Determination of Surface Flaw Parameters for Annealed and Fully Tempered Glass Exposed to NaCl

The popularity of glass as a building material, when compared to other building materials such as steel and concrete, is primarily due to its transparency which allows the passage of natural light. Modern glass applications couple this transparency with energy efficiency, high thermal performance, design flexibility, superior sound insulation, and sustainability. The use of large glazed facades and building envelopes is particularly common in coastal areas, where the incorporation of glass in homes is seen to augment the intake of natural light and offer captivating views. Despite these advantages, glass remains a brittle material and its strength, influenced by microscopic flaws on the glass surface, diminishes over time in the presence of water vapor due to the phenomenon known as static fatigue of glass [1] [2]. In coastal regions, glass faces exposure not only to water vapor but also to sodium chloride (NaCl). While the impact of water vapor on glass strength has been extensively researched over the years, the effect of NaCl on glass strength remains relatively unexplored. This study investigates the effect of NaCl on the strength of glass.

The failure of glass is attributed to randomly distributed microscopic flaws on the surface of glass, resulting in stress concentrations. Due to the unknown factors of the microscopic flaw location, size, and orientation, a probabilistic method is used to accurately determine the strength of glass. The standard to estimate the glass strength, ASTM E1300-16 [3] uses a probabilistic method known as the glass failure prediction model (GFPM) [4] based on a Weibull distribution to describe the strength of glass. The GFPM utilizes two parameters, m_s and k_s, referred to as surface flaw parameters, to quantify the characteristics of the flaws, including their location, size, and orientation of the microscopic flaws on the glass surface. The surface flaws are estimated using the maximum likelihood estimator, (MLE) method tailored to the GFPM [5], utilizing experimental data collected by subjecting glass of similar size to failure.

To investigate the impact of NaCl, a three-point bending test was employed to subject glass beam specimens to a monotonically increasing load until failure. The glass beam specimens were divided into three samples based on the type and thickness of the specimen. One sample consisted of annealed (AN) glass beam specimens with a thickness of 4.77 mm (0.188 in) thickness, while two samples comprised fully tempered (FT) glass beam specimens with thicknesses of 4.77 mm (0.188) and 6.35 mm (0.25 in). Prior to the three-point bending test, a selection of glass beam specimens from each sample were immersed in a 0.2 Molar (M) NaCl solution, 907 gm (2 lb) of NaCl in 75.7 L (20 gal) of water for one year. To compare the strength of untreated glass to glass treated using a 0.2 M NaCl solution, surface flaw parameters for the different samples are calculated using the MLE method. The GFPM uses the MLE-estimated surface flaw parameters to calculate load and stress corresponding to 0.008 and 0.001 probabilities of breakages (pb) for each sample. Results did not exhibit a consistent trend; the AN samples and the 6.35 mm (0.25 in) thick FT sampled both exhibited a decrease in strength at 0.008 and 0.001 probabilities of breakage, while the 4.77 mm (0.188 in) thick FT sample showed an increase in strength. However, due to the limited number of samples tested, to fully understand the effect of NaCl on the strength of glass, the authors recommend further testing.

LITERATURE REVIEW

The glass failure prediction model (GFPM) for a glass plate, proposed by Beason [4] is based on the Weibull distribution [6] and encompasses all factors known to impact the strength of glass. These factors include the surface flaws on the glass, the magnitude and orientation of the stress applied to the glass plate, the duration of the load, the static fatigue constant, as well as the geometry and boundary conditions. Norville et. al. [7] and Morse and Norville [8] observed that flaws on a heat-treated sample must be in tension to create a stress concentration sufficient to initiate fracture. Consequently, the applied load must be large enough to generate a tensile stress exceeding the residual compressive stress strength (RCSS). Thus, a modification to the GFPM was advanced for heat-treated glass. The probability of breakage (pb) for the modified GFPM is expressed as

Latex equation - P_b = 1- exp\left(-k_s \: \sum_{j=1}^{N} \left(c_j \cdot \left( \frac{t_d}{t_{par}} \right)^{\frac{1}{n}} \cdot\left({\sigma}(P)_{max,j}-RCSS\right)\right)^{m_s} \cdot A_j\right)

(Eqn. 1)

where k_s and m_s denote the surface flaw parameters, σ(P )_max,j denotes the maximum principal tensile stress at j^th nodal area as a function of the applied load and location on the glass, c_j denotes the biaxial stress correction factor at j^th nodal area, N denotes the number of nodes, A_j denotes the j^th nodal area, t_d denotes the duration of loading, t_par denotes the load duration used to calculate the surface flaw parameters, and n denotes the static fatigue constant equal to 16 [1]. The GFPM utilizes a finite element model to calculate the maximum principal tensile stress for an applied load, the biaxial stress correction factor, the number of nodes, and the nodal area [5], while RCSS is the measured quantity of the glass sample.

