Toughness evaluation of hard coatings and thin films

Critical reviewToughness evaluation of hard coatings and thin filmsSam Zhang ⁎,Xiaomin ZhangSchool of Mechanical and Aerospace Engineering,Nanyang Technological University,50Nanyang Avenue,Singapore 639798,Singaporea b s t r a c ta r t i c l e i n f o Available online 24September 2011Keywords:Thin films CoatingsFracture toughnessEnormous progress has been achieved over the past decade in evaluating the toughness of hard coatings and thin films.This paper reviews methodologies developed based on indentation,bending,and microtensile testing.In addition,we discuss a recent development in fracture toughness measurement which involves the application of macrotension to a substrate in order to induce microtension in a patterned thin film.©2011Elsevier B.V.All rights reserved.Contents 1.Introduction ..............................................................23752.Qualitative assessment .........................................................23762.1.Indentation plasticity ......................................................23762.2.Scratch toughness ........................................................23763.Quantitative toughness characterization for coatings ...........................................23773.1.Toughness evaluation from radial cracks .............................................23773.2.Toughness evaluation from circumferential cracking and spallation .................................23793.3.Toughness evaluation from channel cracking ...........................................23824.Microtensile testing of fracture toughness for standalone thin films ....................................23844.1.Inchworm actuation .......................................................23854.2.Membrane de flection ......................................................23854.3.Tension by residual stress ....................................................23864.4.Bulging of films .........................................................23874.5.Macrotension of substrate ....................................................23875.Summary ...............................................................23885.1.Hard coatings ..........................................................23885.2.Thin films............................................................2388Acknowledgments..............................................................2388References ................................ (2388)1.IntroductionToughness is one of the important mechanical properties of a ma-terial.The term toughness refers to the ability of a material to absorb energy during deformation up to fracture [1–2],usually measured in terms of fracture toughness.In classical mechanics,fracture toughness refers to the stress resistance of a material to fracture in the presence of a flaw,i.e.the highest stress intensity that the material canwithstand without fracture [3].It is measured as the maximum stress intensity factor under plane strain condition [4–5].To meet the “plane strain ”condition,the specimen dimensions,t ,a and (W −a ),should respectively satisfy the following inequality:t ;a ;W −a ðÞ≥2:5K IC σy!2;ð1Þwhere t and W are specimen thickness and width respectively,a is the flaw size,σy is the yield stress,and K IC is the critical stress intensity that a material can withstand without fracture.In essence,Eq.(1)re-quires that the sample thickness should be large enough,while theThin Solid Films 520(2012)2375–2389⁎Corresponding author.Tel.:+6567904400;fax:+6567911859.E-mail address:msyzhang@.sg (S.Zhang).0040-6090/$–see front matter ©2011Elsevier B.V.All rights reserved.doi:10.1016/j.tsf.2011.09.036Contents lists available at SciVerse ScienceDirectThin Solid Filmsj o u r n a l h o me p a g e :w ww.e l se v i e r.c om /l oc a te /ts finitial crack length and the“no-crack”region(i.e.,W−a)should not be too small.For brittle bulk materials,the minimum thickness t is several to tens of micrometers[5–6].For thinfilms,however,where t usually ranges from nanometers to a few micrometers,the plane strain condition is not met.Measurement of fracture toughness for bulk materials is classical, and is done routinely in research or production using,for example, the Charpy test,four-point or three-point bending,etc.[7–8].Howev-er,these methods barely apply for a thinfilm due to the size limitation of thefilm thickness.Various methodologies have been developed for microtensile testing of freestanding thinfilms,attempting to resolve the technique difficulties in fabricating a standalone thinfilm speci-men and then clamping and testing.For hard coatings bonded to sub-strates,researchers make use of nanoindentation on coatings or bending of ductile substrates to generate different types of coating cracking,based on which various methods for fracture toughness measurement are proposed.In fracture mechanics,coating cracking is a lot more complex.Some hypotheses were put forth for approxima-tion in these methods,resulting in data treatment which is not compa-rable.It is the aim of this paper to provide an overview of the available toughness testing methodologies for hard coatings and thinfilms.Re-cent developments in microtensile testing of thinfilms will be emphasized,especially the application of macrotension to a substrate in order to induce microtension in a patterned thinfimonly used qualitative methods will be describedfirst.In this paper,the term“film”refers to a freestandingfilm(i.e., without substrate),while“coating”refers to afilm attached to a sub-strate.It is necessary to make this distinction because a substrate ren-ders support and also brings constraints on the deformation and fracture of the