atkinson 2012

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REVIEW

Contact mechanics: a review and someapplications

C. Atkinson*1, J. M. Martınez-Esnaola2 and M. R. Elizalde2

A review of indentation contact fracture is given ranging from that resulting from pressure and

percussion flaking of flint by early man to that produced by various shaped indentors on glasses

and ceramics. A recent application is described of controlled damage by nanoindentation on

initially defect free silicon wafer samples.

Keywords: Contact fracture, Indentation, Flaking, Review

This paper is part of a special issue on ‘Hardness across the multi-scales of structure and loading rate’

IntroductionPerhaps the earliest deliberate use of contact fracturehas to be the process of making stone tools by striking aflake from a block of flint or other suitable brittlematerial. The existence of such flints is often described asone of the few tangible evidences of man’s presence forthe past million years or so of what is described ashuman cultural evolution by Speth.1 Two distinctmethods of flake production are pressure flaking, wherea flake is prised from the rock by a steady concentratedforce applied near the edge, and percussion flaking,where this force is applied impulsively. Simplifiedmathematical models of these processes have beenconsidered by Fonseca et al.2 Relevant articles onpercussion flaking are that of Speth1 (referred to above)and Cotterell et al.3 together with the book byWhittaker.4 If one googles flintknapping, one can findmany lectures on YouTube that describe the process andin particular indicate which types of rock are suitable.This topic, although interesting and of great value toarchaeologists, does not lend itself to careful quantita-tive analysis because of the variability of the rocks.Hence, we mainly concentrate here on what has beendetermined via theory and experiment in glasses andceramics, although we make a brief comparison withindentation fracture in rocks.

The plan of the present paper is as follows. We begin(section on ‘Indentation fracture’) by characterising thegeometry and extent of indentation induced cracking interms of contact stress field. Various crack patterns arethen identified: Hertzian cone cracks, median cracks,lateral vents, etc. Expressions are derived for the scale ofmicrofracturing in terms of the indentor load fromwhich the roles of material properties (e.g. hardness H,toughness C and stiffness E) and extraneous variables(e.g. indentor geometry, environment, etc.) may beinferred. Much of this work has been carried out some

time ago. We complete this review (section on‘Applications’) with a brief description of some recentapplications (mostly to nanoindentation) and an exam-ple of controlled damage by nanoindentation on initiallydefect free silicon wafer samples.

Indentation fractureA great deal of study has been made on the indentationfracture of glasses and ceramics, and whereas we mightexpect differences in detail between the phenomenaobserved in these materials and those observed in brittlerock, there are nevertheless fairly close similaritiesbetween them. In this section, we describe the mainfeatures likely to be common to indentation inducedfracture patterns and attempt a quantitative descriptionof them. There is a vast literature on the indentationfracture of ceramics and glasses (see the references at theend of this review for a selection).

The objectives for the first parts of this section will beto:

(i) characterise the geometry and extent of indenta-tion induced cracking in terms of contact stressfield

(ii) derive expressions for the scale of microfracturingin terms of the indentor load from which the roleof material properties (e.g. hardness H, toughnessC and stiffness E) and extraneous variables (e.g.indentor geometry, environment, etc.) may beinferred.

As far as the loading stress field is concerned, a crudedistinction can be made between the types of indentor,i.e. ‘blunt’ or ‘sharp’. For a point indentation, typifiedby a conical indentor, qualitative results are that:

(i) blunt indentors: dominant tension occurs in nearfield just outside the contact area, leading toHertzian contact cracks observed when spheresimpact glass

(ii) sharp indentors: dominant tension develops im-mediately below penetration point.

This distinction can be seen most clearly by compar-ing the Hertzian and Boussinesq stress fields createdby spherical and point indentors respectively (seeAppendixes 1 and 2).

1Department of Mathematics, Imperial College, 671 Huxley Building,South Kensington Campus, London SW7 2AZ, UK2CEIT and TECNUN, University of Navarra, P. Manuel Lardizabal 15, SanSebastian 20018, Spain

*Corresponding author, email [email protected]

� 2012 Institute of Materials, Minerals and MiningPublished by Maney on behalf of the InstituteReceived 11 October 2011; accepted 21 December 2011DOI 10.1179/1743284711Y.0000000123 Materials Science and Technology 2012 VOL 28 NO 9–10 1079

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The brief description given above emphasises theinfluence of the tensile stress field since this is most likelyto directly influence the crack patterns. Note, however,that other stress components, e.g. shear and hydrostatic,exist in the indentation field, and these control theoperation of irreversible deformation modes (plasticflow, densification and crushing) responsible for thehardness impression and residual stress fields that mayoccur during indentation or upon indentor removal.

Evolution of crack patternsThe three stages in the evolution of fracture patterns arecrack nucleation, formation and propagation. Thepropagation stage is very important to chip formationand, in particular, lateral crack system to be describedlater. Nevertheless, the damage created during indenta-tion must affect the effective permeability of the rockduring subsequent indentations and possibly has aninfluence on indexing.

Crack nucleation

A typical brittle solid contains a profusion of pre-existing flaws, and the deformation process duringindentation may itself create crack nuclei by micro-mechanical slip events. The macroscopic inhomoge-neous stress field induced by the indentor interacts withthese so that pre-existing near surface flaws dominatecrack initiation in the case of blunt indentors. This leadsto Hertzian conical cracks for blunt conical indentorsand to prismoidal shaped cracks for blunt chisel shapedindentors. These cracks are indicated in cross-section inFig. 1 by the letters C and P.

For sharp indentors, deformation induced flaws belowthe surface dominate since high tensile stresses arecreated there by the inhomogeneous (sharp) indentorstress field.

Note that more subtle explanations may be relevant ifthe micromechanics of non-elastic deformation beneaththe indentor are looked at in more detail.

Crack formation

The dominant flaws begin to extend into the solid(stably) as the indentor load is increased. This can bedescribed in terms of an energy barrier to full crackpropagation due to the stress cutoff of non-zero contactarea. The depth to which a crack must grow beforeovercoming this energy barrier is usually small com-pared to the dimension of the contact area. This leads tothe crack propagation stage.

