Numerical analysis of brittle materials fractured by sharp indenters

$Page 1: Numerical analysis of brittle materials fractured by sharp indenters$
Engineering Fracture Mechanics 77 (2010) 264–276

Contents lists available at ScienceDirect

Engineering Fracture Mechanics

journal homepage: www.elsevier .com/locate /engfracmech

Numerical analysis of brittle materials fractured by sharp indenters

Alessandro Leonardi a,*, Franco Furgiuele a, Robert J.K. Wood b, Stavros Syngellakis b

a University of Calabria, Mechanical Engineering Department, Ponte P. Bucci, cubo 44-C, 87036 Arcavacata di Rende (CS), Italyb University of Southampton, School of Engineering Science, University Road, Southampton SO17 1BJ, UK

a r t i c l e i n f o a b s t r a c t

Article history:Received 27 August 2008Received in revised form 24 July 2009Accepted 2 August 2009Available online 7 August 2009

Keywords:Indentation fractureFinite element methodStress intensity factorSharp indenterPenny-shaped cracksElliptical cracks

0013-7944/$ - see front matter � 2009 Elsevier Ltddoi:10.1016/j.engfracmech.2009.08.003

* Corresponding author. Tel.: +39 0984 494698; fE-mail address: [email protected] (A. Leonardi).

Indentation can be used to determine the fracture properties of materials. A detailed inves-tigation is here presented on the reliability of finite element simulations of sharp indenta-tions on cracked specimens. Elastic analyses of the stress and deformation fields arisingfrom Vickers, Berkovich and cube-corner indenters generated the stress intensity factoralong the edge of penny-shaped or elliptical cracks. Various materials with a range of prop-erties were analysed and the results compared with published experimental data. Addi-tional measurements from tests on soda-lime glass provided further opportunities forexperimental validation of numerical predictions. The variation of the stress intensity fac-tor indicated trends for crack growth patterns, which were consistent with experimentalobservations. Particular attention was given to the evaluation of the geometry-dependentparameter appearing in the relation yielding the fracture toughness of cracked indentedmaterials. The numerical predictions of this parameter were remarkably consistent withexperimental data and results from other approximate methods.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Indentation methods consist of pressing a hard material into the test specimen and studying the response. These tech-niques have been increasingly used because they are relatively easy to perform, do not require elaborate sample preparationand cause only negligible surface damage. Originally they were used for determining the hardness, which characterises theductile material resistance to penetration. The need to study mechanical properties in micro-volumes has led to thedevelopment of nano-indentation devices which allow the application of loads as low as several mN and provide more infor-mation about material behaviour than traditional optical measurement methods. Computer controlled testing and data pro-cessing makes it possible to determine not only the hardness, but also the apparent elastic modulus, which is directlyproportional to the slope of the load-penetration depth at the beginning of unloading stage, the energy involved in the inden-tation process and the yield strength. Spherical (Brinell) as well as square (Vickers), triangular (Berkovich, cube-corner) andrhomboid (Knoop) base pyramid indenters have traditionally been used for the characterization of metals, glasses, ceramics,polymer composites and coated materials.

The elastic stress field generated by sharp indenters is singular in the vicinity of the indenter tip; this causes irreversibledeformation accounting for both the existence of the residual impression and the initiation of any ensuing micro-fractures.Thus, the indentation test measurements are strongly influenced by cracking and their interpretation for a comprehensivematerial characterization becomes difficult. On the other hand, indentation cracking can be used in order to determinethe fracture toughness of hard brittle materials. Sharp-indenter techniques for the determination of the toughness of ceram-ics have been developed [1] and subsequently improved [2]. Due to the inelastic material response however, it is difficult to

. All rights reserved.

ax: +39 0984 494673.

http://dx.doi.org/10.1016/j.engfracmech.2009.08.003

mailto:[email protected]

http://www.sciencedirect.com/science/journal/00137944

http://www.elsevier.com/locate/engfracmech

A. Leonardi et al. / Engineering Fracture Mechanics 77 (2010) 264–276 265

describe the complex stress and strain fields in closed form and account, in sufficient detail, for the geometric changes thatmay occur at different deformation scales. The application of numerical techniques, such as the finite element method (FEM)to such problems, enhances understanding and promotes further development of existing characterisation techniques.

Bhattacharya and Nix [3] were among the first to apply FEM to the simulation of indentation by axi-symmetric cones,which were assumed equivalent to pyramid indenters with the same penetration volume for a given depth; their predictionsfor hardness compared well with experimental data. It was shown, however, that modelling the Berkovich indenter as a conedoes not always provide valid FEM predictions for the load–indentation curve [4]. Three-dimensional (3D) Vickers FEM mod-els were developed to investigate the indentation response of a coated system [5]. FE analyses in 3D were validated throughcomparison of their predictions with experimental data from Vickers and Berkovich indentation tests [6,7]. Indentation-in-duced tensile stresses were assessed through a detailed 3D numerical investigation [8] from which crack initiation and prop-agation sequences can be inferred; this represented a first step prior to a full fracture mechanics analysis.

