crack propagation in glass coatings under expanding

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Journal of the Mechanics and Physics of Solids

54 (2006) 447–466

0022-5096/$ -

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Crack propagation in glass coatings underexpanding spherical contact

Herzl Chai�

Faculty of Engineering, Department of Solid Mechanics, Materials and Systems, Tel Aviv University,

Tel Aviv 69978, Israel

Received 27 June 2005; received in revised form 10 October 2005; accepted 12 October 2005

Abstract

The growth of transverse cracks under expanding spherical contact in a model system consisted of

soda-lime glass bonded to a polycarbonate substrate is observed in situ from below or from the

polished edge of the bilayer. Abrasion or chemical etching is employed on the coating surfaces to

control the initial fracture. In the limit case of monoliths, the crack mouth becomes fully engulfed by

the expanding contact, which results in a much steeper crack angle compared to the classical Hertzian

cone case. As the coating thickness is reduced, flexure stresses are set in the coating which drive the

cone crack to well away from the contact circle and initiate semi-elliptical-like radial cracks at the

subsurface, right under the contact. Common to all three fracture modes is an initial unstable

propagation phase following by a stable growth, with detrimental failure associated with severe

damage to the top surface and/or delamination at the coating/substrate interface taking place at

loads several times the fracture initiation loads.

LEFM in conjunction with a large-strain FEM contact code is used to study the post-initiation

fracture, with the crack path controlled by the principal stress trajectory or zero-mode II S.I.F. The

analysis exposes the leading geometric and material parameters in each fracture mode, which may be

useful in the design of bilayer structures for optimal mechanical performance. The well-known

Auerbach law governing the initial fracture of monoliths is found to apply also to the bilayer crack

systems within a certain range of the problem parameters. The numerical prediction for the crack

profiles and the fracture envelopes generally collaborate well with the tests.

r 2005 Elsevier Ltd. All rights reserved.

Keywords: Indentation; Coating; Radial cracks; Cone cracks; Auerbach’s law

see front matter r 2005 Elsevier Ltd. All rights reserved.

.jmps.2005.10.004

236408342; fax: +97236406717.

dress: [email protected].

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ARTICLE IN PRESSH. Chai / J. Mech. Phys. Solids 54 (2006) 447–466448

1. Introduction

Indentation fracture is a subject that has received considerable attention due to itsfundamental nature and proven usefulness for probing material surface properties. Interestin this subject has been traditionally focused on monoliths or layered window glass, butmore recently, research on the fracture behavior of thin, hard layer(s) bonded or fused to asofter substrate (bilayer) has proliferated in a variety of bioengineering applications (e.g.,Kelly, 1997; Lawn, 2002; Huang et al., 2005). The majority of indentation fracture studies,whether on monoliths or bilayers, deal with crack initiation. The post-initiation fracturebehavior, while more complex, is of practical interest because the structure can sustainconsiderable load prior to incurring detrimental damage to the top surface and/ordelamination at the coating/substrate interface. To elucidate the evolution of the variousfracture events, some of which may be hidden within the structure or occur interactively,an all-transparent model specimen consisted of a glass layer bonded to a soft(polycarbonate) substrate is employed. To control the initial fracture and inhibit undesiredcompetitive damage, one of two of the glass surfaces is abraded while the other chemicallyetched.Fig. 1 illustrates the test specimen used and the fracture modes observed. The special

case of a very thick coating leads to the classical Hertzian cone crack. Historically, thestudy of this failure mode, which is associated with some unique fracture characteristics,began with Hertz (see account in Hertz’s (1896) Miscellaneous Papers), who introducedanalytical means to evaluate the stress field under the indenter, followed by Auerbach(1891), who determined empirically what is known as Auerbach’s law governing the initialfracture conditions, and Frank and Lawn (1967), who have helped interpret this law byintroducing techniques to determine the crack-tip stress intensity factor using Hertz’s basicsolution. Although extensively researched to the point of exhaustion, this topic is revisitedhere because it bears heavily on the post-initiation fracture behavior and is proveninstrumental in the understanding of some yet unexplained fracture phenomena in bilayerstructures.Upon reducing the coating thickness, tensile flexure stresses are set in the coating which

lead to the migration of the cone crack to well outside the contact circle (Chai et al., 1999)and the initiation of a star-shaped radial crack pattern at the subsurface, right under the

Fig. 1. Schematic of the bilayer structure considered, including the three crack systems observed under expanding

spherical contact.