Crack propagation

Once the formation energy is exceeded for a given crack,rapid crack propagation follows until at a depth much

greater than the contact dimensions, the tensile drivingforce falls below that necessary to maintain growth.Thus, a stable crack is produced, and the near contactstress field is less important.

Cases that have been studied in glasses are as follows.

Hertzian cone cracks

These have been observed as initiating from a surfaceflaw by elastic contact with a normally loaded sphere.They are shown in Fig. 1 indicated by the letter C.

‘Median’ half penny cracks

These are initiated from a deformation induced flaw bythe tip of a sharp cone or pyramid indentor. There couldbe several such cracks depending upon existing inho-mogeneities in the material and the symmetry of theindentor and hence the stress field induced by it.Figures 2 and 3 show a side view of such cracksproduced by wedge and cone shaped indentors respec-tively. The crack will propagate under the influence ofthe tensile stress induced by the indentor.

Lateral vents

The picture that is usually given in the ceramics andglass literature is that, on unloading, the reversal of theindentation load reduces the tensile opening stress on themedian and cone cracks, and they tend to close butrarely heal. However, there may still exist a residualstress due to the mismatch between the irreversibledeformed material and the surrounding elastic matrix,which may still drive median cracks and other radialcracks that may have been initiated in the deformationzone beneath the indentor. In addition, these residualstresses create conditions for the initiation and propaga-tion of the lateral crack system shown in Fig. 4. Inextreme loading cases, these lateral cracks are respon-sible for chip removal as they propagate to the surface orintersect with other crack systems, such as the Hertziancone cracks, to cause craters.

Qualitative relations between scale of crackingand applied loadFor advanced stages of loading, the following geome-trical approximations are made:

(i) for ‘point load indentors’ (i.e. conical indentor),cone and median cracks are considered to be‘penny-like’ (expanding on an ever increasingcircular front)

(ii) for ‘line force indentors’ (i.e. wedge shapedindentors), the median vents and prism shapedcracks are viewed as through the surface planarcracks, as shown in Fig. 2.

Median vents and Hertzian cone cracks

To obtain the basic relationships between the scale ofcracking and the applied load, the balance between themechanical energy UM and the surface energy US isconsidered for virtual displacement dc in the cracksystem. Consider the situations shown in Figs. 2 and 3.For these situations, one gets for the change in thesurface energy dUS

dUS!C(Ldc) for line contact

dUS!C(cdc) for point contact(1)

where C is the fracture surface energy (energy requiredto create a unit area of a new crack surface), and c is acharacteristic crack length, and expression (1) is shown

1 Crack nucleation and propagation: cross-section of

Hertzian conical (C) cracks for blunt conical indentors

and prismoidal (P) shaped cracks for blunt chisel

shaped indentors; M stands for median cracks

Atkinson et al. Contact mechanics

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as proportional since similarity relations are onlyanticipated by this argument. For the median cracksystem shown in Fig. 3, we might, of course, prefer pcdcinstead of cdc in expression (1). To estimate themechanical energy, consider the applied load P withP5PLL if acting over a line such as shown in Fig. 2. Thestress level of the indentation field is estimated as theload divided by a characteristic area. For the stress at adistance c from the load point, the characteristic area isassumed to be the area of the surface everywhere atdistance c from the contact point supporting this load.Hence, the stress level s is given by

s!P=(Lc) for line contact

s!P=c2 for point contact(2)

The strain energy density w (for a linear elastic materialwith Young’s modulus E) varies as s2/(2E), so neglectingthe influence of Poisson’s ratio, this gives

w!P2=(L2c2E) for line contact

w!P2=(c4E) for point contact(3)

The volume of the stressed material (dV, say) associatedwith the crack extension is that traced out by thecharacteristic support area, which is proportional toLcdc for line contact and to c2dc for point contact. Then,the change in strain energy is dUM!2wdV, giving

dUM!{P2:dc=(LcE) for line loads

dUM!{P2:dc=(c2E) for point loads(4)

The negative sign indicates that the mechanical energydiminishes as the crack extends.

Now, the energy balance for crack equilibriumrequires that the total energy change of the system iszero. Hence

dUMzdUS~0 (5)

and from equations (1) and (4), we get

P2L=c!CE for line loads (6a)

P2=c3!CE for point loads (6b)

recalling that P5PLL for line loads. From relation (6a)and (6b), the equilibrium length of median and conecracks can be determined in terms of the applied loadonce the surface energy (or toughness) C of the materialis known as well as its elastic modulus.

An argument similar to that above has been givenby Roesler5,6 for cone cracks and by Lawn andco-workers7–9 in a number of papers. To determine theproportionality constants in expression (6a) and (6b), amore complete analysis is necessary. Such an analysismust take into account the inhomogeneous stress fieldbeneath the indentor and use this to predict the directionin which cracks will propagate and then calculate thestress intensity factors and energy release rates asso-ciated with the fully developed crack. This approach willbe outlined more fully in the next section.

We close this section by deriving a scaling law for thelateral crack system following Swain and Lawn.10

Lateral crack system

Some idea of the crater volume produced by fullydeveloped lateral vents can be obtained if it is assumedthat the residual driving forces produced by the inelasticzone beneath the indentor (Fig. 4) are of sufficientintensity that the lateral cracks will intersect the freesurface. A reasonable hypothesis is then that the size of

2 Wedge indentation with median vent crack (M) and prism shaped crack: PL is line distributed force (force per unit

length)

3 Crack patterns in conical indentor


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the resultant chip should scale with that of the hardnessimpression, since the inelastic deformation associatedwith this hardness impression constitutes the source ofthe residual stress field. At least this is the picturecurrently accepted in the ceramics literature. It seems aplausible starting point for a scaling law for crackformation in rock. For geometrically similar indenta-tions, the characteristic dimension of the residualimpression (Fig. 4) can be related to the peak load P*through the standard hardness (mean contact pressure)relations

H~P�=(2La) for line loads

H~P�=(apa2) for point loads(7)

where a is a factor depending on the indentor geometry(a51 for axially symmetric indentors). The characteristicdimension c9 of the lateral crack, shown in Fig. 4, canthus be written as

c’~lLa~lLP�

2HL~lL

P�L2H

for line loads (8a)

c’~lPa~lPP�

apH

� �1=2

for point loads (8b)

The l terms are scaling factors, which will probablydepend upon the indentor geometry and perhaps alsothe material properties. Note that hardness is the keyparameter in this case since this is assumed to controlthe residual tensile stress field. The fracture toughness isimportant in specifying the resistance to lateral crackformation when the applied loading is low, but here, wehave assumed that the applied load is sufficient to drivethe lateral cracks to the free surface.