Indentation fracture has been investigated numerically using an elastic–plastic-cracking constitutive model implementedthrough the general purpose FEM package ABAQUS [9]; the predicted evolution of crack patterns was consistent with exper-imental observations. A similar study on crack development under both low and high indentation loads was based on thebrittle cracking model available in ABAQUS/explicit and elements with cohesive post-failure behaviour [10]. Indentationcrack growth has also been predicted using FEM with a cohesive interface model in the context of an inverse analysis formaterial characterisation [11].

Despite the substantial amount of previous numerical work in this area, there has not been any attempt to re-assess theanalytical framework proposed by Anstis et al. for the determination of fracture toughness [1] via systematic FEM modellingof stress fields in indented and fractured specimens. This is achieved here through an extensive series of 3D analyses, basedon ABAQUS [12]. Since the specimen material was assumed to be elastic, there was no attempt to model crack growth, whichis strongly influenced by the formation of a plastic zone around the indenter tip. Instead, modelling was applied to alreadycracked specimens subjected to indentation loads. A properly validated FEM model provided reliable and detailed informa-tion on the magnitude and distribution of the stress intensity factor (SIF) along the edge of penny-shaped or elliptical cracksat various loading and crack size stages. Such analyses have the potential of strengthening the validity and expanding thescope of existing semi-empirical approaches for the determination of the fracture toughness, Kc from Vickers as well as Ber-kovich and cube-corner indentations. Indentations on cracked specimens were simulated for a range of material properties,within the small strain formulation. The effect of friction between the indenter and the fractured specimen was assessed.Particular attention was given to the influence of certain geometric parameters, such as penetration depth, crack lengthand shape of the crack front. Comparisons with theoretical [13] and experimental results [1,2,14] were also made.

2. Indentation fracture background

The indentation stress field, although dominated by the shear and hydrostatic components, necessarily contains a smallamount of tension, so the potential for crack exists in any contact event. A necessary condition for cracking to occur duringthe loading phase of a sharp indentation of metals and ceramics is that the materials exhibit high strain hardening, that is, arelatively high hardness (H) to Young’s modulus (E) ratio. As a result of indentation fracture observations [15,16], cracks pro-duced by sharp indenters can be classified into the three major types shown in Fig. 1: (a) radial (or Palmqvist) cracks, gen-erated on planes through the loading axis and emanating from the indentation corners; (b) median, penny-like cracks, whichalso grow on planes through the axis of loading, beneath the plastic zone, truncated at the deformation zone boundary; (c)lateral cracks which can be generated, usually at the unloading stage, beneath the plastic zone, running parallel to the sur-face and being circular in form.

Fig. 1. Common macro-cracking patterns observed in pyramid indentations of hard brittle materials [15]: (a) radial (Palmqvist); (b) median and pennyshaped; (c) lateral.

266 A. Leonardi et al. / Engineering Fracture Mechanics 77 (2010) 264–276

With a Vickers indenter, tension acts on median planes containing the diagonals of the square-shaped surface impression,producing eventually the characteristic half-penny star pattern shown schematically in the surface view of Fig. 2a. In con-trast, the Berkovich and the cube-corner geometry does not permit cracks to join two corners of the impression by passingthrough the centre of the indentation [17]; in such cases, the formation of Palmqvist cracks has been considered more likely[16]. More recently however, Kese and Rowcliffe [18] and Tandon [19] detected, respectively, the formation of quarter-ellip-tical and quarter-penny cracks by cube-corner indentations on soda-lime glass. Thus, the star pattern of such cracks shownin Fig. 2b has been modelled in the present simulations of Berkovich or cube-corner indentations.

It is generally accepted that half-penny or quarter-penny cracks appear during loading at high loads, and only duringunloading at low loads, while lateral cracks are formed during unloading. The aim of the FEM analyses reported here is toreproduce the stress fields likely to develop during loading, therefore penny-shaped and elliptical cracks have beenmodelled.

