Analytical Approaches to Stress Intensity Factor Evaluation forIndentation Cracks

Alessandro Leonardiw and Franco Furgiuele

Mechanical Engineering Department, University of Calabria, 87036 Arcavacata di Rende (CS), Italy

Stavros Syngellakis and Robert J. K. Wood

School of Engineering Science, University of Southampton, Southampton SO17 1BJ, U.K.

In the present paper, an analytical solution for the stress inten-sity factor in the case of cracks produced by Vickers indentershas been extended to the cases of triangular indenters, i.e., Be-rkovich and cube-corner. According to the adopted approach,median/radial cracks produced by indentations are modeled asloaded by either a point-force or a symmetric disk-shapedwedge. The wedge diameter is assumed to be equal to the plas-tic zone size whereas the wedge thickness is evaluated by com-paring the wedge volume with the hardness-impression volume.The point-force and the disk-shaped wedge analyses produce anupper and a lower bound, respectively, of the geometry-depen-dent parameter appearing in the expression for the fracturetoughness. The predictions of the present analysis are in goodagreement with similar experimental and numerical results.

I. Introduction

INDENTATION methods, consisting of pressing a hard materialinto the test specimen and measuring the response, have been

increasingly used for mechanical characterization of metals,glasses, ceramics, polymer composites, and coating materials.1,2

In recent times, sharp pyramid indenters such as Vickers, Be-rkovich, or cube-corner are preferred to spherical ones becausethey maintain geometrical similarity in the deformation zone.The elastic stress field for sharp indenters is singular about theindenter tip; thus, irreversible deformation modes such as plasticor viscous flow or structural densification are produced. Itshould be noted that the main direct experimental results froman indentation test leading to the elastoplastic properties of thespecimen material may be strongly influenced by indentationcracking. On the other hand, indentation cracking is used inorder to determine the fracture toughness of hard brittle mate-rials. Although Quinn and Bradt3 and Kruzic and Ritchie4 haverecommended that the indentation fracture technique is nolonger acceptable for the fracture toughness testing of ceramicmaterials, indentation is nowadays one of the most commonlyused techniques to evaluate the fracture toughness of ceramicsand coating systems because it is easy to perform, does not needspecial samples, and causes only negligible surface damage.

In the present paper, an analytical solution for the stress in-tensity factor in the case of cracks produced by Vickers indentershas been extended to the cases of triangular indenters, i.e., Be-rkovich and cube-corner. To this aim, indentation cracks pro-duced by sharp indenters have been modeled as cracks loaded byeither a point-force or a symmetric disk-shaped wedge.5 In thelatter case, the wedge diameter has been assumed to be equal to

the plastic zone size whereas the wedge thickness has been eval-uated by comparing the wedge volume with the hardness-im-pression volume.5 The point-force and the disk-shaped wedgeanalyses produce, respectively, an upper and a lower bound ofthe geometry-dependent parameter linking the fracture tough-ness of the indented material to the loading and fracture char-acteristics of the crack patterns produced by indentation.5

Moreover, the analytical solution for the stress intensity factorwas applied to two approaches proposed by Anstis et al.6 andGong et al.7 for the evaluation of the toughness of ceramics byindentation.

II. Indentation Fracture Background

The half-penny pattern for a Vickers-indenter system is shownin Fig. 1(a), where c is the surface crack length and a is the halflength of the diagonal of the indentation impression.

Lawn et al.8 presented a detailed fracture mechanics analysisfor the extension of a half-penny-shaped crack centrally loadedby a point-force P, equal to the indentation peak load; in par-ticular, they showed that an equilibrium between the cracklength c and a residual point-force P, resulting from the elasticaccommodation of the hardness-impression volume, exists lead-ing to the relation

P ¼ kc3=2 (1)

with the k term constant to be determined for any given spec-imen/indenter system.

