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Journal of the European Ceramic Society 37 (2017) 381–391

Contents lists available at www.sciencedirect.com

Journal of the European Ceramic Society

jo ur nal home p ag e: www. elsev ier .com/ locate / jeurceramsoc

pherical indentation for brittle fracture toughness evaluation byonsidering kinked-cone-crack

aruppasamy Pandian Marimuthua, Felix Rickheya, Jin Haeng Leeb, Hyungyil Leea,∗

Department of Mechanical Engineering, Sogang University, Seoul, 04107, Republic of KoreaDivision for Research Reactor, Korea Atomic Energy Research Institute, Daejeon, 34057, Republic of Korea

r t i c l e i n f o

rticle history:eceived 30 March 2016eceived in revised form 5 August 2016ccepted 11 August 2016vailable online 17 August 2016

a b s t r a c t

This work aims at evaluating the fracture toughness of brittle materials by spherical indentation. Thecone-cracking is simulated by the extended finite element method (XFEM) in Abaqus. The formation of akinked-cone-crack is observed when the indenter comes into (second) contact with the surface part out-side the ring-crack. The effects of friction, Poisson’s ratio and cone-crack kinking on the Roesler’s constant�c are analyzed. Based on numerical results, the Roesler’s method for evaluating the fracture toughness

eywords:racture toughnesspherical indentation crackinginked-cone-cracksFEM

is enhanced by considering kinked-cone-crack. By performing systematic XFE analyses, a database forenhanced Roesler’s constant �c | kink is provided for the fracture toughness evaluation of brittle materials.Finally, the proposed method is verified by conducting spherical indentation tests on soda-lime glassspecimens.

© 2016 Elsevier Ltd. All rights reserved.

oesler’s constant

. Introduction

Material failure under working condition often causes propertyamage and accidents. Especially brittle materials exhibit catas-rophic failure even under quasi-static loadings. A characterizationf materials based on fracture mechanics helps to prevent theaterial failure. In fracture mechanics, fracture toughness KIC is

n important material property, which describes the resistance of material to fracture. However, evaluation of KIC of brittle mate-ials is a challenging task, as conventional testing i.e. tensile test,ending test, compact tension test is tedious. Preparing those spec-

mens and performing controlled fracture test are difficult due tohe brittleness of materials [1].

Compared with conventional test methods, indentation methodor evaluation of fracture toughness [2] becomes more essentialo predict the lifetime of brittle materials for engineering applica-ions such as bearings, engine components, monoliths and coatings3]. The indentation test offers convenient evaluation of fractureoughness [3–7] on micro/nano scales. The indentation test is rela-ively simple regarding specimen preparation and procedure, and

he indentation test results can be produced at lower cost and time.he indentation cracking test involves developing controlled frac-ure beneath the surface of brittle material by pressing an indenter

∗ Corresponding author.E-mail address: hylee@sogang.ac.kr (H. Lee).

ttp://dx.doi.org/10.1016/j.jeurceramsoc.2016.08.014955-2219/© 2016 Elsevier Ltd. All rights reserved.

tip. As a result, depending on the indenter shape, cone, radial, ring,median, half-penny and lateral cracks form in the specimen. In theevaluation of fracture toughness with formulated simple equations,the dimensions of these cracks and indentation loads are relatedwith other material properties of specimen [3–7]. However, thereis no universal formula applicable to evaluate the fracture tough-ness of all brittle materials. Also indentation test method requiresa clear understanding of the complex deformation processes goingon in the material beneath the indenter.

This present work focuses on spherical indentation cracking,which avoids micro-cracking and phase transformation in thematerials in contrast to the sharp indentation. The test materialsalso show elastic behavior until fracture. Therefore the compli-cations associated with the residual stress caused by inelasticdeformation in sharp indentation can be obviated with sphericalindentation [4].

The aim of this work is to evaluate brittle fracture toughness KIC

by establishing a quickly and directly applicable spherical indenta-tion method. The extended finite element method (XFEM) is used tosimulate the cone-crack formation in Abaqus/Standard (ver. 6.14)[8,9]. The formation of kinked-cone-crack (KCC), which was notaddressed in the spherical indentation studies [3–7], is discussed.The effects of friction, Poisson’s ratio and cone-crack kinking on

the Roesler’s constant �c are analyzed. Based on XFEM, Roesler’slaw method [6] for evaluating brittle fracture toughness is modi-fied by considering kinked-cone-crack. With further XFE analyses,a database for an enhanced Roesler’s constant �c | kink is provided

382 K.P. Marimuthu et al. / Journal of the Europea

Ft

fSi

2

crcar

2

Hfiwt

a

whsccta

2

sticibmrtfl[

cot

erally based on Pc . In the indentation test, Pc tends to vary withfollowing factors: (i) specimen surface condition, especially sur-face roughness, (ii) experimental setup, (iii) friction on the contactinterface and indenter elasticity [11] and (iv) surface residual stress.