The basic equation used to evaluate the material toughness Kc by indentation is:

Kc ¼ nEH

� �1=2 P

c3=20

; c0 � a ð1Þ

where P is the peak load, c0 the critical surface crack length, a a characteristic dimension of the impression and n the so calledgeometry-dependent parameter, a constant dependent only on the indenter shape and not on the material indented.Although Eq. (1) has been widely used, discrepancies between its fracture toughness predictions and those measured byother conventional methods have been frequently reported [14,20,21]. This discrepancy has been attributed to a numberof reasons, but a fundamental factor, that may affect the accuracy of indentation toughness determination, is related tothe definition of hardness H, used in Eq. (1). In fact, the Vickers hardness defined [1] as

H ¼ P2a2 ð2Þ

where a is the half-diagonal of the impression, is not a constant, but rather an apparent hardness depending on the appliedindentation load. Despite such uncertainties, Eq. (1) is so far the most commonly used relationship for evaluating the fracturetoughness from indentation data.

Detailed fracture mechanics analyses of the Vickers indentation process shown in Fig. 2a provide equilibrium relations forthe characteristic dimensions a and c in terms of load P. Thus Eq. (1) has been obtained through explicit models of half-penny

Fig. 2. Idealised deformation and fracture patterns under Vickers (a) and Berkovich (b) indentations.


crack evolution within the elasto-plastic field around the tip of a sharp indenter in an isotropic, homogeneous material. Thebasic assumption is that the residual driving force arising during unloading would be primarily responsible for expanding thecrack system to its ultimate half-penny configuration. It was shown [22] that Eq. (1) applies also to deformation and fracturepatterns produced by Berkovich or cube-corner indenters.

3. Experiments

The purpose of the limited experimentation undertaken was to confirm the published information on indentation fractureand obtain measurements for a specific material for a more direct comparison with the numerical results. The material cho-sen for testing was commercial, easily available soda-lime glass. The specimens were in the form of flat strips with dimen-sions 100 mm � 25 mm � 3 mm. They were initially subjected to three-point bending tests and their Young’s modulus E wasdetermined from the elastic deflections monitored by strain gages; the value obtained was E = 69.5 GPa. The surfaces of thespecimens were mirror-smooth, thus there was no need for any mechanical polishing to produce an optical finish. The spec-imen surfaces were cleaned with acetone before each test.

The indentations were performed using a Vickers diamond pyramid. Care was taken to maintain the indentation axis nor-mal to the surface of each specimen but no attempt was made to constrain the specimen from rotating about this axis. Theindented surfaces were then examined by high-power optical microscopy, in reflected polarized light and the characteristicdimensions measured as averages over the two orthogonal radial directions. The indentations and measurements were car-ried out in ambient air and approximately 1 min elapsed between indenting and measuring; this is necessary because soda-lime glass is known to exhibit environmentally-assisted, post-indentation crack growth [1]. The peak contact loads appliedwere restricted to the range 0.98–9.81 N over which the indentation patterns remained acceptable; at the lower end bymeeting the minimum crack length requirement for the validity of Eq. (1) [1], at the upper end by avoiding chipping orby remaining within the limit imposed by specimen thickness. In order to avoid interactions with the lower free surface,the crack dimensions were not allowed to exceed one-tenth of the thickness. In order to minimize slow crack growth effectsfrom moisture in the atmosphere and to reduce the friction between the diamond indenter and the glass surface, a drop ofimmersion silicone oil was placed on the prospective contact site. Twelve impressions were carried out at each indentationload. Both the diagonal lengths 2a and the crack lengths c were read to an accuracy of about ± 0.1 lm from a micrometerattached to the measuring microscope. Their average values from the 24 readings for each load level and the correspondingstandard deviations are listed in Table 1.

Fig. 3 shows micrographs of Vickers indentations in soda-lime glass for two values of the peak load. Both micrographsshow the half-penny pattern and the coexistence of lateral cracks parallel to the specimen surface. The hardness, H, was

Table 1Experimental indentation results.

P (N) 2a (lm) c (lm) H (GPa) n � 103

0.98 19.9 ± 0.5 20.2 ± 0.2 4.95 ± 0.23 18.57 ± 0.521.96 28.2 ± 0.3 32.5 ± 1.1 4.93 ± 0.12 18.85 ± 0.902.94 34.6 ± 0.6 42.3 ± 2.0 4.93 ± 0.17 18.71 ± 1.344.91 43.0 ± 0.8 61.0 ± 3.2 5.31 ± 0.20 20.14 ± 1.589.81 63.3 ± 1.0 92.7 ± 3.7 4.90 ± 0.16 18.12 ± 0.92

Fig. 3. Micrographs of Vickers indentations on soda-lime glass showing the radial crack pattern: (a) P = 9.8 N; (b) P = 4.9 N.


determined routinely directly from Eq. (2) as the indentation pressure for each measured a and its average values togetherwith the corresponding standard deviations for each applied load are listed in Table 1. Its mean value over the whole range ofloads has been found equal to 5.0 ± 0.3 GPa.

