Fracture toughness evaluation for advanced materials using chevron-notched compact specimens

Materials Science and Engineering, A 143 ( 1991 ) 119-126

119

Fracture toughness evaluation for advanced materials using chevron-notched compact specimens Haruo Nakamura and Hideo Kobayashi Department of Mechanical Engineering Science, Facultyof Engineering, Tokyo Institute of Technology, 2-12 Ohokayama, Meguro-ku, Tokyo 152 (Japan)

Abstract To utilize conventional testing methods and a variety of software appropriate to fracture mechanics tests, it is desired to develop a test method using compact-type specimens for ceramics. For ceramics, however, it is difficult practically to introduce a precrack into the specimen. In this study the utility and applicability of a chevron-notched specimen method and a compression damage method are discussed.

1. Introduction Ceramic materials are quite brittle and contain inherent defects which are produced during manufacturing. Since the fracture toughness is so low in this type of material, the small-scaleyielding condition at fracture initiation is easily satisfied even for such small defects. Thus, in the evaluation of strength, an approach based upon fracture mechanics is especially useful and is needed to accumulate reliable fracture toughness (Klc) data for ceramics. However, because ceramics are so brittle, it is difficult to introduce a fatigue precrack according to the conventional technique. Several simple test methods proposed so far are the controlled surface flaw (CSF) method [1], the indentation strength (IS) method [2], and the indentation fracture (IF) method [3]. All these methods use controlled identation flaws as a crack starter. Several formulations have been proposed to calculate Kic. However, they are not yet well established. To establish the Kic formula, an accurate K m must be given by other reliable methods. Through long experience on metallic materials, fracture mechanics test methods for fatigue, creep and static and dynamic crack growth have been established and fully computerized testing systems and a variety of software are readily available, especially for compact-type (CT) specimens. If CT specimens can be applied to fracture mechanics tests of ceramics, a conventional testing system can be directly used. 0921-5093/91/$3.50

For precise evaluation the introduction of a sharp crack is required. The methods dealt with here are a chevron-notched (CN) specimen method and a compression damage method. In the former it has been shown by using the 3D finite element method that the stress intensity factor K along the crack front for the CN specimen is not constant [4-6]. Thus it is expected that the shape of an actual crack front will not be straight. If this effect can be clarified, an optimum shape design for a simple and accurate Klc test can be made. Concerning the second method, a cyclic compression technique has been proposed [7]. However, it is sometimes difficult to determine the applied load. In this paper it is shown that a precrack can be easily introduced by applying a tensile cyclic load after one compressive load to a through-notched CT specimen until a pop-in sound occurs. 2. Chevron-notched specimen method

2.1. Shearfactorfor CT specimen The stress intensity factor for the CN compact specimen is given by P K = Y*(a) BWI/2

(1)

where Y*(a) is a non-dimensional correction factor, a = a~ W is the relative crack size, a is the crack length, W is the specimen width, P is the load and B is the thickness. As the crack grows, © Elsevier Sequoia/Printed in The Netherlands

120

Y*(a) decreases initially and takes a minimum value Ym* at a crack growth increment Aam; thereafter it increases with increasing crack length. Y*(a) can be easily determined using the slice model [8, 9]. The CN specimen is divided into a number of different slices of thickness B/N, where N is the number of slices, as shown in Fig. 1. Each slice is regarded as a straight throughcracked specimen. The total compliance Ct with a relative crack length a is given by 1

_a-a0

N

1

1

E

a 1- a 0 C(a)

,=m+l

(2)

C(~)i

where ,c is a correction factor considering the shear force acting between adjacent slices, and C(~) is the compliance for a straight throughcrack specimen with a relative crack length ~. By accounting for the necessary and available energy for crack propagation, Y*(a) is given as

y,(a)=(l dCt(a) a~-aoll/2 da a - ao /

(3)

where ao=ao/W and al=al/W are shown in Fig. 1. The values of r in eqn. (2) are given by Bluhm [8, 9] and Munz et al. [10, 11] for a three-point bend specimen and a short rod respectively, but for the CT specimen is not available. Furthermore, the K value along the crack front is not constant for a straight chevron crack [4-6], so the shape of the actual crack front may be not straight. To determine ,¢ for an actual crack, a relation between C and a was determined from experi-