Most experiments on indentation fracture have beencarried out on glasses and ceramics since glasses inparticular allow observations on fracture patterns to berelatively easily. However, some tests of how the aboveexpressions may apply to rock have been given by Swainand Lawn.10 If the proportionality constant in expres-sion (6) is assumed to depend only on the indentorgeometry, then for invariant indentor geometry, equa-tion (6b) should give for wedge indentation

C2

C1

~E2

E1

PL2

PL1

� �2c1

c2

(9)

Comparing these results for granite and glass (they takefor their soda lime glass C53?9 J m22, E573 GPa andn50?25, with Young’s modulus E534 GPa for westerlygranite), they predict from equation (9) a value for thefracture energy of the granite of ,30 J m22, which isof the same order of magnitude as the result (C5

100 J m22) given by Hoagland et al.11 by means of afracture mechanics test. Taking into account thevariability of rock specimens, these results do give adegree of plausibility to the above theoretical results.Swain and Lawn10 also made the observation that thetrend away from median cracking towards cone crack-ing in moving from sharp to blunt indentors is lessevident in granite than in glass, and they attribute this toa difference in crack initiation, in particular to thepresence of pre-existing internal flaws in the granite.

For the lateral crack system, expression (8) can beused in a similar way to that followed in derivingequation (9) if it is assumed that lL and lP depend onlyon the indentor geometry. With this assumption andinvariant indentor geometry acting on two differentmaterials, expression (8b) gives

H2

H1

~P�2P�1

c’1c’2

� �2

for point loads (10)

Swain and Lawn10 claim to use this expression togetherwith data resulting from their experiments to compute H(granite)<7?3 GPa [they take H (glass)<5?5 GPa]. Theyclaim that independent hardness measurements usinga Vickers diamond pyramid (with P*5100 N) gaveH(granite)54?9¡0?3 GPa. They do not give extensivedata for their experiments on rock; most of the data intheir paper refer to glass. Nevertheless, the indentationinduced crack patterns following the tensile stresscreated by the inhomogeneous indentation stress fieldsseem highly plausible, and the influence of hardness onthe lateral crack system is worth consideration.

Fracture mechanics analysis of indentationfractureAlthough the functional dependence between crack size andindentor load described by equation (6a) and (6b) seems tobe verified by the available experimental data, the descrip-tion remains incomplete. To account for the influence ofcontact geometry on the ensuing fracture process, a moredetailed analysis is necessary. In principle, this can becarried out by performing the following sequence of steps:

(i) calculate the inhomogeneous stress field pro-duced by the indentor

(ii) postulate crack initiation of those positions,where the maximum tensile stress is producedprovided it is large enough

(iii) follow the crack propagation process using theindentation induced stress field as a driving forcefor the fracture.

As might be expected, there are a number of complica-tions involved in accomplishing the above steps. For thefirst step, some assumptions must be made about theresponse of the indented material to the applied load. Ifthis response is assumed to be purely elastic, then astress field such as that shown in Appendix 1 forHertzian contact is appropriate for blunt indentors.For sharp indentors, an approximation to the stress fieldfar from the indentor tip can be obtained using theBoussinesq stress field (shown in Appendix 2) evaluated

4 Lateral crack system: residual force Pr exerted by

radially expanded plastic zone determines crack driving

force; broken lines indicate median radial crack system



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for the case of a point or line indentor. Of course, thesestress fields need to be modified when the inelasticdeformation near the point of contact is taken intoaccount.

Hertzian cone cracks

Conical indentation fractures in glasses have received alot of attention since they are stable and the speed atwhich they spread can be controlled. Experiments byRoesler5,6 gave, for fully developed cone cracks, theangle a shown in Fig. 5 to be 21?5¡1u. He also foundthat this crack angle was independent of the indentordiameter. Assuming that the fully developed crackpropagated (as a cone of angle a with the crack tip) ata cone of radius R, as shown in Fig. 5, Roesler hascalculated the crack extension force G per unit width ofcrack front as

G~k(n)P2

ER3(11)

The constant k(n) is dimensionless and is independent ofthe indentor geometry for blunt indentors. Roesler givesthis as

k(n)~f (n)½cos a�n (12)

For n50?25 and a522u, he deduces via numericalanalysis that k52?756103. If we now equate the energyrelease rate G to the surface energy per unit width ofcrack front 2C, we get the equation G52C for theequilibrium position of the crack, i.e.

P2

R3~

2CE

k(n)(13)

which has the same form as equation (6a) and (6b) whenwe note, by comparing Figs. 3 and 5, that c5Rtan a.

A more recent treatment of the cone crack problem isthat of Lawn et al.7 These authors try to analyse thesituation occurring in many test arrangements wherethe indentor load increases monotonically so that thegrowing crack experiences a time varying stress field. Welist here the characteristics of the Hertzian stress fieldbased on the solution of Huber12 given in Appendix 1.These are the following:

(i) within a drop shaped zone beneath the contactcircle, all principal stresses are compressive

(ii) the tensile stress reaches its maximum at thecontact circle and falls off relatively slowly withincreasing radial distance from the contactcentre along the specimen surface

(iii) the tensile stresses decrease rapidly with depthbelow the specimen surface, the stress gradientbeing steepest close to the contact circle; it isthis decreasing tensile stress field that leads tostable cracks

(iv) the trajectories of the minimum (most compres-sive) principal stresses start orthogonal to thespecimen surface and rapidly deviate outwardfrom the contact circle to form a family of nearparallel curves closely resembling the shape ofthe cone cracks (compressive curves shown inFig. 5). The stresses normal to these trajectoriesthen represent the greatest of the principalstresses, which turns out to be tensile.