The GFPM for a glass beam subjected to a three-point bending can be expressed as

Latex equation - P_b = 1- exp\left(-k_s \: \sum_{j=1}^{N} \left(c_j \cdot \left( \frac{t_d}{t_{par}} \right)^{\frac{1}{n}} \cdot\left({\sigma}(P)_{max,j}-RCSS\right)\right)^{m_s} \cdot L_j \cdot w\right)

(Eqn. 2)

where L_j denotes the elemental length, w denotes the width of the sample, and all other variables are the same as described above. The biaxial stress correction factor at each element, c_j, is equal to 1, as the element experiences the same maximum and minimum principal stress. The maximum principal tensile σ(P )_max,j is calculated using the Euler-Bernoulli beam equation for each element. Hausmann [9] in their research studied the effects of aqueous solution on the static fatigue constant and concluded that the effect on the static fatigue constant was minimal. Thus, the value of n used for this study is equal to 16.

The estimation of surface flaw parameters involves subjecting multiple specimens with identical geometry to failure by applying a monotonically increasing load. Given the brittle nature of glass and the uncertainty regarding the location and orientation of the critical surface flaw that initiates fracture, glass breaks at different failure loads and times. To address this variability, the temporal part needs to be normalized. This is achieved by recording the load-time history for each specimen followed by the transformation of the load-time history into a stress-time history. The stress-time history is normalized to 3-second equivalent failure stress (EFS) using the normalization equation proposed by Brown [10] and subsequently modified by Afolabi [11], expressed as

Latex equation - \tilde{\sigma}(P_{t_d})=\left[\frac{\int_{0}^{t_f}(\sigma (P(t))-RCSS_{i})^{16}\: dt}{t_d}\right]^\frac{1}{16}+ RCSS_{i}

(Eqn. 3)

where σ(P(t)) denotes the stress as a function of loading time history P(t), t_f denotes the time to failure, t_d denotes the normalization time (3-sec), σ(P_td) denotes the 3-second equivalent failure stress (EFS) and other variables remain the same as defined above. The equivalent 3-second failure stresses are used to recalculate the 3-sec equivalent failure loads (EFL). The MLE method used the 3-sec EFLs to estimate the surface flaw parameters and the RCSS for a sample [5].

EXPERIMENTAL SETUP AND DATA

In this study, glass specimens were subjected to a 3-point bending test where a monotonically increasing load was applied to each specimen until failure as shown in Figure 1 (a). The distance between the supporting pins is 356 mm (14 in), while the loading pin is at the center of the specimen i.e. at 178 mm (7 in) from the supporting pin, shown in Figure 1 (b). The load-time history for each specimen, including the failure load and time taken for failure, was recorded. Additionally, the location of failure was documented for each specimen. Note that only specimens breaking at the point of maximum stress, specifically at the location where the load was applied, were included in this study. Specimens with different locations of failure were excluded from the analysis.

(a)

(b)

Fig 1 (a): The three-point bending test apparatus (b) Schematic diagram of three-point loading mechanism and associated bending moment/stress diagram induced in a specimen

All glass specimens for the study were 457 mm (18 in) long, and 203 mm (8 in) wide. The specimens were classified into three samples based on the type of glass (i.e., annealed vs. fully tempered) and the thickness of the specimen. Table 1 lists the glass samples categorized for this study.

Table 1 - Sample Properties -

Sample A consists of AN glass specimens with a nominal thickness of 5.0 mm (0.188 in), comprising 10 untreated specimens, denoted as A and 9 treated glass specimens denoted as AT. The treated specimens underwent immersion in a 0.2 M NaCl solution for a year prior to the three-point bending test. Each specimen was 457 mm (18 in) in length and 203 mm (8 in) in width, and the thickness was measured at three locations. Table 1 lists the mean measured thickness of the untreated sample and the treated sample as 4.77 mm (0.188 in.) and 4.74 mm (0.187 in.) respectively, with the standard deviation as 0.102 mm (0.004 in.) and 0.077 mm (0.003 in.) for the untreated and the treated sample, respectively.

