Jonathan A. Salem and Louis J. Ghosn

Glenn Research Center, Cleveland, Ohio

Back-Face Strain for Monitoring Stable Crack 

Extension in Precracked Flexure Specimens

NASA/TM—2010-216341

May 2010
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Jonathan A. Salem and Louis J. Ghosn

Glenn Research Center, Cleveland, Ohio

Back-Face Strain for Monitoring Stable Crack 

Extension in Precracked Flexure Specimens

NASA/TM—2010-216341

May 2010

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135
Acknowledgments

The authors would like to thank Ralph Pawlik for parts of the experimental work.

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Level of Review: This material has been technically reviewed by technical management. 
Back-Face Strain for Monitoring Stable Crack Extension in 

Precracked Flexure Specimens 

Jonathan A. Salem and Louis J. Ghosn 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Abstract 

Calibrations relating back-face strain to crack length in precracked flexure specimens were developed 

for different strain gage sizes. The functions were verified via experimental compliance measurements of 

notched and precracked ceramic beams. Good agreement between the functions and experiments 

occurred, and fracture toughness was calculated via several operational methods: maximum test load and 

optically measured precrack length; load at 2 percent crack extension and optical precrack length; 

maximum load and back-face strain crack length. All the methods gave vary comparable results. The 

initiation toughness, KIi , was also estimated from the initial compliance and load. 

The results demonstrate that stability of precracked ceramics specimens tested in four-point flexure is 

a common occurrence, and that methods such as remotely-monitored load-point displacement are only 

adequate for detecting stable extension of relatively deep cracks. 

1.0 Introduction 

The fracture toughness of ceramics can be measured by a variety of techniques such as the chevronnotched beam (CNB) (Refs. 1 and 2); the single-edge-precracked-beam (SEPB) (Refs. 3 and 4), or the 

surface-crack in flexure (SCF) (Ref. 5). All of these techniques have been standardized by American 

Society for Testing and Materials (ASTM) Committee C28.01 as C1421 (Refs. 6 and 7). One issue 

concerning the techniques is the crack growth stability of the test. This is a fundamental requirement for 

the chevron-notch technique, and may be required to avoid small errors in the SEPB technique (Ref. 8). 

The stability of SEPB (Ref. 9) and chevron-notch (Refs. 2 and 10) test specimens has generally been 

investigated by monitoring load-point displacement (LPD) with the remotely placed, linear variable 

differential transducer (LVDT) used to control the test system actuator or cross head. Alternatively, 

stability is presumed or the matter ignored. As the crack extensions in typical ceramic chevron-notched 

flexure specimens are large (> 0.6 mm), the use of load-point displacement is frequently adequate to 

detect stability. However, for the SEPB specimen, which is relatively unstable because of the 

continuously increasing stress intensity factor coefficient, LPD as monitored with a remotely placed 

LVDT is often inadequate for assessing stability (Ref. 2). 

Results presented in this paper demonstrate that stability is the general rule and instability the 

exception in silicon carbide, silicon nitride, and glass four-point flexure specimens. This brings into 

question conclusions (Ref. 9) regarding the stability of four-point flexure specimens and the effect on the 

measured fracture toughness of ceramics. Other work (Ref. 11) also indicates little effect of stability on 

fracture toughness measured under quasi-static loading conditions. 

Another testing issue concerning fracture toughness determination is the measurement of crack 

length, which has been identified as a major source of error in the fatigue testing of metallic materials 

(Ref. 12). In ceramic materials, crack length measurement is even more difficult for a number of reasons: 

the small specimens typically used; the semi-transparent nature of many glass-based ceramics; the lack of 

plastic deformation at the crack tip which results in poor definition of the crack front; and the high elastic 

moduli and low fracture toughnesses that results in small crack mouth opening displacements. Coarse 

grain structure, as in spinels, can also mask the crack front. Although fracture toughness test specimens 

such as the chevron-notched beam and the double-torsion (Ref. 13) allow the estimation of fracture 

NASA/TM—2010-216341 1 
toughness without crack length measurements, they are not always applicable, and other specimens such 

as the SEPB and the SCF require crack length measurement. 

A variety of optical, electrical and mechanical techniques can be used to indirectly or directly 

measure crack extension in macroscopically cracked specimens. These have been reviewed in detail for 

application to metallic testing (Ref. 14). For ceramic materials, optical techniques are limited by the small 

crack opening displacements and the surface finish. Mechanical techniques such as clip gages apply 

forces to the specimen that can result in significant stress intensities when small test specimens are used. 

A simple and sensitive technique that can be used to monitor crack length and extension in both the 

CNB and SEPB techniques is the back-face strain gage (BFSG). It has been used previously in metallic 

compact tension specimens (Refs. 14 and 15), and in fracture toughness testing of ceramic flexure 

specimens (Refs. 16 to 18). Figure 1 shows a strain gage mounted on the back-face of a flexure specimen. 

Figure 2 shows force as function of LPD and back-face strain (BFS) for a silicon nitride SEPB specimen. 

Although stable crack extension occurred, as demonstrated by the nonlinear region of the force versus 

BFS curve just before maximum load, the force versus LPD curve is completely insensitive to the 

extension. This is a result of remotely placed LVDT’s being relatively insensitive to small changes in 

specimen compliance and therefore unreliable for detecting crack extension in small precracked ceramic 

specimens. 

This paper presents compliance calibration data and new functions for calculating crack length from 

BFS as measured with different strain gage sizes. The associated errors are determined and 

recommendations given. Finally, fracture toughness of ceramics and glass is compared for different 

operational procedures. 

Si 

NASA/TM—2010-216341 2 

Figure 1.—Strain gage mounted on the back-face of a four-point flexure specimen. 