The fracture behaviour is approximated by the followingsteps based on earlier work of Frank and Lawn:13

(i) the flaw first propagates around the circle ofcontact and subsequently grows downward intothe material as a surface ring; this flaw isassumed to have initiated from a perpendicularflaw of length cf located at position (xc,0) in thespecimen surface (Fig. 5)

(ii) the downward propagating surface crack isapproximated by a plane edge crack; this is validprovided that c,,a because the crack curvatureis neglected

(iii) the crack path at any instant is assumed tofollow the direction of the principal stresstrajectories passing through the crack tip.

Having made the above assumptions, Lawn et al.7 definethe stress intensity factor as

K~2c

p

� �1=2ðc

0

s(b)

c2{b2ð Þ1=2db (14)

and the energy release rate or crack extension forcefollows as

G~(1{n2)K2=E (15)

where b is the distance along the crack, s(b) is thenormal stress distribution, as deduced from Huber’ssolution given in Appendix 1, and K and G above aredefined per unit width of crack front.

Equations (14) and (15) are standard results fromfracture mechanics. The principal approximation madehere is to simplify the geometry of the crack by planeapproximation (i.e. neglect crack curvature). Once G hasbeen determined from equation (15), the condition forcrack extension for brittle solids can be taken to be theGriffith condition

G~2C (16)

where C is the energy of unit area of the new fracturesurface. Alternatively, if a chemical environment reactswith stress bounds at the tip of the growing crack, thenthe fracture criterion may be expressed in terms of ratedependent equations, leading to a kinetic equation forthe rate of crack growth of the form

:c~v(G or K), where

the fracture parameters determine the crack velocity,i.e. the crack speed is some function of either the energyrelease rate G or the stress intensity factor K. In each ofthese formulations, it has been assumed that crackgrowth is sufficiently slow that the influence of materialinertia can be neglected. For crack speeds approachingthe speed of Rayleigh waves in an elastic medium,

5 Schematic of Hertzian fracture



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relation (15) should have a multiplicative factor on theright hand side, which depends upon the crack speedand the elastic wave speeds of the body (see e.g.Atkinson and Eshelby14 and Freund15). For slow crackspeeds, this factor does not differ much from unity.

In the computer simulation of Lawn et al.,7 theyfound that they had to use a ‘pseudo’ value of Poisson’sratio of n<0?33 to get good agreement with the observedangle of the cone crack a<22u for soda lime glass eventhough n50?2020?25 is more representative of glass(the correct value of n gave a computed crack angle of.30u). It is possible that a different criterion for thecrack trajectory, e.g. KII50 (no mode II stress in-tensity factor), might have given a more accurateresult for the crack trajectory, but this does not seem tohave been investigated. The conclusion of the above-mentioned authors seems to be that the computermodel is good for predicting trends, and their choice ofa ‘pseudo’ Poisson’s ratio could be used to calibrate themodel. However, more recently, Kocer and Collins16

modelled the problem using finite elements andgrowing the crack incrementally in the direction thatmaximises the energy release rate at each calculationstep. They found that the resulting predictions of crackangle were in excellent agreement with the experimen-tal observations using the real Poisson’s ratio of thematerial, and then concluding that maximising therelease of strain energy is an appropriate criterion topredict crack path.

Median vents

For the median crack system, the situation shown inFig. 6 for sharp indentors has been considered by Lawnand Fuller.8 They treat the situation of the welldeveloped median crack that has propagated to thesurface taking up the approximate half penny config-uration of Fig. 6. This well developed crack is assumedto have been driven by the component of the indentationforce normal to the median plane. Designating this forceas P1 for smooth contact, it can be resolved in terms ofthe half angle y of the indentor to give

P1~P

2 tan y(17)

For rough contact, they modify this equation to

P1~P

2 tan y1

(18)

with y1~y+ arctan m (19)

where m is the coefficient of sliding friction, the plus signrefers to the loading half cycle and the minus sign to theunloading half cycle. In this sense, contact friction canbe regarded as either ‘blunting’ (loading) or ‘sharpening’(unloading) the indentor.

The stress intensity factor for a crack loaded in such away (neglecting the free surface) is evaluated as

K~2P1

(pc)3=2(20)

Noting that G5(12n2)K2/E (for plane strain), thecorresponding equation for the energy balance G52C(with C the fracture surface energy) gives

P2

c3~

2CE

k(21)

where the dimensionless constant k is

k~1{n2

p3tan y1 (22)

Thus, the unknown proportionality constant in expres-sion (6) has been determined for this case. Lawn andFuller8 make comparisons with experiments on sodalime glass, which seem to justify the P2/c3 relationship ofequation (21) for a variety of indentor angles. Thesedata give moderate agreement with equation (19) withthe angle of friction m taken to be zero.

In an earlier paper, Lawn and Swain9 have given amore detailed fracture mechanics analysis of medianvent formation beneath point indentors using theBoussinesq stress field given in Appendix 2. They note,in particular, that for n50?25, the three principal stressess11>s22>s33 nearly everywhere in the field and thatboth s11 and s33 are contained within planes ofsymmetry through the normal load axis with s11

everywhere tensile and s33 everywhere compressive.Thus, the tensile component of the point indentationfield might well be large enough to sustain a brittlecrack. The value of Poisson’s ratio is critical as far asthe tensile stress is concerned: at n50?5, the tensilecomponent in the Boussinesq field disappears, and as nvaries between 0?2 and 0?5, the materials generally tendto vary from highly brittle to highly ductile. The fracturemechanics calculations of Lawn and Swain9 did not,however, bear out the similarity relation (21), and thisis probably due to the fact that their calculations mo-delled the crack as planar, and the distribution of thetensile stress produced by Boussinesq field was thus notcorrectly distributed over the crack plane.