Based on the same considerations, Anstis et al.6 proposed thefollowing relation for assessing fracture toughness, KIC:

KIC ¼ xE

H

� �1=2P

c3=2; c� a (2)

in which E and H are the Young modulus and the hardness ofthe material, respectively, and x is an experimentally calibratednondimensional constant. This constant is regarded as a mate-rial-independent or an indenter geometry-dependent parameter.The values of x, reported in Table I, have been obtained by ex-perimental calibration with known fracture toughness values fora wide variety of ceramics, initially from Vickers6 and subse-quently from Berkovich9 and cube-corner10 indentation data.

In the same table, the numerical results obtained by Leonardiet al.11 are shown.

Although discrepancies between the fracture toughness mea-sured by other conventional methods and indentation toughnessdetermined using Eq. (2) have been frequently reported,12–14 thelatter equation remains the most frequently used relationship toevaluate the fracture toughness of brittle materials.

The origins of these discrepancies are attributed to variousfactors, including (i) the dependence of crack geometry on theproperties of the test materials and the applied load, (ii) errors in

G. Pharr—contributing editor

wAuthor to whom correspondence should be addressed. leonardi@unical.itManuscript No. 25467. Received November 7, 2008; approved March 7, 2009.

Journal

J. Am. Ceram. Soc., 92 [5] 1093–1097 (2009)

DOI: 10.1111/j.1551-2916.2009.03070.x

r 2009 The American Ceramic Society

1093

the measurement of the crack length c, and (iii) nonideal inden-tation deformation/fracture behavior such as subcritical growthof the indentation cracks, lateral cracking, or phase transfor-mation due to indentation. Another important factor that mayaffect the accuracy of indentation toughness determination isrelated to the definition of hardness, H, used in Eq. (2). In fact,the Vickers hardness defined by Anstis et al.6 asH5P/2a2 is themean contact pressure. The actual hardness is not a constant butdepends on the applied indentation load. Such a phenomenonhas been recognized as the indentation size effect.15

Based on the above considerations, Gong et al.7 proposed amodified indentation toughness equation

KIC ¼ kðEH0Þ1=2a2

c3=2; c� a (3)

where k is a constant independent of the test material whileH0 isthe load-independent hardness.15 From known fracture tough-ness values for a wide variety of ceramics, Gong et al.7 foundk5 (4.3070.7)� 10�2, also reported in Table I.

III. Analysis of Vickers Half-Penny Cracks

For the sake of clarity, the analytical approach proposed byShetty et al.5 to evaluate x for Vickers indentations is describedin this section. In this analysis, the half-penny crack has beenconsidered as centrally loaded by either a point-force or awedge.

The stress intensity factor, KI, for a half-penny crack centrallyloaded by a point-force, F, normal to the crack plane, is given by16

KI ¼ OHPF

ðpcÞ3=2(4)

in which OHP is a free-surface correction factor. From numericalsimulations, not reported here for the sake of brevity, it has beenfound that for a point-force-loaded crack of small dimensionsrelative to the specimen OHP5 1.21, which is the same as thefactor for a half-penny crack subject to remote loading.16

The residual force originating from the indentation plasticzone is represented by F in the present problem. As a conse-quence of the elastic accommodation of the hardness-impressionvolume of the plastic zone, a residual pressure pR develops at theelastic–plastic interface in the indentation problem. Lawn et al.8

showed that

pR ¼E

3ð1� 2nÞVi

Vp(5)

where n is the Poisson ratio, while Vi and Vp are the volumesof hardness impression and plastic zone, respectively. For aVickers indenter, it can be shown by reference to Fig. 2(a) that

Vi ¼ffiffiffi2p

a3

3 tanc(6)

where 2c5 1361 is the apex angle of the Vickers indenters.For a variety of materials, which included metallic alloys and

ceramics, experiments have shown that the plastic zones associ-ated with symmetric indenters, like Vickers, Berkovich, andcube-corner, are nearly hemispherical in shape. Thus,

Vp ¼2

3pb3 (7)

in which b is the radius of the hemispherical plastic zone,17 asshown in Fig. 1.