The failure load and failure time for each specimen were recorded, and the temporal part of the failure load-time was normalized. The load-time history for each specimen was transformed into a stress-time history using the Euler-Bernoulli beam equation, by calculating the moment (Figure 1(b)) and dividing by the section modulus of the beam.

Fig 2: Failure load and 3-sec EFL for Sample A

The stress-time history was employed to compute a 3-second EFS using Equation 2, with RCSS as equal to zero (0). Additionally, a 3-second EFL was calculated for each specimen. Figure 2 depicts the failure load for each specimen along with the corresponding 3-sec EFL. All specimens were monotonically loaded at a rate of 106 N/sec (24 lbf/sec). The mean failure load of the untreated specimens was calculated to be 698 N (177 lbf), and the standard deviation was calculated as 82.6 N (18.6 lbf). Similarly, the mean failure load and standard deviation of the treated specimens were calculated as 817 N (184 lbf) and 114 N (25.6 lbf), respectively. It was observed that the normalization of the temporal part of the failure load led to a decrease in the mean of the 3-sec EFL by 11.5%, resulting in a mean of 698 N (157 lbf), while the standard deviation decreased by 14.3%, resulting in a standard deviation of 70.9 N (15.9 lbf) for the untreated sample. The treated samples show a similar trend, with the mean and the standard deviation of the 3-sec EFL calculated as 712 N (160 lbf) and 106 N (23.8 lbf), a decrease of 12.8% and 6.77% compared to the mean and the standard deviation of the failure load, respectively. Table 2 lists the statistical measures of the failure load and the 3-sec EFL of sample A.

Sample B consists of FT glass specimens with a nominal thickness of 5.0 mm (0.188 in), comprising 5 untreated and 5 treated glass specimens denoted as B and BT, respectively. The treated specimens underwent immersion in a 0.2 M NaCl solution for a year prior to the three-point bending test. Each specimen was 457 mm (18 in) in length and 203 mm (8 in) in width, and the thickness was measured at three locations. Table 1 lists the mean measured thickness of the untreated sample and the treated sample as 4.75 mm (0.187 in.) and 4.74 mm (0.187 in.) respectively, with the standard deviation as 0.025 mm (0.001 in.) and 0.030 mm (0.001 in.) for the untreated and the treated sample, respectively.

The RCSS for each specimen was measured in two directions, parallel and perpendicular to the long dimension, using a GASP. Figure 3 displays the two measured RCSS and the average of the measured RCSS for each specimen. The average measured RCSS for the untreated sample was found to be 65.9 MPa (9550 psi), with a standard deviation calculated at 0.764 MPa (111 psi). Similarly, the average measured RCSS and the standard deviation for the treated sample were calculated as 65.9 MPa (9550 psi) and 0.838 MPa (122 psi), respectively.

Fig 3: Measured RCSS for each untreated and treated specimen in sample B

The recorded failure load-time history of each specimen was used to normalize the temporal part. All specimens were monotonically loaded at a rate of 136 N/sec (30.5 lbf/sec). The load-time history for each specimen was transformed into a stress-time history using the Euler-Bernoulli beam equation, by calculating the moment (Figure 1(b)) and dividing by the section modulus of the beam. The stress-time history was employed to compute a 3-second EFS using Equation 2, with RCSS as the mean measured RCSS of the specimen from Figure 3. Additionally, a 3-second EFL was calculated for each specimen. The mean failure load of the untreated specimens was calculated to be 2190 N (493 lbf), and the standard deviation was calculated as 164 N (36.9 lbf). Similarly, the mean failure load and standard deviation of the treated specimens were calculated as 2310 N (519 lbf) and 132 N (29.6 lbf), respectively. The normalization of the temporal part of the failure load led to a decrease in the mean of the 3-sec EFL by 6.87 %, resulting in a mean of 2040 N (459 lbf), and the standard deviation decreased by 1.94%, resulting in 161 N (36.2 lbf) for the untreated sample. The treated samples show a similar trend, with the mean and the standard deviation of the 3-sec EFL calculated as 2150 N (131 lbf) and 483 N (29.5 lbf), a decrease of 6.88 % and 0.42 % compared to the mean and the standard deviation of the failure load, respectively. Table 2 lists the statistical measures of the failure load and the 3-sec EFL of sample B.