B

W 

So 
NASA/TM—2010-216341 3 

Figure 2.—Load versus load-point-displacement (A) and load 

versus back-face strain (B) for silicon nitride. 

Back-Face Strain, ε BF x 106

0 50 100 150 200 250 300

Load, 

P, N

0

50

100

150

200

250

300

350

400

Stable 

Extension 

Closure Opening of Crack 

Load Point Displacement, mm

0.00 0.02 0.04 0.06 0.08 0.10

0

50

100

150

200

250

300

350

400

(A) 

(B) 

Load, 

P, N
2.0 Analysis of Back-Face Strain 

2.1 Finite Element Analysis 

A two dimensional, plane strain finite element analysis (FEA) was performed to determine the BFS as 

a function of normalized crack length (a/W) for the four-point flexure specimen. The FEA results from 

several specimen heights were combined by writing the BFS as 

( )

EBW 2

SSP io

NBF

− = εε (1) 

where P is the applied force, Si and So are the inner and outer spans, E is the elastic modulus, B is the 

thickness, W is the height, a is the crack length, and εN is the normalized strain taken as a function of 

normalized crack length. The normalized crack length as a function of absolute normalized BFS as 

determined from the FEA is shown in Figure 3. For a/W = 0, εN = 3/2, which correspond to that of an 

uncracked beam. The BFS is more sensitive to crack length changes for normalized lengths greater than 

a/W = 0.3. 

The FEA BFS results were fit to 

( )

( ) ( ) 2 3

2

N

ln1 ln ln

ln ln ε

N N N

N

DB F

ECA

Wa

εε ε

ε

++ +

++ = (2) 

where the letters A through F are coefficients listed in Table I. Equation (2) requires εN to determine a/W. 

This can be determined from the compliance observed (εBF/P). 

TABLE I. —COEFFICIENTS FOR EQUATIONS (2) AND (3) ARE GIVEN FOR THE MAXIMUM 

(CENTERLINE) STRAIN AND THE AVERAGE STRAIN IN REGIONS ABOUT THE CENTERLINE 

a/W = f (|εN |)

Region A B C D E F 

Maximum strain –0.07481 –2.13621 0.761818 –0.85086 –1.42381 –0.03027 

± 0.5 mm average –0.0732199 –2.1294083 0.7643304 –0.8732785 –1.4396992 –0.0197908 

± 1 mm average –0.0340202 –2.2671528 0.6392699 –0.7033830 –1.3697006 –0.0253517 

|εN |= f (a/W)

Region A B C D E F 

Maximum strain 1.4953778 0.5095193 –1.1283416 –3.9701723 0.7506841 2.4590004 

± 0.5 mm average 1.4999604 0.4572423 –1.2038010 –3.8075832 0.8494310 2.3526028 

± 1 mm average 1.5000003 0.0583885 –1.5493757 –2.7914306 1.2748560 1.7382084 

For estimation of the normalized back-face strain produced with a given crack length, the results were 

fit to 

( ) ( )

( ) ( ) ( ) 5.0 5.1

5.0

1 WaFWaDWaB

WaEWaCA

N

+ ++

+ +

ε = (3) 

where the coefficients are listed in Table 1. Equations (2) and (3) provide values within 1 percent for 

a/W > 0.15. 

NASA/TM—2010-216341 4 
Normalized Back-Face Strain, εN

0 10 20 30 40 50

Normalized Crack length, a/W

0.0

0.2

0.4

0.6

0.8

1.0

Peak Strain

1 mm Gage Average Strain

2 mm Gage Average Strain

Figure 3.—Normalized crack length as a function of the absolute value 

of normalized back-face strain. 

2.2 Sensitivity to Specimen Geometry 

The mode I stress intensity function for a beam with a straight through crack loaded in four-point 

flexure is 

( ) ( WaF

BW

SSP K io

I 2/3

− = ) (4) 

where F(a/W) is the stress intensity factor coefficient (Ref. 19). By combining Equations (1) and (4), εBF

can be written as a function of 1/√W for a fixed crack length, elastic modulus and stress intensity factor or 

fracture toughness: 

( ) WaF W

K

E

IN

BF

ε 1

ε = . (5) 

Figure 4 shows BFS as a function of normalized crack length for typical properties of silicon nitrides: 

E = 300 GPa and KI = 5 MPa√m. Reducing the specimen height, W, by 50 percent increases the absolute 

back-face strain by ~ 40 percent for a/W = 0.3. For a constant stress intensity, a minimum in back-face 

strain occurs around a/W = 0.25. 

NASA/TM—2010-216341 5 
Normalized Crack Length, a/W

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Absolute Back-Face Strain,

ε

BF

 x 10

6

200

300

400

500

600

700

800

W = 3 mm

W = 6 mm

E = 300 GPa

KI = 5 MPam0.5

W = 4 mm 

Peak Strain

Figure 4.—Absolute back-face strain as a function of normalized 

crack length for an applied stress intensity factor of 5 MPa√m 

and an elastic modulus of 300 GPa. 

3.0 Errors Associated With BFS 

3.1 Uniformity of the Strain Field 

The FEA results were also used to investigate the uniformity of the strain field along the back-face. 

Figure 5 shows the strain distribution for various normalized crack lengths. For normalized crack lengths 

less than approximately 0.6, the strain distribution is relatively uniform over a ± 0.5 mm region about the 

centerline, corresponding to about 25 percent of W. Although strain gages with grid lengths as small as 

0.2 mm are readily available, for deep cracks and larger strain gages, an alternative approach is to derive a 

BFS calibration from the average FEA strain along a specified length, as done by a strain gage. Table 1 

gives coefficients for 1 and 2 mm gage lengths (±0.5 and ±1 mm regions about the specimen centerline). 