In a more recent paper, Lawn et al.17 have recon-sidered this problem by arguing that the complexelastic–plastic stress field beneath the indentor can beresolved into its elastic and residual components. Theelastic component enhances downward median extensionduring the loading half cycle, but since it is reversible,

6 Median crack system associated with sharp indentor:

crack nucleates from plastic contact zone (shaded),

forms into contained penny (broken circle) and ulti-

mately develops into full half penny



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these median vents tend to close during unloading. Theresidual stress component is assumed to provide theprimary deriving force for crack configuration in the finalstages of evolution, where the crack tends to the near halfpenny configuration, which was modelled in equa-tion (21). They adopt the hypothesis that the origin ofthe irreversible field lies in the accommodation of anexpanding plastic hardness impression by the surround-ing elastic matrix. The resulting plastic zone is thenconsidered as the source of an effective outward residualforce on the crack. Assuming a penny-like crack geometryand various approximations to the plasticity analysis,they arrive at the following expression for the stressintensity factor Kr produced by this residual force

Kr~xrP

c3=2(23)

where

xr~jr(Q)E

H

� �1{m

( cot y)2=3 (24)

with m<1/2 (m52/5 is perhaps a better approximation),and jr(Q) is a dimensionless term independent of indentor/specimen system introduced to allow for the effects of thefree surface in their approximate fracture mechanicsanalysis. The angle w is the angle between the radiusdrawn from the point of contact of the indentor to theposition on the crack front and the vertical; jr(Q) is aslowly varying function of value near unity with itsminimum at w50 (median vent orientation, i.e. vertical)and maximum at w5¡90u (radial orientation). Essen-tially, this factor occurs in deriving the approximate stressintensity factor of a half penny crack with a free surfaceand the residual force loading at the centre.

It is worth recalling that for indentors that leavegeometrically similar impressions in homogeneous speci-mens at all loads, the mean indentation pressure remainsinvariant and provides a measure of the hardness of thematerial, i.e.

H~P

apa2(25)

where a is the radius of the surface hardness impression,P is the indentation load and a51 for axial symmetry.

Comparing expression (23) with expression (20), wesee that although the two expressions have the samedependence on crack length c and applied load P, theother factors are different. The dependence on the halfangle of the indentor y is similar, but relation (24) alsoinvolves the ratio of hardness to modulus, an expressionthat does not occur in the simpler analysis leading toequation (20).

The contribution to the crack driving force due to theelastic matrix itself on the loading cycle can be evaluatedin terms of the prior elastic contact stresses over thecrack plane at R>b (b is the radius of the plastic zone)(cf. Fig. 6). For this elastic field, the Boussinesq pointload field of Appendix 2 is a reasonable approximationfor R&a. As shown in Appendix 2, it has the form

s(R,Q)&g(Q)P

R2(26)

where g(w) appropriate to stresses normal to the medianplane is strongly varying, changing from positive(tensile) at w50 to negative (compressive) at w5¡90u.

The stress intensity factor for a half penny cracksubjected to radially distributed stresses over b(R(cis approximated by

K~f (Q)2

c

� �1=2ðc

b

rs(r)

c2{r2ð Þ1=2dr (27)

Formula (27) thus approximates somewhat the effectof g(w) variation in equation (26). Ignoring this andsubstituting for equation (26) in equation (27) gives anelastic stress intensity factor of the form

Ke~xe

P

c3=2(28)

where in the limit c&b

xe~je(Q) ln (2c=b) (29)

Because of the c/b dependence of equation (29), expres-sion (28) deviates from the similarity that we deduced inequations (6b), (21) and (23). However, because of theln (c/b) dependence of equation (29), this variation isnot so great. Furthermore, Lawn et al.17 argue that Ke issecondary in importance to Kr in determining theultimate crack configuration.

The condition for equilibrium growth of these cracksis obtained by equating the net stress intensity factorK to the toughness KC {this can be related to C usedpreviously for example in plane strain; since G5

(12n2)K2/E, one would have KC5[2EC/(12n2)]1/2}.Thus, during the loading/unloading event, one wouldhave K5KezKr5KC, and care must be taken to realizethat equation (28), being the elastic loading component,is reversible, whereas the residual term of equation (23)is irreversible. Since we are interested in what happensafter complete unloading of the crack having thenattained its full half penny configuration, it seemsadequate to use equation (23) and derive

c~(xrPmax=KC)2=3 (30)

as the complete crack dimension Pmax being themaximum load.

For soda lime glass, the evolution of the median/radial cracks has been studied experimentally by theabovementioned authors.

Lateral crack system

Marshall et al.18 have described the mechanics of lateralcrack propagation in a sharp indentor contact field byassuming that the driving force for fracture has its originin the residual component of the elastic–plastic field,which becomes dominant as the indentor is unloaded.The picture is as shown in Fig. 4. A residual tensilestress develops at the nucleation centre near the base ofthe deformation zone so that a residual driving force Pr

acts on the crack as the indentor is withdrawn (Fig. 4).In the absence of reversed plasticity within the centralzone, Pr reaches its maximum at full contact loading andpersists at this value on removal of the indentor. The netmouth opening force for the system is expressed as thedifference between an irreversible residual (opening)component and a reversible elastic (closing) component.The above authors claim that experimental observationsindicate that lateral cracks initiate as the diminishingapplied load approaches zero. The crack geometry is



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taken to be penny-like but with the complication of anadjacent, parallel free surface.

We summarise here the results of the approximateanalysis given in the paper of Marshall et al.18 Thethickness of the material above the crack plane isidentified with the depth of the plastic zone. For theresidual impression formed at peak indentation load P

a~P

a0H

� �1=2

(31)

where H is defined as the hardness, which is an invariant forgeometrically similar indentations, and a0 is a geometricalconstant (a051 for axisymmetry as noted earlier).Calculation of the elastic–plastic boundary based on themodel of an internally pressurised spherical cavity gives

b

a~

E

H

� �1=2

cot yð Þ1=3(32)

[A more accurate model could replace (E/H)1/2 by (E/H)2/5.]Using equations (31) and (32) together gives for the

thickness of the material above the crack plane

h&b&EPð Þ1=2

Hcot yð Þ1=3

(33)

An expression for the equilibrium crack size as a functionof applied load for large contact loads is given as

c~fL

A1=2( cot y)5=6 (E=H)3=4

KCH1=4

" #1=2

P5=8 (34)

where fL is a dimensionless constant independent of thematerial/indentor system. The factor A is a constantdepending on how the lateral crack system is affected bythe other crack system, i.e. the radial crack systemdiscussed in the median/radial section. If the lateralcracks are much larger than the radial cracks, A is takenas A53(12n2)/(4p); on the contrary, if the lateral cracksare smaller than the radial cracks, a quarter plateapproximation is used with A53/4. Note that forchipping to take place, the lateral vents would deviateto intersect the free surface, as shown in Fig. 4. However,the approximations leading to equation (34) assume thatthe crack is propagating horizontally parallel to the freesurface. Furthermore, it is assumed that the cracks arestable and in equilibrium, hence the appearance of thecritical stress intensity factor (toughness) KC in equa-tion (34). Comparing equation (34) with equation (8b)and noting that P*5P, the peak load, we see that thedependence on H is similar but the dependence on P haschanged from P1/2 to P5/8.

