International Journal of Fracture 58:211 222, 1992. © 1992 Kluwer Academic Publishers. Printed in the Netherlands.

211

A new method for evaluating fracture toughness of brittle materials J. TIROSH, E. ALTUS and Y. YIFRACH Faculty of Mechanical Engineering, Technion - Israel Institute of Technology, Haifa, 32000 Israel Received 1 December 1991; accepted in revised form 1 June 1992

Abstract. A new testing procedure is suggested for measuring the fracture toughness of brittle materials as superconductors and ceramics. The idea is to perform a compression test on a subcompact square specimen which contains a central hole. The presence of the hole induces a tensile stress at a certain small region attached to the hole. In this region an artificial notch is introduced such that the fracture path satisfies a pure tensile opening mode (mode I) to which the linear fracture mechanics rules apply. The stress distribution on the fracture plane guarantees a certain amount of stable crack extension. The relationship between the critical compressive load and the stress intensity factor is formulated via an available Green function along with a numerical solution (FEM with ANSYS code). The testing procedure is demonstrated with specimens made of two types of tungsten carbide which differ by their grain size only. Test results are examined via fracture toughness and strength values produced by other conventional methods and the agreement is very good. The geometry and loading direction enable the fracture toughness results to be relatively insensitive to the notch tip radius and the crack length, thereby relaxing the requirements for accurate measurements. The small size of the suggested specimen (12.70mm × 12.70 mm x 5 rnm) and the avoidance of gripping interfaces provide the major cost-wise advantages.

1. Introduction

There is considerable difficulty in characterizing with conventional methods, the fracture toughness of extremely hard materials (like the ceramics family) and very precious materials (like the carbon/carbon composites, the newest super-conductor materials, etc.). The reasons for the difficulty are mainly that with such materials the standard methods (recommended by ASTM E-399) require: 1. Relatively bulky (and thus expensive) specimens to machine for sizing. 2. Gripping fixture and/or COD local gages. 3. Extreme care in handling the specimen during the machine/specimen interface operation. A partial remedy was provided by the Barker measuring method [1], where a short rod with a 'V' shaped slot was suggested. The test is conducted by applying opening pressure in the slot of the specimen. As with the tapered double cantilever beam (TDCB) [2], the compliance of the specimen remains independent of the crack length and hence, results can be inferred simply from measurement of the critical load only. A method with similar features was suggested by Munz, Bubsey and Srawley [3]. They used a short bar with rectangular cross section and a chevron slot. These two methods facilitate considerably the interpretation of the toughness measurements but still leave the preparation of the specimen somewhat complex and relatively expensive, especially the machining of the chevron slot in ceramic materials. In this work we look for a suitable test method which avoids the above difficulties and still maintains reliability and reproducibility of experimental data. The idea is to replace the conventional tensile or bending load on a relatively large specimen (with characteristic length of few centimeters) by a much easier-to-apply compressive load on a subcompact specimen (with characteristic length of few millimeters) containing a hole in its center, as shown in Fig. 1. The

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Fi9. 1. The suggested form of the subcompact square (2b) specimen with a center hole of radius R and a notch of length

a. Two steel plates of thickness h are used to 'spread' the applied load. The force P is applied by a conventional testing machine but no grips are needed. Dimensions of tested specimens are: b = 6.35, R = 2.5, h = 5, q = 0.63, t = 5 (all dimensions are in mm). small 45 ° phase around the edge of the hole, denoted as r/in Fig. 1, will be accounted for. The role of the two hardened plates of thickness h in 'spreading' the load, and the explanation as to how the hole induces tensile stress during the compressive load will be given in the sequel. From the simple features of the suggested test, some advantages seem clear: (a) enabling small specimen dimensions, and (b) obviating the usual gripping difficulties. It remains to be seen how the fracture toughness is actually measured and in what level of reliability as compared to the classical known methods.