3.2 Contact Strains 

Another error associated with the uniformity of the strain field is that due to the contact stresses 

generated at the loading points. The effect of contact loading on the back-face strain can be estimated 

from the derivation of von Karman and Seewald (Ref. 20) which is summarized in Reference 21. The 

contact stress and strain generated in the region between the roller and the crack along the back-face are 

( )

EBW

Pc x

2 ' 12 νβ ε − = BW

Pc x

β σ2 ' = and (6) 

where Pc is the contact force and β is a function of the position and given in Reference 21. The contact 

strain as a fraction of the BFS generated solely from the crack presence is 

( )

N i

BFx

S

W

ε

νβ εε

2 ' 1− = . (7) 

NASA/TM—2010-216341 6 
Distance from Crack Plane, mm

0.0 0.2 0.4 0.6 0.8 1.0

Normalized Back-Face Strain,

εΝ

-30

-25

-20

-15

-10

-5

0

0.2

0.3

0.4

0.5

0.6

0.7

0.8

a/W

Figure 5.—Variation in strain across the back-face for various 

normalized crack lengths. 

Figure 6 shows the strain ratio as a function of the normalized distance between the crack plane and 

the roller along the back-face. For normalized crack lengths and span to depth ratios of a/W ≥ 0.3 and 

Si /W ≥ 2.5, the contact strain is less than 1 percent of the back-face strain given by Equation (1) for a 

± 0.2Si region about the crack plane, or within ± 2 mm for W = 4 mm, implying that the contact strains are 

very localized and of little consequence for typical gages and spans. 

3.3 Sensitivity to Strain Gage Errors 

The error associated with the estimated crack length as a function of the actual crack length is shown 

in Figure 7 for E = 300 GPa, KI = 5 MPa√m and the assumption that a strain gage and conditioner are 

accurate to ± 5 microstrain. For normalized crack lengths ≥ 0.3, the error is less than ± 2.5 percent for 

W ≤ 6 mm. The sensitivity of a circuit using a BFSG can be improved by placing multiple gages on the 

back-face when the specimen size permits, and by using large, high resistance gages. Errors can be 

minimized by good gage alignment. 

4.0 Experimental Verification 

In order to experimentally verify the FEA BFS calibration, notches were diamond ground into silicon 

nitride flexure test specimens. Two notch configurations were used to generate BFS data as a function of 

force: (1) square tip with a 0.76 mm long uniform section preceded by a 0.76 mm long, 60° mouth, and 

(2) a V-grooved tip (Ref. 22) preceded by 0.76 mm straight and 60° sections, shown in Figure 8. The 

notches had an average width of 0.18 mm in the uniform section and the V-grooves had an average radius 

of 0.036 mm. Although such notches are too blunt for fracture toughness measurements, they were 

thought adequate for compliance measurements. The specimens were loaded between 20 and 40 mm 

spans at a rate of 20 N/s. 

NASA/TM—2010-216341 7 
E = 300 GPa

Load

Point

Crack

Plane

5.0 0.6

2x/Si

0.2 0.4 0.6 0.8 1.0

Contact Strain / Back-Face Strain (%)

0

2

4

6

8

10

Si /W a/W

 2.5 0.3

 2.5 0.6

5.0 0.3

Figure 6.—Contact strain normalized to back-face strain as a 

function of normalized distance from the load-point. 

Normalized Crack Length, a/W

Error in Normalized Crack Length, 

0.2 0.3 0.4 0.5 0.6 0.7 0.8

a/W (%)

-8

-6

-4

-2

0

2

4

6

8

E = 300 GPa

KI = 5 MPam0.5

3 3

4 4

6 6 

-5με 

W, mm

+5με

W, mm 

Figure 7.—Error in estimated crack length for a gage accuracy 

of ± 5 microstrain with an applied stress intensity factor of 5 

MPa√m and an elastic modulus of 300 GPa. 

NASA/TM—2010-216341 8 
0.18 mm

0.04 mm R Typ.

Figure 8.—Notch configuration used to verify FEA. 

The resulting, normalized crack lengths predicted from the slope of the force versus BFS curves and 

Equations (1) and (2) are plotted as a function of the optically measured, normalized notch depth, l/W, in 

Figure 9. Overall agreement was good for a/W ≥ 0.4, with a maximum error of 2.9 percent for the square 

notched specimens and 0.2 percent for the V-notched specimens. However, for shorter a/W the BFS data 

tended to overestimate the notch length, and a slight bias in the data is apparent as the BFS estimated 

lengths are usually longer than the optically measured values by an average of 0.035 mm for the square 

notches and 0.012 mm for the V-notches. In order to determine if this was an artifact of the relatively 

blunt notch or machining cracks at the notch root, specimens were heat treated (Ref. 23) and the notch 

modeled with FEA. Annealing did not improve the results; however, modeling of the notch root shape 

produced good correlation between analysis and experiments. 

Three of the saw-notched specimens made from the GPS silicon nitride were loaded to failure at 

0.5 mm/min. No non-linear behavior was exhibited prior to catastrophic failure, indicating good bonding 

of the strain gages and no stable crack extension for the notched specimens. 

5.0 Experiments With Sharp Cracks 

5.1 Materials 

1 The test materials (Table II) were a hot isostatically pressed (HIPed) silicon nitride exhibiting a flat 

crack growth resistance curve (R-curve) (Ref. 24), a hot pressed (HP) in situ toughened silicon nitride2

 

1

GN 10, Allied Signal Ceramic Components, Torrance, California. 

2

NKK Corp., Tokyo, Japan 

NASA/TM—2010-216341 9 
3 exhibiting a rising R-curve (Ref. 25), a gas pressure sintered (GPS) in situ toughened silicon nitride

exhibiting a rising R-curve (Ref. 26), a composite silicon nitride4

 with 20 vol. % SiC whiskers, sintered 

silicon carbide5

, and soda-lime glass. 