As suggested by Hill,18 it is possible to relate the relativeplastic zone radius b/a to the ratio of Young’s modulus andhardness17 as well as c by analogy to a spherical cavity of radiusr0, under internal pressure. In particular,

b=r0 � ðE=HÞm (8)

with m�1/2.Equating the volume of the cavity to the volume of the

indentation,5 it follows that

r0 ¼ cotc=ffiffiffi2p

p� �h i1=3

a (9)

Table I. Geometry-Dependent Parameters x and k

Method

x (� 102) k (� 102)

Vickers Berkovich

Cube-

corner Vickers Berkovich

Cube-

corner

ExperimentalUpper bound 2.006 2.009 4.8010 5.007 — —Mean value 1.606 1.609 4.0010 4.307 — —Lower bound 1.206 1.209 3.2010 3.607 — —

NumericalUpper bound 1.8611 1.8511 4.3811 — — —Mean value 1.6111 1.6011 3.7811 — — —Lower bound 1.3611 1.3511 3.1811 — — —

AnalyticUpper bound 2.305 2.16 4.53 4.60 2.81 5.88Mean value 1.845 1.73 3.63 3.68 2.25 4.71Lower bound 1.385 1.30 2.73 2.77 1.69 3.54

Fig. 1. Median/radial cracks produced by (a) Vickers indenter and(b) triangular indenter. The figures below represent the wedge analogyused in analysis.

Fig. 2. Schematic of the (a) Vickers and (b) triangular (Berkovich andcube-corner) test geometry.

1094 Journal of the American Ceramic Society—Leonardi et al. Vol. 92, No. 5

thus:

b � E

H

� �1=2affiffiffi

2p

p tanc� �1=3 (10)

Comparison with experimental data17 suggests that Eq. (10)slightly overestimates b.

By substituting Eqs. (6), (7),and (10) into Eq. (5), it followsthat

pR ¼H3=2

3ð1� 2nÞE1=2(11)

The integration of this pressure, pR, over the hemisphericalplastic-zone surface in a direction normal to the crack plane,defined by the angle y, gives the point-force, F:

F ¼Z p=2

0

pRpb cos y b sin y dy ¼pb2pR

2(12)

By substituting Eqs. (10) and (12) into Eq. (4) and making useof the Vickers hardness relation, H5P/(2a2), it follows that

KIC ¼OHP

12p1=2ð1� 2nÞðffiffiffi2p

p tancÞ2=3E

H

� �1=2P

c3=2(13)

It is worth noting that Eq. (13) is identical in form to Eq. (2).Thus, the explicit expression for the geometry-dependent pa-rameter, x, for the point-force model is

x ¼ OHP

12p1=2ð1� 2nÞðffiffiffi2p

p tancÞ2=3(14)

By substituting the values of OHP, c, and n5 0.25, a typicalvalue for ceramics, the resulting x for the point-force treatmentis x5 2.30� 10�2. This is above the upper limit for the exper-imentally determined values reported in Table I.

The Vickers indentation cracks are next considered as wedge-loaded half-penny cracks. For a penny-shaped crack opened bya wedge in the form of a smoothly embedded rigid circular diskinclusion, the stress intensity factor is given by19

KI ¼hE

2pð1� n2Þc1=2

� rcþ r2

pc2þ r3

3c3þ r4

3pc4þ r5

5c5þO

r6

c6

� �� �(15)

where r and 2 h are the radius and the thickness of the wedge,respectively. Equation (15) can be extended to a half-pennycrack by multiplying the right-hand side with the free-surfacecorrection, which has been found from numerical simulations tobe the same as that for a half-penny crack subject to remoteloading,16 i.e. OHP5 1.21. The indentation plastic zone acts likea wedge and it is the driving force for the crack; thus, an analogybetween the indentation crack and the wedge-opened half-pennycrack can be made. For well-developed indentation half-pennycracks, i.e., c � r, Eq. (15) reduces to

KI ¼OHPhEr

2pð1� n2Þc3=2 (16)

Based on experimental observations that well-behaved sur-face cracks (c � b) associated with Vickers indentations de-velop predominantly outside the plastic zones,20 it is reasonableto assume that the radius of the wedge for the indenter crackproblem is equal to the outer radius of the plastic zone, i.e.,r5 b.