ments. Using an ASTM A508-3 low alloy steel, the fatigue crack growth test was conducted. The specimen is a chevron-notched 1CT with a 0 = 0.4 and a 1= 0.8. The basic relation between the crack growth rate da/dN and the cyclic stress intensity factor range AK is given in ref. 12. In this experiment, Bluhm's result [8, 9] for *¢ was assumed to be appropriate for the CT specimen. Relations between the specimen compliance C and a and between C and the stress intensity factor K were calculated using this r value. A fatigue crack was grown from the tip of the chevron notch under a constant AK = 24.8 MPa m I/2 at a stress ratio R = 0.05. For every given crack growth length Aa of 0.75 mm (between 0.5 and 1.0 nun), a benchmark was introduced at R = 0.5 keeping the maximum stress intensity factor Kmax constant. A macro-fractograph is shown in Fig. 2. The crack front is the inverse thumbnail shape. The value of da/dN calculated from each benchmark was in good agreement with that for the throughcracked specimen [12]. This strongly suggests that the calculated AK value of 24.8 MPa m 1/2 using the above r is almost correct for the CT specimen. A comparison of crack lengths between experiments and predictions is shown in Fig. 3, where V is the compliance normalized by that at a0. The crack length at a given number of cycles is a maximum at both edge sides, which is denoted by amax (solid circles). The length is a minimum at the mid-thickness and is denoted amin (open circles). Predictions from Bluhm's and Munz's solutions for *¢are shown by triangles and squares respectively. The former solution nearly gives the average crack length for the curved crack. In particular, it agrees with the experiment for

1

z,

-," -7

W Fig. 1. Slice model.

Fig. 2. Macro-fractograph for CN specimen showing curved crack front (A508-3). The crack growth direction is from right to left.

121 t2

0 Experimenl(amin ) @ Experirnent(otma~) A Ar
A508

l

/

/

8

TABLE 1 Chemical composition of epoxide composite Epoxy resin

Bisphenol-A-diglycidyl ether

Curing agent Filler

Tetra acid anhydride, acid anhydride AI203 (60 wt.%)

6

TABLE 2

4

4 ./%.°

0 '{~'~

0's

0'6

Mechanical properties of epoxide composite

~-~

0'7

oB (MPa)

E (GPa)

72.1

9.3

01s

a/w Fig. 3. Relation between normalized compliance and relative crack length. 1.5

a = a , =0.8 where the crack front becomes straight. In the following experiments the ~ value of Bluhm was used for the CT specimen. 2.2. Application to plastic composite To assess the accuracy of K for CN specimens, a comparison of K,c tests on through-cracked (not through-notched) and CN specimens can be carried out. In some ceramics, e.g. partially stabilized zirconia (PSZ), a fatigue precrack can be introduced but the R curve is not horizontal which invalidates the CN method. The correction factor of K in the indentation fracture (IF) method has been developed by referring to the results of other methods, almost all of them being obtained from the IF method. Thus a comparison with KIC from the IF method may not be appropriate. In this study an A1203 particulate-epoxide composite, which is widely employed for insulators in electronic devices, was used to evaluate the accuracy of K. Its chemical composition and mechanical properties are given in Tables 1 and 2 respectively. The specimens were chevronnotched 1CT with a0 = 0.148 and a, = 0.407 and through-notched l CT. For the through-notched specimens a fatigue precrack was introduced with AK = 0.8 MPa m 1/2 and R = 0.05. A typical load (P) vs. load line displacement (V) curve is shown in Fig. 4 for the CN specimen. Every partial unloading line is directed towards the origin and this suggests that the non-linearity is due only to the crack extension. Fracture resis-

1.0

0.5

0

0.I

0.2

0.3

V

Fig. 4. Load composite.

vs.

mm

load line displacement curve for epoxide

+

~2

O¢ O a

O Oe

# I

r o Arrest mark(e In water) t a B.F.S.

Through

Chevron [ o Displacement

v Arrest mark .~!

0

I

I

I

1

0.2

0.4

0.6

0.8

1.0

a/W Fig. 5. K,c values (with arrows) and K R plots for epoxide composite.

t a n c e (KR) plots are shown in Fig. 5 for a number

of specimens. For CN specimens, K R is constant for a given specimen in the chevron area ( a < 0 . 4 0 7 ) as well as in the through-cracked region (a >0.407), although its value is slightly

122

different for each specimen. Furthermore, KR (shown by o , • and zx without an arrow for through-cracked specimens and by [] and x7 without an arrow for CN specimens) agrees with KIC (shown by o, • and zx with an arrow for through-cracked specimens and by [] and ~7 with an arrow for CN specimens) for precracked specimens. Thus it is judged that the accuracy of K for CN specimens is good.

