Mechanical Properties of Brittle Materials
I Purpose/Objective:
Mechanical properties are key attributes of materials. In this unit we will consider the simplest type of mechanical material – an ideally brittle solid, which is very closely represented by ordinary soda lime silicate glass. We will determine the fracture strength of glass using flexural testing rods, and examine how the strength depends upon surface flaws. We will do simple fractography to identify the features on the fracture surface of the rods. We also will use the indentation cracking method to examine the fracture toughness of glass. Since glass is photoelastic, we an visualize stress with polarized light. We have four activities.
II  Experimental Procedure:
In this unit we will determine the fracture strength and toughness of soda lime glass flexural testing of glass rods. Determine the role of surface flaws by testing abraded specimens, and compare results with expectations of Griffith theory. Results from several groups will be combined to make a large enough data set for analysis of fracture statistics. Learn the utility (and limitations) of the indentation technique for inferring the fracture toughness by analyzing the cracks associated with a Vickers indent in glass. Conduct a fractographic analysis of fracture surfaces of the the specimens to identify the fracture origin and relate fracture mirrors to fracture toughness. Use polarized light methods to observe applied and residual stresses.
There will be four major activities, scheduled as below to minimize bottlenecks. These are: flexural testing on MTS Insight 10kN load frame ; examination of fracture surfaces on the Nikon SMZ binocular microscope and the Philips XL30 SEM; producing a series of Vickers microhardness indents at several loads on the Clark Vickers Microhardness tester and measuring the indentation cracks on the Nikon Optiphot microscopes. You also will examine residual stresses using the photoelastic effect, using polarized light.
Activity #1: Flexural Testing
The laboratory instructor will supply each group with 30 specimens of sodalime glass rods. These have been annealed to remove most of the residual stress. The surface condition is “asreceived”. Each group will test 10 rods in the asreceived conditions and 20 rods in an abraded condition. Store all your data in files accessible by your classmates, since each student will conduct statistical analyses of the combined results for all nine lab groups.
The flexural test is conducted according to MILSTD 1942, using a fully articulated fourpoint flexural fixture. The lab instructor will explain the features of this fixture and the details of conducting the test with the Instron software. Identify and test each specimen. Afterwards, tape together and label the fractured specimens, in order to protect the fracture surfaces for later fractographic analysis. Some boxes will be available for storing the fractured specimens. Calculate the strength from the load at fracture and the original dimensions of the specimen.
Abraded specimens:
We will examine the effect of surface condition by testing glass rods after a light abrasion treatment. Each group will test 10 rods abraded with longitudinal scratches, and 10 abraded with transverse scratches. Abrade the rods with SiC grinding paper of your assigned grit size: Group 1 uses 240 grit, Group 2 uses 600 grit, and Group 3 uses 1200 grit paper. With very light pressure, scratch 10 bars along the length to make longitudinal scratches, and 10 additional bars around the circumference to make transverse scratches. Test the rods, again preserving their fracture surfaces.
Activity #2: Fractography
Examine the fracture surfaces of your glass rods on the Nikon or Olympus optical microscopes and using the teaching SEM in the Van Vlack UG Laboratory. Although the SEM gives much higher resolution than the optical microscope, both give valuable information at different length scales. Reflected light also provides contrast for certain features which are not easily visible with secondary electrons. Obtain a few representative optical and SEM images for your report.
Identify the following characteristics of glass fracture: the fracture origin, the fracture mirror, the mist region, and the hackle zone. Use the "shear lip" to infer the orientation of the rod when it broke, and determine if the fracture origin was at the point of peak tensile stress (Note that the most severe flaw might not be at the point of maximum stress.). If possible, determine the size of the fracture origin, which is not always distinct, and relate the fracture origin size to the strength for the individual rods.
Each person in your lab group should individually measure the radius of the fracture mirrors. Note that the location, or even the existence (!), of the mirror can be rather subjective. Hence it will be of interest to compare what each person sees.
Activity #3: Vickers Indentations
In brittle materials, a Vickers diamond indenter makes a square impression of diagonal 2a (where ‘a’ is the edge length.), which is related to the hardness. A pattern of cracks  the "radial/median crack system"  also develops. The surface traces of the cracks appear along the diagonals of the indent. Anstis et al have related the length of these cracks to the fracture toughness [Kc] with the following expression:^{1}
where P is the indentation load in Newtons, ‘c’ is the distance from the center of the indent to the tip of the crack in meters, E is the modulus [70 GPa for soda lime glass] and H is the hardness [5.5 GPa for this glass]. It is common to use this equation to infer fracture toughness from simple measurements of indentation cracks.