Sample C consists of 6.35mm (0.25 in) thick FT glass specimens comprising 5 untreated samples denoted as C and 5 treated glass specimens denoted as CT. The treated specimens underwent immersion in a 0.2 M NaCl solution for a year prior to the three-point bending test. Each specimen is 457.2 mm (18 in) in length and 203.2 mm (8 in) in width, and the thickness was measured at three locations. Table 1 lists the mean measured thickness of the untreated sample and the treated sample as 5.81 mm (0.229 in.) and 5.80 mm (0.228 in.) respectively, with the standard deviation as 0.020 mm (0.001 in.) for both the untreated and the treated sample.

The RCSS for each specimen was measured in two directions, parallel and perpendicular to the long dimension, using a GASP. Figure 4 displays the two measured RCSS and the average of the measured RCSS for each specimen. The average measured RCSS for the untreated sample was found to be 65.3 MPa (9490 psi), with a standard deviation calculated at 0.393 MPa (57.0 psi). Similarly, the average measured RCSS and the standard deviation for the treated sample were calculated as 65.7 MPa (9530 psi) and 0.519 MPa (75.3 psi), respectively.

The recorded failure load-time history of each specimen was used to normalize the temporal part. All specimens were monotonically loaded at a rate of 224 N/sec (50.3 lbf/sec). The load-time history for each specimen was transformed into a stress-time history using the Euler-Bernoulli beam equation, by calculating the moment (Figure 1(b)) and dividing by the section modulus of the beam. The stress-time history was employed to compute a 3-second EFS using Equation 2, with RCSS as the mean measured RCSS of the specimen from Figure 4. Additionally, a 3-second EFL was calculated for each specimen. The mean failure load of the untreated specimens was calculated to be 3510 N (789 lbf), and the standard deviation was calculated as 178 N (40.1 lbf). Similarly, the mean failure load and standard deviation of the treated specimens were calculated as 3370 N (758 lbf) and 206 N (46.3 lbf), respectively. The normalization of the temporal part of the failure load led to a decrease in the mean of the 3-sec EFL by 7.29 %, resulting in a mean of 3250 N (731 lbf), and the standard deviation decreased by 10.3 %, resulting in 160 N (36.0 lbf) for the untreated sample. The treated samples show a similar trend, with the mean and the standard deviation of the 3-sec EFL calculated as 3110 N (700 lbf) and 195 N (43.9 lbf), a decrease of 7.71 % and 5.19 % compared to the mean and the standard deviation of the failure load, respectively. Table 2 lists the statistical measures of the failure load and the 3-sec EFL of sample B.

Fig 4: Measured RCSS for each untreated and treated specimen in sample C

Table 2 - Statistical measures of failure load and 3-sec equivalent failure load (EFL) -

ANALYSIS AND DISCUSSION

The 3-sec EFL for the treated and the untreated specimen for each sample were used to estimate the surface flaw parameters using the maximum likelihood estimator method [5]. The surface flaw parameters for each sample are listed in Table 3.

Table 3 - Estimated surface flaw parameters -

To facilitate a meaningful comparison of the estimated surface flaw parameters, a numerical analysis alone may not provide insightful conclusions. Therefore, graphs are employed, and loads corresponding to different probabilities of breakage (pb) are utilized for comparison. To assess the strength of both untreated and treated samples, the EFLs for each sample are arranged in ascending order. Rank estimators for each EFL are then assigned as [12]:

Latex equation - E_i = \frac{i-0.3}{n_{sam} + 0.5} (Eqn .4)

where, i is the rank of the EFL for a specimen, and n_sam is the number of specimens in the sample.

The estimated surface flaw parameters from Table 2 are employed to calculate the probabilities of breakage for a range of loads, which are then plotted against the rank EFLs. ASTM E1300-16 (ASTM, 2016) defines the load resistance as a 3-second uniform lateral load associated with a probability of breakage (pb) set at 8 in 1000 (0.008). Additionally, AAMA GDSG-1-87 (1987) recommends a load resistance with a pb of 1 in 1000 (0.001) specifically for overhead sloped glazing. The small value of the compared pb prevents a comprehensive depiction of trends through a standard x-y graph. Consequently, the x-axis scale was retained, while the y-axis scale was changed to a scale of lnln11-pb, to enhance clarity of the trends. In Figures 5, 6, and 7, the rank EFLs for the untreated and treated samples are represented by the dot marker and the diamond markers, respectively. The fit of the untreated EFL is depicted by the continuous line, whereas the fit of the treated EFL is indicated by the dashed line. The fits for both the untreated and the treated samples are generated using the estimated surface flaw parameters presented in Table 3 for the corresponding sample. The loads corresponding to 0.008 and 0.001 pb for each treated and untreated sample, are enumerated in Table 4