Crack sizes were measured for all imprints which gave clearly defined radial traces as a function of load. Then, the quan-tity P/c3/2 was calculated at each load level and its standard deviations for a minimum of five indentations per load indicatedthat, within the experimental scatter, P/c3/2 is effectively invariant with respect to the load, as anticipated by Eq. (1). Intro-ducing the material parameters previously obtained and the fracture toughness for soda-lime glass Kc = 0.75 [23] into Eq. (1)gives a ‘‘calibration” constant for each indentation leading to a mean value n = 0.0189 ± 0.0017. The sensitivity of this exper-imentally determined geometry-dependent parameter for soda-lime glass to likely measurement variations of E and Kc hasnot been assessed but its subsequent comparison with the corresponding numerical value is still valid because almost iden-tical values for these material parameters were used in FEM modelling.

4. Model description

The aim of the present work is to investigate the validity of current fracture characterisation techniques based on inden-tation of brittle solids. This is achieved through extensive finite element simulations of the loading stage of pyramid inden-tations using ABAQUS [12]. As pointed out in the Introduction, there is no attempt here to simulate crack growth for whichan elasto-plastic analysis would have been essential. Instead, an elastic analysis is performed on already cracked specimenssubjected to indentation loads up to or above those originally causing the assumed crack. It is thus implicitly assumed thatneglecting the plastic zone formed during the original indentation fracture process would have a small effect on the predic-tion of SIF magnitude and distribution around the crack front. This assumption can be justified by the constrained nature ofplastic deformation whose influence is further reduced for crack sizes much larger than those of the plastic zone. Anotherobvious limitation of the proposed methodology is the need to assume crack profiles, which may not be physically realisable.The parametric nature of this study allows however the identification of trends towards natural crack shapes and sizes, thus

Fig. 4. Schematic of the Vickers (left) and Berkovich (right) test geometry: (a) side view; (b) top view.


justifying the numerical validation of Eq. (1). The elastic approach has the great advantage of relative simplicity comparedwith complex elasto-plastic simulations required for the prediction of the plastic zone and the subsequent crack growth.However its validity needs to be re-assessed through a critical evaluation of its output.

The Vickers indentation exhibits the eightfold symmetry of deformation depicted in Fig. 4. However, to simplify the spec-ification of symmetry constraints, one fourth of the Vickers indentation configuration was modelled. The resulting finite ele-ment model of the specimen is shown in Fig. 5; it consists of about 30,000 twenty-node brick elements. A non-uniform meshwas developed by employing a relatively fine mesh along the crack front and over contact planes, and a coarse mesh awayfrom these critical zones. The minimum element size at the crack front was 50 nm. The indenter was assumed to be rigid,therefore only its interface with the specimen was modelled. Contact in ABAQUS is modelled by simply specifying the sur-faces that are or could be in contact with each other. Friction can be accounted for through the classical isotropic Coulombfriction model, which requires the input of an appropriate value for the static friction coefficient l. The modelling of Berko-vich and cube-corner indentations was very similar apart from accounting for the sixfold symmetry of their configurations asindicated in Fig. 4. Apart for the symmetry conditions applied to the symmetry planes, the models were also constrainedvertically over their bottom face and left free of traction over their lateral cylindrical surfaces. Analyses performed with mod-el dimensions greater than those shown in Fig. 5 confirmed that the results were not sensitive to such changes.

Contact and the subsequent indentation was simulated as shown in Fig. 4, that is, by applying in the negative x2-directiona quasi-static displacement, equal to the penetration depth d of each analysed case, to a rigid surface representing a face ofthe indenter. For a given displacement, the total load was extracted from the output as the total reaction force acting on therigid indenter. The indentation is a geometrically non-linear deformation process, since the extent of the contact area is notknown a priori. Therefore, a penalty approach method is used for the determination of contact pressure [12]. Initially, frictionwas not considered between the indenter and the specimen material but was introduced in later parametric studies.

The material of the specimen was considered to be soda-lime glass and assumed homogeneous and isotropic, having thefollowing mechanical properties: E = 70 GPa, Poisson’s ratio m = 0.25, H = 4.5 GPa and Kc = 0.75 MPa m1/2 [23]. The adoptedvalue for E is very close to the experimental value reported in Section 3. Bulk constitutive behaviour is assumed for the spec-imen, which essentially means that the obtained results are meaningful only when the indentation depth is much greaterthan the characteristic micro-structural size of the indented material.

The patched multi-block technique for FEM meshing, shown in Fig. 5, was used, where the physical domain was dividedinto a set of sub-domain blocks joined precisely together along their common boundaries. The effectiveness of the resultingmesh was first tested on a benchmark problem, the Vickers indentation on a semi-infinite linear elastic, isotropic materialwith no friction at the interface and for a penetration depth of 30 lm. The numerically predicted P–d relation proved to be inexcellent agreement with the empirical formula based on previous simulations of Vickers indentations [6], the maximumrelative difference being around 1%. Moreover, the model provided converged solutions for up to a maximum penetrationdepth of about 30 lm. Above this depth, convergence within the specified tolerances could not be achieved and further pro-gress of the solution towards larger depths would have required higher tolerances resulting in less accuracy or smaller solu-tion steps resulting in considerable computational cost.