2. Experimental 2.1. General approach

Vertical notch (or a crack) attached to a pre-fabricated hole is used to induce local tensile stress at the notch tip. The hole in the specimen is introduced by the manufacturer during the sintering process. In the case of ceramics made of tungsten carbide, such specimens with a hole are provided 'from the shelf'. Two types of a commercial series of these ceramics, designated as IC-10 and IC-21, were used in this work to study the suggested approach to measure K~c. Their chemical composition and mechanical properties are summarized in Table 1. The results were compared to the Klc values reported by Porat and Malek [4]. A sharp notch is implanted in the specimen hole by an Electro-Discharge-Machine (EDM) using a copper electrode of 0.07 m m thickness. The shape and dimensions of the notch were intended to serve as parameters for fixing 'the best' conditions for the tests. It was found, however, as will be shown, that the geometry of the notch has a surprisingly small effect on the stress concentration factor in the compressive load. The compressive tests were video-filmed throughout the fracture process. A typical photograph showing the notch area is seen in Fig. 2. By replaying the film in slow motion, the exact load at which fracture or 'jumps' in crack growth initiate are visually detected. The critical load

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Table 1. Chemical, microstructural and mechanical data for the materials tested (based on manufacturer's data Material property

Units

Co Ta(Nb)C WC

(%w) (%w) (%w)

Density, 7 Grain size, de Mean free path ,;. Coercive force elasticity modulus, Hc Young's modulus, E Poisson ratio, v Transverse rupture stress, TRS Toughness, K~c

(gr/cm 3) (~tm) (p.m)

IC-10

IC-21

Compositio~l 8 2 90

8 2 90

Properties

(kA/m) (GPa)

(M Pa) (MPa.m 12 )

14.74 3.127 0.477 14.95 610 0.22 2400 14.2

14.74 5.82 0.885 9.65 610 0.22 2500 17.7

Fig. 2. A typical video picture of the near notch tip area and the 'natural' crack emanating from the notch due to a vertical compressive load. The thickness of the notch is 0.07 mm, which is the thickness of the copper electrode employed by the electro discharge machine to produce the notch.

and the critical crack length are viewed simultaneously on the video screen and thus generate single experimental data related to the material behavior. These data, repeated with small scattering at the same test conditions, are used to resolve the fracture toughness of the material.

2.2. Description The circular hole in the compressed specimen causes a stress concentration factor (SCF) of the order of unity at the upper and lower points along the free edge of the hole (x = 0, y = + R). For an infinite body the SCF is exactly unity (denoted as the Kirsch solution). For a finite body with given dimensions (Fig. 1), the SCF is different, depending on the boundary conditions which prevail on the areas in contact with the press. From a technical viewpoint, one should note that the ceramic specimen, due to its high hardness, may damage the ram and the bottom die of the press by local plastic indentation. To avoid this occurrence, two hardened steel plates were used to transmit the compressive load from the press to the specimen. The effect of the hardened steel plates on the stress field, especially on the stress concentration factor (SCF) was accounted for in