TABLE II.—PROPERTIES OF THE TEST MATERIALS 

ElasticA modulus HardnessB Number of 

Material GPa GPa fracture tests 

Soda Lime Glass 70.6 6.3 ± 0.5 4 

Alpha Silicon Carbide 397 ------------ 5 

HIPed Silicon Nitride (GN-10) 285 14.4 ± 0.1 6 

Composite Silicon Nitride/20% SiC 298 17.4 3 

GPS Silicon Nitride (AS-800) 299 14.3 ± 0.7 5 

HP Silicon Nitride (NKK) 298 14.5 ± 0.2 4 

A. By strain gaging a four-point flexure specimen. 

B. With a Vickers microhardness indentor. 

Normalized Saw Notch Length, l/W

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Normalized BFS Crack Length, a/W

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

SLOPE = 1

Kryptonite: Square Notch, Annealed 

Ideal line 

Kryptonite: Square Notch

GN10: Sharpened Notch, Notch FEA 

Kryptonite: V-Notch

Figure 9.—Normalized notch length determined from optical 

measurements and back-face strain measurements. 

 

3

AS-800, Allied-Signal Ceramic Components, Torrance, California. 

4

Kryptonite, Japan Metals and Chemicals Co., Tokyo, Japan. 

5

Hexoloy SA Sintered alpha α-SiC, St. Gobain (Carborundum), Niagara Falls, New York.

NASA/TM—2010-216341 10 
NASA/TM—2010-216341 11 

 

5.2 Test Geometries 

The HIPed and HP silicon nitride specimens measured 3 by 6 mm and 3 by 4 mm in thickness and 

height, respectively, and were tested between 10 and 20 mm spans at a rate of 0.05 mm/min. The GPS 

and composite silicon nitride specimens measured 3 by 4 mm in thickness and height, and were tested 

between 20 and 40 mm spans at a rate of 0.5 mm/min. The silicon carbide specimens measured 3 by 4 

mm in thickness and height, and were tested between 10 and 20 mm spans at a rate 0.01 mm/min. The 

glass specimens measured 5 by 8 mm in thickness and height and were tested between 20 and 40 mm 

spans at a rate of 0.5 mm/min. Test system stiffness ranged from 3,000 to 10,000 N/mm. 

5.3 Precracking and Loading Mode 

The specimens were precracked using the bridge indentation technique (Refs. 3 and 4). After 

precracking, a uniaxial strain gage (typically less than 1 mm gage length) was centered on the back-face 

opposite the precrack mouth. The precrack was then marked by applying a drop of oil based dye 

penetrant6

 as described in the next section. 

Specimens were loaded with one of the following modes: monotonic loading in displacement control 

until final fracture occurred, or monotonic loading until an increment of crack extension occurred, 

followed by unloading and reloading until another increment of crack extension occurred. This cycle was 

repeated several times until catastrophic failure. After failure, the crack length, as indicated by the dye 

penetrant, was immediately measured in an optical microscope at approximately 30x magnification. The 

specimens were then washed with acetone and the physical crack lengths optically measured. 

5.4 Crack Marking 

The precracks were marked by applying a drop of dye penetrant6 to the crack mouth and allowing it 

to penetrate for a minimum of several minutes. Prior to testing, excess penetrant was swabbed from the 

surfaces with cotton and a solvent. 

Penetrants are designed to infiltrate the crack and be wiped from the surface while retaining the 

ability to bleed out of a defect upon application of a developer. As a result, they tend to remain fluid and 

to creep or spread in the presence of a developer, making them a less than ideal agent for marking the 

length of a crack that is to be broken open. Thus, any crack length measurements must be made 

immediately after specimen failure and viewed with some suspicion. The tendency of a penetrant to 

spread away from the defect can be reduced by thinning with solvent (Ref. 27). 

Cracks in ceramics are very narrow and the degree of penetration is an issue. Studies on glass slide 

plates indicate that most penetrants and water will fill gaps less than 0.3 μm (estimates are as low as 0.1 μm) 

(Ref. 27). The results presented in the results section indicate that penetration is typically sufficient. 

Ideally, a penetrant that will penetrate very tight cracks, dry without adhering the crack faces, and 

leave an indication is needed. The penetrant used in this study can be dried by baking in an oven at about 

160 °C, however, a sticky residue that bonds the crack faces together results. If higher temperatures are 

used, the residue is hardly visible. An alternative is to use a lead acetate solution and a wetting agent. The 

disadvantage is that the dried lead acetate is not readily visible with optical methods, and thus scanning 

electron microscopy with back scattered electron imaging is required. 

5.5 Corrosive Effects of Penetrants 

The fatigue crack growth rates and threshold stress intensity factors of ceramics and glasses are 

affected by moist or humid environments. Thus, any penetrant used to mark the crack may enhance or 

retard such stress corrosion crack growth and thereby affect the properties being measured. Many 

6

DP-40, Sherwin Inc., South Gate, California.
NASA/TM—2010-216341 12 

penetrants, though oil based, contains emulsifiers that may form an alkaline solution if water gets into the 

container (Ref. 27). Such a water based alkaline solution could enhance corrosion of glasses and some 

ceramics. 

The corrosive properties of the penetrant used in this study were investigated by running constant 

stress rate tests on glass specimens that were precracked by using a Vickers indenter with a 49 N load. 

After indenting, the specimens were dried for various times by placing them in a desiccator attached to a 

vacuum pump. The specimens were then removed, marked with the penetrant or silicon oil, and loaded at 

displacement rates that resulted in stressing rates from 10 to 1000 MPa/min. 