Table 1�c values for soda-lime glass (� = 0.23) in the literature.

� (N/m) 103 × �c method

Lardner et al. [6] 8.0 14.0 FEMRoesler [18] – 27.1a experimentLawn and Wilshaw [21] 8.0 12.8 experiment

ig. 1. Example of evolved cone-crack in soda-lime glass under spherical indenta-ion: indenter radius R, ring-crack radius ro , cone-crack base radius rc .

or the fracture toughness evaluation of diverse brittle materials.pherical indentation tests are conducted on soda-lime glass spec-mens to validate the cone-crack kinking observed in XFE analyses.

. Prior spherical indentation methods

In brittle materials, spherical indentation generates ring andone-cracks beneath the indenter as shown in Fig. 1. Numerousesearch works describe the spherical indentation system and theone-crack formation [3–7,10–18]. Nonetheless, there are limitedrticles which explain the complete cone-cracking in brittle mate-ials under spherical indenters [3–7].

.1. Spherical indentation (Hertzian theory)

By assuming a frictionless contact between two elastic solids,ertz [10] proposed an elastic theory to describe the complex stresseld in the contact materials. When an elastic spherical indenterith radius R indents an elastic half-space with indentation load P,

he contact radius a and the indentation depth h are given as [11]

=(

3PR

4Ee

)1/3; h = a2

R;

1Ee

= 1 − �2

E+ 1 − �2

I

EI(1)

here E, EI and �,� I are Young’s modulus and Poisson’s ratio of thealf-space and the indenter, respectively. The above equations areufficient to specify the loading condition for all kinds of spheri-al contact [12]. By extending Hertz’s theory, Huber [12] providedomplete stress field solutions for spherical contact. Later, Hertzianheory was used to explain the initiation and propagation of ringnd cone-cracks in brittle materials [3–7].

.2. Cone-crack formation

Fig. 2 schematizes the formation of the cone-crack under apherical indenter. In the specimen surface, radial stresses �r areensile outside the contact circle (with radius = a) and reach a max-mum �max

r at the contact periphery (r = a) [11]. Therefore, it isonvenient to understand the stress field in the surface as a local-zed tensile skin outside the contact circle [13]. A ring-crack maye presumed to initiate at the contact periphery by satisfying theaximum tensile stress fracture condition. But in experiments, the

ing-crack radius ro is always larger than a; the spherical indenta-ion tests resulted in the ratio ro/a = 1.1 to 1.4, which depends onaw size [3], indenter radius, high stress gradient below the surface7] and friction in the contact regions [14].

At a critical indentation load Pc , the ring-crack grows into a cone-rack. Auerbach [15] extensively studied the effect of indenter sizen cone-crack initiation in glass specimens, and gave a linear rela-ionship between Pc and R as Pc∝R (Auerbach’s law), which holds

n Ceramic Society 37 (2017) 381–391

only for small indenters. Experimental observations show that Pc

is proportional to the square of R (Pc∝R 2) for large indenters [16].Argon [17] also noted that the strength of glass depends on indentersize.

In the quasi-static spherical indentations, if P > Pc , the cone-crack substantially grows with an angle � to the surface. Whenthe cone-crack becomes relatively larger than ring-crack, furtherpropagation of cone-crack is not affected by the contact zone [18].By conducting the spherical indentation test on soda-lime glass,Roesler [18] reported that � depends only on Poisson’s ratio �, andthus � is a sort of invariant for a given �; soda-lime glass (� = 0.25)resulted � = 22◦ [18].

2.3. Prior fracture toughness evaluation methods

Under plane strain condition, the fracture toughness KIC isrelated to the critical strain energy release rate or fracture energy� as follows

� =(

1 − �2)

EK2

IC = 2� (2)

where surface energy � is the energy required to create a unit areaof new surface in the materials. To evaluate fracture toughness,the spherical indentation variables (P, Pc , a, ro, rc) were relatedto � based on following three basic methods: (i) Auerbach’s lawmethod, which uses the critical load Pc [5,19] (ii) minimum fractureload method, which uses a value of minimum load Pmin

c necessary topropagate cracks from at least 25 indentation tests [4,20] and (iii)Roesler’s law method [6,21], which uses cone-crack base radius rc

and corresponding P.At larger indentation loads P, the cone-crack base radius rc varies

as the 2/3 power of indentation load P [6]. Based on this observation,Roesler [18] related P, rc and � as

P2

r3c

= �E

�(3)

where � is a Roesler’s dimensionless �-dependent coefficient forwell-developed cone-cracks (rc/ro > 5). Lardner et al. [6] re-wrotethe Roesler’s law in terms of normalized energy release rate as

G = �2E�r3c

P2= �c (4)

For well-developed cone-cracks, G saturates to a constant valuei.e. �c (Roesler’s constant) for a given �; that is, �c is also depen-dent on �· Based on FE analyses, Lardner et al. [6] obtained thenormalized energy release rate and validated the Roesler’s law.For soda-lime glass, �c values from the literature are listed inTable 1. Due to the different experimental procedures used by var-ious researchers, experimental scatter can be seen in �c values forthe same material.

