

1000

10000

Time (min)

Fig. 11. Effect of the machine stiffness on the compressive stress relaxation (520 C, 100 MPa).

0 1

10

100 Time (min)

1000

Fig. 12. Compressive stress relaxation tests at different temperatures (initial stress: 100 MPa).

J. Shen et al. / Journal of Non-Crystalline Solids 324 (2003) 277–288

285

Table 5 Model parameters from uniaxial compressive stress relaxation tests Experimental conditions 450 475 500 520 535 550 500 500 500 500

C, C, C, C, C, C, C, C, C, C,

100 MPa 100 MPa 100 MPa 100 MPa 100 MPa 100 MPa 200 MPa 150 MPa 80 MPa 60 MPa

C1

C2

s1 (min)

s2 (min)

g1 (GPa Ms)

g2 (GPa Ms)

E2 (GPa)

0.303 0.191 0.173 0.089 0.065 0.046 0.125 0.099 0.164 0.192

0.697 0.809 0.827 0.911 0.935 0.954 0.875 0.901 0.836 0.808

1770 1490 611 329 77.8 12.5 1320 1440 523 398

87.0 34.7 28.7 11.6 3.83 0.84 35.5 40.2 14.9 16.0

2.37 1.24 0.512 0.158 0.0343 0.00544 0.774 0.712 0.390 0.354

0.609 0.196 0.186 0.0734 0.0306 0.00973 0.202 0.234 0.0856 0.0994

39.4 19.7 22.8 12.9 14.8 21.2 14.1 12.1 18.0 23.3

calculated from the compressive stress relaxation data by Eq. (12), and these two methods give the same result. From the viscosity values at different temperatures, the activation energy for the relaxation process can be calculated according to the following equation [8]:
 Q g ¼ g0 exp ; ð13Þ RT where g0 is a constant, and Q is the activation energy for viscosity, R is the ideal gas constant. The viscosity values at different temperatures were plotted against inverse absolute temperature in Fig. 13. The viscosity values are compared to those obtained in the 3PB tests and from previous work [3]. The viscosity data from the various tests are in reasonable agreement. It can be seen, however,

Temperature (˚C) 500 450

550

400

Log10 (uniaxial viscosity/Pa s)

20 18 16 14

Compressive relaxation (as-received) 3PB relaxation

12

3PB creep De Bast & Gilard

10 8 1.20

Compressive relaxation (annealed)

1.25

1.30

1.35

1.40

1.45

1.50

1000/T (1000/K)

Fig. 13. Viscosity data from creep and stress relaxation tests (initial stress: 100 MPa).

that the log viscosity is a non-linear function of inverse absolute temperature in the range of 450– 550 C. The temperature dependence deviates significantly from the Arrhenius behavior as the test temperature decreases. Surprisingly, the activation energy decreases with decreasing temperature. The calculated activation energy for uniaxial compressive stress relaxation decreases from 609 kJ/mol at the higher temperatures to 142 kJ/mol at the lower temperatures.

5. Discussion 5.1. Temperature dependence of the uniaxial viscosity Fig. 13 shows the viscosity values from different tests and DeBast and GilardÕs work [3]. The dashed line in the figure is the extrapolated viscosity values according to the Arrhenius equation. These data are all below Tg . As shown in Fig. 13, the viscosity values from the current compressive stress relaxation tests agree well with the literature data at the higher temperatures. The glass used in DeBast and GilardÕs work was stabilized glass, while as-received glasses were used in the current study. Thus, one possible reason for the deviation from the Arrhenius behavior could be related to the thermal history. The thermal equilibrium time at each temperature before the stress relaxation tests was about 2 h. At high temperatures, the stabilization time for the glass is short and the glass has already been stabilized before the stress relaxation

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J. Shen et al. / Journal of Non-Crystalline Solids 324 (2003) 277–288

test. Therefore, the viscosity values from the current study agree well with those from stabilized glass. However, at low temperatures, the stabilization time for the glass is possibly longer than 2 h, and the glass has not been stabilized before the stress relaxation test. As a result, the stress relaxation at low temperatures in the current study is expected to be faster than that for stabilized glass at the same temperature because of the more open structure. Fig. 14 shows the stress relaxation curve for the as-received and annealed glass sample under the same experimental conditions. The stress relaxation for the annealed glass is slightly slower than that for the as-received glass. It is, however, shown that the viscosity value from the compressive stress relaxation test for annealed glass at 450 C (6.30 GPa Ms) is only slightly higher than that from the stress relaxation test for as-received glass at the same temperature (2.37 GPa Ms), and it is still much lower than the extrapolated viscosity value at the same temperature. Therefore, it implies that even for stabilized glass, the viscosity value does not follow the Arrhenius behavior in the temperature range 450–550 C. It is hypothesized that the deviation from the Arrhenius behavior at low temperatures is caused by the change of stress relaxation mechanisms with the change of temperature. At high temperatures, viscous flow is the dominating mechanism for the stress relaxation, and the activation energy in this temperature region (450–500 C), 609 kJ/mol, is close to that for viscous flow, 690 kJ/mol [3], and