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contact (Lardner et al., 1997; Grant et al., 1998; Chai et al., 1999). When the coatingthickness is further reduced to a thin-film, the ‘‘remote’’ cone crack migrates back to thecontact circle, resembling the classical cone crack but with superimposed plate flexurestresses. In the case of glass/polycarbonate bilayer, the transitions from a monolith to abilayer and a bilayer to a thin-film configurations occur when the ratio d/r is reduced fromapproximately 0.5 and 0.02, respectively, where d and r are the coating thickness andsphere radius, respectively (Chai et al., 1999). The fracture analysis for the thin-filmconfiguration is especially involved because the linear stress vs. load relationship that holdstrue for more common bilayers becomes inapplicable. The subsurface crack is generally thedetrimental failure mode in bilayers except for ultra-thin coatings, where it may becompletely suppressed (Chai, 2003a, 2005; Chai and Lawn, 2004) or when the stiffness ofthe substrate becomes comparable to that of the coating (Davis et al., 1991; Momber,2004). As for monoliths, the majority of works on bilayers are limited to the crackinitiation phase, with the crack profile typically confined to a cylindrical surface, whetheremanating from the upper coating surface (Weppelmann and Swain, 1996; Chai, 2003b;Sriram et al., 2003) or the subsurface (Chai, 2005). The post-initiation crack problem isconsiderably more involved owing to the complex three-dimensional (3-D) geometry in thecase of the subsurface crack or the crack mouth engulfment in the case of the top surfacecone crack. Available post-initiation fracture analyses usually rely on the FEM techniqueand invoke some simplified assumptions concerning the crack profile (e.g., Davis et al.,1991; Chen et al., 1995) or are carried out on a case study basis (Cao, 2002). It is thereforeprudent to explore the feasibility of accessible analytical relations in light of hardexperimental evidence and rigorous numerical analyses.

In this work, spherical indentation tests are carried out on the glass/polycarbonatebilayer shown in Fig. 1. The glass thickness is varied, but always kept large compared tothe contact radius. The damage evolution is observed in situ from below or from thepolished edge of the specimen using a zoom optical microscope. LEFM in conjunctionwith the FEM technique is used to model the fracture behavior in the brittle coating. Thetop surface cracks (Hertzian or remote cones) are modeled as axisymmetric truncatedcones while the subsurface damage is treated as a semi-elliptical surface crack, with thestress intensity factors in this case evaluated via available 3-D empirical solutions. Section2 provides some useful preliminary discussions on the pre-crack stress fields in monolithsand bilayers. The experimental apparatus is detailed in Section 3 while the test results andthe FEA are presented in Sections 4 and 5, respectively. The crack initiation conditions areanalyzed in depth in Section 6 while Section 7 discusses the bilayer crack systems in thecontext of Auerbach’s law.

2. Pre-crack stress fields

The stress distribution in the undamaged coating provides valuable insight into theinitial fracture stage, where the crack-triggering flaws are usually small compared to thescale of the stress gradients. FEM is well suited for this purpose, and Fig. 2 summarizesour results for a 1mm thick glass bonded to a thick polycarbonate slab, the whole of whichis indented by a 1.57mm radius W/C sphere. The load (P ¼ 130N) corresponds to theonset of radial crack for this configuration. Fig. 2a shows contour plots for the radial (sxx)and hoop (syy) stresses in the coating while Figs. 2b and c detail the variations of thesestresses along the radial direction (x) or the symmetry axis (z). As shown, the tensile stress

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Fig. 2. FEM stress fields in a 1mm thick glass bonded to a thick polycarbonate slab due to indentation by a

1.57mm radius W/C sphere at load P ¼ 130N: (a) stress contours, (b) variations of the hoop (syy) and radial (sxx)

stresses along the radial (x) direction, and (c) variation of the radial (or hoop) stress along the symmetry axis; the

dotted line is a plate flexure approximation.

H. Chai / J. Mech. Phys. Solids 54 (2006) 447–466450

is locally maximized either on the top or the bottom surfaces of the coating. The hoopstress at the subsurface peaks up at x ¼ 0 and rapidly decay with that coordinate. Alongthe symmetry axis the tensile stress resembles plate flexure conditions (dotted lineaccessory), but this is limited to the lower half of the coating thickness, d. The radial stresson the top surface is more complex, exhibiting the highly inhomogeneous Hertzian fieldnear the contact edge and a smaller, flexure-induced stress peak, yet on occasions moredetrimental, at a distance R ¼ 3.5d to the contact axis. Analytic relations for the peakstresses in this system, which are responsible for the onset of radial, remote cone and, tosome extent, cone cracks, are discussed below.For a monolith indented by a sphere of radius r, the Hertzian tensile stress along the

symmetry axis is given by (e.g., Johnson, 1996)

sðzÞ ¼ 3P=ð2pa2Þfð1þ ncÞ½�1þ ðz=aÞtan�1ða=zÞ� þ 0:5=ð1þ z2=a2Þg, (1)

where P is the load and a the contact radius, given as

a ¼ ð4Prk=3EcÞ1=3, (2)


where k�(9/16)[1�nc2+(1�ni

2)(Ec/Ei)], E and n are the elastic constants, and subscripts iand c stand for indenter and coating, respectively. As shown by the dashed line in Fig. 2c,along the contact axis the Hertzian field dominates nearly up to the middle section of thecoating before plate flexure takes over. The tensile Hertzian stress on the top coatingsurface, a distance R from the contact axis, is given by s0 ¼ (1�2nc)P/(2pa2a2], a � ðR=aÞ2

(Johnson, 1996), or, using Eq. (2):

s0 ¼ f cðP=r2Þ1=3, (3)

where fc ¼ [(1�2nc)/(2pa2)](3Ec/4k)2/3.

