1995 Warren Determining the Fracture Toughness of Brittle Mtls by Hertz Indent

8/2/2019 1995 Warren Determining the Fracture Toughness of Brittle Mtls by Hertz Indent

1/7

Journal of the European Ceramic Society 15 (1995) 201-207Elsevier Science LimitedPrinted in Great Britain09552219/95/$9.50

Determining the Fracture Toughness of BrittleMaterials by Hertzian IndentationP. D. WarrenDepartment of Materials, Oxford University, Parks Road, Oxford OX1 3PH, UK(Received 21 December 1993; revised version received 14 July 1994; accepted 1 August 1994)

AbstractThe use of Hertzian indentat ion as a meth od fordetermi ning the fracture tou ghness of any britt lemat erial is described. The advantage of t he met hoddescribed here i s that the only quanti ty to be measuredis a fracture load. Experimental determina tions ofK,, for soda-l ime glass and high-purity aluminagive results 0.71 and 3.7;? MPa m, respectively.

Introduction

The fracture toughness of a brittle material is oneof its most important mechanical properties.Determination of this value is frequently a compli-cated process involving preparation of specimenswith well-defined sharp cracks of known length. Arecent review article has presented a summary ofavailable techniques. However, one method wasconspicuous by its absence-that of Hertzianindentation. It is the purpose of this paper to sug-gest that this method should be looked at afresh.Indentation techniques in general have certainadvantages compared with the more conventionalmethods: the experimental procedure is straight-forward, involving minimal specimen preparation,and the amount of material needed is small. TheVickers indentation method2- is well known.However, it is not without drawbacks. There arenow a very large numbe.r of theoretical models inthe literature (19 are reviewed by Ponton andRawlings5,6) relating the isurface crack length mea-sured after indentation to the indenter load andmaterial parameters-Youngs modulus, hardnessand fracture toughness. There is still considerabledebate as to the nature of the cracks observedaround a Vickers indentation-the idealised half-penny characteristics of -the median/radial system,as opposed to Palmqvist cracking. Cook andPharr7 have presented a considerable amount ofevidence to suggest that radial cracks form almost

immediately on loading in a wide variety of crys-talline solids. There is also debate on how todetermine the type of cracking from the surface-crack length vs applied load relation. For aspirited discussion of these matters the articles byLi et al.89 and Srinivasan and Scattergood1oshould be consulted. It should be noted, however,that Lawn asserts that Palmqvist cracks . . .retain the essential half-penny character.Many attempts have been made to use Hertzianindentation-where a hard sphere is pressed intothe flat surface of a brittle substrate-to determineK,, for brittle materials: work by Frank andLawn-glass,12 by Powell and Tabor-TiC,i3 byWilshaw-glass, l4 by Nadeau-vitreous carbon,15by Warren-Tic, ZrC, VC and WC,r6 by Matzkeet al.-U02,17 by Matzke-Th02,18 by Inoue andMatzke-ThO,, by Matzke and Warren-Th02,20 by Matzke and Politis-NbC, Nb(C,N)and NbN,21 by Matzke et al.-(U,Pu)C and(U,PU)(C,N),~~ by Laugier-TiN coatings on WC-Co 23 Al,O,-ZrO 24 variouspoiites and Sialo&25 Al,O,-ZrO, com-various WC-Co compos-ites26 and finally, by Zeng et al.-glass, Al,O, andAI,O,-SIC composites.27j28

Broadly speaking, the early work, prior to 1978,used the thebry of Frank and Lawn,12 whichstrictly only applies for ring-cracks initiatingexactly at the edge of the contact zone. The pa-pers by Warren and co-workers,1G19,21,22 (andthose of Laugier23-26)used Warrens theory,16 whichinvolves measuring the radius of the ring-crackseen on the surface after fracture: Matzke andWarren and the recent work by Zeng et a1.27,28involved the cross-sectioning of the specimensafter fracture to determine the depth of the conecrack. The results obtained so far have been variable.Almond et aZ.29point out that the values obtainedby Warren16 on various metal carbides are signifi-cantly lower than conventional values; Laugier,26on the other hand, for various WC-Co compos-ites, found values considerably higher. This poor201