The Auerbach’s law and minimum fracture load methods are gen-

Lawn and Fuller [22] 7.8 11.7 experimentChaudhri and Phillips [23] 7.8 14.5 experimentDavis et al. [24] 8.0 10.4 experiment

a Overestimates the cone-crack radius at a given indentation load [6].

K.P. Marimuthu et al. / Journal of the European Ceramic Society 37 (2017) 381–391 383

Fig. 2. Schematic representations of cone-crack formation.

Ft

Wco

Efia�eutc

3

3

r9(iifunws

esAcwiadfdit

tt

ig. 3. Cut view of FE model; enlarged view shows the refined inner region beneathhe indenter.

hereas Roesler’s law method is based on well-developed cone-racks, the above mentioned factors do not affect the propagationf well-developed cone-cracks.

By conducting spherical indentation test, � can be calculated byq. (4) provided � and �c are known. However, �c values are hard tond in the literature for brittle materials. Only for soda-lime glass,

large number of experimental resources are found to calculatec , which is thus limited to � = 0.23. It would thus be worthwhile tonhance the Roesler’s law method for an easy and quantitative eval-ation of fracture toughness. Therefore, this present work improveshe Roesler’s law method by analyzing the spherical indentationracking for a wide range of brittle materials.

. Spherical indentation cracking

.1. Finite element modelling

Spherical indentation cracking is simplified as an axisymmet-ic problem. An axi-symmetric XFE model, which contains about8000 nodes and 52000 4-node bi-linear axisymmetric elementsCAX4) [9], is used (Fig. 3). The inner region, beneath the indenter,s refined with square elements in order to maintain the same min-mum element size e; the model thus can facilitate the cone-crackormation. The inner region elements are enriched with discontin-ous function for cracking analysis in XFEM. In the FE model, theodes on the bottom are allowed to move only in the r-direction,hile the axisymmetric boundary conditions are applied to the left

ide nodes.The material is assumed to be isotropic and exhibit purely linear

lastic behavior. For preliminary studies, the material properties ofoda-lime glass (E = 70 GPa, � = 0.23) are assigned to the FE model.s the well-developed cone-cracks are independent of the contactondition [18] i.e. indenter deformation, a rigid spherical indenterith R = 500 m is employed. The appropriateness of the FE mesh

s confirmed by comparing FE solutions of radial stress distributionnd load-depth curve with Hertzian solutions. Since we considerissimilar material properties for indenter and the specimen, theriction plays significant roles in determining the ring-crack radiusuring loading and initiating secondary ring-cracks during unload-

ng [25]. Therefore, a non-zero friction coefficient is considered in

he contact region for cracking analysis.

XFEM is used to simulate the fracture. A more detailed descrip-ion about XFEM can be found elsewhere [9,26–28]. We applyhe bilinear traction-separation (T−ı) law (Fig. 4) [9] to gov-

Fig. 4. Schematic representation of traction-separation law for damage initiationand propagation [29].

ern the damage initiation and propagation on a continuum level.The traction-separation law governs damage initiation througha damage-initiating stress threshold �c . The damage is initiatedwhen the applied tensile stress reach � c . Upon further loadingthe stiffness tensor is degraded according to the softening branch.Once the separation ı reaches a critical value ıc , a crack has formed.For linear softening, �c and ıc are related to the fracture energy �through� = 1

2 �c ıc . Since the considered fracture energy and othermaterial properties are identical, we assumed that the variation of�c (thus ıc for a given � ) has a negligible effect on the crack propa-gation of well-developed cracks [29]. In fracture analysis, materialsoftening behavior and stiffness degradation may lead to severeconvergence difficulties [30]. Hence, we use viscous regularizationby adding a small artificial viscosity (1 × 10−5) to the XFE model.As a result, tangent stiffness matrix becomes to be positive definitefor sufficiently small time increments [9].

3.2. Finite element analysis

The formation of cone-cracks under rigid spherical indenters issimulated in XFEM by considering soda-lime glass properties withf = 0.1 [25], �c = 0.3 GPa and � = 8 N/m [7]. As the indenter indentsthe specimen surface, damage initiates ahead of the contact circlein the surface. The ring-crack is then formed with radius ro, whichis observed to be always larger than contact radius a. The ring-crackthen propagates downwards with further loading. At critical loadPc , the crack assumes a skirt-like shape and then grows very rapidlyas a cone-crack as shown in Fig. 5. In experiments, this rapid cone-crack formation is accompanied by acoustic emission [5]. Whenmaintaining P > Pc , the cone-crack propagates continuously withan angle � to the surface. Note that the simulated cone-crack angle� = 18◦ ± 0·5◦ for � = 0.23 is close to the experimental value for glass[4].