For bilayers, exact solutions are difficult to obtain. Tests and FEA (Chai et al., 1999;Chai, 2003b) show that for sufficiently thick coatings (a=d51), the peak tensile stresses onthe coating surfaces (excluding the Hertzian field) are well approximated as

Upper coating surface : s0 ¼ f uP=d2; R ¼ 3:5d for glass=polycarbonate; (4a)

Lower coating surface : s0 ¼ f lP=d2; R ¼ 0, (4b)

where fu and fl are functions of the modulus ratio Ec/Es, being equal to 0.075 and 1.1,respectively, for the present glass/polycarbonate system. Over their applicability range,Eqs. (3) and (4) imply a stress-controlled fracture, with the fracture initiation loadproportional to r2 (monolith) or d2 (bilayer). This simplification breaks down, however, forsmall indenter radii in monoliths (Auerbach, 1891) or small coating thicknesses in bilayers(Chai, 2005). The results of the present work suggest that the latter breakdown may be amanifestation of Auerbach’s law in bilayers.

2.1. Fracture mechanics considerations

For a crack of length c under a uniform tensile stress s0, the stress intensity factor maybe expressed as

K ¼ Fs0ðpcÞ0:5, (5)

where F is a boundary correction function, typically determined from FEA. Using Eqs. (2),(3) and (4a), one has

Cone crack ðmonolithÞ : P=Ka1:5 ¼ 2a2ðpa=cÞ0:5=½F ð1� 2ncÞ�, (6)

Remote cone crack ðbilayerÞ : P=Kd1:5¼ ðd=cpÞ0:5=ð f uF Þ. (7)

The case of radial crack is more involved. These non-dimensional relations are provenuseful when presenting the post-initiation fracture data.

3. Experimental

The preparation and testing of the specimens follow closely a previous study by Chai etal. (1999). Indentation tests are carried out on an 8mm thick monolith window glass(Ec ¼ 70GPa, nc ¼ 0.22) resting on a thick steel plate or on standard microscope glassplates bonded to a clear polycarbonate block (Es ¼ 2.35GPa, ns ¼ 0.35) by means of a thinepoxy adhesive. Coating thicknesses ranging from 50 mm to 3mm are produced from theglass plates by means of polishing and/or chemical etching. To simulate a half-space


support, the substrate dimensions H and L are kept at least 50 times as large as the coatingthickness. The indenter is a W/C sphere (Ei ¼ 610GPa, ni ¼ 0.22) of radius r. Threedifferent radii are used, namely 1.57, 3.97 and 12.7mm. Initial, randomly distributed flawsare introduced into one of the two glass surfaces by abrasion with slurry of 600 SiCparticles. The opposing glass surface is always etched using hydrofluoric acid to eliminateor reduce interactive fracture.Indentations are made at a slow crosshead speed such that fracture initiates within

10–30 s. The evolution of damage in the coating is observed in situ from the polished edgeor from below (i.e., upward direction) the samples using a video camera that is connectedto a high-resolution zoom telescope (Questar, Inc.). In the case of the edge views, a diffusedmonochromatic light source, projected toward the viewing axis, is used in order to improvecontrast in the naturally occurring COD fringe pattern.

4. Test results

In this section we are primarily concerned with the stable crack propagation phase. Theissue of crack initiation is discussed in detail in Section 6, following the FEA. Each of thethree failure modes observed are discussed separately below. In reducing the fracture data,the commonly cited fracture toughness value Kc ¼ 0.75MPam1/2 is used.

4.1. Monolith (cone crack)

Fig. 3 typifies the damage evolution observed. The crack initiation phase is bestvisualized using the largest indenter (r ¼ 12.7mm), and corresponding, separate resultsfrom upward and edge views are depicted in Figs. 3a and b, respectively, where the firstprint in each sequence immediately follows the first detectable fracture event. The crackpops in from a small spot slightly ahead of the contact circle before quickly runningaround to complete the ring (Fig. 3a). The embryonic crack then develops into a fullytruncated cone, with typical collar and very shallow angle (Fig. 3b). The process ofcracking that leads to the observed instability is discussed in detail by Frank and Lawn(1967). Here we are concerned only with the latter event. Accordingly, let P0 and cF be theload and the crack length associated with the observed onset of cracking. The ratio R/a0,where a0�a(P ¼ P0) and R the ring radius, is found as (1.26, 1.23, 1.22) for r ¼ (1.57, 3.97,12.7) mm, with scatter typically less than 10%. Similar ratios are also reported byAuerbach (1891) and Tillett (1956), although somewhat smaller (Warren, 1978) or larger(Mouginot and Maugis, 1985) values are also cited. The tendency for the ring to initiatesome distance outside the contact circle in a surface containing a wide distribution of flawshas been deduced analytically from energy release rate considerations in the latter source.The variation of the normalized P0 with r is depicted as unfilled circles in Fig. 4a; the loadis normalized according to Eq. (3) so that the vertical axis represents the peak tensile stressfor this type of crack. The post-initiation fracture phase is exemplified in Fig. 3c, where theprint shown corresponding to just after the emergence of a median-radial vent crack (Lawnand Fuller, 1975) whose plane happened to be orthogonal to the viewing axis. Such cracks,which cause extensive damage to the top surface upon unloading, may signal the limitserviceability of the structure. The initial and the post-engulfment crack profiles from Figs.3b and c are depicted as dash–dot lines in the insert and the main print of Fig. 5,