The resulting failure strengths are plotted in Figure 10. For specimens that are sufficiently dried (≥3.5 

hr), the silicon oil (which acts as an inert environment) and the penetrant result in similar strengths and a 

large crack growth parameter (n = 80). However, for specimens dried less than 3.5 hr, a loss in strength 

(18 percent) and a crack growth parameter typical of glass (n = 22) is exhibited. This result is in 

agreement with crack growth parameters measured statically and dynamically with indentation flaws in 

water (n = 18 to 22 (Refs. 28 and 29)), indicating that the penetrant does not affect slow crack growth 

behavior unless significant vacuum drying is performed. 

6.0 Results 

6.1 Capabilities of Penetrant 

Figure 11 shows crack lengths determined from the same specimens with penetrant and after washing 

the penetrant off. Agreement was excellent except when the crack front could not be readily delineated 

and measured at three points along the crack front: the mid-thickness and the two quarter-thickness 

points. Such results are plotted with zero values on the abscissa. Measurements without penetrant were 

particularly difficult for the in situ toughened materials because of the coarse microstructure. 

It was noted during the optical precrack measurements that when the precrack segment and the fast 

fracture segment were well aligned, the crack could not be detected optically without using penetrant, as 

shown in Figure 12(A) and (B). This occurred in about 25 percent of the specimens. For such specimens, the 

crack length as measured with the penetrant was used for any further calculations requiring a crack length. 

Evidently, optical measurements without penetrant detect the line of intersection formed by the 

precrack and fracture planes, as shown in Figure 12(C) and (D). The misalignment is the result of the 

precrack tip turning out of plane or being “hooked.” 

The specimen showing the worst agreement was a HIPed silicon nitride specimen cycled multiple 

times before failure occurred. The BFS indicated crack growth of 0.34 mm prior to failure, corresponding 

to three-quarters of the discrepancy shown in Figure 11. Evidently the penetrant, to a large extent, follows 

the crack when it is extended by cycling. 

Crack lengths determined without penetrant measure the initial, physical length of a crack that is 

frequently out of plane with the final fracture plane. Thus both the optical and penetrant measurements 

are good approximations of the initial, physical crack length for monotonic loading if little crack 

extension occurs. However, the penetrant provides a better, though possibly inadequate estimate of the 

final, physical crack length when the crack is cycled. 

6.2 Crack Length Estimation From BFS 

Silicon nitride specimens exhibited three distinct regions: a short steep region; a large semi-linear 

region; and a short, highly nonlinear region (Fig. 2). Glass specimens exhibited only the second two 

regions and the saw notched specimens exhibited only a single linear region. 

The initial region represents the effective stress intensity factor or load to begin opening the precrack 

and overcome the “closure” caused by wedging of grains and asperities in the precrack wake. The middle 

region represents the compliance of the precrack and its slope was used to estimate the precrack lengths. 

The third region represents stable crack extension. 
Stressing Rate, MPa/min

1 10 100 1000 10000

Failure Stress, MPa

20

30

40

50

60

70

80

2 Hr in Vacuum & Penetrant

3.5 Hr in Vacuum & Penetrant

3.5 Hr in Vacuum & Silicon Oil

n = 80

n = 22= 2 2

Figure 10.—Fatigue strength of glass slide plates subjected to 

vacuum drying and marking with penetrant or silicon oil. 

a/W With Penetrant

0.0 0.2 0.4 0.6 0.8 1.0

a/W Without Penetrant

0.0

0.2

0.4

0.6

0.8

1.0

NKK, Cycled

GN10, Cycled

AS800

NKK

GN 10

KRYPTONITE

Slope = 1

SiC

Figure 11.—Normalized crack length measured with and 

without penetrant. 

NASA/TM—2010-216341 13 
(A) (B) 

(C) (D) 

Growth 

Direction 

Figure 12.—Effect of precrack shape on optical visibility: (A) and (B) are 

side and top views of a coplanar precrack and failure plane; (C) and (D) 

are views of a precrack with a curved tip. 

Figure 13 compares the precrack length measured optically (generally without penetrant) with 

estimates made from the slope of the second region in the load-BFS curves. Excellent agreement is 

indicated for the HIPed material, which has a fine grain size, and the glass, which is amorphous. For the 

coarse grained, in situ toughened materials, the BFS slightly underestimates the optically measured 

precrack length. This is probably due to grain bridging in the crack wake that results in crack growth 

resistance and the appearance of a shorter crack. 

In order to determine final compliance and the amount of stable crack extension occurring in the tests, 

a compliance line was drawn on the load-BFS curve from the final fracture point to the zero load point of 

the initial compliance line. Figure 14 compares the precrack length measured optically with estimates 

made from the load-BFS curves. All but two SiC test specimens exhibited measurable crack extension. 

On average, the silicon nitride materials extended 5 percent while the SiC extended 2.3 percent. 

Specimens subjected to several load-unload cycles exhibited substantial stable crack extension (> 0.20 

mm), and the corresponding crack growth resistance curves based on BFS from a single specimen are 

shown in Figure 15. 

NASA/TM—2010-216341 14 
Slope = 1

Normalized Precrack Length, a/W

0.0 0.2 0.4 0.6 0.8 1.0

a/W via Back-Face Strain

0.0

0.2

0.4

0.6

0.8

1.0

SiC

NKK 

AS800 

KRYPTONITE

GN10 

GLASS

Figure 13.—Comparison of precrack lengths determined 

by optical measurements and by the initial compliance 

as measured with back-face strain. 

Slope = 1

Normalized Precrack Length, a/W

0.0 0.2 0.4 0.6 0.8 1.0

a/W via Back-Face Strain

0.0

0.2

0.4

0.6

0.8

1.0

SiC

NKK Cycled

GN10 Cycled

NKK 

AS800 

KRYPTONITE

GN10 

GLASS

Figure 14.—Comparison of precrack lengths determined by 

optical measurements and by the final compliance as 

measured with back-face strain. Arrows show extension due 

to cycling. 