As P (>Pc) increases, a also increases. If ro is not much larger thana, the indenter may come into (second) contact with the surfacepart outside the ring-crack as shown in Fig. 5. The correspond-ing indentation load is defined as second critical load Psc . As a

result, the stress field ahead of the crack tip changes as comparedin Fig. 6(a)-(b). The cone-crack extends along the new minimumprinciple stress �3 directions. Therefore, the direction of cone-crackpropagation changes and a kink appears in the cone-crack. It is evi-

384 K.P. Marimuthu et al. / Journal of the European Ceramic Society 37 (2017) 381–391

Fig. 5. Alteration of Von Mises stress �v distribution due to cone-crack formation in soda-lime glass. �v is normalized by using maximum contact pressure po . �v and po arefunction of P.

ss [ar

dsastwp

t

Fig. 6. Directions of maximum principle stre

ent from Hubers’s stress field solutions that � affects the entiretress field [12]; hence it also affects the angle of cone-crack prop-gation. The kink angle �K becomes functions of �. In fine-grainedilicon nitride (Si3N4) monoliths and bilayers, the kink appeared inhe cone-crack as shown in Fig. 7 [31] under WC spherical indenterith R = 1.98 mm at load P = 3 kN. But Lee et al. [31] didn’t state the

henomena of kinking in their work.

Upon further loading (P > Psc), the kinked-cone-crack continueso propagate with an angle �K to the surface. Due to the second

rows] before (left) and after kinking (right).

contact, �1 becomes tensile outside the ring-crack as shown inFig. 6(b). If the fracture criterion is satisfied, the secondary (addi-tional) ring-cracks may form in the surface outside the ring-crackand extend along in loading direction as shown in Fig. 8. Thesesecondary ring-cracks do not spread out as secondary cone-cracks,since the primary cone-crack prevents the development of tensile

stress in the direction normal to the primary cone-crack [32]. Inexperiments, the secondary ring-cracks may merge with the pri-

K.P. Marimuthu et al. / Journal of the European Ceramic Society 37 (2017) 381–391 385

Fig. 7. Hertzian contact damage in Si3N4 monoliths (left) and bilayers (layer thickness =

diameter is indicated with A-A [31].

mo

4

aXs�neo

t|esuirFstsKlt

4

(mrs

Fig. 8. Formation of secondary ring-cracks in spherical indentation.

ary cone-crack to produce a ‘detachable collar’ around the neckf the cone (Fig. 8) [32].

. Results and discussion

Eq. (4) was developed based on a single cone-crack regime byssuming that � is invariant for a given material. However, theFE analyses support the formation of kinked-cone-crack underpherical indenters. Due to kinking, the cone-crack angle changes

(before the kinking) to �K (after the kinking). Therefore, one can-ot accurately evaluate the � from Eq. (4) without considering theffect of kinking on �c . In the following sections, the effect of kinkingn �c is investigated.

To analyze the effect of kinking on �c , a comparison betweenhe values of �c calculated from a kinkless cone-crack (KLCC) i.e. �c

kinkless and from the KCC i.e. �c | kink are needed. To obtain KLCCven at higher loads P » Pc , Lawn and Fuller [22] used truncatedpherical indenter with a base radius of 1 mm, and Lardner et al. [6]sed Hertzian pressure distribution (in FEM) instead of spherical

ndenter to ensure that the contact radius be less than ring-crackadius. Here, the indenter shape is slightly modified as shown inig. 9(b) and KLCC is produced even at higher loads by avoiding theecond contact of indenter with cracked surface. Fig. 9(c) compareshe load-depth curves (P-h) from kinked and kinkless cone-crackystem. At higher loads, P-h curve of KCC deviates from those ofLCC. The value of �c | kinkless, which is calculated from the simu-

ated KLCC, is compared with the �c values in Table 1 to validatehe XFE results.