Two quarter-penny cracks were modelled into the specimen, labelled as I and II in Fig. 6a where b is the crack depth. Inthe case of Vickers indentation, these cracks lie on the x1–x2 and x2–x3 planes, respectively. In the case of Berkovich or cube-

Fig. 5. FE mesh for the simulations of Vickers indentations.

Fig. 6. Detail of the contact region and crack fronts; (a) schematic representation of the cracks inserted into the Vickers model; (b) mesh detail around thecracks.


corner indentations, the planes of two cracks make an angle of 120� while the two symmetry planes make an angle of 60�,therefore only one crack was included in the model. Considering independently the characteristic crack lengths c and b, willhelp the assessment of the effect of the crack front shape on the SIF.

The use of ABAQUS proved advantageous since it provides numerical procedures [12] for evaluating the SIFs using theinteraction integral method. Due to the symmetry of the present problem, only the mode I SIF (KI) is generated by this pro-cess. About 60 equally-spaced nodes were created along each crack front, as can be seen in Fig. 6b, in order to guaranteeconvergence for the calculation of the interaction integral as well as SIF.

5. Results and discussion

In this section, the numerical results obtained are presented in detail and compared with relevant analytical and exper-imental results retrieved from the literature. The simulations were performed for various surface crack lengths c, crackdepths b and penetration depths d, evaluating KI at each node of the crack fronts. Due to the symmetry of the Vickers inden-tation problem, only the results for crack I in the x1–x2 plane, shown in Fig. 6a, are presented. The position of the nodes alongthe crack front was specified relative to a polar frame of reference, r � h, with origin at the centre of the crack and h = 0 cor-responding to the surface node A shown in Fig. 6a.

As reported in Section 2, Eq. (1) is valid if c0� a; for soda-lime glass, in particular [23], the relationship is valid if c0/a > 2.5, where c0/a is occasionally referred to as the dimensionless crack length. The maximum applied penetration depthd, for a specific surface crack length c, is thus calculated from the relation between penetration and indentation size so thatthe condition min (c/a) = 2.5 is satisfied. As shown in Fig. 4, for a Vickers indenter,

a ¼ffiffiffi2pðtan /Þd;

where 2/ = 138, so that d = 3.70 lm for c = 30 lm.Initially, penny-shaped cracks were considered, that is, b = c = 30 lm. Since ABAQUS has the capacity to evaluate KI at

each node along the crack front, it is possible to plot its variation with the angular co-ordinate h of each crack front node.The evolution of KI patterns with increasing penetration depth is shown in Fig. 7 for both Vickers and Berkovich indenters.In this as well as subsequent figures, the node angle h represents the angular co-ordinate of a crack front node relative to thepolar frame of reference defined above.

The stress intensity factor reached its maximum value near the surface, at h � 2�. However, this maximum is only 2%higher than the KI value at the surface end of the crack front (h = 0�) and can be attributed to a numerical error also notedin ABAQUS modelling of the classical problem of a semi-infinite solid with a semi-elliptical surface crack, subjected to re-mote uniform tension in the direction normal to the crack faces. The Vickers results are consistent with experimental[16] and numerical [8] evidence that radial cracks (corresponding to low h values) may be initiated, followed by mediancracks (corresponding to high h values) during the early loading stages and these two patterns eventually merge into thefinal half-penny shape at an advanced loading stage. In contrast, the Berkovich results indicate initiation of only radial cracks,which gradually develop into quarter-penny shapes. The maximum numerical KI value becomes equal to the material tough-ness Kc at d = 3.5 lm for Vickers and d = 3.1 lm for Berkovich indentations. The KI distribution at these depths indicates thatthe natural crack profile should be very close to circular for Vickers but slightly elliptic for Berkovich indentations. The cube-corner KI results followed patterns very similar to those obtained for Berkovich but, as expected, the fracture toughness of

Fig. 7. Variation of KI along the penny-shaped crack front for a range of penetration depths.


the material was reached at the much lower penetration depth d = 1.2 lm. It was thus confirmed that a cube-corner indenterwould fracture a specimen at a much lower load than that required for Vickers or Berkovich indentations.