Because the residual driving force for indenter cracks is de-rived from an elastic accommodation of the hardness-impres-sion volume and that the influences of indenter geometry and

the indentation load on the residual driving force are completelycharacterized by the hardness-impression volume, the wedgethickness can be evaluated by equating the volume of the half-disk wedge to the volume of the impression. Thus,

h ¼ffiffiffi2p

a3

3pb2 tanc(17)

By substituting Eqs. (10) and (17) in Eq. (16) and making useof the Vickers hardness relation, H5P/(2a2), it follows that

KIC ¼OHP

6pð1� n2Þðffiffiffi2p

p tancÞ2=3E

H

� �1=2P

c3=2(18)

It is worth noting that Eq. (18) is identical in form to Eq. (2).Thus, the explicit expression for the geometry-dependent pa-rameter, x, for the wedge-loaded crack model is

x ¼ OHP

6pð1� n2Þðffiffiffi2p

p tancÞ2=3(19)

By substituting the values of OHP, c, and n, the resulting x forthe wedge-loaded crack model is x5 1.38� 10�2.

This value is within the range of the experimentally and nu-merically obtained x reported in Table I. It seems that thewedge-load analysis leads to a better approximation for x thanthe point-force assumption and, hence, the assumptions made inthe former analysis can be considered reasonable.

Based on the theoretical assumptions leading to the respectivesolutions, Shetty et al.5 concluded that the point-force modelrepresents an upper bound for the geometry-dependent param-eter while the wedge-loaded crack model is a lower bound. Thisconclusion is confirmed by the comparison of the analytical so-lutions with the available experimental and numerical data.

IV. Analysis of Berkovich and Cube-CornerQuarter-Penny Cracks

The analysis presented by Shetty et al.5 is here extended totriangular indenters, i.e., Berkovich or cube-corner.

As reported by Dukino and Swain,21 the Berkovich and thecube-corner geometry does not permit quarter-penny cracks tojoin two corners of the impression and pass through the centerof the indentation. However, recently, Kese and Rowcliffe22 andTandon23 observed, by cross-section of cube-corner imprints insoda-lime glass, the formation of quarter-elliptical and quarter-penny cracks, respectively. Thus, in this study, it is assumed thatquarter-penny cracks are induced by Berkovich and cube-cornerindenters, as shown in Fig. 1(b).

(1) Point-Force Model

For triangular indenters, it can be shown by referring toFig. 2(b) that

Vi ¼ffiffiffi3p

a3

4 tan b(20)

where the meanings of a and b are given in Fig. 1(b) and 2(b),respectively.

Equating the volume of the cavity to the volume of theindentation, it follows that

r0 ¼ffiffiffi3p

=2ðcot b=pÞ1=3a (21)

Thus, the approximate relation (8) for plastic zone radiusbecomes

b �ffiffiffi3p

2

E

H

� �1=2a

ðp tan bÞ1=3(22)

May 2009 Analytical Approaches to SIF Evaluation for Indentation Cracks 1095

By substituting Eqs. (20), (7), and (22) into Eq. (5), it followsthat

pR ¼H3=2

3ð1� 2nÞE1=2(23)

By substituting Eqs. (22) and (12) into Eq. (4), replacingOHP for the half-penny shape with OQP for the quarter-pennycrack and making use of the triangular hardness relation,H ¼ 4P= 3

ffiffiffi3p

a2� �

, it follows that

KIC ¼OQP

6ffiffiffi3p

p1=2ð1� 2nÞðp tanbÞ2=3E

H

� �1=2P

c3=2(24)

It is worth noting that Eq. (24) is identical in form to Eq. (2).Thus, the explicit expression for the geometry-dependent pa-rameter, x, for the point-force model is

x ¼ OQP

6ffiffiffi3p

p1=2ð1� 2nÞðp tan bÞ2=3(25)

Based on the Ouchterlony analysis24 of stress intensity factorsfor the expansion-loaded star crack, Dukino and Swain21

showed that OHP/OQP5 1.073 and, thus, OQP�1.13. This valueof the free-surface correction factor for quarter-penny crackshas been confirmed by numerical simulations carried out by theauthors but not reported herein.