.1. Applying the calibration techniques to XFE analysis

The element size has substantial effects on damage initiation

i.e. critical load Pc). When we use the same �c for different ele-

ent size e in the inner zone, we obtain different critical load anding-crack radius. In order to avoid this mesh-dependency, we use acaled damage initiation strength �sc for particular e to simulate the

600 m) (right) under WC spherical indenter with R = 1.98 mm at P = 3 kN. Contact

cone-crack, which is equivalent to the real size and shape of cone-crackobserved in experiment at particular load. For that, we use calibra-tion techniques of Cao and Zhang [33] and mean strength theoryof Bao et al. [7], which average the maximum principle stress in anelement for stress-based crack initiation in the element-failure FEanalyses. �sc is calculated by following Cao and Zhang’s [33] relationas follows

�sc = KIC/√

�e = (

�E′)1/2/√

�e (5)

where E′ = E/(1−� 2). Cao and Zhang [33] calibrated the dimension-less parameter as 1.2 for e = 8.17 m by comparing FE resultswith those of Lardner et al. [6] for soda-lime glass. XFEM is usedhere to calibrate for different e by comparing � c | kinkless withthe literature data [6,21–24], and thereby, to select an appropriate�sc for considered e. Note that, �sc is not an actual material con-stant, but equivalent to �c for considered e. From mesh refinementstudy, the XFE model with e = 5 m is found to be appropriate tosimulate the cone-cracking with better convergence at relativelylow computational cost. By simulating KLCC, is then calibrated as0.83 for e = 5 m (based on Table 2) by selecting soda-lime glass asa characteristic material, which has �c ≈ 12 × 10 −3 (Table 1) as acharacteristic value.

4.2. Analyzing the characteristics of Roesler’s constant �c

To see that �c depends only on �, we use two different � = 7,8 N/m and � = 0.21, 0.23 in the XFE analysis. �sc is calculated basedon Eq. (5). Fig. 10(a) shows the variation of G as a function ofrc/ro. We use ro to normalize rc , since ro is constant in sphericalindentation. Above rc/ro > 7, G reaches a saturated value, which isrepresented as �c . Fig. 10(a) also confirms that the saturated valuesvary with �, not with � . When kinking occurs at Psc , � changes to �

K . As a result, G decreases [Fig. 10(b)]; however, G reaches a satu-rated value for well-developed KCC. But the saturated G values forKCC are smaller than those for KLCC as shown Fig. 10(b). It confirmsthat the kinking has significant effect on �c .

The effect of friction on �c is analyzed by considering variousf = 0, 0.1, 0.12, 0.15, 0.2 in the contact interface between the rigidindenter and the specimen. When f increases, Pc and ro increaseas listed in Table 3. The changes in Pc and ro change Psc and thelocation of kinking in the cone-crack. As a result, the G and theload-depth curves are significantly affected when the cone-crackbecomes large as shown in Fig. 11(a) and (b), respectively. Weobserve negligible effect of friction on load-depth curve when thecone-cracks are relatively small. Unlike �, �K is not always functionof Poisson’s ratio. It gently depends on ring-crack and kinking loca-tion in most cases. This is the reason for the small variation in �K

for the change in friction coefficient from f = 0 to f = 0.1 in Table 3.The friction has a negligible effect on �c as the change in f has trivial

effects on � and � K . Therefore, G curves reach almost similar satu-rated values for well-developed KCC [Fig 13(a)] even with differentf. For further analysis, the non-zero friction coefficient is thus takenas f = 0.1.

386 K.P. Marimuthu et al. / Journal of the European Ceramic Society 37 (2017) 381–391

Fig. 9. Formed (a) KCC under spherical indentation and (b) KLCC under modified spherical indentation at Pmax = 0.60 kN. (c) Comparisons of load-depth curves of kinked andkinkless cone-crack system.

Table 2Calibration of for KLCC in XFEM (E = 70 GPa, � = 0.23, f = 0.1, R = 500 m, rc/ro > 7).

e (m) � (N/m) �sc (GPa) Pc (kN) 103 × � c |kinkless

55555

88888

0.1850.1800.1700.1650.160

0.950.930.880.850.83

0.0830.0800.0650.0600.058

10.010.110.911.311.8

Bold values are calibrated values.

Fig. 10. Comparison of G variation (a) for KLCC with different � = 0.21, 0.23 and � = 7, 8 N/m and (b) between KLCC and KCC.

Table 3Effect of friction on cone-cracking (E = 70 GPa, � = 0.23, � = 8 N/m, � sc = 0.16 GPa, rc/ro > 7, KCC).

f Pc (kN) ro (m) �(deg.) �K (deg.) 103 × �c |kink

0.000.100.120.150.20

0.0410.0610.0680.0810.096

87.5107.5112.5127.5142.5

1818181818

2926262626

9.339.409.509.609.39

K.P. Marimuthu et al. / Journal of the European Ceramic Society 37 (2017) 381–391 387

Fig. 11. Comparison of (a) Gvariation and (b) load-depth curves for simulated KCC with different f = 0.0, 0.1, 0.12, 0.15, 0.2.