The evolution of KI with crack size is shown in Fig. 8 from simulations of both Vickers and Berkovich indentations. Theseresults confirm the eventual forming of half- or quarter-penny shapes expected at higher loads. Again the cube-corner re-sults show similar patterns to those obtained for Berkovich but the quarter-penny shape is predicted at the lower crack sizeof c = 80 lm. This is consistent with experimental evidence that the cube-corner indenter reduces the indentation crackingthreshold [2].

During an indentation, with the penetration depth continuously increased, Fig. 7 provides an insight into the evolution ofindentation cracks until critical conditions are reached. In all cases, it is predicted that a sort of Palmqvist crack pattern willbe initiated and continue to grow in the radial direction. This is consistent with experimental determinations of materialtoughness when only what happens at the surface crack tip of the specimen is considered. The critical condition for crackgrowth is therefore based here on the value of KI at point A, shown in Fig. 6. The numerically evaluated load P, for which max-imum KI is equal to Kc is entered into the expression (E/H)1/2 P/c3/2, in order to obtain the geometry-dependent parameter, n,from Eq. (1) using the material properties of soda-lime glass as well as those of other brittle materials with known elastic,hardness and fracture properties [23]. This calculation was performed for various crack sizes ranging from c = 5 lm toc = 280 lm in order to assess the numerical stability of the result and its possible dependence on the penetration depth d.The material properties and the respective numerical results obtained for the three indenter types considered are listed inTable 2.

It should be noted that the n values listed in Table 2 are the averages of those calculated over the whole range of adoptedcrack sizes and respective penetration depths. For soda-lime glass in particular, the standard deviations of these averageswere equal to 1.02%, 0.67% and 0.94% for Vickers, Berkovich and cube-corner indentations, respectively. Almost identicalstandard deviations were obtained for each other material input. This strong correlation between a broad range of fracturetoughness and the quantity (E/H)1/2 P/c3/2 over a wide range of penetration depth enhances the validity of the numericalanalysis. Referring, in particular to the Vickers result for soda-lime glass, its excellent agreement with the experimental valuen = 0.0189 ± 0.0017, reported in Section 3, should be noted. It was shown that the prediction of n remains stable for penetra-tion depths higher than that for which the critical condition is first reached.

Plotting Kc versus the calculated value of (E/H)1/2 P/c3/2 for each indenter and performing a regression analysis providedthe respective values of n as the slopes of the regression lines, which are listed in Table 3 with the maximum deviations of theresults listed in Table 2 from these values. Some typical experimental and analytical values of n are also included in that tablefor direct comparison. The analytical upper and lower bounds of n listed in Table 3 are provided by approximate relationships

Fig. 8. Variation of KI along penny-shaped crack fronts for a range of crack sizes.

Table 2Material properties of analysed cracked specimens and respective predictions of parameter n appearing in Eq. (1).

Material E (GPa) H (GPa) Kc (MPaffiffiffiffiffimp

) n

Vickers Berkovich Cube-corner

Soda-lime glass 70 4.5 0.75 0.0188 0.0186 0.0441Al2O3 hot pressed sintered 390 19.7 3.66 0.0166 0.0165 0.0391Al2O3 normal 272 9.3 3.8 0.0137 0.0136 0.0321SiC normal sintered 410 22.2 2.2 0.0172 0.0171 0.0405Si3N4 normal sintered 314 14.1 6.1 0.0157 0.0155 0.0368TZP 210 12.5 5.5 0.0181 0.0179 0.0424SCF5 120 6.8 2.02 0.0176 0.0175 0.0414

Table 3Analytical and experimental values of n.

Indenter type n

ABAQUS Tests Analysis

Vickers 0.0161 ± 0.0025 0.016 ± 0.004 (in oil) [1] 0.014 [13]0.022 (in air) [2] 0.023 [13]

Cube-corner 0.0378 ± 0.006 0.04 ± 0.016 [22] 0.027 [24]0.058 (in air) [19] 0.045 [24]


based on a wedge and a point-force model, respectively [13]. Due to the equivalence between Vickers and Berkovich inden-ters, the experimental value for the former is also quoted for the latter in the literature. The present ABAQUS modelling con-firmed that equivalence by producing almost identical n values for the two indenters.

It clear from Table 3 that the predicted n values are consistent with published results. There is, of course, a degree ofuncertainty concerning the values of the material properties entered in Eq. (1) for the calculation of n. Their variation is ex-pected to have some effect on the upper and lower limits of the calculated n shown in Table 3 but not much on its mean


value. In the case of hardness, in particular, a value higher by as much as 47% than that shown in Table 2 has been reportedfor glass [19] corresponding to an increase of 25% for the computed n. However, the recent experimental measurements of H,reported in Section 3, gave values much closer to the adopted one and thus provided confidence in the reliability of the pres-ent results.

