MECH 321 LAB 5 Experiment- FLEXURAL TESTING OF CERAMICS (Properties & Failure of Materials) 2026 Con

MECH 321 LAB 5 Experiment- FLEXURAL TESTING OF CERAMICS (Properties & Failure of Materials) 2026 Concordia University

Experiment 5 Flexural Testing Of Ceramics

MECH 321 Lab Section MI - X Group A Winter 2025

Prof. Sam Eskadarian

Concordia University Montreal, QC, Canada

TABLE OF CONTENTS TABLE OF CONTENTS ............................................................................... 2 LIST OF TABLES AND GRAPHS ..................................................................... 2 OBJECTIVE .......................................................................................... 3 INTRODUCTION ..................................................................................... 3 EQUIPMENT.......................................................................................... 5 PROCEDURE ......................................................................................... 5 RESULTS ............................................................................................. 6 SAMPLE CALCULATIONS ....................................................................... 10 DISCUSSION........................................................................................ 10 CONCLUSION ...................................................................................... 11 REFERENCES....................................................................................... 12

LIST OF TABLES AND GRAPHS Figure 1. 4-point flexure test Figure 2. (a) Weibull distribution; (b) Cumulative plot on log-log scale Table 1. Tabulated Results for Borosilicate Clear rod (glass) - as received Figure 3. Weibull distribution for Borosilicate Clear rod (glass) - as received Table 2. Tabulated Results for Borosilicate Clear Rod (GLASS) - Sanded with sic paper Figure 4. Weibull distribution for Borosilicate Clear rod (glass) - Sanded with Sic Paper Table 3. Tabulated Results for Borosilicate Clear Rod (GLASS) - Polished with Diamond Paste Figure 5. Weibull distribution for Borosilicate Clear rod (glass) - Polished with Diamond Paste Table 4. Tabulated Results for Borosilicate Clear Rod (GLASS) - Etched in HF Acid Figure 6. Weibull distribution for Borosilicate Clear rod (glass) - Etched in HF Acid Table 5. Weibull Modulus for each set of bars

OBJECTIVE The objective of this experiment is to determine the flexural strength of borosilicate glass rods using a 4-point bend test, analyze the statistical variability of ceramic strength via Weibull modulus calculations, and investigate the effect of surface treatments (abrasion, polishing, and hydrofluoric acid etching) on the modulus of rupture (MOR).

INTRODUCTION Ceramics, renowned for their high hardness and compressive strength, often exhibit brittle failure under tensile stresses due to inherent flaws such as microcracks or surface imperfections. Unlike ductile materials, ceramics lack significant plastic deformation, making their strength highly sensitive to flaw distribution. This experiment evaluates the flexural strength of borosilicate glass rods using a 4-point bending test, a method preferred over tensile testing due to the challenges of gripping brittle materials without inducing premature failure.

Figure 1. 4-point flexure test

The modulus of rupture (MOR), defined as the maximum tensile stress at the specimen’s outermost fiber during bending, is calculated using the formula:

𝑀𝑂𝑅 =

16𝐹𝑎 𝜋𝐷3

(for round specimens)

where F is the fracture load, a is the distance between supporting and loading pins (10 mm), and D is the specimen diameter. The 4-point test distributes stress more uniformly than 3-point

bending, minimizing the influence of localized flaws and providing a representative measure of bulk material strength.

Despite the 4-point test’s ability to mitigate localized stress concentrations, ceramic strength remains inherently variable due to the stochastic nature of flaws. Traditional Gaussian statistics, which describe ductile materials with narrow strength distributions, fail to capture this variability. To account for the inherent variability in ceramic strength, Weibull statistics are applied, relating the cumulative probability of failure (P) to the MOR:

ln (

1 1−𝑃

) = 𝑚 ∙ ln(𝑀𝑂𝑅)

Here, m, the Weibull modulus, quantifies material reliability, with higher values indicating a narrower distribution of flaw sizes. As shown in Figure 2, the Weibull curve reveals three regions: low-stress failures (large flaws), intermediate-stress failures (moderate flaws), and highstress survival (minimal flaws). This statistical approach is critical for designing ceramic components in applications such as bearings and cutting tools, where predictable failure behavior is essential.