Note that the crater volume produced by the lateralcracks can be estimated as (cf. Fig. 4)

V~pc2h (35)

and hence from equations (33) and (34)

V~pfL

A1=2( cot y)7=6 E5=4

KCH2P7=4 (36)

ApplicationsIn the section on ‘Indentation fracture’, we havedescribed the main features of contact fracture in a

‘broad brush approach’. Here, we describe some recent(and not so recent) applications; these are merely asample of a very large literature on the subject. We beginwith an example of sensing subsurface flaws (or fracturecreated by contact) by the surface displacement method.Here, a heat source is produced by a laser, which heatsup the subsurface and any defects there. A second laserrunning along at a prescribed distance behind measuresthe surface displacement and hence images the subsur-face. This method was first introduced by Ameri et al.19,Martin and Ash20 and Martin et al.,21 and Atkinson andMartınez-Esnaola22 provided a suite of calculationswhereby this thermal field interacted with surface andsubsurface cracks, producing surface displacement dueto the thermal expansion of the material/crack interac-tion. In particular, these calculations indicated theoptimal spacing of the two lasers. This is just oneexample of how subsurface imaging can be produced. Atechnique for controlled crack growth and brittlefracture toughness determination has been suggestedby Martin-Meizoso et al.,23 in which they use a dual tipindentor to extend cracks previously generated by aVickers indentation on soda lime glass. They show thatcontrolled crack growth is feasible.

To describe the precise prediction of plastic flowpatterns during indentation from the point of view ofmicroscopic flow and complex dislocation interactions,Brown24 has presented a model for a wedge shapedindentor, and Czecski and Brown25 have adapted thisfor a cylindrical indentor. They use a slip circleconstruction and assume a rigid plastic flow field andalso apply this construction to a spherical indentor.However, the present work is confined to describinghardness rather than contact fracture, although theindentation flow behaviour will ultimately influence thecontact fracture behaviour.

We discuss below a situation that affects the integrityof silicon wafer manufacture and show how some of theissues described in the section on ‘Indentation fracture’,such as both lateral and median cracks, can occur anddescribe a number of non-destructive techniques fordetermining them.

Practical exampleThe original theory of Hertzian fracture in isotropicelastic materials was extended by Lawn26 to the case ofbrittle single crystals, analysing the inhomogeneousstress field and crack paths developed in the indentationof a flat surface with a spherical indentor. He found thatthe Hertzian crack passes through four equilibriumstages before reaching its fully developed length, whichis experimentally confirmed in indentation tests on singlecrystals of silicon.

A compilation of indentation results on silicon crys-tals for several indentor types and a large range in loadvalues covering elastic, plastic and fracture behaviourshas been reported by Armstrong et al.27 In the case ofsharp indenters, they attribute crack initiation to dis-location reaction mechanisms that develop during mi-croslip deformation interaction, in agreement with theTEM results reported for damage of silicon by variouscontact testing experiments.

The role of dislocations in indentation (and nanoin-dentation) processes has been analysed by Chaudhri,28

mainly based on elastic behaviour, as well as the phase



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transformation of silicon associated with stresses gener-ated during indentation hardness experiments.

Here, we describe briefly a practical example aimed atunderstanding silicon wafer handling damage.

In order to replicate defect generation in realprocesses, controlled damage was introduced on initiallydefect free silicon wafer samples using nanoindentationwith a Berkovich tip. The tested samples consisted ofsquare pieces of 20620 mm size. They were prepared byfirst cleaving a 300 mm silicon wafer with (100) surfacecrystallographic orientation along the [011] direction

indicated by the notch (see Fig. 7) and then creating thelittle squares in the figure (also using cleavage planes).Loads between 100 and 200 mN were applied at rates of10 mN s21 using a Nanoindenter II (Agilent, formerlyNano Instruments Inc.) with a Berkovich diamond tip.The indents were oriented so that one of the imprintsides was parallel to the [011] direction.

Direct observation of the crack systems beneath theindents was performed using a Quanta three-dimen-sional (3D) dual beam (FEI) focused ion beam (FIB)system. This gives an idea of the type and extension of

7 Scheme of indented wafers and square samples prepared (highlighted in green): location of indents in wafers is also

shown with one side of Berkovich tip parallel to [011] direction (note that scale of imprint is not real)

8 Sequence of field emission gun SEM images from top–down fibbing on 110 mN Berkovich indent on Si, showing coex-

istence of radial and median cracks close to surface and further development of median cracks under indentor imprint



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damage generated by nanoindentation. The micrographsobtained from top–down fibbing, and shown in Fig. 8,suggest a double crack system, i.e. radial and me-dian cracks, coexisting under indents ,100 mN (seeGorostegui-Colinas et al.29). Radial (or Palmqvist)cracks emerge from the corners of the indentor imprintand have a maximum depth on the order of 0?5 mm.These close surface cracks have been observed in brittlematerials and sharp indentors at low loads (see, forinstance, Tang et al.30). In addition, median cracksoriginate beneath the remaining plastic region under theindentor imprint at higher loads and propagate alongthe median axis upon unloading.

Furthermore, direct observation through cross-sec-tional fibbing indicates that, in addition to the men-tioned crack systems, lateral cracking appears under theindentor imprint at peak loads between 120 and150 mN. According to Fig. 9, lateral cracking runsacross a region of well oriented dislocations and abovethe intersection with a median crack. Another example isshown in the 3D reconstruction (using Amira 4?0software) shown in Fig. 10, where, in addition to themedian crack following a vertical (011) plane, a lateralcrack is also initiated.