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Fig. 3. The evolution of Hertzian cone cracks in glass monoliths; (a) and (b) separate upward and edge views of

the initial stage of fracture; the first print in each sequence immediately follows the onset of visible damage, and

(c) an edge view taken just after the emergence of a median-radial vent crack.

H. Chai / J. Mech. Phys. Solids 54 (2006) 447–466 453

respectively. The crack angle, f, measured from the free surface, approaches 301 in the farfield, as compared to 221 for spherically ended punch indenters (e.g., Fett et al., 2004).

Fig. 6 (symbols) summarizes the variation of load P with crack length c for all the threesphere radii studied; the data are terminated when a vent crack pops in. The initial contactradii a0 needed to reduce these data are determined using Eq. (2) together with the P0

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Fig. 4. Crack instability data for monolith (a) and bilayer (b). The vertical axes represent corresponding

normalized crack instability loads or alternatively peak surface tensile stresses. Symbols and curves correspond to

tests and analytic predictions, respectively, points A and B define the Auerbach’s range (see also Figs. 6, 9 and 13).

Fig. 5. Cone crack profiles in a glass monolith subject to spherical indentation. The dash–dot lines in the insert

and the main print correspond to the tests of Figs. 3b and c, respectively, while the dash–dot–dot and solid lines

are FEM predictions for fixed and expanding contacts, respectively.


values presented in Fig. 4a. The so normalized load seems little sensitive to the indenterradius, increasing monotonically with the crack length.

4.2. Bilayer

4.2.1. Remote cone crack

Fig. 7 shows the evolution of this type of damage in a 1mm glass coating whose topsurface is abraded while its bottom etched. The first frame immediately follows the onset of

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Fig. 6. Normalized P vs. c relationships for cone crack in a glass monolith whose top surface is abraded; symbols

are test results, dash–dot–dot and solid lines are FEM predictions for fixed and expanding contacts, respectively,

dotted lines are asymptotic solutions. Points A1 and B1 define the Auerbach’s range.

Fig. 7. The propagation of remote cone crack in a 1mm thick glass bonded to a polycarbonate substrate due to

indentation; upper glass surface abraded, lower one chemically etched. Print (a) immediately follows the pop-in

crack while print (c) just precedes the onset of a detrimental radial crack at the subsurface.


cracking while the last just precedes the emergence of a large radial crack at the subsurface,which caused severe delamination at the coating/substrate interface. The crack profile forthis test is depicted as a dash–dot line in Fig. 8. As shown, the crack initiates normal to the

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Fig. 8. Remote cone crack profiles in a glass/polycarbonate bilayer subject to spherical indentation. The dash–dot

and solid lines correspond to the test of Fig. 7 and the FEA, respectively.


free surface at RE4d before curving out, ultimately grazing near, but not quite at theinterface. The pop-in crack extends to about three-quarters of the plate thickness beforearresting (Fig. 7a).Fig. 4b (triangles) summarizes the variation of the fracture instability load P0 with d;

again, the load is normalized according to Eq. (4a) so that the vertical axis represents thepeak tensile stress for this type of crack. The data pertain to a 1.57mm radius indenterexcept for relatively thick coatings, where the larger radii are used in order to precludeunwanted Hertzian cone fracture. The data for d4200 mm seem fairly constant, indicatinga failure stress sF ¼ 80MPa. For thinner coatings, this simplification obviously breaksdown, necessitating a more sophisticated fracture criterion (Section 6.1). Fig. 9 (circles)details the variation of load, normalized according to Eq. (7), with crack length c for a1mm thick glass coating.

4.2.2. Radial crack

Fig. 10 exemplifies the evolution of the subsurface damage for a 1mm thick glass coatingfor which the bottom surface is abraded while the top one etched. The fringe pattern inthese micrographs represents crack opening displacements at 0.33 mm intervals. The pop-indamage (first print) is characterized by two orthogonal radial cracks, with the plane of thevisible one happening to be perpendicular to the viewing axis. This crack grewconsiderably before arresting. Thereafter, the crack grew only laterally, never reachingthe top surface until a detrimental cone crack emerged right under the contact (last frame).It is evident from the fringe pattern that the coating/substrate interface is completelydelaminated at the base of the crack during the fracture process.

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Fig. 9. Normalized P vs. c relationships for the glass/polycarbonate crack systems; symbols are test results, solid

and dotted lines are FEM and asymptotic solutions, respectively. Points A and B define the Auerbach’s range.