2/7

20 2 P. D. W arrenagreement may be one reason why the test is notmore commonly used. This is a pity because theHertzian indentation test has at least one consid-erable advantage over Vickers indentation: untilfracture occurs around the indenter the deforma-tion of the substrate is entirely elastic (unlessspheres of very small radius, ~1 mm, are used).This means that the complications associated withthe residual stresses in the Vickers indentationtechnique do not exist. Furthermore, the elasticstress field is well known.30The disadvantages in the method may be due tothe following factors: (i) the stress gradients in theHertzian field are very steep so that it is difficultto obtain accurate estimates for stress-intensityfactors for cracks driven by Hertzian loading; (ii)the results of the analysis are extremely sensitiveto the value of Poissons ratio of the substrate,and (iii) a number of previous experimental deter-minations of fracture toughness have ignored theeffects of elastic mismatch (and friction at theindenter-substrate interface) when the sphere andthe substrate are made of elastically dissimilarmaterials. Much work has been done using eithersteel or tungsten carbide spheres on glass sub-strates and there is good experimental evidence3that the effects of this significant elastic mismatchon the observed fracture loads are not negligible.Below, we summarise the main aspects of thetheory of Hertzian fracture and present a methodwhich enables the value of Kit to be found for anybrittle material, as long as it is indented by asphere made of the same material. The methoddescribed here is extremely easy to use, does notrequire measurement of the ring-crack size andgives reliable results.

TheoryFigure 1 shows a sphere (radius R, elastic constantsvi, EJ pressed by a load P into a flat substrate(with elastic constants v, E). The load is supportedover a circular area whose radius, a, is given by:32

where:1 1 - 9 + 1 - ZJ:-=_

E* E EI

(1)

(2)The peak pressure under the sphere, p,,, is given by:

3PPO = && (3)The stress field outside the contact zone has atensile radial stress near to the surface which

Fig. 1. The geometry of Hertzian contact.

rapidly decreases, and soon becomes compressive,with depth. Thus calculating the stress intensityfactors for surface-breaking cracks is not trivial.For short cracks (i.e. those whose depth is lessthan a/10) lying perpendicular to the free surfacethe method of Nowell and Hills33 (employing dis-tributed dislocations) can be used to derive expres-sions for the mode I (and mode II) stress intensityfactors. This approach differs from previouswork12~4,16,27,28,34here it was assumed that thecrack path initially lay along the trajectory of theminimum principal stress; here, the pre-cursorflaws are assumed to be perpendicular to the sur-face; this will tend to reduce the Ki values derivedhere compared to previous results. The use of themethod of distributed dislocations enables theeffects of the free surface to be accounted for; thiswill tend to increase the Kr values. In this method,a short plane crack of depth c, normal to the freesurface, is placed close to the contact zone. Thestate of stress in the cracks absence is found.When the crack is inserted, unsatisfied tractionsappear along the line of the crack. These may becancelled by the application of equal and oppositetractions along the crack faces, which may be gen-erated by installing a distribution of dislocations.(These dislocations are not real dislocations in acrystalline lattice, but a mathematical device.) Thestate of stress induced by one of these dislocationsis known-the expressions are given explicitly inRef. 33. By applying a distribution of dislocations,of unknown density B,(z), the requirement thatthe crack faces be traction-free may be ensured bywriting

0 = gj,(z, r) + G IT(K+l)O Nz)Wz, r) dz (4)G is the shear modulus of the material, K isKolosovs constant (= 334~ in plane strain), thecoordinates r, z are as in Fig. 1; the functionK(z,r) is given in Ref. 33. The unknown quantity,B,(z), is defined by B,(z) = db,ldz, and 6 repre-sents the state of stress induced by the contact inthe cracks absence, i.e. in this case the radial


3/7

Fracture toughness of brittle materials by Hertzian indentation 203

AL 0.02P,& 0.016

0.001 0.01 0.1c/a

Fig. 2(a). Normalized mode I stress intensity factors as afunction of normalized crack size for six normalized crackpositions. Poissons ratio = 0.24.

: : (1 1.2 1.4 1.6 1.8 2

r/aFig. 2(b). Normalized mode I stress intensity factors as afunction of normalized crack position for six normalizedcrack sizes. Poissons ratio = 0.24.