The laboratory instructor will discuss the recommended procedures for estimating the fracture toughness by the indentation method. The sample will be a sodalime glass slide. Use the Buehler microhardness tester in 2235 Dow, with a Vickers diamond indenter. Produce five sound indents each at four different loads between 1 and 10 N [Newtons, not kilograms! ]
Examine the residual stress fields around the indents using one of the Nikon microscopes in polarized transmitted light.
Using the Nikon Optiphot metallographic optical microscope in the Van Vlack Undergraduate Laboratory, for each indent, measure the size of the Vickers indent and the length of all four crack traces. The position of the crack tip is rather subjective, so have several individuals measure some of the cracks, and discuss any variation in interpretation. Obtain a few nice (!) representative images for your report.
 SOP:
 Image Capturing:
III  Theory/Background Information:
Activity #1: Flexural Strength Testing
The flexural strength technique we use is based upon the standard ASTM C1161 and MILSTD 1942 for fixture dimensions, loading rates, and other pertinent variables. Typically brittle materials are tested in bending as machined bars. The strength results are quite sensitive to the quality of the machining, and the literature has guidelines for recommended grinding techniques. The edges on the tensile side of the bar should be ground flat, or chamfered, to eliminate artifacts from stress concentrations. However, for this lab, we adopt rodshaped samples in an asmanufactured condition. This is primarily for convenience and low cost, since we avoid the bother of machining.
Recall what you learned in ME 211 and MSE 420 about the mechanics of a beam in bending. The normal stress varies from compressive on one surface, to zero at the neutral axis, and tensile at the opposite surface. Shear stress varies from zero at the outer fibers to a maximum at the neutral axis. Bending beams are useful despite their complex stress state only for quite brittle materials. In very brittle materials, cracks grow in response to the tensile stress, typically from preexisting flaw which are much smaller than the specimen size. For a rod, the maximum tensile stress exists only along a narrow line in the "outer fiber" of the rod. Typically we assume that the fracture originates at the point of maximum tensile stress and estimate the flexural strength from the maximum tensile stress at the point of fracture(Sometimes there is a more severe flaw in another location. With fractographic analysis, one can determine whether the fracture did indeed originate in the 'outer fiber'. If it did not, it is possible to calculate the stress at the location of the originating flaw, although this is rarely done.). For any beam, the tensile stress at a distance 'c' from the neutral axis is:
where I is the moment of inertia and M is the bending moment applied to the beam. For a rod, c=R, the radius. For a rod with circular crosssection, the moment of inertia is:
For a beam loaded in fourpoint flexure, with characteristic lengths linner and louter (both with respect to the center of mass of the beam), bearing a load F, the moment is:
For our particular fixture, linner = L and louter = 2L.
In a flexural test, we increase the deflection of the beam at a constant rate by moving the crosshead while measuring the load with a load cell. The peak load at the point of fracture is related to the maximum stress at the outer fiber and gives the fracture strength as:
For a proper test, one must assure that the only stresses on the sample come from the bending moment. If the sample is not straight and flat (for bars), the sample will twist in the fixture. This torsion can cause major errors. The fixture must apply only normal loads. To avoid errors, the fixture must have loading pins which are fully articulated. Take the time to examine the Instron fixture we are using, and ask your lab instructor to explain its features.
Activity #2: Fractography
Refer to Michalske's article "Quantitative Fracture Surface Analysis" from the ASM Engineered Materials Handbook, Vol. 4: Ceramics and Glasses, 1987. There is also a brief discussion in the article by Amin entitled“Toughness, Hardness, and Wear”.
Fracture Statistics
If the strength variability of a set of specimens (This form is strictly for samples of a standard volume, subject to uniform tensile stress. This is not strictly applicable for flexural tests, but this formula is commonly applied. Data for one set of samples (of a certain size and test type) can be related to another set (different size and test type) are reviewed by Geoge Quinn in "Strength and Proof Testing" and Curt Johnson "Advanced Statistical Concepts" in ASM Engineered Materials Handbook, Vol. 4: Ceramics and Glasses, 1987.) fit a Weibull distribution, the probability that a sample will fail at a particular strength (σ_{i}) is P_{i}
where σo is the characteristic strength, σu is the threshold stress (below which there is no failure) and 'm' is the Weibull modulus. This threeparameter Weibull expression is actually rarely used, because there does not seem to be a threshold stress for brittle materials (Physically, a threshold strength implies a maximum possible flaw size. Why should there be a limit to flaw size, other than the trivial limit set by the size of the sample itself?) , and because it is mathematically inconvenient. A more practical form is the twoparameter Weibull expression:
Thus, we characterize the central tendency of strength with the characteristic strength σo (strength at 63.2 % probability of failure) and the scatter in the strength data with the Weibull modulus. For many brittle ceramics, 'm' falls in the range 520.