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the numerical stress analysis. Two kinds of boundary conditions on the plate/specimen interface were used in the numerical (FEM) computation. These boundary conditions are: (a) A 'smooth' interface: No friction exists on the contact areas and therefore no resistance to interfacial sliding. (b) A 'clamped' interface: The interfacial areas are 'welded' and no sliding is allowed. The above boundry conditions, representing the two extreme situations, prevail in compressive tests and thus generate the upper and lower bounds on the real frictional conditions. Figure 3 shows the stress field (axx) ahead of the hole, along the anticipated crack growth path (x = 0, y > R) for 5 cases: The solution for infinite and finite plate, with and without steel plates for the above two boundary conditions. From this figure one may conclude: (a) The four solutions with a finite specimen size show a compression region ( a = negative) beyond the near tension field. A test specimen whose local tension region is surrounded by a compressive environment has the potential of stabilizing the crack growth, so desired for controlling fracture of brittle materials. (b) The two extreme boundary conditions ('smooth' interface and 'clamped' interface specimen) yield relatively close stress solutions. Therefore there is no need to pay for an experimental effort to control the friction conditions in the plate/specimen interface. In practice, any commercial lubricant (i.e. MoS2) might be adequate to get a representative solution which stays between the upper and the lower stress solutions. (c) The numerical stress solution indicates that the steel plates on the top and bottom of the specimen have a substantial effect on the SCF. In our case it corresponds to the value of the stress at the edge of the hole axx(O, R) normalized by the nominal compressive stress applied by the press, - ao. It is about 50 percent higher than for the case of the infinite plate solution and almost twice the value of the solution without plates as shown in Fig. 3. The relatively 'soft' plates (around ½ of the Young's modulus of the ceramic material) helps to 'smear' the contact stresses more equally, causing a somewhat higher portion of the load to be concentrated around the center line of the specimen (x = 0, y = b). Moreover, higher SCF Oxx(O,y) C5~1

1.5 L {il 1.0 i. :~i, ,

0.5

{,

: <, × -

'

~

Smooth - no plates Clamped - xlo plates Smooth - with plates ~ . (lamped - with plates l Infinite (Kitsch)

~.~, 0.0 -0.5 2.5 3.0 p - y/R Fig. 3. The stress distribution a:,~(0,y) normalized by the average nominal compressive stress (-ao) [press load/(specimen contact area with the plate)]. The FEM solutions are shown for the two extreme boundary conditions (smooth and clamped surfaces) with and without steel plates, as compared with an exact solution appropriate for an infinite body. 1.0

1.5

2.0

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215

means lower failure loads, which adds some convenience in the laboratory measurements. Therefore it is our suggestion to conduct the tests with steel plates and with a light interface lubrication. In the sequel, however, solutions for both of the two extreme boundary conditions (BC's), mentioned above, will be given for better assessments of the results.

3. Analytical

3.1. Stress distribution near an elliptical hole. Since a circular hole and a crack-like slit are two extreme cases of an elliptical hole, it is instructive to get the general stress solution which encompasses all crack shapes. For an infinite body containing an elliptical hole subjected to a remote unidirectional compressive load, an exact elastic solution can be found in handbooks (i.e. Savin [5]), or be reached by employing, for example, Muskhelishvili-Kolosov stress-potential approach (Muskhelishvili, [6]). For an infinite plate it was found that the hoop stress distribution at the stress-free boundary of the hole is reduced to

a00 = p

1 - m 2 -b 2m cos 2~ - 2 cos 2(0 - ~) 1 - 2m cos 20 + m 2

0 ~< m ~< 1,

(1)

where ~ is the angle between the load direction and the short axis of the ellipse, 0 is the angle of the point on the stress free boundary, p is the remote compression stress and the parameter 'm' represents the 'eccentricity' of the hole according to the following relation R a -

m - - -

R b

Ra + Rb'

0~
where Ra, Rb a r e the long and short radii of the ellipse, respectively. When one axis of the ellipse shrinks to zero (say, Rb ~ 0), then m --* 1 and the hole becomes a crack-like slit. In the other extreme, when Rb = Ra then m ~ 0 and the hole becomes circular. From (1) one can conclude that the compressive load induces tension along a portion of the hole boundary of maximum value according to tr00max(at ~ = 0 = +½x)=p(.1--mZ--2m--2-) -i- + 2m + my = -- p

(2)

(for any m).