F TiB2,

F upple

lEiei

ig. 12. (a)−(c) P vs. rc plots and (d)−(f) variation of G as a function of rc /ro for KCC inor all considered brittle materials, the same plots are separately presented in the S

To check the effects of E and �sc on �c , we have also simu-ated the KCC and KLCC formation for following material properties

= 410 GPa, � = 50 J/m2, � sc = 1 GPa and � = 0.14, 0.23. Compar-ng �c values for � = 0.23 in Tables 3 and 4, we observe negligibleffect of E, � sc on �c . For different �, the effect of kinking on �c

s listed in Table 4. A smaller � enhances the crack propagation by

ZrB2, PcBN (These plots are presented here for a part of considered brittle materials.mentary materials).

accommodating smaller elastic deformation and increases differ-ence between � and �K . As a result the effect of kinking is significant

for smaller �. Whereas a larger � enhances the failure-resisting abil-ity i.e. toughness of the material by accommodating larger elasticdeformation [34] and decreases the difference between � and �K .Hence, the effect of kinking on �c is relatively small. Henceforth, it’s

388 K.P. Marimuthu et al. / Journal of the European Ceramic Society 37 (2017) 381–391

Fig. 13. Variation of (a) �c and (b) cone-crack angles as a function of �.

Table 4Effect of � and kinking on �c (E = 410 GPa, f = 0.1, � = 50 N/m, �sc = 1 GPa, rc/ro > 7).

� � (deg.) �K (deg.) 103 × �c |kink

103 × �c |kinkless

�c

difference

0.14 23 – – 20.9 9.65 × 10−3

sor

4

�tthUalvgKFoTddtBtl

a��rattIdbcrKd

0.14 23 350.230.23

1818

–26

uggested to use the enhanced Roesler’s constant � c | kink insteadf � c or � c | kinkless in Eq. (4), especially for the materials withelatively small Poisson’s ratio.

.3. Generating database of � c (� c | kink and � c | kinkless)

To generate database of �c , we collect material properties (E,, KIC ) of various brittle materials from several literatures. Ashe microstructure of ceramics plays an important role in brit-le responses under spherical contact, the material properties ofomogenous and fine-grained ceramics are used in this analysis.sing Eq. (2) and Eq. (5), we then calculate the corresponding �nd �sc ( = 0.83), respectively, for cracking analyses in XFEM asisted in Table 5. Spherical indentation cracking are analyzed forarious brittle materials. Since we use �sc in XFEM, we achieveood convergence for considered brittle material properties. ForCC system, increasing rc is plotted against increasing P as shown inig. 12. The plot P vs. rc can be regressed with 2nd order polynomialr exponential functions for well-developed cone-cracks (rc/ro > 5).he regression functions for particular material can be used to pre-ict rc under applied P, and thereby, to validate the experimentalata. We then plot the variation of G against increasing rc/ro, and wehus get saturated G values, that is � c | kink, for each brittle material.y following the same procedure, � c | kinkless are calculated fromhe KLCC. Obtained � c | kink values are smaller than � c | kinkless asisted in Table 5.

The variation of � c | kink and � c | kinkless with � has two regions Ind II as shown in Fig. 13(a). In region I, �c | kink is rather sensitive toK which depends on � and kinking-location, thus giving scatteredc | kink. Also the phenomena of kinking play an important in this

egion. In the region II, �c | kink is more sensitive to Poisson’s ratio,nd exponentially decays with �. �c | kinkless variation appears noto change in Region II, when compared with Region I· Regressinghe plot �c |kink vs. � with two regression functions for each region

and II, we can obtain KIC with ± 5% accuracy. Also, both � and �K

epend on � as shown in Fig. 13(b). The �c | kink values in Table 5 cane used to evaluate KIC from spherical indentation by considering

one-crack kinking. For all considered brittle materials, the P vs.c plots are separately presented in the supplementary materials.nown fracture energies are used in XFEM analyses only to developatabase of � c for a wide range of � as listed in Table 5. By knowing

11.7 ––9.50

11.4–

1.9 × 10−3

material’s �, this method provides the fracture toughness based onreverse engineering analysis. Therefore one can apply this methodto a brittle material whose fracture toughness is not known a priori.

4.4. Measurement of fracture toughness

In this section, we explain the method to evaluate fracturetoughness of brittle materials by considering well-developed KCC.By substituting maximum indentation load Pmax and �c | kink insteadof P and �c, respectively, we re-write Eq. (4) as follows

� = 1E�2

P2max

r3c

�c |kink (6)

By combining Eq. (2) and Eq. (6), KIC is then given as

KIC (or) KIC |kink = (�E

1 − �2)1/2

= [1

�2

P2max

r3c

�c |kink

1 − �2]1/2

(7)