coating.In all equations in this paper,K is the stress intensity factor of a crack in a coating or afilm and K C is the fracture toughness(critical stress intensity factor);for the opening mode of cracking,the sub-script‘I’is used;σis the stress in a coating or afilm;and t denotes film thickness.E,H,υ,n are the Young's modulus,hardness,Poisson's ratio,and the hardening exponent of a coating or afilm;in some cases,subscripts‘f’and‘s’are added to these symbols to distinguish coating and substrate.The terms a,b,c,W,l,etc.are used to describe geometric sizes of testing specimens,and their meanings are defined in the text.2.Qualitative assessmentIndentation plasticity[9–11]and scratch toughness or“load-bearing capacity”[11–14]are,due to their operational simplicity[15],the two most widely used qualitative methods for determining the toughness of coatings.2.1.Indentation plasticityIndentation plasticity is defined as the ratio of the plastic displace-ment divided by the total displacement in the load–displacement curve of a nanoindentation measurement[16](see,for example, Fig.1),plasticity¼εpε¼OAOB;ð2Þwhereεp is the plastic deformation andεis the total deformation.OA and OB are displacements defined in Fig.1.Nanoindentation has found a wide application in evaluating coating“toughness”.As reported, nc-TiC/a-C coatings(nc=nanocrystalline and a=amorphous)have an indentation plasticity of40%[17],while that of nc-TiC/a-C(Al)coatings is55%[10].In a related approach,Fox-Rabinovich et al.[11,18]proposed the “microhardness dissipation parameter”(see Fig.1)to express the plasticity in terms of the mechanical work done during different stages of indentation measurement,MDP¼plastic work=plastic workþelastic workðÞ:ð3ÞHowever,plasticity is not toughness.Plasticity is the capacity to resist plastic deformation(dislocation movement),while toughness measures the ability of a material to resist crack propagation.2.2.Scratch toughnessScratch testing is most widely used in evaluating the adhesion strength of hard coatings.During the test,a diamond stylus subjected to a linearly increasing load is drawn across the coated surface until ad-hesion failure is induced at a critical load.Generally(but not necessari-ly),for hard coatings,microcracks appear in the scratch track before failure occurs[19].The minimum load at which thefirst crack occurs is termed the“lower critical load”L c1,and the load corresponding to complete delamination peeling of the coating is the“higher critical load”L c2(Fig.2).Some researchers have directly used the lower critical load to indicate crack resistance,or even termed it“scratch toughness”[9,12,14,20–23].Zhang et al.[24]pointed out that the coating toughness should be proportional to both the lower critical load and the difference between the higher and the lower critical load.Obviously,a coating can have an early crack,but if it fractures or peels off at very high load,it means that the coating has a very high“toughness”because,during the measurement,the coating has successfully resisted the propagation of the crack.How long the coating can resist delamination and with-stand further loading before catastrophic fracture occurs is as importantPlastic Elasticdeformation deformationFig.1.Schematic plot of a nanoindentation load–displacement curve.Plasticity is calcu-lated as OA/OB=plastic work/(plastic work+elastic work)[15–16].Cracking occursTotal peelingFig.2.Typical scratch adhesion profile for nc-TiN/a-SiNx coatings deposited on silicon wafers[15].2376S.Zhang,X.Zhang/Thin Solid Films520(2012)2375–2389as crack initiation.A new parameter termed “scratch crack propagationresistance ”is thus proposed to directly use the scratch results to indi-cate coating toughness,CPR S ¼L c 1L c 1−L c 2ðÞ:ð4ÞHoehn et al.[25]proposed a simpli fied model of a scratch in order to formulate an expression for fracture toughness of coatings (see Fig.3):K IC ¼2pf g R cot θa π1=2sin−1R;ð5Þwhere p is the pressure required to open the crack,R is the radius ofthe indenter cone into the groove,2a is the total crack length,and f g is the coef ficient of grooving friction,which depends on the cone angle 2θand can be obtained from the scratch track width and the depth of penetration.However,the model is oversimpli fied,and the actual state of forces in the groove ahead and right below the tip is much more complex and has to be taken into account for a better description of the process.Holmberg et al.[26]designed a 3-D finite element model for the determination of fracture toughness via calculation of the tensile stress induced in the coating during scratch operation (see example in Fig.4),K I ¼σffiffiffib p f a ;b ðÞð6Þσis the tensile stress which induces the crack,a is the crack length,and b is the crack spacing.f (a ,b )is a nondimensional function depen-dent on crack length a ,and crack spacing b .Practical application of this method is dif ficult as there is no general expression for σobtained through a three-dimensional finite element model.3.Quantitative toughness characterization for coatingsFor a hard coating well bonded with the substrate,three types of cracking patterns may be introduced through indentation of the coating or bending of the substrate:radial cracking,circumferential cracking and spallation,and channel cracking.All these modes of cracking are used for quantitative analyses of the fracture toughness of the coating.3.1.Toughness evaluation from radial cracksRadial cracks may be introduced at the surface of a ceramic mate-rial when indenting with a sharp edge indenter,e.g.a Vickers or a Berkovich indenter (Fig.5a).The radial cracking indentation method was initially proposed for bulk materials [27].The relationship be-tween the fracture toughness and the length of radial cracks was established decades ago [28–29]:K c ¼αE 1=2Pc ;ð7Þin which P is the peak load at indentation;c is the crack length;and αis the empirical constant which depends on the geometry of the in-denter,α=0.016for both a Berkovich and Vickers type indenter.Elastic/plastic indentation fracture mechanics requires a median and radial crack pattern being well developed (see Fig.6for de finition of median and radial).To ensure the complete formation of a “half-penny ”cracking pattern,geometrically,it is required that c ≥2a (a is the radius of the impression (Fig.6c)).The derivation of Eq.