By substituting the values of OQP, n5 0.25, and b in Eq. (22),the resulting x from the point-force treatment is x5 2.16� 10–2

and x5 4.53� 10–2 for Berkovich (b5 77.04) and cube-corner(b5 54.74) indenter, respectively. Both these values are close tothe upper limit of the respective, experimentally obtained resultsgiven in Table I.

(2) Wedge-Loaded Model

For well-developed indentation half-penny cracks, i.e., forc � r5 b, and taking into account the correcting factor OQP,Eq. (15) reduces to

KI ¼OQPhEb

2pð1� n2Þc3=2 (26)

Equating the volume of the quarter-disk wedge to the half-volume of the triangular impression, it follows that

h ¼ffiffiffi3p

a3

4pb2 tanb(27)

By substituting Eqs. (22) and (27) into Eq. (26), and makinguse of the triangular hardness relation, H ¼ 4P= 3

ffiffiffi3p

a2� �

, itfollows that

KIC ¼OQP

3ffiffiffi3p

pð1� n2Þðp tan bÞ2=3E

H

� �1=2P

c3=2(28)

It is worth noting that Eq. (28) is identical in form to Eq. (2).Thus, the explicit expression for the geometry-dependent pa-rameter x, obtained from the wedge-loaded crack model is

x ¼ OQP

3ffiffiffi3p

pð1� n2Þðp tan bÞ2=3(29)

By substituting the values of OQP, b, and n, the resulting xfor the wedge-loaded crack model is x5 1.30� 10�2 andx5 2.73� 10�2 for Berkovich and cube-corner, respectively.

The Berkovich value obtained from the wedge-loaded crackmodel is quite close to the lower limit of x obtained from frac-ture toughness calibration and numerical simulations reportedin Table I. This cannot however be said for the respective cube-corner value, which seems to be less reliable than that obtained

from the point-force model probably due to the sharpness of thecube-corner indenter. Another source of error is the empiricalrelation for the evaluation of the plastic zone b, which is over-estimated. Thus, the assumptions made in the wedge-load anal-ysis for Vickers appear to also be reasonable for Berkovich butnot for cube-corner indenters. As for Vickers, the wedge-loadedcrack model provides a lower bound.

It is interesting to note that the averages of the upper andlower bounds obtained from this analytic approach are reason-ably close to the mean values of Table I obtained experimen-tally, as shown in Table I.

V. Analysis for a Modified Approach

In analogy to x, it is possible to find the upper and the lowerbounds for k from the point-force and the wedge-loaded models,respectively.

Starting from the point-force model, the Vickers peak loadP5 2a2H0 is substituted in Eq. (13) to give

KIC ¼OHP

6p1=2ð1� 2nÞðffiffiffi2p

p tancÞ2=3EH0ð Þ1=2 a2

c3=2(30)

It is noted that Eq. (30) is identical in form to Eq. (3). Thus,the explicit expression for the geometry-dependent parameter k,obtained from the point-force model for Vickers indenters is

k ¼ OHP

6p1=2ð1� 2nÞðffiffiffi2p

p tancÞ2=3(31)

Comparison of Eqs. (31) and (14) shows that the analyticallypredicted indentation fracture parameters satisfy k5 2x and,thus, k5 4.602� 10�2, which is slightly higher than the calibra-tion constant found by Gong et al.7

For the wedge-loaded model, it follows from Eq. (18):

KIC ¼OHP

3pð1� n2Þðffiffiffi2p

p tancÞ2=3EH0ð Þ1=2 a2

c3=2(32)

thus:

k ¼ OHP

3pð1� n2Þðffiffiffi2p

p tancÞ2=3(33)

As for the point-loaded model, comparison of Eqs. (33) and(19) shows that k5 2x. Then, it follows that k5 2.77� 10�2.