The fracture toughness of the indented material can be calcu-lated from Eq. (7) by knowing �, �c | kink, Pmax and rc . By consideringKCC, this research provides �c | kink values for a wide range of � aslisted in Table 5. By conducting spherical indentation test up to aspecified Pmax, one can measure corresponding rc by using opticalmicroscope for transparent specimens or the serial cross-sectioningmethods for opaque materials (see reference [48]). In addition, theexperimental values of rc can be compared with those calculatedbased on regression in Fig. 12 (also see Supplementary data). Whenwe considered � c | kinkless in Eq. (7), there are some error involvesin fracture toughness evaluation as shown in Fig. 14. The errors areestimated quantitatively as error (%) = (KIC | kink−KIC | kinkless)/KIC |kink. KIC | kinkless are evaluated by substituting � c | kinkless insteadof � c | kink in Eq. (7), and Pmax and rc of kinked-cone-crack. As weexplained in the Section 4.2, the effect of kinking on KIC is relativelylarge when the materials Poisson’s ratio is small. The percentage oferror gradually decreases with increasing of �. Hence, this researchwork suggests to use � c | kink in Eq. (7) while evaluating the fracturetoughness of brittle materials by spherical indentation.

4.5. Experimental validation of proposed method

Spherical indentation tests are conducted to verify the forma-tion of kinked-cone-crack and validate the proposed method. A

K.P. Marimuthu et al. / Journal of the European Ceramic Society 37 (2017) 381–391 389

Table 5Database of � c (� c | kink and � c | kinkless) values from XFE analysis.

materials literature data XFE results

E (GPa) � � (N/m) 103 × � c | kink 103 × � c | kinkless � | �K (deg.)

Titanium diboride (TiB2) [35] 565 0.11 67.2 14.1 26.0 24.0 | 37.5Zirconium diboride (ZrB2) [36] 346 0.11 16.5 15.5 25.0 24.0 | 36.3Polycrystalline cubic boron Nitride (PcBN)[37] 894 0.12 51.0 14.3 24.1 23.4 | 36.1Fused silica [38] 70 0.14 9.0 13.2 20.7 22.6 | 35.4Silicon carbide (SiC) [39] 410 0.164 61.7 11.8 18.2 21.3 | 33.4Boron carbide (B4C) [40] 465 0.17 17.6 13.2 18.1 20.7 | 32.1Magnesium oxide (MgO) [41] 317 0.17 19.2 13.0 18.8 20.7 | 32.1Borofloat glass [42] 64 0.18 16.0 11.5 16.4 20.4 | 32.1Titanium carbides (TiC) [43] 400 0.19 38.6 11.4 16.1 19.8 | 31.0Vanadium carbide (VC) [19] 320 0.19 5.5 12.7 16.3 19.4 | 29.3Starphire [42] 73 0.20 14.8 11.0 14.0 19.2 | 29.3Pyrex (borosilicate glass) [41] 64 0.20 9.0 11.2 14.1 19.4 | 30.0Zirconium carbides (ZrC) [41] 407 0.20 9.43 12.2 14.5 18.9 | 29.1Tungsten carbides (WC) [19] 680 0.22 12.3 11.5 12.7 18.1 | 27.1Soda-lime glass [43] 70 0.23 7.61 9.65 11.3 a18.0 | 27.0Aluminium oxynitride (AlON) [44] 323 0.24 15.4 8.50 10.0 16.9 | 25.9Aluminium nitride (AlN) [45] 308 0.25 42.9 8.30 10.2 16.3 | 26.5Silicon nitride (Si3N4) [46] 335 0.27 42.1 7.90 7.40 a15.5 | 22.8Alumina (Al2O3) [4] 393 0.27 47.8 7.50 8.00 15.4 | 23.1Calcium fluoride (CaF2) [41] 108 0.29 5.43 6.50 6.60 14.2 | 21.6Strontium fluoride (SrF2) [41] 90 0.29

Yttria (Y2O3) [47] 176 0.30

a Only for soda-lime glass and Si3N4 [31], � and �K are compared with experimental ob

F

mKmtctTfitgmuii

ssstpcyK

part outside the ring-crack. The element size effect on cone-crack

ig. 14. Error involved in fracture toughness evaluation with � c | kinkless in Eq. (7).

icro indentation machine (Indentation system, R&B Inc., Southorea) with a spherical indenter tip is used. The indentationachine has a load resolution of ± 5 N. The indenter tip is a

ungsten-carbide sphere with R = 500 m. The experiments arearried out on soda-lime glass specimens with 100 mm diame-er and 20 mm thickness (Seoul special glass Ind., South Korea).he glass specimens are used in an as-received condition with nourther annealing or surface treatment. The indenter is pressednto the specimen surface with a loading rate of 5 m/min andhe load-depth data are recorded. The indentation machine is pro-rammed in a way of displacement control. If P reaches the specifiedaximum indentation loads P max (= 0.1, 0.2, 0.3, 0.4, 0.5 kN), the

nloading process starts with an unloading rate of 5 m/min. 2–5ndentations are made on the smaller specimens, and minimum 5ndentations are made on the larger specimens.