Figure 2. (a) Weibull distribution; (b) Cumulative plot on log-log scale

The experiment also examines how surface treatments: abrasion, polishing, and hydrofluoric acid (HF) etching, modify flaw populations. For instance, abrasion introduces controlled surface defects, while HF etching removes surface irregularities, altering the failure stress distribution.

These treatments aim to narrow the Weibull distribution, enhancing material reliability by reducing flaw variability.

EQUIPMENT The experiment utilized a 4-point flexural testing rig with supporting pins spaced 20 mm apart and loading pins spaced 10 mm apart, as illustrated in Figure 5. Specimens consisted of borosilicate glass rods (5 mm diameter, 60 mm length), divided into four groups: as-received, abraded with 120-grit SiC paper, polished with 5-micron diamond paste, and etched in 5% hydrofluoric acid (HF).

Surface treatments were conducted in plastic beakers, with HF handled under strict safety protocols, including nitrile gloves, goggles, and a lab coat. A plexiglass screen was employed during fracture testing to protect against debris. Load data was recorded using a calibrated load cell connected to a data acquisition system.

Post-etching, specimens were rinsed with deionized water for 5 minutes, followed by an alcohol rinse and air-drying to preserve surface integrity. Calipers were used to verify specimen dimensions prior to testing, ensuring consistency across all samples.

PROCEDURE The experiment began with the preparation of four sets of borosilicate glass rods. Five specimens were tested as-received, while others underwent surface treatments: 1. Abrasion: Specimens were lightly abraded with 120-grit SiC paper to simulate controlled surface damage.

2. Polishing: Rods were polished with 5-micron diamond paste to reduce surface roughness.

3. HF Etching: Specimens were immersed in 5% HF for 20 minutes to chemically remove surface flaws, then rinsed and dried without physical contact to avoid contamination.

Each specimen was mounted on the 4-point testing rig, ensuring a 20 mm span between supporting pins and a 10 mm span between loading pins. A constant crosshead speed of 1 mm/min was applied until fracture, and the failure load (F) was recorded.

Following testing, the MOR for each specimen was calculated using Equation 1. For statistical analysis, failure stresses were ranked, and the cumulative probability of failure (P) was determined by dividing each specimen’s rank by n+1, where n=5. A Weibull plot of ln(1/1-P) versus ln(MOR) was constructed to determine the Weibull modulus (m). Post-test comparisons evaluated the impact of surface treatments on MOR variability, and the advantages of 4-point bending over 3-point testing were discussed in the context of stress distribution and flaw sensitivity.

RESULTS sample number 1 2 3 4 5

BOROSILICATE CLEAR ROD (GLASS) - AS RECEIVED fracture load Cumulative Probability of X MOR [MPa] Y (Ln[ln(1/1-P)]) [N] Failure (ln(MOR)) 202.570 82.534 0.167 -1.702 4.413 228.690 93.177 0.333 -0.903 4.534 471.240 192.000 0.500 -0.367 5.257 379.370 154.569 0.667 0.094 5.041 384.710 156.745 0.833 0.583 5.055

Table 1. Tabulated Results for Borosilicate Clear rod (glass) - as received

Figure 3. Weibull distribution for Borosilicate Clear rod (glass) - as received

BOROSILICATE CLEAR ROD (GLASS) - SANDED WITH SIC PAPER sample number

fracture load [N]

5

240.910

98.156

0.167

-1.702

4.587

2

288.400

117.505

0.333

-0.903

4.766

4

291.650

118.829

0.500

-0.367

4.778

3

301.020

122.647

0.667

0.094

4.809

1

363.630

148.156

0.833

0.583

4.998

MOR [MPa] Cumulative Probability of Failure

Table 2. Tabulated Results for Borosilicate Clear Rod (GLASS) - Sanded with sic paper

Figure 4. Weibull distribution for Borosilicate Clear rod (glass) - Sanded with Sic Paper