2.3. Application to brittle fracture of steel Using an ASTM A533B-1 steel, a fracture toughness test at - 196 °C was conducted on CN specimens to obtain the Kr~ curve for cleavage fracture. The fracture toughness obtained was Kc = 52 MPa m ~/2, which is higher than the value of 35 MPa m 1/2 obtained by the conventional K~c test according to ASTM standard E399-83. The reason is that no stable crack growth took place in the CN specimen. For fatigue-precracked CN specimens with a < am, KIC = 32 MPa m ~/2 was obtained, but again no stable crack growth occurred. This suggests that either (i) K R decreases as the crack grows (see Fig. 6(a)) or (ii) K c at the fracture nucleation site (the grain boundary or the inclusion) is much higher than that for the matrix (see Fig. 6(b)). The Kr~ curve for cleavage fracture can therefore not be obtained with CN specimens. In many ceramics the R curve is believed to be flat but this is not the common characteristic in brittle materials.

For the straight-notched specimen, stable crack growth did not occur. On the fracture surface there were many steps (so-called ratchet marks) near the notch tip. These marks suggest that many microcracks initiate at several locations of the notch root and that the planes of those cracks are not unique. As these cracks grow, they become one main crack. Thus the fracture surface was relatively rough and this roughness disappeared as the crack advanced. The maximum distance 6 where the surface is rough was measured. The relation between K c and 6 is shown in Fig. 7. It is clear that K c increases with increasing 6. Thus there is a possibility that K~c is overestimated in these specimens. The K R curve for a 0 = 0 . 4 and a~=0.8 is shown in Fig. 8. The trend is quite different from that of Fig. 7. The higher value of K R for the smaller Aa region may be due to the finite notch tip radius. Outside this region K R is almost constant. Pc indicates where the unstable fracture occurred and is very near to the calculated am. The results for a0 = 0.8 and a~ = 1.0 are shown in Fig. 9. The crack growth increment up to unstable fracture is relatively small. A leading feature of the CN specimen method is the simplicity of its test procedure. A displacement record is not required. If the crack resistance (KR) curve is horizontal, Kic is given by TABLE 3

2.4. Application to ceramics The material used was hot-pressed silicon nitride (Si3N4). The mechanical properties are given in Table 3. Specimen types were 1/2CT, 1CT and three-point bend specimens ( W= 20-25 mm). The notch tip radius p was 0.05 mm. For comparison, a straight through-notched specimen was also used.

Mechanical properties of hot-pressed Si3N 4 Young's modulus Density Flexure strength Compressive strength

314 GPa 3.26 g cm -3 980 MPa 4410 MPa

8 ,&

00 0

.o

Kic

z~

K ,c

o

~' o

o

~6

ell •

o~ o

0 3PointBend A l l Z CT a ICT

z~

• o

R-curve

~a (a)

Aa (b)

Fig. 6. Schematic illustration of K R for cleavage fracture in steel.

T

0

~

I

0.2

i

I

0.4

i

I

0.6 fi ( m m )

J

I

0.8

i

[

,

1.0

Fig. 7. Relation between K¢ and 6 for hot-pressed Si3N4.

1.2

123

6

~m o_ ~" 4

K,c*

,

2

4

OG:0.4 0~, =08 i

0.40

I

i

0A5

050

0.55

R.T.

O(

II I PW

I

800

900

I

|

!

i000

ii00

1200

Temperature

Fig. 8. K R curve for hot-pressed Si3N 4 ( a o = 0.4, a i = 0.8).

L~ ~2 1£

K[c. for hot-pressed

A

6 m~

°C

Fig. 11. Temperature dependence of Si3N4.

"~Pc 4

2 O'o= 0.8 06=I.0 085 or

Q80'

[]

ii00 °C

/~

1160 °C

KIc ~ o

09O

(R.T.)

Fig. 9. KR curve f o r h o t - p r e s s e d Si3N 4 ( a , = 0.8, a l = 0. ! ).

I

l 0. i0

0.05

0.15

6~

Fig. 12. K R curves at 1100 and l l60°C for hot-pressed Si3N4.

8.C 3 Point Bend

t

0{o=0.8

aLm

(XI= 1.0

6.C ~v >-

4£ e.o :0.4

/~,

2.C

=o.j. C?.o=0

'

0.1'

'

o:2

'

o13

'o14

'

0.5

(I-C1 o Fig. 10.