O.CO8 0.012 0.016r/a 1.51.4 0017

0.0171.30.016

1.2 0.0120.0081.1 O.W4

10.01 0.025 0.05 0.075 0.1

daFig. 2(c). Contours of normalized mode I stress intensity fac-tors Ktl(p,-\l~~~) s a function of crack depth (O@OS c/u < 0.1)and crack position (1.0 < r/lz < 1.5). The filled circle showsthe maximum value for c/a = 0.025; at this position (r/u _1. ), K, - 0.0182 p,dma. The filled square marks the positionof the absolute maximum: K, - 0.01937 p,d~u at c/a - 0.046,r/u *-1.18.

component of the Hertzian stress field. This inte-gral equation is effectively represented by a set oflinear algebraic equations, typically 20: the (nor-malized) mode I stress intensity factor can then beexpressed in terms of th(e unknown coefficients inthese linear equations. A standard computerlibrary routine for solution of simultaneous equa-tions is then employed. This technique is very welladapted to problems wh.ere cracks exist in a steepstress gradient. If the radial Hertzian stress (= a)

is expressed in terms of the peak Hertzian pressure,pO, (eqn (3)) then the result of the calculation is anumber p which is related to the (dimensional)mode I stress intensity factor, K,, by:

(5a)Normalizing with respect to a (rather than c) wehave:

KI-_ =P&a j.dc/a (5b)

The rationale for this second normalization is thatthe dependence of KI on the crack size is nowexclusively in the two terms on the RHS of theequation. All subsequent results presented beloware in terms of this second normalization. Warrenet al.35 present results for the case where thesphere and the substrate are made of the samematerial. In this case the stress field in the sub-strate (and hence the values of KJ depends ononly one material parameter-Poissons ratio ofthe substrate. Figures 2(a), (b) and (c) show somebasic results from this analysis.Figure 2(a) shows the normalized mode I stressintensity factor as a function of normalized cracklength (c/a) for six normalized crack positions (r/a),for Poissons ratio = 0.24. Figure 2(b) shows val-ues for the normalized KI as a function of r/a forsix values of c/a; again v = 0.24. Figure 2(a) showsthat KI has a maximum value as a function of c/afor each da; furthermore for large cracks situatedclose to the contact radius the implied value of KIis negative (because the crack tip is now deepenough that it lies within the compressive region ofthe Hertzian radial stress). Figure 2(b) shows thatfor each c/a the maximum in KI occurs for r/a > 1.This explains the well-known fact that ring cracksare always observed to form outside the contactzone, even though the surface radial stress has itslargest value at r/a = 1. This point has been notedpreviously. 16134$36igure 2(c) shows a contour plotof K,l(p,dTa) as a function of crack position andcrack depth (both normalized). We note twothings. For each c/a, we can mark the position ofthe largest normalized K,this is shown by thefilled circle for c/a = 0.025. At this position (r/a -1. l), KI - O.O182p,dma); 0.0182/210.025 is thereforethe maximum value of p (= Jo,_) for this value ofc/a, (c/a = 0.025). h is thus just a function of c/aand u, h = g(& v). Secondly KI has an absolutemaximum-marked by the filled square, at c/a -0.046, r/a - 1.18, KI - 0.01937pod7ra.The above expressions are in terms of stress in-tensity factors; we will now re-write them in termsof loads. When KI for one small surface flawreaches K,, (at a load P = PF), it begins to grow-


4/7

204 P. D. W arren

0 . 2 80.240.20.16

Fig. 3. Minimum normalized fracture load necessary to prop-agate a crack of normalized size c/u. Results for five differentvalues of Poissons ratio. The value of the absolute minimumfor v = 0.28 is shown.to form either a ring crack or the well-knownring-cone system. Using eqns (1) and (3), equa-tion (5b) becomes:

E*PF -IT-=RG 3p2(c/a) (6)

We define a normalized fracture load, P,,:PFN = yg

IC(7)

Assuming that the flaw distribution is denseenough so that there will always be a flaw (of sizec/a) situated close to the position where the cracktip stress intensity experienced is a maximum (forthis crack size) eqn (6) can be writtenP,, = =3&n&a)

These values of PFN are therefore the minimumnormalised loads necessary to propagate cracks ofsize c/u. Plots of PFN as function of c/u (for fivedifferent values of V) are shown in Fig. 3. For allvalues of Poissons ratio there is an absolute mini-mum in the normalized fracture load for c/a val-ues in the range 0.02-0.06. For a given V, thisabsolute minimum m P,, corresponds to theabsolute maximum in K,l(p,~~u). For example,

millPFN16000 ~1400012000100008000600040002000

0

(C/a) mT OO,+ 0 . 0 6

0.1 0.14 0.18 0.22 0.26 0.3 0.34

P oi s so n ra t i o

Fig. 4. Absolute minimum normalized fracture loads (P$)and the corresponding values of (c/u) as a function ofPoissons ratio.