NASA/TM—2010-216341 15 
Crack Extension Δa, mm

0.0 0.1 0.2 0.3 0.4

2

4

6

8

10

12

2

8

10

12

NASA/TM—2010-216341 16 Fracture Resistance, 

KIR , MPa m0.5

4

6

NKK

GN-10

Figure 15.—Crack growth resistance curves determined from 

BFS measurements. 

6.3 Fracture Toughness Estimation 

Fracture toughness can be calculated by a variety of methods: from the maximum load and optically 

measured precrack length; from the load at the onset of crack extension and the corresponding precrack 

length as measured with BFS; or from the maximum load and the corresponding crack length as estimated 

with BFS. Typically the first of these techniques has been used (Refs. 6 and 7) for very brittle failure in 

which little or no crack extension occurs. The second technique is complicated because the onset of crack 

extension and associated load are difficult to identify. However, this complication can be avoided by 

applying the technique used in ASTM E399 (Ref. 30). A slope that corresponds to a fixed crack extension 

(e.g., ~ 2 percent) is drawn on the loading diagram and the load corresponding to the intersection used to 

calculate the fracture toughness. Values calculated by various methods are listed in Table III, and the first 

and second methods are plotted in Figure 16. Only data from monotonically loaded specimens were used. 

Fracture toughnesses calculated with the maximum load and optical precrack length agree reasonably 

with those calculated from the load for 2 percent crack extension, PQ, and the optical precrack length. 

This occurs because the crack extension is small and occurs under nearly constant load. 

TABLE III.—STRESS INTENSITY FACTORS (MPa√M) CALCULATED WITH VARIOUS OPERATIONAL METHODS 

KIpb KIQ KIBFS KIi KIR

Load Maximum PQ Maximum Initiation Final 

Crack length Optical precrack Optical precrack BFS precrack BFS precrack BFS final crack 

Glass 0.70 ± 0.01 0.70 ± 0.01 0.71 ± 0.03 0.61 ± 0.03 0.72 ± 0.04 

Alpha SiC 2.59 ± 0.15 2.57 ± 0.19 2.53 ± 0.10 2.40 ± 0.11 2.53 ± 0.10 

HIPed Si 5.42 ± 0.17 3N4 5.38 ± 0.20 5.53 ± 0.19 5.20 ± 0.15 5.81 ± 0.11 

Si3N4/SiC 6.62 ± 0.17 6.56 ± 0.16 6.53 ± 0.10 5.92 ± 0.33 6.54 ± 0.10 

GPS Si3N4 7.68 ± 0.15 7.52 ± 7.58 ± 0.35 7.18 ± 0.40 7.91 ± 0.13 0.19 

HP Si3N4 10.3 ± 0.6 10.2 ± 9.45 ± 0.18 8.62 ± 0.24 10.3 ± 0.3 0.6 
Slope = 1

Fracture Toughness, KI P2% , MPa m

0 2 4 6 8 10 12

Fracture Toughness, KI Pmax , MPa.

m0.5

0

2

4

6

8

10

KRYPTONITE

AS800 

NKK 

GN10 

GLASS

SiC

 Figure 16.—Comparison of the fracture toughnesses 

determined with maximum load and the load for 2 percent 

crack extension. 

6.4 Effects of Crack Configuration on Stable Extension 

SEPB precracks exhibit varying degrees of crack front straightness, and the cracks frequently curve 

out of the initial crack plane near the crack tip. The amount of stable crack extension exhibited in 

ceramics may be related not only to the materials crack growth resistance and sensitivity to stress 

corrosion, but also to the degree of crack tip hook and crack front straightness. 

The effect of crack front straightness on the degree of crack extension is shown in Figure 17. No 

particular trend is exhibited. The straightness was defined as the maximum of the differences between the 

crack lengths measured at the center and the two mid-centers of the specimen thickness. 

The effect of precrack length on the stable crack extension is shown in Figure 18. Again, no trend can 

be discerned. 

The effect of out-of-plane curvature on the fracture toughness of the composite material, as calculated 

with the maximum load and optically measured crack length, is shown in Figure 19. No significant effect 

is exhibited. The hook angle was defined as the angle between lines drawn tangent to the initial crack 

plane and the crack tip. 

As mentioned in the previous section, two SiC test specimens did not exhibit stable extension, but 

crack jumps followed by unstable failure. The cracks in these specimens were noted to be relatively 

twisted, with fracture toughness of 2.73 as compared to 2.59 MPa√m for ideal cracks. 

A more systematic study with a wider parameter range may be needed to distinguish the factors 

controlling the amount of stable extension. 

NASA/TM—2010-216341 17 
Crack Front Straightness, %

0 4 8 12 16 20

Normalized Crack Extension, Δa/W , %

0

2

4

6

8

10

12

NKK 

GN10

KRYPTONITE

AS800

Glass 

 Figure 17.—Normalized crack extension as a function of 

crack front straightness. 

Normalized Precrack Length, a/W

0.0 0.2 0.4 0.6 0.8 1.0

Normalized Crack Extension, Δ a/W , %

0

2

4

6

8

10

12

NKK 

GLASS

GN10 

KRYPTONITE

AS800 

Figure 18.—Normalized crack extension as a function of 

normalized precrack length. 

NASA/TM—2010-216341 18 
Kryptonite

Maximum Hook Angle, degrees

0 10 20 30 40

Fracture Toughness, MPa.

m0.5

6.0

6.2

6.4

6.6

6.8

7.0

Figure 19.—Fracture toughness as function of crack hook angle. 