Y*(a)-A a curves for three-point bend specimen.

substituting Ym* and the fracture load Pc into eqn. ( 1 ). However, if K R increases as Aa increases, K~c is overestimated. In this sense it is desirable that Aa m ( = a m- a0) be as small as possible. Typical Y*(a)-Aa curves for the three-point bend specimen are shown in Fig. 10. W h e n a 0 = 0 and a l = 0 . 5 , Y*(a) becomes constant over a wide range of A a around Aam, indicated by an arrow, as is also the case for a 0 = 0 . 4 and al =0.8. On the other hand, when a notch is deeply introduced as for a 0 = 0 . 8 and a ] = l . 0 , Y*(a)

increases rapidly after am. Furthermore, the crack increment A a m up to Ym* is relatively small. T h e C T specimens show a similar trend. From the above results the optimum shape for the K]c test is summarized as follows. (1) A a m must be relatively large (greater than 2p) to avoid the influence of the finite notch tip radius. (2) Y*(a) must increase rapidly after A a m so that K]c is not overestimated when K R increases with increasing Aa, i.e. the chevron notch must be deep. (3) Stable crack growth must occur easily, so the notch tip angle must be relatively sharp. From these results it is concluded that the deeply notched specimen with a 0 = 0 . 8 and a ~= 1.0 is optimum for the K~c test. T h e temperature dependence of K[c is shown in Fig. 11. T h e material is another heat of hotpressed Si3N 4. It is clear that K=c is almost constant up to 1160 °C. T h e K R plots at 1100 and 1160 °C obtained by the multiple-specimen method are shown in Fig. 12. T h e R curves at both temperatures are the

124

same. Also, they agree with the value of K]c at room temperature shown by a solid circle.

"1111111111

d

0

3. Compression damage method The machining of CN specimens is relatively difficult. Therefore an easier way to introduce a fatigue precrack in a straight through-notched specimen has been developed.

The material used was sintered Si3N 4. The specimen type was 1/2CT of thickness B= 10.0 ram, relative notch length a = 0.6 and notch tip radius/9 = 0.5 and 1.0 mm. The three types of compressive loading shown in Fig. 13 were attempted. In Fig. 13(a) one of the pin-holes was broken by a high tensile stress. In Fig. 13(b) a crack was initiated from the back face due to a tensile stress. In Fig. 13(c) a specimen was loaded keeping the upper and lower faces parallel. At Pi = - 130 to - 1 3 5 kN (/9=0.5 ram) and - 1 5 5 to - 1 6 0 kN (/9 = 1 mm) a microcrack initiated from the notch root and this was easily detected by its popin sound. The crack length was about 0.1 mm and debris from the broken matrix was dropped from the specimen surface at the notch root. The maximum stress Oym~~ at the notch root can be given by

[13] 2K (4)

By assuming the CT specimen to be a half of the centre-cracked specimen with a width of 2 x 1.25 W and a crack length of 2 x (a + 0.25 W) (see Fig. 13(c)), the K value in eqn. (4) can be approximately given by [14]

K=°[x(a+O'25W)]l/2f( a+0"25W1.25 W '

1.250"5H)w

(5) where o is the normal applied stress and H is the specimen height (see Fig. 13(c)). According to eqns. (4) and (5), the Oymax/O0 values at Pi are 1.8 (p = 0.5 mm) and 1.5 (p = 1 mm), where cr0 is the compressive strength. The compressive stress exceeds o 0 over a distance of 0.20-0.25 mm ahead of the notch tip. The final failure load Pc is twice Pi, thus damage can be introduced easily. Hereafter this specimen is called the damaged specimen.

Z

711///////I

5W

(b)

(o)

3.1. Materialand experimentalprocedure

Oymax - - ( g p ) l / 2

////i//i/,

(c)

Fig. 13. Three compressive loading methods.

0.8 EEO.6

o0.4 <~ 0.2

Kmo,=4.7MPo,/-m R=0.1

J I

10 Number of cycles N

1'5 XlO 5

20

Fig. 14. Relation between number of cycles and Aa for sintered Si3N4.