Figure 11 shows a schematic of the finite elementmodel designed for stress computations and fractureanalysis of the indentations. The simplest model consistsof a 48648696 mm piece of Si, which is assumed tobehave as an anisotropic elastic material but plasticallyisotropic. The central part is discretised using astructured mesh of eight-node hexahedral elements ofsize 0?25 mm. A finer meshing of the silicon piece usingsubmodelling techniques was also used in combinationwith cohesive zone models to investigate the initiationand propagation of cracks during indentation. In thelatter model, cohesive elements of zero thickness wereplaced in the prospective planes of cracking develop-ment. A triangular traction–separation law was used,characterised by the peak stress smax and the separationenergy (area below the traction–separation law) C0 asthe main parameters (the initial slope of the traction–separation load will not be discussed here). Basedon literature data (see Sopori31), the values smax5

1200 MPa and C054 J m22 were used in the calcula-tions although better approximations could be obtainedif one takes into account the anisotropic toughnessproperties of the silicon. Details of the numericalanalyses performed can be found in the study ofGorostegui-Colinas et al.29

One way of validating the models is through micro-Raman spectroscopy experiments carried out in siliconwafers. Details of this experimental technique can befound in Allen et al.32 The Raman measurements agreequite well with the frequency shifts resulting from thestress field predicted in the finite element model si-mulations, as shown in Fig. 12. The discrepancies can beattributed to the lack of consideration of the phasetransformations in silicon during unloading. Note alsothat the resolution of the Raman measurements islimited in the central part of the indent imprint due tothe inclination of the surface.

Figure 13 shows contour plots of the damage para-meter associated with the cohesive zone approach and a3D reconstruction of a vertical real crack. The resultsindicate that during unloading of the indentationprocess fracture continues propagating. The shape ofthe crack agrees qualitatively with the experimentalresults.

ConclusionsWe have concentrated in the section on ‘Indentationfracture’ on outlining the consequences of the inhomo-geneous stress field produced by the indentor, the maintheme being that tensile stresses can initiate cracksand subsequently sustain crack propagation. We haveemphasised the crack propagation stage since suchmacrocracks lead to chip formation. However, if

9 Image (STEM) of cross-section cut of median and lat-

eral cracks under 150 mN Berkovich indent on Si

10 Three-dimensional reconstruction from FIB tomogra-

phy of median and lateral cracks under 120 mN

Berkovich indent on Si

11 a geometry of silicon piece to be indented and b, c structured mesh of eight-node hexahedral elements for half of

model



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we consider a material like rock, which is such aninhomogeneous material, there will inevitably also be anumber of microcracking events. The existence ofmicroscopic flaws is, of course, a well known ingredientin theoretical models that attempt to explain the failureof rock under compression. Costin and Holcomb33,34

and Costin35 have considered microcrack models for thedeformation and failure of brittle rock. They representthe rock by an essentially elastic constitutive equationwith material behaviour due to microcrack growthaccounted for by the inclusion of an internal state

variable, which is a measure of the crack state. The formof the evolutionary equation for this crack stateparameter is determined from fracture mechanicsanalysis of single cracks and experimental results. Thismodel has been used by the above authors to simulateuniaxial and triaxial compression tests and comparisonsmade with laboratory tests on westerly granite. The ideathat a microcracked solid can be represented by anequivalent continuum model is, of course, a naturalconsequence of modern continuum mechanics, but imposedon any such model is the deliberate inhomogeneous stress

12 Comparison of 2D Raman spectrum map obtained experimentally (left) and prediction based on stress field resulting

from finite element model simulations (right)

a at maximum indentation load; b after indentation process (broken material has been removed from plot for clarity); c3D reconstruction of vertical experimental crack in same plane

13 Crack in vertical plane: contour plots of generated damage



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field imposed by the contacting indentor, and it is this actionthat most concerns us here. If we did have a goodcontinuum model of the underlying material microstructure,then we would calculate the resulting inhomogeneous stressfield produced by the indentor and then rationalise theresulting crack field as outlined in the section on‘Indentation fracture’. In the section on ‘Applications’, thisprocess has been carried out effectively by means of avariety of techniques: finite element stress analysis, crackgrowth analysis plus Raman spectroscopy, TEM, etc. torationalise a practical problem.

References1. J. D. Speth: Am. Antiquity, 1972, 37, 34–60.

2. J. G. Fonseca, J. D. Eshelby and C. Atkinson: Int. J. Fract., 1971,

7, 421–433.

3. B. Cotterell, J. Kamminga and F. P. Dickson: Int. J. Fract., 1985,

29, 205–221.

4. J. C. Whittaker: ‘Flintknapping: making and understanding stone

tools’; 2003, Austin, TX, University of Texas Press.

5. F. C. Roesler: Proc. Phys. Soc. B, 1956, 69B, 55–60.

6. F. C. Roesler: Proc. Phys. Soc. B, 1956, 69B, 981–992.

7. B. R. Lawn, R. Wilshaw and N. E. W. Hartley: Int. J. Fract., 1974,

10, 1–16.

8. B. R. Lawn and E. R. Fuller: J. Mater. Sci., 1975, 10, 2016–2024.

9. B. R. Lawn and M. V. Swain: J. Mater. Sci., 1975, 10, 113–122.

10. M. V. Swain and B. R. Lawn: Int. J. Rock Mech. Min. Sci., 1976,

13, 311–319.

11. R. G. Hoagland, G. T. Hahn and A. R. Rosenfield: ‘Influence of

microstructure on fracture propagation in rocks’, Semiannual

research report, ARPA contract no. H0210006, Battelle Columbus

Laboratories, Columbus, OH, 1971.

12. M. T. Huber: Ann. Phys., 1904, 319, 153–163.

13. F. C. Frank and B. R. Lawn: Proc. R. Soc. London A, 1967, 299A,

291–306.

14. C. Atkinson and J. D. Eshelby: Int. J. Fract., 1968, 4, 3–8.

15. L. B. Freund: ‘Dynamic fracture mechanics’; 1998, New York,

Cambridge University Press.

16. C. Kocer and R. E. Collins: J. Am. Ceram. Soc., 1998, 81, 1736–

1742.

17. B. R. Lawn, A. G. Evans and D. B. Marshall: J. Am. Ceram. Soc.,

1980, 63, 574–581.

18. D. B. Marshall, B. R. Lawn and A. G. Evans: J. Am. Ceram. Soc.,

1982, 65, 561–566.