Y (Ln[ln(1/1-P)]) X (ln(MOR))

BOROSILICATE CLEAR ROD (GLASS) - POLISHED WITH DIAMOND PASTE MOR Y (Ln[ln(1/1sample number fracture load [N] Cumulative Probability of Failure [MPa] P)]) 2 310.700 126.591 0.167 -1.702 4 332.980 135.668 0.333 -0.903 3 371.230 151.253 0.500 -0.367 1 384.910 156.826 0.667 0.094 5 418.090 170.345 0.833 0.583 Table 3. Tabulated Results for Borosilicate Clear Rod (GLASS) - Polished with Diamond Paste

Figure 5. Weibull distribution for Borosilicate Clear rod (glass) - Polished with Diamond Paste

X (ln(MOR)) 4.841 4.910 5.019 5.055 5.138

BOROSILICATE CLEAR ROD (GLASS) - ETCHED IN HF ACID MOR Y (Ln[ln(1/1sample number fracture load [N] Cumulative Probability of Failure [MPa] P)]) 5 719.830 293.285 0.167 -1.702 2 1099.780 448.091 0.333 -0.903 3 1175.800 479.064 0.500 -0.367 1 1360.950 554.501 0.667 0.094 4 1702.450 693.641 0.833 0.583

X (ln(MOR) 5.681 6.105 6.172 6.318 6.542

Table 4. Tabulated Results for Borosilicate Clear Rod (GLASS) - Etched in HF Acid

Figure 6. Weibull distribution for Borosilicate Clear rod (glass) - Etched in HF Acid

Glass Rod Condition Weibull Modulus As Received 1.95 Sanded with Sic Paper 5.71 Polished with Diamond Paste 7.45 Etched in HF Acid 2.74 Table 5. Weibull Modulus for each set of bars

SAMPLE CALCULATIONS Sample Calculation for Modulus of Rupture (MOR) of Borosilicate Clear Rod (Glass) - As Received sample 1: 2 ⋅ 𝐹 ⋅ 𝑎 2 ⋅ 202.57 ⋅ 10 = 𝑀𝑂𝑅 = = 82.534 𝜋 ⋅ 𝑟3 𝜋 ⋅ 2.53 For Weibull modulus: 𝑖 𝑃𝑖 =

𝑛+1

1 = 𝑃1 =

5+1

= 0.167

1 ln (ln ( 1 )) ln (ln ( )) 1 − 𝑃 1 − 0.167 𝑚= = = 1.95 ln(𝑀𝑂𝑅) ln(82.534)

DISCUSSION Due to the cost of processing near flawless materials, most processed materials generally retain their flaws and even gain some in the machining process [2]. For ceramics in particular, pores are created during the processing and sharp edges/cracks/scratches can be created during the surface machining [2]. Another flaw may be the inclusion of foreign particles within the ceramic matrix [2]. Each flaw has a different underlying mechanism leading to the failure of the material. When looking at pores, the strength of load bearing decreases due to a reduction in effective crosssectional area [3]. Sharp edges/cracks/scratches create stress concentrations that have a much higher likelihood to propagate cracks leading to abrupt material failure [2]. Inclusions can induce either innate tension or compression of the material creating a stress concentration but also can provide a nucleation site given the correct conditions [2]. Since all these flaws are virtually impossible to perfectly account for it is very likely that a large source of error and variation in the strength of the ceramic stems from them and how they vary between each sample.

Looking at the results of the experiment, the glass samples that were abraded and polished performed similarly, with the diamond polished samples performing slightly better for the modulus of rupture. This is likely due to the diamond polish having a higher hardness and leaving a finer finish on the surface of the sample compared to the SiC paper. Ceramics having a high hardness need harder materials to leave better surface finishes, hence the diamond paste is superior. Some of the variations between samples might be explained by each sample being abraded/polished by different students who certainly used varying techniques or levels of pressure. The HF acid treated samples showed a significant increase in flexural strength compared to the abraded and polished samples. This is due to HF acid being able to dissolve

glass and completely remove outer layers with their flaws without forming new flaws [4]. This removes stress concentrations from the surface to greatly increase the flexural strength. Although all samples were etched in HF for the same amount of time, it is not known whether some had much deeper cracks or scratches upon the surface, and thus some samples may not have had sufficient etching time to reach the same surface finish as other samples.