3.2. Introduction ofprecrack Within the first several tens of cycles the specimen was broken by applying cyclic compression with load Pi and it was difficult in practice to control the crack extension. However, by applying cyclic tension to a damaged specimen, a fatigue crack can be easily introduced. The relation between the number of cycles, N, and Aa for Kraal=4.7 MPa m 1/2 ( < K ~ c = 5.5 MPa m 1/2) at R = 0 . 1 is shown in Fig. 14 ( p = 0 . 5 mm). The crack shows the delayed retardation behaviour due to a damaged effect. On the fracture surfaces there were many steps near the notch root, which suggests that many microcracks initiate within the damaged area. With crack advance they become one main crack, as is the case in Fig. 7, which should have caused the above delayed retardation. Thus in fracture toughness or fatigue crack growth tests this effect must be clarified.

3.3. Fracturetoughness test In ASTM standard E399-83 the Kma x value during fatigue precracking must be smaller than 60% of K~c. In the IF method or the CN specimen method Kmax is equal to Km. The effect of Kmax on KIc is shown in Fig. 15 for Aa > p , where Aa is the precrack length. When KmaX is at the

125 7

~6

2~ 5

1.0 ~- - - L ~ - -m- e-t-h°-d- - -

oo

/

~og0.5 - - a %

z l 0 -8

I

I

1

2

I

I

I

3

4

o

5

.........

o

o

I

0 3.5

MPo~

4.'5

5 MPoJm

5.5

Fig. 17. Relation between Kop/Kmax and Kma× for sintered Si3N4.

a~ 10-6

O R=O, 1 D R=O,8 nu n

I

4

Kmox

Z~ R=O,5

• O R=0.1

@

z~x

~.10 -7 E

zx~

•

A R=05 (SN3) (SN4)

•

- [] R = 0 8

z 10-8

D

Z~

10-9

0

A~

0

©

~ 1 0 -9

0

Si3N4

10-1o 10 -11

Open SN1 Solid SN2

Foti 9ue threshold

Fig. 15. Effect of Kmax during preloading on K~c for sintered Si3N4.

"~" "~-10 -7

Specimen

~,~,,,~wa

a

Si3N4

Kmox

_~ 10 -6

0 R=0.1 A R=O5(SN3)

E

-""

/z.4_¢,~}.."" %~?-""

-~

.""

0 Pre-cracked

~ 4 - ~ CN

3

oO~

I , I, I 0.8 I

I

2 AK

,S°lild6u, -
3

4

5

0

10-1c

OpenA• > p 6

10-11

I,Iii

I

1

0.8

MPo./-m

2 AKeff

Fig. 16. Relation between da/dN and AK for sintered Si3N 4.

fatigue threshold value, Km.JKIc is 0.65. Within the range 0.65 ~
3.4. Fatigue crack growth test The fatigue crack growth tests were conducted with a cyclic frequency of 25 Hz at R = 0.1, 0.5 and 0.8 under AK-decreasing and AK-increasing conditions. The relation between the crack growth rate da/dN and AK is shown in Fig. 16. When Aa < p, where the delayed retardation took place, da/dN shown by solid circles is lower than that for Aa > p shown by open circles (R = 0.1). The data for Aa > p show a strong R dependence. The relation between the opening stress intensity factor Kop and Kmax is shown in Fig. 17

i

Specimen Open SN1 Solid SN2 I I

3

MPoJ-m

¢

Fig. 18. Relation between da/dN and AKcf f for sintered Si3N4.

o 10 -6

"-~.10 -7

R=O.1 0 R=0.5 [] R = ~ o

z 10 -8 10-9

#o

10-1c 10-11

i

I

,

4

I

t

I

I

5

6 Kmax MPa~

Fig. 19. Relation between da/dN and Km~x for sintered Si3N~.

for Aa > p. At R = 0.1, Kop/Kmax decreases with decreasing Kmax. At R = 0.5 and 0.8, no crack closure was observed. The relation between da/dN and the effective stress intensity factor AKeff (=Kmax-Kop) is shown in Fig. 18. The R dependence cannot be explained by AKeff, The data are replotted against Kmaxin Fig. 19, where the scaling of both axes is the same as in Figs. 17 and 18. The results at R = 0 . 1 and 0.5 agree with each other and those at R = 0.8 are located at the slower side. However, the scatter band in Fig. 19 is relatively narrow compared

126 4. Conclusions

_~ 10 -6

4tKic

o lOHz

"~10 -7

,', 1Hz

z" o 10 -8

0

Y

o

-o 10-9

121

10-I(

~redictedfrom 10-11

SCCtests - -

10Hz

..... 1Hz 10-12 3

,

I

Iscc ,

i

I

,

4 5 6 Kmox UPaJ-m

Fig. 20. Relation between da/dN and Kmax for sintered Si3N4 (effect of cyclic frequency).