Fig 5: Comparison of treated and untreated samples for sample A

Fig 6: Comparison of treated and untreated samples for sample B

Fig 7: Comparison of treated and untreated samples for sample C

The treated samples in both sample A and sample C exhibit a decrease in strength at both 0.008 and 0.001 pb, as depicted in Figures 5 and 7, respectively. In sample A, the treated specimens showed a reduction of 9.63% and 14.4%, while in sample C, a decrease of 3.02% and 2.57% was observed at 0.008 and 0.001 pb, respectively, compared to the corresponding untreated specimens.

Contrastingly, in sample B, a different trend emerged between the treated and untreated specimens. It was observed that at both 0.008 and 0.001 pb, the treated specimens exhibited an increase in strength by 14.7% and 18.4%, respectively.

Table 4 - Load corresponding to different pb

CONCLUSION

This study was designed to understand on a macroscopic level the effect of NaCL on the strength of soda lime glass. The research employed a three-point bending test, subjecting various glass samples to monotonically increasing loads. Three types of samples were examined: annealed glass samples with a nominal thickness of 5.0 mm (0.188 in), fully tempered glass samples with a nominal thickness of 5.0 mm (0.188 in), and fully tempered glass samples with a nominal thickness of 6.35 mm (0.25 in). A 0.2 molar NaCl solution was used to immerse a few specimens from each sample for a full year before subjecting them to failure loading. Notably, a decrease in strength at 0.008 and 0.001 probability of breakage was observed in the treated specimens of the 5.0 mm (0.188 in) thick annealed and 6.35 mm (0.25 in) fully tempered glass samples. Conversely, the treated specimens in the 5.0 mm (0.188 in) thick fully tempered sample exhibited an opposite trend. The authors acknowledge the challenge of establishing a definitive cause for these observed trends due to the limited number of specimens tested.

In light of this limitation, the authors recommend further exploration through additional testing, incorporating multiple variables such as different concentrations of NaCl solution and varying loading rates. This approach is anticipated to provide a more comprehensive understanding of the influence of NaCl on the strength of soda lime glass.

REFERENCES

1. Charles, R. J. (1958). Static fatigue of glass. II. Journal of Applied Physics, 29(11), 1554-1560.
2. Wiederhorn, S., & Bolz, L. H. (1970). Stress corrosion and static fatigue of glass. Journal of the American Ceramic Society, 53(10), 543-548.
3. ASTM Standard. (2016). ASTM E1300 Standard Practice for Determining Load Resistance of Glass in Buildings.
4. Beason, W. L., & Morgan, J. R. (1984). Glass failure prediction model. Journal of Structural Engineering, 110(2), 197-212.
5. Goswami, N., Schultz, J. A., Zhang, K., Dowden, D. M., Swartz, R. A., & Morse, S. M. (2023). Estimation of flaw parameters for holes in glass using maximum likelihood estimator. Glass Structures & Engineering, 8(3), 405-421.
6. Weibull, W. (1939). A statistical theory of strength of materials. IVB-Handl.
7. Norville, H. S., Bove, P. M., & Sheridan, D. L. (1991). The strength of new thermally tempered window glass lites. Glass Research and Testing Laboratory, Department of Civil Engineering, Texas Tech University.
8. Morse, S., & Norville, H. (2011). An analytical method for determining window glass strength.
9. Hausmann, B. D., & Salem, J. A. (2017). Effects of Aqueous Solutions on Slow Crack Growth of Soda Lime Silicate Glass. Mechanical Properties and Performance of Engineering Ceramics and Composites XI: Ceramic Engineering and Science Proceedings Volume 37, Issue 2, 37, 45-52.
10. Brown, W. G. (1974). A practicable formulation for the strength of glass and its special application to large plates. National Research Council Canada.
11. Afolabi, B., Norville, H. S., & Morse, S. M. (2016). Experimental study of weathered tempered glass plates from the Northeastern United States. Journal of Architectural Engineering, 22(3), 04016010.
12. Fothergill, J. C. (1990). Estimating the cumulative probability of failure data points to be plotted on Weibull and other probability paper. IEEE Transactions on Electrical Insulation, 25(3), 489-492.