7.0 Discussion 

Most noteworthy of the results is that crack stability during fracture of ceramics is the rule rather than 

the exception, regardless of the test material or crack length. However, adequate detection techniques 

must be used as demonstrated by Figure 2, otherwise stable tests may be interpreted as unstable. Further, 

use of a very large load cell in order to promote system stiffness and stability, as done in Reference 9, is 

ill advised as the precision is diminished. The tests in this work used 500 to 2000 N load cells. 

The apparent lack of stability reported in Reference 9 is the result of using a remotely placed LVDT 

to detect compliance changes. For long cracks the compliance and its change are large and can marginally 

be detected; however, for short cracks, the changes are too small for detection via the LVDT. 

The lower fracture toughness observed in Reference 9 for specimens that had long cracks and 

appeared to be stable is probably a result of the crack extension changing the actual SIFC (stress intensity 

factor coefficient) in long cracks more than for short cracks, and the use of the SIFC corresponding to the 

precrack. For long cracks the SIFC increases more rapidly than for short cracks. Thus for any crack 

extension the error between the real SIFC and that associated with the precrack length is exacerbated as 

the crack length becomes greater. Another other source of error associated with crack stability is that 

resulting from poor crack configuration. For example, twisted cracks exhibit little or no stable crack 

extension. The instability probably results from the divergent paths along the crack front causing pinning 

so that a larger load, which is usually unstable, is required for propagation. This led to larger apparent 

fracture toughness measurements in Reference 2. 

Fracture toughness measurements made with maximum load and the precrack length are comparable 

to those made with 2 percent extension for the materials tested. Initiation stress intensities and maximum 

resistive intensities are significantly different. Crack closure was exhibited by the materials with coarse 

microstructures, and thus the initiation toughness values are likely higher than the initiation toughness 

associated with a traction-free crack or possibly a fatigue crack. Fatiguing of precracked specimens or use 

of the SEVNB specimen may provide better measurements of initiation values. 

NASA/TM—2010-216341 19 
NASA/TM—2010-216341 20 

8.0 Summary 

The back-face strain gage is a simple, inexpensive method for monitoring crack extension in ceramic 

flexure specimens during fracture toughness testing, and indicates that four-point flexure is adequately 

stable for accurate fracture testing of ceramics. It is recommended that the crack length be between 

0.3 ≤ a/W ≤0.6 with 1 ≤ W/B ≤ 2 and Si/W ≥ 2.5. For small flexure specimen, strain gages should have a 

gage length less 1 mm. 

All silicon nitride specimens tested exhibited stable extension prior to fast fracture, despite the use of 

four-point flexure. Unstable extension occurred only for two silicon carbide specimens that contained 

twisted cracks. An average stable crack extension of 5 percent, as measured with the BFSG, occurred in 

the silicon nitrides tested. For silicon carbide an average extension of 2.3 percent occurred. Precrack 

lengths are difficult to optically measure when the precrack plane and the final fracture plane are well 

aligned. This occurred in approximately 25 percent of the specimens tested. Also, the apparent precrack 

lengths estimated from force versus BFS data reasonably estimate the physical precrack length for 

materials without a microstructure (e.g., glass) or with a fine microstructure. For in situ toughened 

materials, force versus BFS data slightly underestimated the physical precrack length. Common 

operational procedures gave similar estimates of fracture toughness. 

References 

1. D. Munz, R. T. Bubsey, and J.L. Shannon, Jr., “Fracture Toughness Determination of Al2O3 using 

Four-Point-Bend Specimens with Straight-Through and Chevron Notches,” J. Am. Ceram. Soc., 63 

[5–6], 300–305 (1983). 

2. J.A. Salem, L. Ghosn and M.G. Jenkins “Stress Intensity Factor Coefficients for Chevron-Notched 

Flexure Specimens and a Comparison Fracture Toughness Methods,” Ceram. Eng. Sci. Proc. 20 [3], 

503–512 (1999). 

3. T. Nose and T. Fujii, “Evaluation of Fracture Toughness for Ceramic Materials by a Single-EdgePrecracked-Beam Method,” J. Am. Ceram. Soc., 71 [5], 328–33 (1988). 

4. R. Warren and B. Johannesson, “Creation of Stable Cracks in Hardmetals Using ‘Bridge Indentation’, 

Powder Metallurgy, 27 [1], 25–29 (1984). 

5. J.J. Petrovic and M.G. Mendiratta, “Fracture from Controlled Surface Flaws,” in Fracture Mechanics 

Applied to Brittle Materials, ASTM STP 678, S.W. Freiman ed., 83–102, 1979. 

6. J.A. Salem, G.D. Quinn, M.G. Jenkins, “Measuring the Real Fracture Toughness of Ceramics: ASTM 

C1421,” pp. 531–553 in Fracture Mechanics of Ceramics: Active Materials, Nanoscale Materials, 

Composites, Glass, and Fundamentals, R.C. Bradt, D. Munz, M. Sakai and K. White, eds., Springer, 

2005. 

7. “Standard Test Methods for Fracture Toughness of Advanced Ceramics,” Test Method C1421, Annual 

Book of ASTM Standards, Vol. 15.01, West Conshohocken, PA, 2000. 

8. F.I. Baratta and W.I. Dunlay, “Crack Stability in Simply Supported Four-Point and Three-Point Loaded 

Beams of Brittle Materials,” Mechanics of Materials, 10, 149–159 (1990). 

9. I. Bar-on, F.I. Baratta, and K. Cho, “Crack Stability and its Effect on Fracture Toughness of Hot Pressed 

Silicon Nitride Beam Specimens,” J. Am. Ceram. Soc., 79 [9], 2300–2308 (1996). 