for v = 0.24 Fig. 2(c) shows that the absolutemaximum in Krl(p,~7ru) is 0.01937; therefore theabsolute minimum in normalized fracture load,P*, is 7r/3/(0.01937)2 - 2790. The values of theseabsolute minima and the corresponding values ofc/a as a function of Poissons ratio are shown inFig 4 and also listed in Table 1. We emphasisethat P,$ is a dimensionless quantity. (For all val-ues of Poissons ratio the absolute minimum infracture load occurs at a relative crack position r/u- 1.20.)

Determination of K,,The principle of the method for determining K,, isnow clear. For a material with a given E* and Krcone searches for the minimum value of P,/R. Ifone performs a series of Hertzian tests with asphere of given R (25 tests should be sufficient),noting the fracture loads PF, there exists a definiteminimum load for fracture, as shown in Fig. 5. Ifone assumes that the crack propagated in this test(by a load PFmin) is of a size within the rangewhere P,, shows a minimum with c/u then thefracture toughness can be determined from thefollowing equation:

K,, = (E;%&)li (9)where the number P,$, can be read from Table 1assuming that Poissons ratio for the substrate isaccurately known. Similar results, but for specificr/u values (r/u = 1.05, 1.10, 1.20) were published byMatzke et ~1.~Since typical Hertzian contact radiiare of the order of a few hundred microns (forspheres with radii l-5 mm), to obtain the appropriatec/a values shown in Table 1 one needs flaw sizes,c > 10 pm. Therefore the specimen surface should beprepared by coarse diamond polishing or abrasionwith fine SIC grits. Severe grinding of the surfaceTable 1. P$j and the corresponding c/a values as a functionof v

V pm i nFN (cWmi, V PFk (c/a)mi,0.10 789 0 . 0 6 7 9 0.23 2490 0.04720.11 850 0.0664 0.24 2790 0.04560.12 917 0.0648 0.25 3131 oGI440.13 991 0.0632 0.26 3530 0.04230.14 1074 0.0616 0.27 4001 0.04070.15 1.167 0.0600 0.28 4560 0.03910.16 1270 0.0584 0.29 5229 0.03740.17 1386 0.0568 0.30 6037 0.03570.18 1517 0.0553 0.31 7022 0.03410.19 1665 0.0537 0.32 8235 0.03240.20 1883 0.0521 0.33 9748 0.03070.21 2025 0.0504 0.34 11658 0.02900.22 2247 0.0488 0.35 14106 0.0273


5/7

Fracture toughness of brittle materials by Hertzian indentation 205Probability of 6acture

10.90.80.70.60.50.40.30.20.1

0 I400 500 600 700

Fracture load (N)800 900

Fig. 5. Experimentally determined fracture loads, showing aminimum fracture load at P, = 470 N. Material (substrateand sphere) - alumina; sphere radius = 2.5 mm.

may introduce flaws of suitable depth. but it is wellknown37 that such treatment can introduce consid-erable surface residual dresses. The tests can thenbe repeated with spheres of different radii and theminimum value of P,,,IR found. This methodrequires no knowledge of the initial crack size, nomeasurement of the radius of the ring-crack seenon the surface after fracture, nor any measurementof the final depth of the cone-crack. It is importantto note that the average fracture load found willgive an overestimate of K,, because not all crackswill be situated at the particular r/a value for whichK,, is a minimum-most cracks will be found atgreater r/a values, since in reality the flaw density isnot infinite.It is appropriate here to discuss some of thelimitations of this approach. With the sharpindentation method, part of the uncertainty in theanalysis lies in the assumption of a particularcrack geometry (median--radial or Palmqvist) afterindentation. With the Hertzian method, the uncer-tainty lies in the value of the constant P$, whichin turn depends on the assumed shape of the pre-cursor flaw before indentation. Warren et a1.35assumed that the pre-cursor flaws were of a givendepth c, but large in extent (with respect to thecontact radius) across the surface, i.e. the sort offlaw produced by a sharp point dragged across thesurface. Work is currently in progress3 to evalu-ate stress intensity factors for different crackgeometries, such as semicircular or semi-ellipticalflaws. The analysis presiented here assumes thatthe sphere and the substrate are made of the samematerial. The method also assumes that the mate-rial does not show a substantial T-curve, i.e. in-creasing fracture resistance with crack extension.Finally, both the Vickers and the Hertzian methodare affected by the presence of near-surface resid-* Engineering Systems (Nottm), 1 Loach Court, RadfordBridge Road, Nottingham NG8 INB