19. S. Ameri, E. A. Ash, V. Neuman and C. R. Petts: Electron. Lett.,

1981, 17, 337–338.

20. Y. Martin and E. A. Ash: ‘Imaging of micro-cracks by

photodisplacement microscopy’, Proc. IEEE Ultrasonics Symp.,

Dallas, TX, USA, November 1984, IEEE. 647–650.

21. Y. Martin, E. A. Ash, G. Busse and R. S. Gilmore: Philos. Trans.

R. Soc. London A, 1986, 320A, 257–269.

22. C. Atkinson and J. M. Martinez-Esnaola: Int. J. Fract., 1991, 49,

79–122.

23. A. Martin-Meizoso, J. M. Martinez-Esnaola and J. Gil Sevillano:

Int. J. Fract., 2001, 110, 15–22.

24. L. M. Brown: Mater. Sci. Forum, 2007, 550, 105–117.

25. H. Czecski and L. M. Brown: J. Phys. D: Appl. Phys., 2008, 41D,

1–5.

26. B. R. Lawn: J. Appl. Phys., 1968, 39, 4828–4836.

27. R. W. Armstrong, A. W. Ruff and H. Shin: Mater. Sci. Eng. A,

1996, A209, 91–96.

28. M. M. Chaudhri: ‘Dislocations and indentations’, in ‘Dislocations

in solids’, (ed. F. R. N. Nabarro and J. P. Hirth), Vol. 12, 447–550;

2004, Oxford, Elsevier B.V.

29. E. Gorostegui-Colinas, J. Garagorri, M. R. Elizalde, D. Allen and

P. J. McNally: ‘Nanoindentation induced silicon fracture and 3D

modelling’, An. Mec. Fract., 2010, 27, 559–564.

30. Y. Tang, A. Yonezu, N. Ogasawara, N. Chiba and X. Chen: Proc.

R. Soc. London A, 2008, 464A, 2967–2984.

31. P. R. A. B. Sopori: Int. J. Fract., 2009, 155, 67–74.

32. D. Allen, J. Wittge, A. Zlotos, E. Gorostegui-Colinas, J. Garagorri,

P. J. McNally, A. N. Danilewsky and M. R. Elizalde: Nucl.

Instrum. Methods Phys. Res. B: Beam Interact. Mater. Atoms, 2010,

268B, 383–387.

33. L. S. Costin and D. J. Holcomb: Tectonophysics, 1981, 79, 279–296.

34. L. S. Costin and D. J. Holcomb: ‘A continuum model of

inelastically deformed brittle rock based on the mechanics of

microcracks’, Proc. Int. Conf. on ‘Constitutive laws for engineering

materials’, Tucson, AZ, USA, January 1983. SAND-82–1633c,

Conf-830104–1, Sandia National Labs., Albuquerque, NM, USA

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36. B. R. Lawn and R. Wilshaw: J. Mater. Sci., 1975, 10, 1049–1081.

Appendix 1

Hertzian stress fieldThe Hertzian stress field has the property of geometricalsimilarity when all stress components are normalised tothe mean contact pressure p05P/(pa2) and all spatialcoordinates normalised to the contact radius a. Thestresses in the OXZ plane (see Fig. 5) are given byHuber12 as

sxx

p0~

1{2n

2

a

r

� � 2

1{z

u1=2

� � 3" #

z1:5z

u1=:2

� � 3a2u

u2za2z2z

1:5z

u1=2(1{n)

u

a2zuz(1zn)

u1=2

aarctan

a

u1=2

� �{2

� �

szz

p0~{1:5

z

u1=2

� � 3a2u

u2za2z2

sxz

p0

~{1:5xz2

u2za2z2

a2u1=2

a2zu

where r25x2zz2 and

u~1

2(x2zz2{a2)z x2zz2{a2

� 2z4a2z2

h i 1=2 �

The principal normal stress across the crack path (at anangle a to the horizontal) is

sN(x,z)~sxx sin2 azszz cos2 a{2sxz sin a cos a

The angle a between the crack path and the specimensurface is found from

tan 2a~{2sxz

sxx{szz

Appendix 2

Boussinesq problemThe stress field in an isotropic elastic half spacesubjected to a normal point load P (the Boussinesqproblem) can be written in terms of the curvilinearcoordinates of Fig. 14 as

srr~P

pR2

1{2n

4sec2 (Q=2){

3

2cos Q sin2 Q

� �

shh~P

pR2

1{2n

2cos Q{

1

2sec2 (Q=2)

� � �

szz~P

pR2{

3

2cos3 Q

� �



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srz~P

pR2{

3

2cos2 Q sin Q

� �

srh~shz~0

Two of the principal normal stresses, s11 and s33, arecontained in the symmetry plane h5constant, with theirangles with the specimen surface given by

tan 2a~2srz

szz{srr

The third principal normal stress s22 is everywhereperpendicular to the symmetry plane. The principaldirections are labelled such that s11>s22>s33 generally(s.0 for tension). Then, noting that srz(0 for all0(w(p/2, we have

s11~srrsin2azszz cos2 a{2srz sin a cos a

s22~shh

s33~srr cos2 azszz sin2 az2srz sin a cos a

The principal shear stresses are given by

s13~s11{s33

2

s12~s11{s22

2

s23~s22{s33

2

inclined at p/4 to the principal directions. In addition,the magnitude of the component of hydrostatic com-pression is

p~{s11zs22zs33

3

Figure 15 shows a comparison of the shh stress term forBoussinesq and Hertzian fields and their divergence inthe vicinity of the contact zone (see for example Lawnand Wilshaw36): at zR0, shhR‘ in the Boussinesq field,whereas shhR21?125P0 in the Hertzian field.

14 Coordinate system for axially symmetric point loading

P: expressions for stress components indicated are

given in Appendix 2

15 Comparison plot of shh(z) stress term for Boussinesq

and Hertzian fields (plotted for n50?25): note diver-

gence of curves in vicinity of contact zone: at zR0,

shhR‘ in Boussinesq field, whereas shhR21?125P0 in

Hertzian field



atkinson 2012

Documents

extent of indentation

pressure flaking

controlled damage

review section onapplications

toughness c

median cracks

hardness h

wherea flake