10. M. Mizuno and H. Okuda, VAMAS Round Robin on Fracture Toughness of Silicon Nitride at High 

Temperature, Technical Report No. 16, Japan Fine Ceramics Center, Nagoya, Japan, December 1993. 

11. I. Bar-on and K. Cho,” Crack Stability Analysis and Fracture Toughness of Ceramic Bend Bars with a 

Modified Circular Cross Section,” Experimental Mechanics Vol. 36 104–111, June (1995). 

12. W.G. Clark and S.J. Hudak, “Variability in Fatigue Crack Growth Rate Testing,” J. Testing and 

Evaluation, 3 [6], 454–476, (1975). 

13. J.A. Salem, M. Radovic, E. Lara-Curzio, G. Nelson, “Fracture Toughness of Thin Plates By the 

Double-Torsion Test Method,” pp. 63–73 in Proceedings of the 28th International Cocoa Beach 
NASA/TM—2010-216341 21 

Conference on Advanced Ceramics and Composites: B, Ceramic Engineering and Science 

Proceedings, Vol. 25, No. 4, Waltraud M. Kriven and H.T. Lin, eds. (January 2006). 

14. W.F. Deans and C.E. Richards, “A Simple and Sensitive Method of Monitoring Crack Extension and 

Load in Compact Fracture Mechanics Specimens Using Strain Gages,” J. Testing and Evaluation, 7 [3], 

147–154 (1979). 

15. W.T. Riddell and R.S. Piascik, “A Back Face Strain Compliance Expression for the Compact Tension 

Specimen,” NASA/TM—1988-208453 (1988). 

16. J.A. Salem, L.J. Ghosn, and M. G. Jenkins “A Strain Gage Technique to Measure Stable Crack 

Extension in Ceramics,” pp. 1–8 in the Proc. of the Soc. for Experimental Mechanics Spring 

Conference, June 2–4, 1997, Bellevue, WA. 

17. J.A. Salem, L. Ghosn and M.G. Jenkins “Back-Face Strain as a Method for Monitoring Stable Crack 

Extension,” Ceram. Eng. Sci. Proc. 9 [3], 587–594 (1998). 

18. T. Fett, D. Munz, and G. Thun, “Method to Estimate Crack-Tip Toughness from Bending Tests on 

Prenotched Bars,” J. Am. Ceram. Soc. 83, 421–423 (2000) 

19. J.E. Srawley and B. Gross, in Cracks in Fracture, ASTM STP No. 601, American Society for Testing 

and Materials, West Conshohocken, PA., 1976. 

20. T. von Karman and F. Seewald, Abh. Aerodynam. Inst., Tech. Hochs, (Aachen), 7, 3, 1927. 

21. S. Timoshenko and J.N. Goodier, Theory of Elasticity, 2nd ed. McGraw-Hill, N.Y., 117–119, 1951. 

22. J. Kubler, “Fracture Toughness of ceramics Using the SEVNB Method: Preliminary Results,” Ceram. 

Eng. Sci. Proc., 18 [4], 155–162 (1997). 

23. J.A. Salem, N.N. Nemeth, L.M. Powers, and S.R. Choi, “Reliability Analysis of Uniaxially Ground 

Brittle Materials,” J. Eng. for Gas Turbines and Power, Vol. 118, 863–871 (1996). 

24. J.A. Salem, S.R. Choi, W.A. Sanders, D. Fox, “Elevated Temperature Mechanical Behavior of 

Monolithic and SiC Whisker Reinforced Silicon Nitrides, NASA Technical Memorandum 105245, 

1991. 

25. S.R. Choi and J.A. Salem, “Crack-Growth Resistance of in Situ-Toughened Silicon Nitride,” J. Am. 

Ceram. Soc., 77 [4] 1042–46 (1994). 

26. C.W. Li, and J. Yamanis, “Super-Tough Silicon Nitride with R-Curve Behavior,” Ceram. Eng. Sci. 

Proc., 10, 632–645 (1988). 

27. C. Betz, Principals of Penetrants, Magnaflux Corp., Chicago, Ill., 1969. 

28. E.R. Fuller and B.R. Lawn, “Theory for Brittle Flaws Originating from Residual Stress 

Concentrations,” J. Am. Ceram. Soc. 66 [5] 314–321 (1983). 

29. J.A. Salem and R. Tandon, “Test Method Variability in Slow Crack Growth Properties of Sealing 

Glasses,” International Journal of Fatigue, 32, 557–564, 2010. 

30. Standard Test Method for Plane-Strain Fracture Toughness of Metallic Materials.” Test Method E399, 

Annual Book of ASTM Standards, Vol. 03.01, ASTM, West Conshohocken, PA, 1990. 
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Back-Face Strain for Monitoring Stable Crack Extension in Precracked Flexure Specimens 

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Salem, Jonathan, A.; Ghosn, Louis, J. 

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Submitted to the Journal of the American Ceramic Society. 

14. ABSTRACT

Calibrations relating back-face strain to crack length in precracked flexure specimens were developed for different strain gage sizes. The 

functions were verified via experimental compliance measurements of notched and precracked ceramic beams. Good agreement between the 

functions and experiments occurred, and fracture toughness was calculated via several operational methods: maximum test load and 

optically measured precrack length; load at 2 percent crack extension and optical precrack length; maximum load and back-face strain crack 

length. All the methods gave vary comparable results. The initiation toughness, KIi , was also estimated from the initial compliance and 

load.The results demonstrate that stability of precracked ceramics specimens tested in four-point flexure is a common occurrence, and that 

methods such as remotely-monitored load-point displacement are only adequate for detecting stable extension of relatively deep cracks. 

15. SUBJECT TERMS

Fracture mechanics; Strain; Crack length; Stiffness; Stability; Silicon nitrides; Ceramics 

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