The two values previously determined represent an upper anda lower bound of k obtained from fracture toughness calibrationreported by Gong et al.7

As done previously, this analysis can be extended to triangu-lar indenters, i.e., Berkovich and cube-corner.

Substituting the triangular indenter peak load P ¼ 3ffiffiffi3p

a2H0=4in Eq. (24) gives

KIC ¼OQP

8p1=2ð1� 2nÞðp tan bÞ2=3EH0ð Þ1=2 a2

c3=2(34)

It is noted that Eq. (34) is identical in form to Eq. (3). Thus,the explicit expression for the geometry-dependent parameter, k,for the point-force model and for triangular indenters is

k ¼ OQP

8p1=2ð1� 2nÞðp tan bÞ2=3(35)

Then, it follows that k5 2.81� 10�2 for Berkovich whilek5 5.88� 10�2 for cube-corner.

For the wedge-loaded model, it follows from Eq. (28):

KIC ¼OQP

4pð1� n2Þðp tan bÞ2=3EH0ð Þ1=2 a2

c3=2; (36)

1096 Journal of the American Ceramic Society—Leonardi et al. Vol. 92, No. 5

thus:

k ¼ OQP

4pð1� n2Þðp tanbÞ2=3(37)

Then, it follows that k5 1.69� 10�2 for Berkovich whilek5 3.54� 10�2 for cube-corner.

It is interesting to note that for triangular indenters and boththe point-force and the wedge-loaded model, k ¼ 3

ffiffiffi3p

x=4.To our present knowledge, k has not been calibrated for Be-

rkovich and cube-corner; thus, we are not able to conclude thatthese values for triangular indenters are in good agreement withexperimental results. However, because the ratio k/x is a con-stant and equal to the ratio P/a2H0 for all the indenters, it ispossible to presume that the point-force model represents anupper bound for the geometry-dependent parameter while thewedge-loaded crack model is a lower bound for the Gong rela-tion as well as for the Anstis relation.

VI. Discussion

For all the indenters, the crack analyses under point-force andwedge-loading produce, respectively, an upper and a lowerbound of the geometry-dependent parameter appearing in thefracture toughness formula. This is because the evaluation of thelatter has been based on an overestimation of the plastic radius;thus, the resulting geometry-dependent parameter is underesti-mated. On the other hand, in the point-force model, the hard-ness-impression volume is elastically accommodated in theplastic zone, i.e., the point-force model has an infinite elasticstiffness relative to the plastic zone. Thus, the point-force modelrepresents an upper bound because the plastic zone acts withfinite compliance relative to the crack.

VII. Conclusions

In the present paper, an analytical solution for the stress inten-sity factor in the case of cracks produced by Vickers indentershas been extended to the cases of Berkovich and cube-cornerindentations. In particular, the median/radial pattern has beenmodeled as cracks loaded by point-force and symmetric disk-shaped wedges. The wedge diameter has been assumed to beequal to the plastic zone size while the wedge thickness has beenevaluated by comparing the wedge volume with the hardness-impression volume.

Two different approaches to the indentation toughness of ce-ramics have been considered.

The resulting values obtained by the present analysis are ingood agreement with upper and lower limits of similar experi-mental and numerical results. Thus, the assumptions made inthe point-force and wedge-load analyses for Vickers indentersappear to be reasonable for triangular ones as well. The onlyexception is the wedge-load result for cube-corner indenters,which is rather low in comparison with the respective range of

experimental and numerical results. It seems that a refinementthat would account for the greater sharpness of the indenter isnecessary in this case.

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