Equation (2) indicates that the maximum tensile stress is always equal to a00max --p and does not depend on the shape factor m. The peculiar fact that under remote compression load the SCF at the edge of an elliptical hole of any shape is always unity, allows us to select, for simplicity, a circular hole rather than a crack as a tensile stress generator in the suggested specimen. However, some effects on the SCF result from the finite dimensions of the specimen and the proximity of the hole to the location of the applied load which will be assessed later by F E M solutions. Therefore, inserting a notch from the boundary of the hole is still helpful for controlling the crack =

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initiation. In general, due to (2), the SCF is not sensitive to the form of this notch. It relaxes somewhat the need for precision in the machining process. The associated stress intensity factors will be formulated next. 3.2. The Stress I n t e n s i t y Factor (SIF)

Calculation of SIF for the present geometry and loading condition can be performed either by implanting cracks of different sizes in the finite element calculation, or by some analytical approximation. To achieve sufficient accuracy by the first method, a singular element is needed and the mesh has to be changed for each crack length. The finite size of the specimen causes difficulties in designing a 'constant accuracy mesh' for a whole set of crack lengths. To get high reliability, the sensitivity of the SIF has to be checked for mesh changes for each crack length. It seemed worthwhile to 'cross-check' the FEM results with the Green Shwartz (GS) analytical approximation. By this method, the elastic stress distribution surrounding the hole is searched first for the no-crack case. Next, a fictive notch of characteristic crack length a is introduced normal to the maximum tensile stress. In order to satisfy the traction free condition on the surfaces of this notch, the normal stress along this notch is forced to vanish by superimposing a solution of the same problem but with an opposite traction (known from the previous 'no crack' solution). The K factor is then evaluated by employing the Green function in conjunction with the above distributed traction. The specific details are as follows: For a unit pair of loads acting at a distance y'( = y - R) on the face of the crack of length 2a, the associated expression of the stress intensity factor (the 'Green function') is given by Sih [7]. It reads

G(y', a) =

1 y'+~]1,'2

~a a - y ' j

(3)

"

Due to the linear nature of elastic solutions, (3) can be superimposed for any distributed load on the crack faces. In particular they may serve to approximate the overall SIF generated by the normal load ax~ (0, y) dy (per unit thickness) distributed along the flank, as shown in Fig. 4, according to R+a

K1 =

I

axx(O, y ) G ( y , a ) d y

for

R <~ y <~ (R + a),

(4)

dR where the 'weight function' axx(0, y) is solved beforehand as an elastic problem without a crack. The superimposed solutions (4), though leaving the faces of the crack stress-free, do not comply with the boundary conditions along the hole unless a different Green function is devised for this particular case which is a difficult task by itself. Therefore an additional solution needs to be added to the previous one in such a manner that the sum of them will leave the boundary of the hole traction free. The additional solution, however, might cause disagreement with the boundary conditions along the crack, so that an additional correction is needed. This alteration of superimposed elastic solutions (known as the 'Shwartz alteration method') may go on until approaching a tolerable accuracy in satisfying the whole boundary condition.

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217

Fiq. 4. The Green-Shwartz (GS) model of the solution for the Kt through an equivalent problem with a known Green function from [7]. The stress field normal to the anticipated crack line activates the Green function for an equivalent crack (a correction factor of 1.12 to the K t was added to account for the adjacent free edge [7]).

A sequence of such corrections, when using (3), was shown by Tirosh and Tetelman [8] to converge rapidly in the case of a crack approaching a hole. In the case of a crack emanating from a hole, akin to the present situation, the corrections were indeed unnecessary (Tirosh I-9]) and will be used here without further elaboration. As an example, consider an infinite plate with a hole of radius R subjected to a remote compressive stress p. The induced tensile stress along the y axis for y >~ R yields [ 1 3 ] oxx(0, y) = p ~2p2 2~,~

,~ for p ~> 1 where p - R'

(5)

Substituting (5) in (4), the integration gives the solution

K~ = p

~ [

l

2(1 + c~)3/2

3

2(1 - c07/2

(1 +½~+ ~2)]

where = a/R.

Notice the high convergent rate for sufficiently small ~.

(6)

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J. Tirosh et al.