Table 2. Material propertiesMaterial v E(GPa) P$Glass 0.25 70 3131Alumina 0.24 390 2790

ual stresses. The Hertzian method may be particu-larly affected because the surface finish shouldspecifically not be a fine diamond polish; the sur-face preparation method needs to be carefullychosen to produce 5-10 pm flaws without intro-ducing residual stresses.

Experimental Technique and ResultsHertzian indentation tests were performed using amodified ET500 testing machine (Engineering Sys-tems (Nottm)*). A wide-band acoustic emissiontransducer was mounted in the loading train todetect the growth of a small surface flaw into thecharacteristic ring- or ring-cone-crack. Twodifferent substrate materials were used-glass, andhigh purity (99.99%) alumina. The assumed elasticconstants and the expected minimum normalisedfracture load, P$/, for the materials are listed inTable 2. In each case two different sphere radiiwere used-2.5 and 5 mm. The spheres were madeof glass and alumina. The specimens had a varietyof surface treatments: the alumina was eitherground with 600-grit SIC slurry on a cast iron lap,or ground and then polished with 6 pm polycrys-talline diamond powder; the glass was tested inthe as received condition, and also after grindingwith 600-grit SIC. In each case about 25 tests wereperformed. The results are shown in Table 3 forglass and Table 4 for alumina. The average frac-ture loads (k one standard deviation) are alsolisted.We therefore estimate the values of K,, (derivedfrom the minimum fracture loads from the SiC-abraded surfaces) to be 0.71-0.74 MPa m12 forglass and 3.72-3.80 MPa ml* for alumina. Thesevalues are in good agreement with those in theliterature-for example Zeng et al.* found valuesO-8 and 3.77 MPa m12 for glass and alumina,

Table 3. Glass resultsSurface Sphere radius Av erage Min imu m K ICtreatment (mm) fractu re fracture (MPa m)load (N) load (N)

As-receivedSIC grit

2.5 290 f 110 127 0.785.0 638 f 135 340 0.902.5 158 f 95 105 0.715.0 348 f 107 231 0.74


6/7

206 P. D. WarrenTable 4. Alumina results

Surface Sphere radius Average Minimum K,,treatment (mm) fracture fracture (MPa mli2)load (N) load (N)Sic gritDiamond

2.5 739 f 160 470 3.7250 1600 f 280 982 3.802.5 849 f 145 540 3.995.0 1751 f 260 1273 4.33

respectively. It can be seen that in every case theaverage fracture load is at least 35% bigger thanthe minimum load, and as expected the calculatedvalue of K,, is most accurate for the most badlydamaged surface indented with the smallestsphere. This is because the more damaged the sur-face the larger c/a because of the increase in c; thesmaller the sphere the larger c/a because of thereduction in a. In either case, the more likely it isthat a crack of the appropriate size will be found.Some of the experimental limitations are nowdescribed. (1) The crack extension must be sulhcientlyrapid to be detected by the acoustic emission trans-ducer-.39 n all cases here, the detection of an acous-tic signal corresponded to the appearance of acircular crack seen on the surface; in a very few cases,examination of the surface after detection of a sig-nal revealed the presence of two ring cracks, sug-gesting that the first fracture event was not pickedup. Because the sphere and substrate were made ofidentical materials, it is unlikely that crackingoccurred on unloading; Johnson et a131 showed thatif the substrate is more compliant than the indenterthen the frictional tractions at the substrate-inter-face, which reduce the radial tension on loading,enhance the radial tension on unloading. (2) If anacoustic emission transducer is not available thenthe detection of the minimum load must be doneby trial and error. (3) Obviously, the substrate mustcrack before the sphere does, this was not found tobe a problem but the fact that it may occur sug-gests that tough spheres with the appropriate elasticconstants (to minimise the extent of elastic mis-match between the sphere and the substrate) mightbe used-for instance, glass-ceramic spheres totest glass, or silicon nitride spheres to test otherengineering ceramics. The effects of elastic mismatchon Hertzian fracture have been analysed,40 the useof different indenter materials can substantiallyaffect the value of P$.