Fig. 14. Effect of thermal history on the compressive stress relaxation.

the activation energy (585–669 kJ/mol) for the creep and stress relaxation of other soda lime silicate glasses in the glass transition range as reported in the previous investigations [3,10]. At low temperatures, however, viscous flow may not be the major stress relaxation mechanism because of the low temperature, while the diffusion of alkaline ions in the glass is expected to dominate the stress relaxation process. This can be supported by the fact that the activation energy for the stress relaxation in the high temperature range (500–550 C), 142 kJ/mol, is close to that for Naþ –Kþ interdiffusion process in the soda lime silicate glass, 152 kJ/mol [25]. 5.2. Effect of loading geometry on stress relaxation The viscosity data from the 3PB stress relaxation tests are also shown in Fig. 13. It was found the viscosity data from the 3PB and uniaxial compressive stress relaxation tests have a good agreement. This agreement implies the stress relaxation in tensile and compression have the same mechanism in the studied stress range. This conclusion agrees with the earlier work on stress relaxation in other studies [26–28]. 5.3. Effect of applied strain on stress relaxation Fig. 15 shows the stress relaxation curve under different initial applied strain at the same temperature. It can be seen that the stress relaxation at low initial stresses (60, 80 MPa) is faster than that at high initial stresses (100–200 MPa) at the same temperature. This difference can be explained by the free volume theory [23]. At low stress levels, the glass has more free volume, so the viscous flow and ion diffusion in the glass is easier, and the stress relaxation is faster. However, the quantitative relationship between the stress relaxation and applied initial strain is still unknown. It has been reported that in the range of moderate stresses (0.3–10 MPa), and within the error limits (usually ±10–15%), viscosity does not depend on either the stress values or geometry of the applied stresses [16]. At higher stresses, however, some authors found the viscosity sharply increases when the applied tensile stress is higher than a

J. Shen et al. / Journal of Non-Crystalline Solids 324 (2003) 277–288

Table 6 Glass densities before and after compressive stress relaxation

1.2 60 MPa

1 Normalized stress

287

80 MPa

Glass

Density (g/cm3 )

Density change (%)

As-received As-received, 500 C, 200 MPa Annealed glass A Annealed glass A, 520 C, 100 MPa, 26 h Annealed glass B Annealed glass B, 520 C, no stress, 26 h

2.4929 2.5003 2.4966 2.4984

– 0.30 – 0.072

2.4968 2.4980

– 0.048

100 MPa

0.8

150 MPa 200 MPa

0.6 0.4 0.2 0 1

10

100

1000

Time (min)

Fig. 15. Initial strain dependence of the uniaxial compressive stress relaxation (500 C).

critical stress (usually about 100 MPa) [16]. In the current study of compressive stress relaxation, a slight increase of viscosity or relaxation time with the increase of stress has been found for the SLS glass in this temperature range. 5.4. Glass density and stress relaxation The data in Fig. 14 show that the thermal history can influence the stress relaxation behavior. It is possible, therefore, that density change could be a mechanism of stress relaxation. The densities for these two glass samples (as-received and annealed) prior to testing were 2.4966 and 2.4929 g/cm3 respectively. It can be seen that stress relaxation in the annealed glass is slower than that in the asreceived glass, although the density change between these two glasses is only 0.1%. By measuring the glass density before and after the stress relaxation test, it was found that the glass can be densified during the compressive stress relaxation test. Table 6 shows the glass densities before and after stress relaxation. For as-received glass, the density change after compressive stress relaxation is about 0.3% and for annealed glass, this change is less than 0.1%. In order to see the importance of the stress relaxation in the densification process, two annealed glass samples with similar densities were chosen for a comparison experiment. As shown in Table 6, the densities for glass A and B are 2.4966 and 2.4968 g/cm3 re-

spectively. The glass A was stress relaxed at 520 C for about 26 h, and the initial applied stress is 100 MPa. The glass B was left in the same furnace without any applied stress, and the experimental conditions are the same except the stress. The densities of both glass samples were measured after the test, and both of them were densified. However, the density increase for glass A is about 1.5 times the density increase for glass B. Therefore, this experiment implies the densification can be caused by both the structural and stress relaxation.

6. Conclusions For the temperature range under study (450– 550 C), soda-lime silicate glass shows the following behavior: (1) The Burger model can be used to fit both the creep and stress relaxation data. (2) In uniaxial compressive stress relaxation, the relaxation time and the calculated viscosity are not a linear function of inverse absolute temperature. The deviation from Arrhenius behavior was due to the change of stress relaxation mechanisms with temperature, and also influenced to a small extent by the thermal history of the glass. (3) Glass can be densified during compressive stress relaxation but there is a competition with structural relaxation, in which the glass densifies simply by thermal activation.

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(4) The compressive stress relaxation showed a non-linear behavior at high stresses.

Acknowledgements The authors wish to acknowledge NSF Center for Glass Research for the financial support of this work and allowing the publication of this work.

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