For a finite specimen with dimensions as in Fig. 1, the elastic stress distribution is solved by numerical methods (FEM). The discrete FEM stress distribution for the different cases are presented by the following polynomial function of degree n

ax~(O, y) = i AiP i

1 <. p <~ 2.52,

(7)

i=l

where Ai are best fit coefficients for the numerical solution. The formulation of K~ is now a straightforward integration of (4) using the polynom of (7) as the 'weight function', namely

K~ =

IR+ai JR

A~p~G(y, a)dy

for R ~< y ~< (R + a).

(8)

i=l

The simplicity of getting an expression for K~ stems from the ability to integrate analytically (8) for any polynomial of degree i. It is significantly facilitated by rewriting the variables as z 2 = (p - 1 + ~)/(1 + g - p),

p = 1 + c~- 2~/(z 2 + 1), dp = 4gzdz/(z 2 + 1)2.

(9)

Now, the integration involves only recursive operations of the following integrands Z2/(Z 2 "k- 1)i+2

1/(Z 2 + 1)i+2

i : 1. . . . . n.

The solution for our particular specimen (using a polynom of degree 7) involves extended algebraic expressions which, for brevity, are skipped. Alternatively, K~ can also be extracted directly from the FEM solution of the loaded pre-cracked specimen. A convenient way to do so is to use the crack opening displacement (COD) approach, the relationship of which is

KI =

E(2~) 1/2 4n--~-=~5) i

j=l

(u~)jr]-~/z,

(10)

where n is the total check point along the face of the crack at which the opening displacement ux and the distance r (from the notch tip) are associated at each point j. E is Young's modulus and v is the Poisson ratio. The change in K~ for various crack lengths is given in Fig. 5 for the four boundary conditions discussed above. For comparison purposes, results are normalized by the stress intensity factor tro~ induced by an infinite plate having a central crack of length 2R. The considerable influence of the steel plates is demonstrated, as well as the minor effect of the two extreme boundary conditions, especially for the specimens with the plates (8%). As expected, the K~ for 'smooth' type boundary condition is higher, since the specimen is allowed to expand horizontally, as compared to the clamped case. The fact (shown in Fig. 5) that K~ is a function which undergoes maximum value, indicates that a stable crack growth is expected in the decreasing range of the function (for a/R > 0.2 and a > 0.5 mm). This has a clear advantage over other standard testing methods, (three or four point bend specimen), where the crack grows unstably, thus producing only one data point for each test, whereas in the suggested test method one may

Fracture toughness of brittle materials

219

Smooth -with plates Clamped-with plates ¢, Smooth -no plates × Clamped-no plates

KI

o0A-ff 0.5 0.4 0.3 0.2

7,." 0.1

~'

0.0

.... 0.0

i

0.1

. . . . . . .

0.2

0.3

13( ~ a / R

0.4

0.5

Fig. 5. The variation of the stress intensity factor normalized by a 0 x ~ with respect to the current crack length a normalized by hole radius R for 4 cases, calculated by the COD method (via FEM). It shows a stable crack growth for a/R > 0.2.

o o q~--~

0.5 0.4 0.3 0.2 0.1

o GS

Smooth

× GS

Clamped =a/R

0.0 0.0

0.1

0.2

0.3

0.4

0.5

Fig. 6. The variation of the stress intensity factor normalized by a o x / ~ with respect to the current crack length a, normalized by hole radius R for specimens with steel plates, using two different calculation methods: direct FEM results using singular element and the approximate Green-Shwartz method.

get a handful of d a t a p o i n t s (critical l o a d at various crack lengths) by one single specimen. In Fig. 6 the K~ values o b t a i n e d b y F E M for specimens with steel plates based on (10) are c o m p a r e d to the results of the analytical G r e e n - S h w a r t z (GS) solution based on (8). The a g r e e m e n t is sufficient for engineering p u r p o s e s a n d s u p p o r t s the use of either of them as convenient.