As for the other two crack systems, the crack initiation load is best determined fromupward views. The results, depicted in Fig. 4b (square symbols) analogously to the remotecone crack, are similar to the latter except for an order of magnitude load reduction and alarger failure stress. Fig. 9 details the crack propagation history analogously to the remotecone, with c denoting half the lateral extent of the crack. The fracture load seems littlesensitive to the coating thickness while increasing monotonically with the crack length.Qasim et al. (2005) have reported a similar trend in their testing of curved-surface coatings.

5. Analysis

LEFM is used to evaluate the fracture behavior in the brittle coating. The indenter,coating and substrate are assumed linearly elastic, with elastic constants as specifiedearlier. The cone and remote cone cracks are modeled with the aid of a commercial FEMcode (ANSYS, 6.0), specified to large deformation and axisymmetric conditions. The loadtransfer between indenter and coating is modeled using a built-in frictionless contactalgorithm. The latter is also employed between the crack faces to prevent self-penetration.The dimensions of the substrate are kept sufficiently large to insure a half-space typesupport. Following the discussion of Fig. 10, the radial crack is treated as a semi-ellipticalsurface crack in an infinitely wide plate.

5.1. Monolith

As shown in Fig. 6, following the onset of unstable cracking a stable growth ensuesstarting at cEa0, during which time the crack mouth is subjected to compressive contact

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Fig. 10. The evolution of subsurface damage in a 1mm thick glass bonded to polycarbonate due to indentation;

upper glass surface abraded, subsurface chemically etched. The damage in this particular test comprised of two

orthogonal cracks, with the plane of the visible one being normal to the viewing axis. Print (a) immediately follows

the pop-in crack while print (c) just precede the emergence of a detrimental cone crack at the top surface of the

coating.


stresses. Accordingly, the analysis is partitioned into two parts, the first at a fixed loadP ¼ P0 and the second at increasing load, starting with the crack profile for c ¼ a0 fromthe previous analysis. The second analysis is very similar to that used in studyinghydraulic-pumping-assisted propagation of inner cone cracks under cyclic loading (Chaiand Lawn, 2005). The crack path is constructed of small, straight-line segments, theorientations of which conform to the instantaneous principal stress trajectory. The stressintensity factors are determined from Irwin’s COD, as detailed previously (Chai, 2005).The analysis is implemented for r ¼ 1.57mm, although dimensional considerations suggestthat the results may be of a broader significance.For the first part, we take P ¼ P0 and position a starting crack c0 ¼ 1 mm normal to the

free surface, a distance R ¼ 1.23a0 from the contact axis, consistent with the tests. Theanalysis starts with a small crack increment Dc5c0 orientated at a certain angle f. Thelatter is varied in small steps until the mode II S.I.F., KII, vanishes, thus establishing KI andf for this increment. This process is repeated for each additional increment, all the whilekeeping the load fixed at P0. The resulting crack profile and the normalized KI(�K) vs. c

plot are shown as a dash–dot–dot line in Figs. 5 and 6, respectively. The initial crack pathseems to collaborate well with the tests. The cone angle increases somewhat with load inthe far field, i.e., from 231 at z=a ¼ 2 to 251 at z=a ¼ 10. This compares with the value of221 reported by Kocer and Collins (1998), and Fett et al. (2004), for z=a � 2. The fracture


envelope asymptotes to the small-crack solution (Eq. (6)) or the far-field approximation(i.e., load is proportional to 1/c1.5, see Roesler, 1956) given as dotted lines in Fig. 6,exhibiting a broad, nearly flat range over a region between these extremes. Other authorshave constructed similar fracture envelopes based on the pre-crack stress field (e.g.,Mouginot and Maugis, 1985), although it is noted that such curves are roughly 10% higherthan the present one.

The second analysis starts with the crack profile for c ¼ a0 from the first part andcontinues similarly except for the additional constraint KI ¼ Kc. Specifically, for a givencrack increment KI and KII are calculated as a function of load P. This is repeated forvarious orientations of the crack segment until the dual condition KII ¼ 0 and KI ¼ Kc issatisfied. The resulting crack trajectory and normalized P vs. c plot are depicted as solid linesin Figs. 5 and 6, respectively. The fracture load is nearly indistinguishable from its non-expanding contact counterpart, both that seems to correlate well with the tests. The crackpath significantly deviates from the classical Herzian case, however, approaching 381 at c/a0 ¼ 17. As evident from Fig. 5, the predicted cone angle increasingly deviates from theexperimental path (dash-dot line) as the crack length is increased. This departure may beexplained by noting that in practice the surface outside the crack mouth incurs secondaryring cracks which effect is to relieve somewhat the contact force beyond the crack mouth.Consequently, the crack angle would shift toward the dash-dot-dot line representing thelimit case where the surface outside the mouth of the main crack is completely removed.