Three methods are used to measure the flexural strength of ceramics: 3-point bending, 4-point bending, and biaxial flexural strength [5][6]. Three-point bending is the more common method and is practical for ductile materials or materials with higher shear strength as the single load also applies a higher bending moment as well as shear at the midpoint of the material [5]. Due to the brittle nature of ceramics, 3-point bending is not ideal as ceramics do not handle shear stresses very well. Four-point bending tests are slightly more difficult to set up but distribute the load more evenly and remove the shear stress between the loading points, allowing for more accurate testing [5]. The biaxial uses a ceramic disc with 3 points of support in a triangular configuration [6]. Multidirectional stresses are applied to the disc and have the advantage of a more realistc simulation of real-world conditions [7]. Although a better estimation of flexural strength, biaxial tests are significantly more difficult to set up and require specific equipment [7].

When it comes to strengthening ceramics, or glass, there are several different methods besides chemical etching that can be used. As seen in the experiment, abrading/polishing the surface can increase the strength by reducing flaws. Another method is to prestress the material during processing [8]. By adding residual compressive stress to the material, higher tensile loads can be achieved [8]. Other methods include dispersion strengthening, grain size and porosity reduction, and reduction of crystal anisotropy [8]. These methods are used to strengthen metallic materials and have proven to be effective in ceramics as well.

CONCLUSION In conclusion, the 4-point bending test proved to be an effective method to find the Weibull distribution and modulus. By plotting the natural log of the modulus of rupture vs the natural log of the probability of failure, a statistical approach for ceramic failure was successfully found. It can also be concluded that treating the ceramic surface makes a significant difference as the treated samples have a lower scatter compared to the untreated samples. It can also be concluded that different methods of surface treatment will improve the modulus of rupture by varying amounts.

REFERENCES [1] MECH 321 LABORATORY MANUAL – PROPERTIES & FAILURE OF MATERIALS – WINTER 2025 [2] Eric Martin, Dominique Leguillon, Oldřich Ševeček, Raúl Bermejo. Understanding the tensile strength of ceramics in the presence of small critical flaws. Engineering Fracture Mechanics, 2018, 201, pp.167- 175. 10.1016/j.engfracmech.2018.06.021. hal-01906008, March 29th 2025 [3] D. Munz, “Failure criteria in ceramic materials,” Journal of Nuclear Materials, vol. 155–157, pp. 77–81, Jul. 1988, doi: 10.1016/0022-3115(88)90229-2.

[4] R. W. Rice, “Relation of tensile strength-porosity effects in ceramics to porosity dependence of Young’s modulus and fracture energy, porosity character and grain size,” Materials Science and Engineering: A, vol. 112, pp. 215–224, Jun. 1989, doi: 10.1016/0921-5093(89)90361-4.

[5] P. M. Moreira, G. L. M. Carvalho, R. de Castro Albuquerque, and C. B. André, “Effect of hydrofluoric acid and self-etch ceramic primers on the flexural strength and fatigue resistance of glass ceramics: A systematic review and meta-analysis of in vitro studies,” Japanese Dental Science Review, vol. 60, pp. 198–210, Dec. 2024, doi: 10.1016/j.jdsr.2024.06.001.

[6] tec-science, “Bending flexural test | tec-science.” Accessed: Mar. 29, 2025. [Online].

Available: https://www.tec-science.com/material-science/material-testing/bending-flexural-test/ [7] “Biaxial Test - TestResources.” Accessed: Mar. 29, 2025. [Online]. Available: https://www.testresources.net/applications/test-types/biaxial-test [8] Y. Bao et al., “A simple way to make pre-stressed ceramics with high strength,” Journal of Materiomics, vol. 5, no. 4, pp. 657–662, Dec. 2019, doi: 10.1016/j.jmat.2019.06.001.