An optical microscope is used for taking the images. The crossection images of KCC [Fig. 15(c)] are captured after sectioning thepecimens. The sectioning process includes scratching the indentedurface with a glass cutter and then applying 3-point bending loado the specimen. The sectioned surfaces are polished by using emeryapers with grit size 320 and 600 to make smooth surfaces. Fig. 15

ompares the experimental cone-crack with that from XFEM anal-ses. The experimental observations also confirm the formation ofCC when the indenter comes into contact with the surface part

10.2 6.90 5.90 14.2 | 21.38.74 6.32 5.70 13.4 | 20.0

servations.

outside the ring-crack. However KCC may not occur in a flat-endedcylindrical indentation because the contact radius a is constant, andalways smaller than the ring-crack radius rc .

It is hard to discern the cone-crack-tip because the cone-crackcloses during unloading. Therefore, we put a drop of commercialred ink on the surface before starting unloading (test A) and afterunloading (test B). Due to the capillary effect, the red ink is drawninside the crack up to the cone-crack-tip; doing so the cone-cracktips become visible as shown Fig. 16. Radial and secondary ring-cracks are also observed in the experiments. The dimensions ofradial crack may be used to evaluate the fracture toughness. Dis-cussing the radial cracks is beyond the scope of present work. It willbe studied in the future work.

Propagated cone-cracks at different indentation loads Pmax = 0.1,0.2, 0.3, 0.4 and 0.5 kN are compared in Fig. 16. The experimen-tal values of rc is also compared with those obtained in XFEanalysis. The rc = 745 ± 5 m at Pmax = 0.5 kN is used to validatethe proposed approach as Pmax = 0.5 kN produces well-developedcone-cracks (rc/ro > 7) in soda-lime glass. By using Eq. (4), theexperimental value of normalized energy release rate is calculatedasG = 9.14 × 10−3, which is in close agreement with the modeledvalue of G = 8.96 × 10−3 at Pmax = 0.5 kN; XFE model is validated.Hence the modeled value of the �c | kink = 9.65 × 10−3 in Table 5is used to evaluate the fracture toughness of soda-lime glass asKIC = 0.79 MPa m1/2, which is in good agreements with those in liter-atures KIC = 0.7 ∼ 0.8 MPa m1/2 for soda-lime glass [4]. This, in somesense, validates the generated database for � c | kink.

5. Summary and conclusions

In this research work, we established a quickly and directlyapplicable spherical indentation method for evaluation of brittlefracture toughness KIC by considering kinked-cone-crack (KCC). Byselecting soda-lime glass as a reference material, we simulated thecone-crack formation in XFEM. The formation of KCC was observedwhen the indenter comes into (second) contact with the surface

propagation was reduced by scaling the damage initiation strength.The effects of friction and Poisson’s ratio and cone-crack kinkingon the Roesler’s constant �c were analyzed. To apply this method

390 K.P. Marimuthu et al. / Journal of the European Ceramic Society 37 (2017) 381–391

Fig. 15. (a) ring and (b) simulated and (c) experimentally observed cone-cracks (section A−A) in soda-lime glass under spherical indenter.

F . Conev

fr�df

o

o

e

nad

A

tR

A

prccj

[

[[

[

[

ig. 16. Top view propagated cone-cracks at different Pmax = 0.1, 0.2, 0.3, 0.4, 05 kNalues.

or fracture toughness evaluation to a wide range of brittle mate-ials with diverse �, we provided a database for �c (�c | kink andc | kinkless) with further XFE analyses. Indentation tests were con-ucted on soda-lime glass to confirm the cone-crack kinking. Theollowing conclusions can be drawn from this study,

→ Both XFEM and experimental studies support the formationf kinked-cone-crack under spherical indenters.

→ For a given material, � c | kink is smaller than �c | kinkless.→ Friction has significant effect on cone-crack initiation, but not

n � c .→ P vs. rc plot can be regressed with 2nd order polynomial or

xponential functions for developed cone-crack (rc/ro > 5).Based on the experimental data, we obtained the fracture tough-

ess of soda-lime glass as KIC = 0.79 MPa m1/2, which is in goodgreements with KIC = 0.7 ∼ 0.8 MPa m1/2 in literature. This vali-ates the generated database of �c for other materials.

cknowledgment

This research was supported by Basic Science Research Programhrough the National Research Foundation of Korea (No. NRF-20151A2A1A 15056163).

ppendix A. Supplementary data

To get more insight into the proposed method, we provided Sup-lementary data, which can be found in the online version, of P vs.

c and Gvs. rc/ro plots (Fig. 12) for brittle materials in Table 5 byonsidering KCC. Supplementary data associated with this articlean be found, in the online version, at http://dx.doi.org/10.1016/j.eurceramsoc.2016.08.014.

[

[

-crack edge is clarified by dyeing. Experimental values of rc is compared with XFE

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