5.2. Bilayer

5.2.1. Remote cone crack

This configuration is similar to the cone crack case except that crack mouth engulfment isnever an issue. An initial small crack c0 ¼ 1mm is placed normal to the free surface of thecoating (d ¼ 1mm), a distance R ¼ 4d (per tests) from the contact axis. The calculations ofthe crack path and the fracture envelope are carried out similarly to the second stage analysisin the monolith case. The resulting crack profile and normalized K vs. c plot are shown assolid lines in Figs. 8 and 9, respectively, where they seem to collaborate well with the tests.

5.2.2. Radial crack

A fully-fledged analysis of this failure mode poses a formidable task owing to the 3-Dcrack geometry and the highly inhomogeneous stress field. One such study is due to Cao(2002), who have applied the FEM technique in conjunction with a crack velocity law todetermine the evolution of the crack front under a fixed loading condition. We here seek amore simple approach by approximating the crack front as a semi-elliptical surface crackof minor axis b and major axis c, and assuming that the crack resides in an infinitely wideplate of thickness d, see Fig. 11. Partial symmetry dictates that the fracture is purely modeI, with KI (�K) locally maximized at D or S, the deepest and the sidewise farthest points onthe crack front, respectively. Perhaps the most effective approach for calculating the stressintensity factors for this problem is due to Newman and Raju (1981), who studied thespecimen of Fig. 11 under the action of a uniform bending moment applied along the x-axis. Although not providing for the stress decay along this coordinate or the complexstress distribution right under the indenter shown in Fig. 2, this approach seem to be asound first approximation for the behavior at point D, as long as it is not too close to thetop surface, as well as at point S upon matching the surface bending stress in this analysis

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Fig. 11. The semi-elliptical surface crack configuration used for modeling the subsurface radial crack.


with our hoop stress at that point. Using empirical relations provided by the authors, theboundary correction function F could be calculated for each of the two fracture sites as afunction of the problem parameters, from which the corresponding mode I stress intensityfactors are found. Finally, by virtue of Eq. (4b), the surface stress in this analysis can beexpressed in terms of P0/d

2. The resulting variations of the normalized load with b/d areshown as light solid or dashed lines in Fig. 12 for various choices of c/d. To implementthese results, we define the onset of fracture as the start of a simultaneous unstable growthin the two crack axes. (In practice, one expects the initial growth to start in one of the twoflaw axes, but we ignore this complication). Accordingly, the intersections of all pairs ofsuch curves, shown as a heavy solid line in Fig. 12, would constitute the fracture envelope.Note that in accord with Fig. 10, once b=d � 0:8 the crack at point D is locked up so thatonly lateral growth is possible. Accordingly, the fracture envelope in this range is taken tobe the intersection of the traces of the side point with the vertical line b=d ¼ 0:8 rather thanthe intersections of the traces for points D and S. The fracture envelope, alternativelypresented in Fig. 9 (lower curve) as a function of c/d, seems to agree well with the tests(symbols) up to c=d � 2:5. Thereafter, a systematic departure becomes evident, indicatingthe need for a more appropriate model capable of accounting for the complex stressdistribution at the contact vicinity.With the fracture envelopes for all three fracture modes thus determined, the fracture

initiation data presented in Fig. 4 can be analyzed in more detail. It is advantageous tobegin the discussions with the bilayer due to its relative simplicity.

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Fig. 12. The dependence of the normalized load needed to initiate the subsurface radial crack on the crack axes as

predicted by the model of Fig. 11. The light solid lines and the dashed lines correspond to points D and S on the

crack front, respectively. The fracture envelope (heavy solid line) is the intersection of Ks and KD. The insert shows

the interrelationship between the two crack axes on the fracture envelope.


6. Crack initiation

6.1. Bilayer crack systems

The crack length at onset of instability, cF, is obtained by intersecting the normalizedfracture initiation load P0/Kcd

1.5 for each test point in Fig. 4b with the correspondingfracture envelope in Fig. 9. The results are presented in Fig. 13 as a normalized cF vs. d

plot; as in Fig. 4b, the triangle and square symbols represent the remote cone and radialcracks, respectively. It is apparent that the critical crack length asymptotes to a fixed valuecF ¼ cmax � 23mm as the coating thickness becomes sufficiently large. This trend isconsistent with conventional LEFM, where the largest flaw in the population (cmax)controls the fracture. Using Eq. (7) with c ¼ cmax and F ¼ 1:12, in the case of the remotecone crack, and the small-crack asymptote to the fracture envelope of Fig. 12 in the case ofthe radial crack, one obtains

Remote cone crack : P0 ¼ 0:46Kcd2=f uc0:5max; d=cmaxb1, (8a)

Radial crack : P0 ¼ 0:89Kcd2=f lc

0:5max; d=cmaxb1, (8b)

where f u ¼ 0:075 and f l ¼ 1:1 for the glass/polycarbonate combination (Eq. (4)). Asindicated previously, these relations are limited to do0:5r because for thicker coatings theHertzian cone precedes the bilayer cracks. In view of Eq. (4), these relations represent a

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Fig. 13. Critical pop-in crack length cF vs. coating thickness (bilayer) or initial contact radius (monolith), as

determined by intersecting the normalized crack initiation loads in Fig. 4 with the fracture envelopes of Fig. 6 or 9.