_~ 1 0 - 6 o , KIc

"~10

0 I

-7

0 0

10-8

In this study the utility and applicability of a chevron-notched specimen method and a compression damage method were discussed. T h e results obtained can be summarized as follows. (1) T h e crack tip front for the C N specimen is not straight: it shows an inverse thumbnail shape. T h e stress intensity factor was calibrated considering this effect by experiments. (2) A n optimum shape design of the CN specimen was defined for Kic tests. By using a deeply notched CN specimen, K~c is obtained accurately without measuring the displacement. (3) By applying one compressive load to a straight through-notched specimen until a pop-in sound occurs, a fatigue crack can be easily introduced by a tensile cyclic load which is below the fracture load. (4) If the precrack length is well beyond the notch tip radius in the above specimen, the effect of damage during compression can be neglected. References

0 0

1 J.J. Petrovic and M. G. Mendiratta, in Fracture Mechanics Applied to Brittle Materials, ASTM Spec. Tech. Publ., 678, American Society for Testing and Materials, Phila-

10-9 o

10-~(

~

Klscc

10-11

i

I

,

4

i

5

.

i

,

6

Kmax MPa/m

Fig. 21. Relation between da/dtand

Kmax for sintered Si3N 4.

delphia, 1979, p. 83. 2 P. Chantikul, G, R. Anstis, B. R. Lawn and D. B. Marshall, J. Am. Ceram. Soc., 64 ( 1981 ) 539. 3 A. G. Evans and E. A. Charles, J. Am. Ceram. Soc., 59 (1976)371. 4 I. S. Raju and J. C. Newman Jr., in Chevron Notched Specimens: Testing and Stress Analysis, ASTM Spec. Tech. Publ., 855, American Society for Testing and Materials,

with those in Figs. 17 and 18. Thus it is concluded that Kmaxhas an important effect on d a / d N . T h e effect of the cyclic frequency is shown in Fig. 20. At higher Krnax, the data for the cyclic frequency f = 1 Hz (o) are faster than those for f = 10 Hz ([]). To examine the effect of stress corrosion cracking (SCC), an SCC test was conducted. T h e result is shown in Fig. 21. T h e threshold value of SSC, Kmcc, is higher than the fatigue threshold Kth. From Fig. 21 the crack growth rate d a / d N due to SCC for a given cyclic frequency can be predicted by integrating the growth rate d a / d t for one cycle. T h e predictions are compared with experimentally obtained data of d a / d N in Fig. 20. Except near K~c, the crack growth due to SCC is almost negligible. Thus the Kmax dependence in Fig. 19 should not be attributed to the occurrence of SCC during the fatigue test. A study on the interaction effect between fatigue and SCC is now under way.

Philadelphia, 1984, p. 32. 5 A. R. Ingraffea, R. Perucchio, T. Han, W. H. Gerstle and Y. Huang, in Chevron Notched Specimens: Testing and Stress Analysis, ASTM Spec. Tech. PubL, 855, American Society for Testing and Materials, Philadelphia, 1984, p. 49. 6 A. Mendelson and L. J. Ghosn, in Chevron Notched Specimens: Testing and Stress Analysis, ASTM Spec. Tech. Publ., 855, American Society for Testing and Materials,

Philadelphia, 1984, p. 69. 7 L. Ewart and S. Suresh, J. Mater. Sci. Lett., 5(1986) 774. 8 J.I. Bluhm, Eng. Fract. Mech., 7(1975) 593. 9 J. I. Bluhm, Proe. 4th Int. Conf. on Fracture, Vol. 3, University of Waterloo Press, Waterloo, Ont., 1977, p. 409. 10 D. Munz, R. T. Bubsey and J. L. Shannon Jr., J. Am. Ceram. Soc., 63 (1980) 300. 11 D. Munz, R. T. Bubsey and J. E. Srawley, Int. J. Fract., 16 (1980) 359. 12 H. Kobayashi, T. Ogawa, H. Nakamura and H. Nakazawa, Trans. Jpn Soc. Mech. Eng. Ser. A, 50(1984) 309. 13 M. Creager and P. C. Paris, Int. J. Fract. Mech., 3(1967) 247. 14 M. Ishida, Int. J. Fract. Mech., 7(1971) 301.