To fit a set of strength data to the Weibull distribution, one first rankorders the samples in order of increasing strength and assigns an index: i=1 for the weakest with strength σ1, i=2 for the second weakest with strength σ2, and so on, up to i=N for the strongest sample. For each sample, assign an estimate for the probablity of failure by
Plot the data in the following linearized form of the Weibull distribution:
The Weibull modulus is obtained from the slope of a plot of lnln(1/1P) vs lnσ, and the characteristic strength obtained from the intercept. These are often determined by least squares, although this might exaggerate the importance of the lowstrength data points.
The error in the estimate of the Weibull parameters, strictly from mathematical factors (not experimental), can be appreciable, particulary for small sample sizes. The figures below show confidence interval bands for the slope and characteristic value versus the number of samples. Notice that 4060 samples are required for reasonably small errors.
(a) 
(b)
The possible error in (a) the predicted Weibull slope versus the sample size and (b) the Weibull characteristic strength value versus the sample size. 
One can extract the Weibull parameters from a set of data only if it fits the Weibull distribution. Maybe your data will not fit. Be skeptical  avoid a foolish forcefit. Plot all of your data in the linear form. Look for the "outliers" atypically strong or weak samples. Treating them properly requires your judgment. Outliers may be telling you something important (such as hinting at the existence of a second flaw population, which is rare but severe when it happens) or they may be telling you nothing (maybe a bad measurement or an error in data logging). Some people strictly include every data point, which is honest but can distort the "true" data with a false data point. Others cull their data of "suspicious" outliers (Manufacturers sometimes do this. Often it is valid, since occasional errors exist. However, it IS rare to see someone reject a specimen because it was too strong!) before fitting the Weibull. This can be done legitimately if there is some reason to suspect that the outlier is not a reliable data point. The danger of this approach is obvious. It is your data, so you have to decide. Another way to assess the sensitivity of Weibull parameters is by calculating them from the complete data set, and then recalculating them after rejecting a few of the weaker and stronger samples. If the parameters change little, then once might have more confidence in them. If they vary substantially, then one might have much less confidence in them.
If you have a good linear regression line, you can get the slope 'm' value, from least squares. Most regression packages also report the "correlation coefficient", R2, which is a goodnessoffit parameter. It reports if the data are wellcorrelated, but does not give an estimate of the error of the slope. Johnson and Tucker discuss "maximum likihood estimators",^{2} but these involve rather complicated analysis.
You can get an estimate of the standard deviation of the slope from the residual sum of squares, following a procedure from Rektorys.^{3} To fit a set of N data points (yi,xi) with a linear equation of the form, one can estimate the intercept by:
and the slope by:
An estimate of the standard deviation of the intercept (sa) and the standard deviation of the slope (sb) is obtained from the "residual sum of squares" Σo, with:
with the residual sum of squares from:
The probable error on the slope is 0.674 sb. It is useful to calculate the probable error on your slope (Weibull 'm'), and report the Weibull moduli as "m +/ the probable error".
Activity #3: Indentation Fracture Toughness
Refer to Amin's article "Toughness, Hardness, and Wear" from the ASM Engineered Materials Handbook, Vol. 4: Ceramics and Glasses, 1987. This article reviews many fracture toughness techniques, including the widely used indentation fracture technique pioneered by investigators such as Anthony Evans, Brian Lawn, and David Marshall. Lawn’s text, “Fracture of Brittle Materials” (second edition), Cambridge University Press, 1993 [TA409.L.37 1993] is an excellent reference for these methods and their analysis.
http://www.scielo.br/img/fbpe/jbsms/v23n1/a02fig02.gif
Left: Residual Stresses in glass under Vickers indent, as revealed by transmitted polarized light
From Figure 3 of D.B. Marshall and B.R. Lawn, “Residual stress effects in sharp contact cracking” Journal of Materials Science 14(1979) 20012012
Right: A rather more decorative version of the same, entered in the Princeton “Art of Science” competition as Blue Indentation, by Elizabeth Allaway ‘07, Ryan Rimmele ‘07, Richard Li ‘07, Nate Beck ‘06, and George Scherer of the Princeton Department of Civil and Environmental Engineering “This is an indentation in glass made with a pyramidal diamond tip (called a Vickers indenter) as part of a lab in CEE 364 Materials in Civil Engineering. The goal was to make a flaw of controlled size and measure its effect on the strength of the glass—but the result was a thing of beauty. The color results from crossed polarizers that indicate the shape of the stress distribution.”