Symbols are test data, dashed lines are linear fits, where cF/d or cF/a0 is fixed. Points A and B define the

Auerbach’s range, cmax is the largest effective flaw in the population.


stress-controlled fracture criterion. As is apparent from Fig. 13, this simplification breaksdown once d is decreased from point A2 (remote cone) or A3 (radial crack), where atransition toward a thickness-dependent critical crack length, i.e., cF/d ¼ 0.16 and 0.073for the remote cone and the radial cracks, respectively, represented by the upper twodashed lines, takes over. The position of these cF/d values on the fracture envelopes isidentified by point A2 or A3 in Fig. 9. Within the range bounded by points A and B in thisfigure, the growth is unstable, following by a stable one starting from point B. Over thisrange, the normalized load P0/Kcd

1.5 is fixed, i.e., 20.7 (remote cone crack) or 3.3 (radialcrack). It immediately follows that

Remote cone crack : P0 ¼ 1:6Kcd1:5=f u; dB2ododA2, (9a)

Radial crack : P0 ¼ 3:6Kcd1:5=f l; dB3ododA3, (9b)

where

dA2=cmax ¼ 1=0:16 ¼ 6:3; dB2=cmax ¼ 1=0:7 ¼ 1:4,

dA3=cmax ¼ 1=0:073 ¼ 13:7; dB3=cmax ¼ 1=1:5 ¼ 0:67.

Note that P0 in Eqs. (9) is proportional to d1.5 as opposed d2 in Eqs. (8). It is interestingto note that Eqs. (9) have been previously deduced on the basis of maximizing the energy


release rate with respect to all permissible flaw sizes present on the coating surface, i.e.,cocmax (Chai, 2003b, 2005), analogously to deducing the fracture conditions in themonolith case (Mouginot and Maugis, 1985). Eqs. (8) and (9) are shown over theirapplicability ranges as dotted and dashed lines in Figs. 4b and 9. (Note the correspondencebetween the dotted lines, dashed lines and points A and B in Figs. 4b, 9 and 13.) As shownin Fig. 4b, the analytic predictions asymptotically fit the experimental data from each sidereasonably well. It is also evident from this figure that for d4dA, the results conform to afailure stress criterion, i.e., 80 and 126MPa for the remote cone and the radial cracks,respectively; the departure between these two values reflect the differing crack geometry.

To illustrate the results more graphically, it may be beneficial to resort to a case study.Consider the radial crack case with d ¼ 50 mm. According to Fig. 9, the crack will growunstably from point A3, where c ¼ cF ¼ 0.073d ¼ 3.7 mm, before arresting at point B3,where c ¼ 1.5d ¼ 75 mm. Upon increasing the load, the crack follows the fracture envelope.In addition to the main crack, secondary radial cracks will continue to develop, albeit at arelaxed stress field, leading to the well known phenomenon of simultaneous proliferationand extension of the subsurface damage.

6.2. Monolith

Using the crack instability loads P0 depicted in Fig. 4a together with Eq. (2), the initialcontact radius a0 could be determined for each of the three sphere radii used. Thecorresponding critical crack lengths cF are next obtained by intersecting the normalizedloads P0/Kca0

1.5 with the dash-dot-dot line curve in Fig. 6 (or, even more simply, the dotted-line asymptotic solution from Eq. (6)). The results, depicted as circles in Fig. 13, seem toconform to a fixed ratio cF/a0 ¼ 0.007 (lowest dashed line). This implies that cF scales withr, as has been first proposed by Frank and Lawn (1967). If presented in Fig. 6, all threedata points would converge on a single point, A1. It follows from extension to the bilayerthat this conformation is limited to a0B1oa0oa0A1 or rB1ororA1, where the latter aregiven below. As indicated in Fig. 6, in this range the cracks grow unstably before arrestingat point B1, where c/a0�1, consistent with the tests pertaining to r ¼ 12.7mm (triangles).Using c/a0 ¼ 0.007 in Eqs. (2) and (6), and taking a ¼ 1.23 per the tests, one has

P0 ¼ qkrGc; rB1ororA1, (10)

rA1 ¼ 0:63c1:5maxðEc=qk2GcÞ1=2,

rB1 ¼ 0:0041rA1,

where q�4443/(1–2nc)2(1�nc

2), Gc�Kc2(1�nc

2)/Ec and ðrA1; rB1Þ ¼ ð200; 0:82Þ mm for thepresent surface treatment. Eq. (10), known as Auerbach’s law, implies that the fractureload is proportional to the indenter radius. The coefficient q in this relation ( ¼ 15� 103

for nc ¼ 0.22) compares with the value 20� 103 reported by Lawn and Wilshaw (1975).The range enclosed by points A1 and B1 is generally known as the Auerbach’s range. Byextension from the bilayer results of Fig. 13, it is apparent that for very large contact (orsphere) radii, cF ¼ cmax so that Eqs. (2) and (6) together with F ¼ 1.12 and a ¼ 1:23 lead to

P0 ¼ 6:77� 10�4ðkrÞ2ðqGc=cmaxÞ1:5=E0:5

c ; rbrA1. (11)

Eqs. (10) and (11) are plotted in Fig. 4a as dashed and dotted lines, respectively.Note that the present tests do not cover the range specified by Eq. (11) (r4200mm).