ConclusionsUse of a refined stress intensity factor formulationfor surface-breaking cracks in steep stress gradi-ents3 has enabled accurate estimates to be made of

the minimum loads necessary to propagate cracksby Hertzian indentation. At present the analysisonly applies for the case where the sphere and thesubstrate are made of the same material. However,the analysis is comprehensive in that it can beapplied to sphere-substrate systems with any valueof Poissons ratio. By measuring this minimumload (and that is the only quantity that must bemeasured) an accurate estimate of Krc may easilybe made. The only other requirement is that thesurface should be coarsely polished and a sphere ofrelatively small radius (~5 mm) should be used.Two further points are worthy of note. Becausethe only quantity that must be measured is a frac-ture load, it is possible that this method of deter-mining Krc can be automated. Also, the existenceof an absolute minimum fracture load, for a givensphere size, suggests that the Hertzian indentationtest could find use as a localized proof-test.

AcknowledgementsThe author gratefully acknowledges discussionswith Professor Sir P. B. Hirsch, Dr S. G. Robertsand Dr D. A. Hills.

References1.2.

3.

4.

5.

6.

7.

8.

9.

10.

Sakai, M. & Bradt, R. C., Fracture toughness testing ofbrittle materials. Znt. Met. Rev., 38 (1993) 53-78.Evans, A. G. & Charles, E. A., Fracture toughness deter-minations by indentation. J. Am. Gram. Sot., 59 (1976)371-2.Lawn, B. R., Evans, A. G. & Marshall, D. B.,Elastic/plastic indentation damage in ceramics: the me-dian/radial system. J. Am. Ceram. Sot., 63 (1980) 574-81.Anstis, G. R., Chantikul, P. Lawn, B. R. & Marshall, D.B., A critical evaluation of indentation techniques formeasuring fracture toughness. J. Am. Ceram. Sot., 64(1981) 533-8.Ponton, C. B. & Rawlings, R. D., Vickers indentationfracture toughness test-Part 1: Review of literature andformulation of standardised indentation toughness equa-tion. Mat. Sci. Technol., 5 (1989a) 86572.Ponton, C. B. & Rawlings, R. D., Vickers indentationfracture toughness test-Part 2: Application and criticalevaluation of standard&d indentation toughness equa-tions. Mat. Sci. Technol., 5 (1989b) 961-76.Cook, R. F. & Pharr, G. M., Direct observation andanalysis of indentation cracking in glasses and ceramics.J. Am. Ceram. Sot., 73 (1990). 787-817.Li, Z., Ghosh, A., Kobayashi, A. S. & Bradt R. C.,Indentation fracture toughness of sintered silicon carbidein the Palmqvist regime. J. Am. Ceram. Sot., 72 (1989)904-11.Li, Z., Ghosh, A., Kobayashi, A. S. & Bradt, R. C.,Reply to Comment on Indentation fracture toughness ofsintered silicon carbide in the Palmqvist regime. J. Am.Ceram. Sot., 74 (1991) 889-90.Srinivasan, S. & Scattergood R. O., Comment on Inden-tation fracture toughness of sintered silicon carbide inthe Palmqvist regime. J. Am. Ceram. SOL, 74 (1991)887-8.


7/7

Fracture toughness of brittle materials by Hertzian indentation 20711. Lawn, B. R., Fracture oj Brittle Solids, Second Edition.Cambridge University Press, Cambridge, 1993, p. 259.12. Frank, F. C. & Lawn, B. R., On the theory of Hertzianfracture. Proc. R. Sot. Lond., A229 (1967) 291-306.13. Powell B. D. & Tabor, D., Fracture of TiC under staticand sliding contact. J. 1hy.s. D: Appl. Phys., 3 (1970)783-8.14. Wilshaw, T. R., The Hertzian fracture test. J. Phys. D:Appl. Phys., 4 (1971) 1567-81.15. Nadeau, J. S., Hertzian fracture of vitreous carbon. J.

Am. Ceram. Sot., 56 (1973) 467-72.16. Warren, R., Measurement of the fracture properties ofbrittle solids by Hertzian indentation. Acta Metafl., 26

(1978) 1759969.17. Matzke Hj., Inoue, T. & Warren R., The surface energyof UO, as determined by Hertzian indentation. J. Nucl.