It is also noted in Table 3 that the experimental n values for Vickers indentations in air, are quite different from those insilicone oil; this must be attributed to the reduction of friction in the latter case. The coefficient of friction for contact be-tween glass and diamond can be as high as 0.6 [25]. The effect of friction was investigated numerically by introducing arange of l values into the FE modelling. The results shown in Fig. 9 were obtained for soda-lime glass; their variability asa result of friction can be as high as 37%, which is very close to that of the experimental results presented in Table 3. Inthe case of cube-corner indentations, an experimental result for n, obtained from tests on tempered glass in air [19], washigher by 45% than the mean value quoted by Pharr [22]. However, the effect of friction may be underestimated by the pres-ent elastic analysis since its prediction of the contact area should be smaller than the actual one developing between theindenter and the plastic zone during experiments as can be seen in Fig. 3.

The influence of the crack front shape on KI was assessed by adopting semi- or quarter-elliptical cracks of various crackdepth to length ratios, b/c. The minimum ratio b/c considered was 0.2 based on experimental evidence from cube-cornerindentations, which produced cracks with b/c ffi 0.3 [18]. In Figs. 10–12, KI is plotted versus h for Vickers, Berkovich andcube-corner indentations, respectively. The penetration depths d in each case corresponded to the critical solution stagewhen maximum KI becomes equal to Kc. The assumed crack profiles are not necessarily realisable; therefore the predictedKI distribution is not expected to be uniform along the crack front. However the position of maximum KI can point to thedirection of crack growth towards the crack’s natural profile.

It is evident from Fig. 10 that, for b/c 6 0.7, KI first reaches the critical value Kc at h = 90�, that is, directly below the in-denter tip. This indicates a tendency towards a half-penny crack growth. For b/c > 0.8, the results point to a radial crackgrowth. A slight initial ellipticity of the crack is therefore predicted for Vickers indentations. This may be difficult to detectexperimentally especially if the crack quickly evolves to the half-penny shape at higher loads, as implied by the earlier re-sults with the half-penny cracks shown in Fig. 8. The results for the Berkovich indenter in Fig. 11 clearly show a trend to-wards radial crack growth and a likely quarter-ellipse shape with b/c = 0.3. This could be achieved at a lower penetrationdepth than that implied by the respective quarter-penny result shown in Fig. 8. The KI variations for the cube-corner inden-tations shown in Fig. 12 do have distinct patterns compared with the previous sets. The most obvious characteristic of theseresults is the persistence of radial crack growth for any crack ellipticity. This confirms experimental evidence of Palmqvistcracks dominating such indentation events [16], which may develop into quarter-penny cracks at much higher loads.

Fig. 9. Friction dependence of parameter n for Vickers indentation fracture.

Fig. 10. Vickers KI patterns for various depth-to-surface ratios b/c.

Fig. 11. Berkovich indenter-induced KI variations for various depth-to-surface ratios b/c.


Fig. 12. Cube-corner KI patterns for various depth-to-surface ratios b/c.


6. Concluding remarks

The experimental results reviewed by Ostojic and McPherson [15] as well as Cook and Pharr [16] suggest that no singlegeneralization can be made concerning crack initiation and propagation sequences. The variation of SIF along the entire crackfront obtained by elastic FEM analysis for soda-lime glass confirmed that, below a certain penetration depth and crack size,the natural crack profile due to indentation cannot be perfectly circular or elliptical. In the case of Vickers indentations, thenumerical results are consistent with experimental evidence that median and radial cracks are initiated during the loadingstage and then grow to the final half-penny shape at an advanced loading stage. In the case of Berkovich and cube-cornerindentations, the SIF predictions confirmed the plausibility of quarter-ellipse and Palmqvist cracks, respectively.

The results for the geometry-dependent parameter n are remarkably consistent with those obtained experimentally or byapproximate analytical methods. The effect of friction on n was assessed and found to be significant, in agreement withexperimental evidence. Such consistency reinforces the validity of Eq. (1) and provides the basis for its refinement throughmore systematic parametric studies involving other experimentally observed crack patterns, especially in the case of thesharper cube-corner indenter.

An essential premise for the development of Eq. (1) is the consideration of plastic deformation under the indenter and itsinfluence on crack growth. It is therefore important to extend the analysis to the elasto-plastic range so that crack develop-ment during both the loading and unloading stages could be simulated and the effects of plastic deformation on SIF inves-tigated through comparison with elastic analysis predictions and relevant experimental output. One critical issue in such ananalysis is the identification of realistic plastic properties for the indented material. This can be achieved by a methodologybased on nano-indentation experiments [26].

References

[1] Anstis GR, Chantikul P, Lawn BR, Marshall DB. A critical evaluation of indentation techniques for measuring fracture-toughness. 1. Direct crackmeasurements. J Am Ceram Soc 1981;64(9):533–8.