For polished surfaces, cmax is considerably reduced so that a stress-controlled fracture,Eq. (11), could be realized for more common sphere radii (Tillett, 1956; Mouginot andMaugis, 1985).

7. Discussions

In view of the similarities in the fracture characteristics and the fracture envelopes forthe monolith and the bilayer seen in Figs. 4, 6, 9 and 13, it is proposed that Eq. (9)represents an analogous Auerbach’s law for bilayers. The power knockdown in d in the P0

vs. d relationship in Eqs. 8 and 9, from 2 to 1.5, compares with the likewise knockdown in r

in the P0 vs. r relationship for the monolith, from 2 to 1. (Additional support for thisproposition is provided by results on bilayers loaded by a cylindrical surface indenter(Chai, 2003a), where a transition from P0pd to P0pd 0.5 occurs when d becomessufficiently small). The bilayer seems to offer a simplified version of Auerbach’s lawbecause the critical flaw now scales with the coating thickness, a purely geometric quantity.This configuration also facilitates a convenient means for determining cmax, i.e., byemploying a relatively thick coating, thus helping identify rA in Eq. (10). Auerbach’s lawmay thus be regarded as a general phenomenon common to all crack systems exhibiting thedual condition of a concave K vs. c fracture envelope and high stress gradients. It is possiblethat a third condition need be imposed on this law, namely that the surface must contain adistribution of flaws. However, the validation of this involves a very complex set ofexperiments.An intriguing issue concerning the fracture envelope in Fig. 6 (or 9) is that the critical

load in the Auerbach range well exceeds the lowest point on the fracture curve,corresponding to c/a0E0.04 (or c/dE0.4 in the bilayer case), where one expects a crack toinitiate. Upon increasing the load, this crack will grow stably before a detectable unstablegrowth takes over (Frank and Lawn, 1967). (It is the latter event that defines our criticallength cF). The detail of the initial growth process leading to cracking instability should beof little consequence, however, given that the surface already contains a wide range offlaws. To get beyond this initial process, one may expect point A1 in Fig. 6 to intersect thefracture envelope at c/a0E0.015, corresponding to a normalized load level of about 80 orapproximately 20% less than the experimental value. This discrepancy may be due in partto the idealization of the initial critical flaw as a cylindrical surface crack, which may not bethe case in practice. The imperfect crack would increase the apparent toughness, thuslowering the position of point A1 in Fig. 6. Accordingly, the above noted discrepancywould be largely circumvented if one assumes a 20% increase in Kc over the crackinitiation regime as compared to the toughness value in the stable growth regime.

8. Summary and conclusions

The propagation of transverse cracks in selectively abraded soda-lime glass platesbonded to polycarbonate due to an expanding spherical contact is followed in situ. In theextreme case of monoliths, the resulting cone crack is much steeper than the classicalHertzian cone. Reducing the coating thickness leads to flexure stresses in the coating,which drive the cone crack away from the contact region (with due effects on the crackprofile) and produce radial cracking at the subsurface. The fracture is conclusivelycharacterized by an unstable growth phase following by a stable one, with detrimental


failure associated with severe damage to the top surface and/or delamination at thecoating/substrate interface taking place when the load becomes several times the crackinitiation load.

LEFM in conjunction with the FEM technique is used to predict the evolving crackprofiles. The trajectories of the top surface cracks are determined incrementally under thedual condition KII ¼ 0 and KI ¼ Kc. The subsurface damage is treated as a semi-ellipticalsurface crack in an infinitely wide plate. The fracture envelope for all the three fracturemodes studied is concave, implying an initial stable growth following by a detectable pop-in fracture and finally stable crack propagation. For relatively thick coatings (yet not toothick such that a Herzian cone fracture occurs), the fracture is governed by a failure stresscriterion, with the critical load proportional to the square of the coating thickness. Thissimplification breaks down, however, when the latter becomes sufficiently small, in whichcase the size of the flaw that triggers unstable cracking scales with the coating thickness.Consequently, the critical load becomes proportional to d1.5 instead of d2, analogously tothe well-known transition from P0pr2 to P0pr in monoliths. In view of these similarities,it is proposed that Auerbach’s law should be common to all crack systems possessing aconcave K vs. c relationship, large stress gradients, and possibly a surface containingdistributed flaws.

The analytic predictions generally collaborate well with the tests. The results presentedprovide direct information on the geometric and material aspects of the three competingfracture modes, which may be useful in the design of layered structures against indentationfailure.

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