Jonathan A. Salem and Louis J. Ghosn

Glenn Research Center, Cleveland, Ohio

Measuring Crack Length in Coarse Grain Ceramics

NASA/TM—2010-216242

March 2010
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Jonathan A. Salem and Louis J. Ghosn

Glenn Research Center, Cleveland, Ohio

Measuring Crack Length in Coarse Grain Ceramics

NASA/TM—2010-216242

March 2010

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135
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Level of Review: This material has been technically reviewed by technical management. 
NASA/TM—2010-216242 1 

Measuring Crack Length in Coarse Grain Ceramics 

Jonathan A. Salem and Louis J. Ghosn 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Abstract 

Due to a coarse grain structure, crack lengths in precracked spinel specimens could not be measured 

optically, so the crack lengths and fracture toughness were estimated by strain gage measurements. An 

expression was developed via finite element analysis to correlate the measured strain with crack length in 

four-point flexure. The fracture toughness estimated by the strain gaged samples and another standardized 

method were in agreement. 

Introduction 

Three techniques for measuring the fracture toughness of ceramics were standardized by the 

American Society for Testing and Materials (ASTM) (Refs. 1 to 3): the chevron-notched beam (VB), the 

single edged precracked beam (SEPB), and the surface crack in flexure. One issue concerning the 

techniques is the measurement of crack length, which has been identified as a major source of error in 

fatigue and fracture testing of metallic materials (Ref. 4). In ceramics, crack length measurement can be 

particularly difficult for a number of reasons: the small specimens typically used; the semi-opaque 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 high elastic modulus and low fracture toughness that result in small crack opening 

displacements. Ceramics with coarse grain structure, such as ZnSe, alumina and MgAlO4 (spinel), present 

an additional difficulty because the crack front is difficult to delineate from the grain boundaries. 

A simple and sensitive technique that can be used to estimate crack length and extension in SEPB 

specimens is the back-face strain gage (BFSG), as illustrated in Figure 1. BFS was previously used with 

metallic compact-tension specimens (Ref. 5), and in fine gain, opaque ceramics (Ref. 6). An additional 

benefit of strain gaging specimens is the ability to easily monitor the stability of the test, which is a 

requirement for the chevron-notch method. 

This communication illustrates how BFS was employed to measure crack length and fracture 

toughness in a transparent, coarse grain spinel. 

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

− ε=ε (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 2. For a/W = 0, εN = 3/2, which corresponds to that of an 

uncrack beam. The BFS is more sensitive to crack length changes for normalized lengths greater than a/W

= 0.3. 
NASA/TM—2010-216242 2 

The FEA BFS results were fit via least squares regression: 

( )

( ) ( ) 2 3

2

1294083.21 8732785.0 0197908.0

7643304.00732199.0 4386992.1

N N N

N N

Ln Ln Ln

Ln Ln

Wa

− −ε −ε ε

− + −ε ε = (2)

with the strain taken in absolute terms. Equation (2) resulted in correlation better than r

2

 = 0.999. 

Equation (2) requires knowledge of εN to determine a/W. This can be estimated from the compliance 

observed (ε/P) prior to stable crack extension, as shown in Figure 3. Then the mode I stress intensity 

factor at any point during the test can be estimated from Equation (3), where F(a/W) is the appropriate 

stress intensity factor coefficient (Ref. 3): 

( ) ( ) WaF

BW

SSP K io

I 2/3

− = . (3) 

For fracture toughness measurements (KIpb), the maximum load observed and the initial crack length are 

employed in Equation (3). 
NASA/TM—2010-216242 3 

Experimental Application 

During recent testing of a transparent spinel (MgAlO4), crack length could not be measured to the 

operational requirements of ASTM C1421. The coarse structure and poor crack front definition, even with 

indicial illumination, can be seen in Figure 4. Thus BFS was employed to estimate the effective crack 

length and thereby estimate the fracture toughness KIpb. 

Precracking of 3.0 by 4.0 mm2

 cross-sections was performed using a 4 mm bridge span with three, 

5 kg Vickers indentations. A 1 mm strain gage was employed, and the precracked beams loaded between 

20 and 40 mm spans at a rate of 0.2 mm/min in air or dry nitrogen. Chevron-notched tests were also 

conducted using the same spans and a 1 mm strain gage. Example P-BFS curves are shown in Figure 3 
NASA/TM—2010-216242 4 

for SEPB and VB tests, and the results are summarized in Table 1. Good agreement occurs between the 

techniques. The elastic constants in Table 1 were determined in accordance with ASTM C1259 (Ref. 7). 

TABLE 1.—FRACTURE TOUGHNESS AND ELASTIC CONSTANTS 

Fracture toughness 

(MPa√m) 

Poisson’s 

ratio, v

Elastic 

modulus, E

Configuration Air (GPa) 

(75o

F, ~60% RH) 

Dry N2

SEPB 1.32 ±0.05 1.52 ± 0.07 0.26 265 ± 4 VB 1.48 ± 0.14 1.58 ± 0.08 

Conclusion 

Back-face strain is a simple, inexpensive method for monitoring crack length in ceramic flexure 

specimens when the usual optical techniques are inadequate. The fracture toughness of a transparent 

spinel, as measured with BFS applied to the SEPB specimen, was comparable to that measured using the 

chevron-notch in flexure. Monitoring BFS in chevron-notched specimens is also very useful, as the 

stability is clearly determined. 

References 

1. J. A. Salem, L. J. Ghosn, M. G. Jenkins and G. D. Quinn, “Stress Intensity Factor Coefficients or 

Chevron-Notched Flexure Specimens and a Comparison Fracture Toughness Methods,” Ceramic 

Engineering and Science Proceedings, 20 [3] 503–512 (1999). 

2. 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, Edited by R.C. Bradt, D. Munz, M. Sakai and K. 

White, Springer, 2005. 

3. ASTM C1421–07 “Standard Test Methods for Fracture Toughness of Advanced Ceramics,” American 

Society for Testing and Materials Annual Book of Standards, Vol. 15.01, 2007. 

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

3 [6] 454–476, (1975). 

5. 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. Test. and Eval., 7 [3] 147–154 

(1979). 

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

Crack Extension,” Ceramic Engineering and Science Proceeding, 9 [3] 587–594 (1998). 

7. ASTM C1259-08 “Dynamic Young’s Modulus, Shear Modulus, and Poisson’s Ratio for Advanced 

Ceramics by Impulse Excitation of Vibration,” American Society for Testing and Materials Annual 

Book of Standards, Vol. 15.01, 2008. 
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Measuring Crack Length in Coarse Grain Ceramics 

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14. ABSTRACT

Due to a coarse grain structure, crack lengths in precracked spinel specimens could not be measured optically, so the crack lengths and 

fracture toughness were estimated by strain gage measurements. An expression was developed via finite element analysis to correlate the 

measured strain with crack length in four-point flexure. The fracture toughness estimated by the strain gaged samples and another 

standardized method were in agreement. 

15. SUBJECT TERMS

Ceramics; Fracture toughness; Crack growth; Strain gages; Grain size 

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