4. Test results and discussion T h e p r o c e d u r e starts with m e a s u r i n g the tensile strength of the considered brittle m a t e r i a l via the suggested compressive test. U n n o t c h e d specimens with a hole (10 for each material, IC-10 a n d IC-21) were tested u n d e r compressive l o a d until fracture. The n o m i n a l compressive stresses

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J. Tirosh et al.

at fracture (i.e. the critical average stress applied by the press on the specimen at fracture) were a0c = 1370 and 1270 MPa for the above materials with standard deviations of 8 and 9 percent, respectively. Due to the presence of the hole in the specimens the induced maximum tensile stress (which triggers the fracture) was calculated for the two extreme boundary conditions (smooth and clamped interfaces) as explained before. The results are summarized in Table 2. As shown in the table, the strength values resulted from the suggested compressive test are relatively close to the TRS values given by the manufacturer (ISCAR) as measured by a classical method (three point bending). The scatter in our fracture data (and hence the strength values) reflect a typical statistical dispersion in strength of very brittle materials as in [4]. The validity of the experimental setup and measurement techniques have been therefore established. By the basic principle of linear fracture mechanics, stating that a crack initiates when its SIF reaches the critical value of K~c, the associated critical compressive stress, aoc [that is in our case the press load P/(specimen contact area with the plate) at fracture] can be assessed for a notched material if the toughness K,c is known. Conversely, if experimental data provide information on aoc at various crack lengths, the material property K~c can be calculated. This is the procedure undertaken here. Experimental crack growth results are shown in Figs. 7 and 8 based on the test of 3 to 6 specimens for each material and for five initial crack lengths between 0.1 and 0.5 mm. Each data point in the figures corresponds to the load at the initial crack length before its incremental growth rather than at the arrest point (crack arrest points were omitted, since they do not reflect the static K~c values). Results for crack lengths less than 0.63 mm (the value of the q) were corrected to account for the in-situ thickness reduction associated with the cutting phase at the edge of the hole. Each figure contains two solid curves, both obtained using (8). The first curve is obtained for the known Kic value given by the manufacturer, and the second curve is the best fit solution for the experimental data shown. The K~c value which parameterizes the best-fit-curve provides the suggested fracture toughness of the tested material. The deviation of the presented K~c values for the two materials (13.33 and 17.75 M p a x / m ) from the values provided by the manufacturer (14.20 and 17.68 M p a x / m ) is less than 7 percent. Some conclusions can be derived from this investigation. 1. The agreement between the theoretical based curves and the experimental data, especially for crack sizes over 0.4mm where a stable crack growth is predominant, are seen to be satisfactory for both materials (Figs. 7 and 8). This agreement indicates that the suggested method to assess K~c is feasible. 2. Since all data points are gathered close to the theoretical curves, it is evident that there is not an apparent difference in the nominal load between artificial notches (done by EDM, in our case) and natural cracks of the same length. This supports the theoretical prediction, (2), stating Table 2. Tensilestrength predictions for the current specimen, based on experiments and FEM analysis, compared to manufacturer's data (ISCAR). ao = P/2bt~ is the average compressivestress applied on the specimen and %c is its critical value at failure

Material

fl = I%x(0,R)/aol Analytical predictions smooth

%c (current specimen tests)

clamped

fl%c(tensile strengthprediction) smooth

IC-10

1.38

1.49

{MPa) 1830

IC-21

1.38

1.49

1662

TRS (ISCAR data)

clamped (MPa)

2525

2727

(MPa) 2400

2294

2476

2500

Fracture toughness of brittle materials

2.0

I

I

I

I

~

'

~

'

I

~

~

I

221

'

~

GOC ( GPa ) 1.5

initial crack K = 14.2 MPa.m ~/2"

1.0

..