[2] Morris DJ, Cook RF. Radial fracture during indentation by acute probes: I, description by an indentation wedging model. Int J Fract 2005;136(1–4):237–64.

[3] Bhattacharya AK, Nix WD. Finite element analysis of cone indentation. Int J Solids Struct 1991;27(8):1047–58.[4] Swaddiwudhipong S, Hua J, Tho KK, Liu ZS. Equivalency of Berkovich and conical load-indentation curves. Model Simul Mater Sci Engng

2006;14(1):71–82.[5] Wang HF, Bangert H. 3-Dimensional finite-element simulation of Vickers indentation on coated systems. Mater Sci Eng A – Struct Mater Prop

Microstruct Process 1993;163(1):43–50.


[6] Zeng K, Giannakopoulos AE, Rowcliffe DJ. Vickers indentations in glass. 2. Comparison of finite-element analysis and experiments. Acta Metall Mater1995;43(5):1945–54.

[7] Larsson PL, Giannakopoulos AE, Soderlund E, Rowcliffe DJ, Vestergaard R. Analysis of Berkovich indentation. Int J Solids Struct 1996;33(2):221–48.[8] Larsson PL, Giannakopoulos AE. Tensile stresses and their implication to cracking at pyramid indentation of pressure-sensitive hard metals and

ceramics. Mater Sci Eng A – Struct Mater Prop Microstruct Process 1998;254(1–2):268–81.[9] Zhang W, Subhash G. An elastic–plastic-cracking model for finite element analysis of indentation cracking in brittle materials. Int J Solids Struct

2001;38(34–35):5893–913.[10] Muchtar A, Lim LC, Lee KH. Finite element analysis of Vickers indentation cracking processes in brittle solids using elements exhibiting cohesive post-

failure behaviour. J Mater Sci 2003;38(2):235–43.[11] Maier G, Bocciarelli M, Bolzon G, Fedele R. Inverse analyses in fracture mechanics. Int J Fract 2006;138(1–4):47–73.[12] ABAQUS Analysis User’s Manual. Providence, RI: SIMULIA; 2008.[13] Shetty DK, Rosenfield AR, Duckworth W. Analysis of indentation crack as a wedge-loaded half-penny crack. J Am Ceram Soc 1985;68(2):C65–7.[14] Ponton CB, Rawlings RD. Vickers indentation fracture–toughness test. 2. Application and critical-evaluation of standardized indentation toughness

equations. Mater Sci Technol 1989;5(10):961–76.[15] Ostojic P, McPherson R. A review of indentation fracture theory – its development, principles and limitations. Int J Fract 1987;33(4):297–312.[16] Cook RF, Pharr GM. Direct observation and analysis of indentation cracking in glasses and ceramics. J Am Ceram Soc 1990;73(4):787–817.[17] Dukino RD, Swain MV. Comparative measurement of indentation fracture–toughness with Berkovich and Vickers indenters. J Am Ceram Soc

1992;75(12):3299–304.[18] Kese K, Rowcliffe DJ. Nanoindentation method for measuring residual stress in brittle materials. J Am Ceram Soc 2003;86(5):811–6.[19] Tandon R. A technique for measuring stresses in small spatial regions using cube-corner indentation: application to tempered glass plates. J Eur Ceram

Soc 2007;27(6):2407–14.[20] Ponton CB, Rawlings RD. Vickers indentation fracture–toughness test. 1. Review of literature and formulation of standardized indentation toughness

equations. Mater Sci Technol 1989;5(9):865–72.[21] Liang KM, Gu KF, Fantozzi G. Mechanical analysis of indentation cracking in transformation toughening ceramics. J Chinese Ceram Soc 1994;22:29–37.[22] Pharr GM. Measurement of mechanical properties by ultra-low load indentation. Mater Sci Engng A – Struct Mater Prop Microstruct Process

1998;253(1–2):151–9.[23] Gong JH, Wang JQ, Guan ZD. Indentation toughness of ceramics: a modified approach. J Mater Sci 2002;37(4):865–9.[24] Leonardi A, Furgiuele F, Wood RJK, Syngellakis S. Analytical approaches to stress intensity factor evaluation for indentation cracks. J Am Ceram Soc

2009;92(5):1093–7.[25] Ashby MF, Jones DRH. Engineering materials 1: an introduction to properties, applications and design. 3rd ed. Oxford: Elsevier Butterworth-

Heinemann; 2005.[26] De Fazio L, Syngellakis S, Wood RJK, Fugiuele FM, Sciume G. Nanoindentation of CVD diamond: comparison of an FE model with analytical and

experimental data. Diam Relat Mater 2001;10(3–7):765–9.

Documents

Numerical analysis of brittle materials fractured by sharp indenters