IV  Theory/Background References:
 G.R. Anstis, P. Chantikul, D.R. Lawn, and D.B. Marshall, J. Amer. Ceram. Soc. 64 [9] p. 533538 (1981).
 C.A. Johnson and W.T. Tucker, "Advanced Statistical Concepts" in ASM Engineered Materials Handbook, Vol. 4: Ceramics and Glasses, 1987.
 K. Rektorys, Survey of Applicable Mathematics, MIT press 1969, page 1291. This is in the Media Union Library QA 37.R373 1969.
 2. W.D. Callister, Materials Science and Engineering, An Introduction, 5th edition. John Wiley & Sons, Inc. New York, 2000.
V Activity Schedule:
Date/Time  Group 1  Group 2  Group 3  Group 4 

Day 1 first half 1:503:40 
Activity #3 IndentationToughness 
Activity #4 Residual Stress 
Activity #1 Flexural Testing 

Day 1 second half 3:405:30 
Activity #3 IndentationToughness 
Activity #4 Residual Stress 
Activity #1 Flexural Testing 
Activity #2 Fractography 
Day 2 first half 1:503:40 
Activity #4 Residual Stress 
Activity #1 Flexural Testing 
Activity #2 Fractography 
Activity #3 IndentationToughness 
Day 2 second half 3:405:30 
Activity #1 Flexural Testing 
Activity #2 Fractography 
Activity #3 IndentationToughness 
Activity #4 Residual Stres 
Day 3 first half 1:503:40 
Activity #2 Fractography 
Finish Tasks  Finish Tasks  Finish Tasks 
Day 3 second half 3:405:30 
Finish Tasks  Finish Tasks  Finish Tasks 
Finish Tasks 
VI Format and Important Questions for Lab Report:
The Discussion and Result chapters should include:
Activity #1
1) Flexural strengths of your group's asreceived rods
2) Use the combined asreceived strength data for the entire class [there should be about 120 specimens]. Examine the combined data set and infer if the various group's data can lumped together. If lumping the data is justified, fit a 2parameter Weibull distribution and calculate the Weibull modulus and characteristic strength. Can the data be fit to a 3parameter Weibull?
3) Flexural strengths of your group's abraded rods
4) Combine the flexural strengths for the abraded of the four groups for your lab day. Test the assumption that the dominent flaw size is proportional to the size of the abrasive particles[1], assuming that the strength and flaw size obey a Griffith relationship.
5) Use the combined abraded strength data for the entire class [there should be about 16 specimens for each grit size]. Examine the combined data set an infer if the various group's data can lumped together. If lumping the data is justified, fit a 2parameter Weibull distribution and calculate the Weibull modulus and characteristic strength for each grit size.
Does the abrasivesize dependence of the characteristic strength fit your expectation? Do you expect the Weibull modulus to change with grit size? Use the combined asreceived strength data for the entire class [there should be about 60 specimens]. Examine the combined data set an infer if the various group's data can lumped together. If lumping the data is justified, fit a 2parameter Weibull distribution and calculate the Weibull modulus and characteristic strength.
[1]use the average SiC particle size in microns, not grit size. The relationship between particle size and grit size is available in standard handbooks.
Activity #2
1) Accurate photograph of one specimen, identifying origin, mirror, mist, hackle, and shear lip
2) Fracture origin size plotted against strength, to test for Griffith behavior.
3) Mirror radii for each specimen, with discussion of discrepancies between yourself and other observers. Estimate the fracture mirror constant of the glass with the Mecholsky equation (eq. 5, p. 655 of Michalske handout).
Left: T.A. Michalske, “Quantitative Fracture Analysis”, Right: J.R. Varner “Descriptive Fractography” from Vol 4, Engineered Materials Handbook, ASM International (1991) (see resources)
Activity #3
1) Plot the crack length C3/2 vs. the indentation load and determine the slope with linear regression
2) Use the slope of this curve, with the Anstis et al. equation to calculate the fracture toughness of the glass.
3) Calculate a toughness estimate for every individual indent, and compare these values with the one obtained in (2)
4) Discuss the residual stress patterns seem in polarized light.