Mater., 91 (1980) 205-20.18. Matzke, Hj., Hertzian indentation of thorium dioxide,ThO,. J. Mater. Sci., 15 ( 1980) 73946.19. Inoue, T. & Matzke, Hj., Temperature dependence ofHertzian indentation fracture surface energy of ThO,. J.Am. Ceram. Sot., 64 (1981.) 355-9.

20. Matzke Hj. & Warren R., Hertzian crack growth in Th02observed by serial sectioning. J. Mater. Sci. Lett., 1, (1982)4414.21. Matzke, Hj. & Politis, C., The fracture properties ofNbC, Nb(C,N) and NbN by Hertzian indentation. Phys.Stat. Sol., A69 (1982) 269980.

22. Matzke, Hj., Meyritz, V. It Routbort, J. L., Fracture-sur-face energy and fracture toughness of (U,Pu)C and(U,Pu)(C,O). J. Am. Ceram. Sot., 66 (1983) 183-8.23. Laugier, M., Load bearing capacity of TiN coated WC-Co cemented carbides. J. Mater. Sci. Lett., 2 (1983) 419-21.24. Laugier, M., Hertzian indentation of sintered alumina. J.Mater. Sci., 19 (1984) 254-8.

25. Laugier, M., Toughness determination of some ceramictool materials using the method of Hertzian indentationfracture. J. Mater. Sci. Left., 4 (1985) 154224.26. Laugier, M., Hertzian indlentation of ultra-fine grain sizeWC-Co composites. J. Mater. Sci. Lett., 6 (1987) 841-3.27. Zeng, K., Breder, K. & Rowcliffe, D. J., The Hertzianstress field and formation of cone cracks-I. Theoreticalapproach. Acta Metall. Mater., 40 (1992a) 25955600.

28. Zeng, K., Breder K. & Rowcliffe D. J., The Hertzianstress field and formation of cone cracks-II. Determina-tion of fracture toughness. Acta Metall. Mater., 40(19926) 260145.

29. Almond, E. A., Roebuck, B. & Gee, M. G., Mechanicaltesting of hard materials. In Science of Hard Materials,Inst. Phys. Conf. Ser. Vol. 75, pp. 155-77.30. Huber, M. T., On the theory of contacting solid elasticbodies. Ann. Phys., 14 (1904) 153-63.31. Johnson, K. L., OConnor, J. J. & Woodward, A. C.,The effect of the indenter elasticity on the Hertzian frac-ture of brittle materials. Proc. R. Sot. Land. A., 334(1973) 95-117.32. Hertz, H., On the contact of elastic solids. Zeitschrzjt fur

die Reine und Angewandte Mathematik, 92 (1881) 15671.English translation in Miscellaneous Papers (translated byD. E. Jones and G. A. Schott); pp. 14662. Macmillan,London, U.K., 1896.33. Nowell D. 8~ Hills, D. A., Open cracks at or near freesurfaces. J. Strain Anal., 22 (1987) 177-84.34. Mouginot, R. & Maugis, D., Fracture indentationbeneath flat and spherical punches. J. Mater. Sci., 20

(1985) 435476.35. Warren, P. D., Hills, D. A. & Roberts, S. G., Surfaceflaw distributions and Hertzian fracture. Submitted to J.Mater. Res.

36. Finnie, I., Dolev, D. & Khatibloo, M., On the physicalbasis of Auerbachs law. J. Engng Mater. Technol., 103(1981) 1834.37. Marshall, D. B., Evans, A. G., Khuri-Yakub, B. T., Tien,J. W. & Kino, G. S., The nature of machining damage inbrittle materials. Proc R. Sot. Lond A., 385 (1982)461-75.38. Dai, D. N., Hills, D. A., Warren, P. D. & Nowell, D.,The propulsion of surface flaws by elastic indentationtesting. Accepted for publication, Acta Metall. et. Mater.39. Scruby, C. B., Quantitative acoustic emission techniques.

In Research Techniques in Non-destructive Testing, vol.8, ed. R. S. Sharp. Academic Press, London, 1985,p. 141.40. Warren, P. D. & Hills, D. A., The influence of elasticmismatch between indenter and substrate on Hertzianfracture. J. Mater. Sci., 29 (1994) 2860-6.

Documents

1995 Warren Determining the Fracture Toughness of Brittle Mtls by Hertz Indent