0.5

,~,,¢H ~

_

g

~

~

~ ~ -

13.33 MPa.m ~/2 i

0.0

i

~

0.0

I

0.2

i

~

,

I

i

i

0.4

i

)

I

~

l

0.6

0.8 o~ = a / R

Fig. 7. Critical stress aoc as a function of the current crack length a normalized by hole radius R for IC-10. The solid lines parameterize the K m of the material. One line is based on the K~c given by the manufacturer (ISCAR). The second line is the predicted K~c resulting from the best fit of the K~ solution employing the Green-Shwartz technique, (8) to the experimental data. Each data point reflects a different initial notch length ranging from 0.1 to 0.5 ram.

GOC

2.0 No initial crack

(GPa) 1.5

KIC = 17.68 MPa.m ~/2 (ISCAR) 1.0

0.5

17.75 MPa.m (Best Fit)

0.0

i

0.0

L

i

i

i

]

0.2

0.4

I

~

i

1/2

i

0.6 0~=

0.8 a/R

Fig. 8. Critical stress aoc as a function of the current crack length a normalized by hole radius R for IC-21. The solid lines parameterize the K~c of the material. The first line is based on the K~c given by the manufacturer (ISCAR). The second line is the predicted K~c resulting from the best fit of the K~ solution employing the Green-Shwartz technique, (8) to the experimental data. Each data symbol reflects a different initial notch length, ranging from 0.1 to 0.5 mm.

222

J. Tirosh et al.

that the local tensile stress is insensitive to the crack tip radius when compression load is applied. This fact has a practical advantage in relaxing the extreme precision in manufacturing and machining of the suggested compact specimen. 3. A large stable crack growth region is clearly observed experimentally for ~/> 0.2. It reconfirms that K~c is relatively insensitive to the crack length (except for small cracks). This is a basic demand needed to ensure a reliable fracture data, since real measurement of crack length is encountered generally with technical difficulties, especially for very small cracks. For ceramics and similar materials, this point is noteworthy. 4. For small EDM notches (a < 0.3 mm), experimental results of fracture toughness are lower than the predictions for both materials. No solid explanation has been found for this phenomena, and it may be related to a size effect, i.e. small cracks should be analyzed in a different way.

Conclusions (a) The suggested small size specimen for testing fracture toughness by compression seems suitable since results for TRS and K~c agreed well with tests done by other methods. (b) The specimen exhibited a relatively large region of stable crack growth, for which there is no need to measure the crack length accurately (provided ~ ~> 0.2). (c) Results were insensitive to the notch tip configuration. Fracture data seem to be reproducible in spite of the fact that the notches, produced by EDM, have an inconsistent shape to their notch tip. Analytical support of this observation is given in (2) based on the stress solution around an elliptical notch. (d) No special fixtures were used (beside the two free steel pads) and hence the experimental set-up is relatively cheap. (e) The simple (but approximate) Green Shwartz method employed here to formulate the K~ function seems to be in good agreement with the more precise FEM solution (using the ANSYS code).

Acknowledgement The authors thank the Ministry of Science and Technology of the Israel Government for funding this work via the Technion, Israel Institute of Technology.

References 1. 2. 3. 4. 5. 6.

L.M. Barker, Engineering Fracture Mechanics 9 (1977) 361-369. S. Mostovoy, P.B. Crosley and E.J. Ripling, Journal of Materials 23 (1967) 661~81. D. Munz, R.T. Bubsey and J.E. Srawley, International Journal of Fracture 16 (1980) 359-374. R. Porat and J. Malek, Material Science and Engineering A105/106 (1988) 289-292. G.N. Savin, Stress Concentration Around Holes, Pergamon Press (1961). N.I. Muskhelishvili, Some Basic Problems in Mathematical Theory of Elasticity, P. Noordhoff Ltd., Groningen, The Netherlands (1964). 7. G.S. Sih, Handbook of Stress-lntensity Factors, Lehigh University (1974). 8. J. Tirosh, Journal of Applied Mechanics, ASME Transaction 99 (1977) 449-454. 9. J. Tirosh and A.S. Tetelman, International Journal of Fracture 12 (1976) 187 199.