(7)as-sumed “unlimited ”sample thickness.In practice,the application ofGroove trackPF2R 2aFig.3.Schematic diagram of a microscratch fracture toughness measurement with a pressure P opening a crack of maximum width 2a out of a groove width 2R [25].Fig.4.Schematic illustration of a stylus drawn along a coated sample.The material loading and response can be divided into three phases:ploughing,interface sliding,and pulling a freestanding coating [26].2377S.Zhang,X.Zhang /Thin Solid Films 520(2012)2375–2389Eq.(7)requires that the depth d of the half-penny crack beneath the surface be less than one-tenth of the thickness of the sample [30].An ultralow load is applied during nanoindentation of coatings [27](Fig.5b).For a coating with residual stress σr ,the following rela-tionship is commonly used [31–32]K IC ¼αE H 1=2P c3=2 þZ σr c 1=2;ð8Þwhere Z is the crack shape factor given by [32]Z ¼1:12ffiffiffiπp d =c 3π8ÀÁþπ8ÀÁd =c ðÞ:ð9ÞZ =1.26for an idealized half-penny,i.e.the depth d of the crack isequal to the crack length c ,making the half-penny an ideal semicircle.To meet the geometrical requirements of Eqs.(7)or (8),the in-dentation depth (smaller than the depth d of the crack induced)should be much less than 10%of the coating thickness.However,a load threshold exists for the occurrence of the radial crack during in-dentation.For most ceramic materials,the threshold load of a Vickers or Berkovich indentation is 250mN or more,and the corresponding impression produced is several micrometers in depth [27,33].A sharper angle indenter greatly reduces the threshold load for radial cracks.For example,indenting with a cube-corner indenter reducesthe load threshold by at least an order of magnitude compared with that with a Vickers indenter [27].Yet even here,the indentation depth is still a few hundred nanometers for many brittle materials in order to induce a radial crack [34].Therefore,for coatings,to intro-duce well-developed radial cracks,the depth limitation of nanoinden-tation to exclude substrate effects is usually very dif ficult to meet.Unlike standardized tests with a single well-de fined crack in a well-de fined loading con figuration (like uniaxial tensile testing)[35],indentation induces a complex crack network and residual dam-age around the impression.This makes mechanical analysis extreme-ly dif ficult [29,36].The “expanding cavity ”model was adopted to depict the damage zone of an indentation [28,37],but its reliability is questionable [38–39]:experiments on bulk ceramic materials revealed that the details of indentation cracking are extremely mate-rial dependent.That is,the crack patterns are often not the idealized half-penny shape as assumed in the model [40–41].In bulk ceramicRadialaMedianHalf pennybcFig.6.Crack patterns in a brittle material upon Vickers indentation:(a)a radial crack;(b)a median crack;(c)half-penny cracking (a combination of a radial crack and a me-dian crack)[41].Fig.5.(a)Schematic illustration of radial cracking upon Vickers indentation,and (b)ultra-low load nanoindentation radial cracks [27].2378S.Zhang,X.Zhang /Thin Solid Films 520(2012)2375–2389materials,the actual crack pattern is obtained through observing the cross-section of the median/radial crack (Fig.6c)[41–42],but it is al-most impossible to do the same on coatings.Therefore,it is very dif-ficult to judge the reliability of Eqs.(7)and (8)for coatings.A substrate indentation method was proposed to tackle the problem of substrate in fluence [43]:indentation is conducted on the uncoated side of the substrate surface such that the radial crack in the substrate propagates into the coating.In this way,a single through-thickness crack is induced in the coating (Fig.7a).The tougher the coating,the shorter the crack in the coating (Fig.7b).Based on the energy balance principle,the following equation is obtained A f G f þA s G s ¼A sf A s þA ss G s :ð10ÞG f and G s are the strain energy release rates for coating and substrate,A f and A s are true cracked film and substrate areas.A sf ,and A ss are cracked areas when coating properties are identical to those of the substrate.Through comparing the crack lengths of coated and uncoated sides,an expression is obtained for the toughness of the coatingK f ¼K 2s 1þλϕb −a ðÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE f s 1−υ2s ÀÁ1−υ2f v u u t 2643752þ2ψc σr ffiffit p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE f s 1−υ2s ÀÁ1−υ2fv u u t 26437528><>:9>=>;1=2;ð11Þwhere the subscript f ′refers to the film and s ′to the substrate;K s is the fracture toughness of the substrate;a and b are crack lengths as shown in Fig.7a;the dimensionless factors λand ψc are 0.45and 0.95respec-tively,obtained from finite element model (FEM)calculations;and ϕis a geometry term obtained from FEM.3.2.Toughness evaluation from circumferential cracking and spallation Circumferential cracking and spallation describe peeling of the coating around the indentation (Fig.8a).For a very brittle coating,it is generated upon nanoindentation.At spallation,a plateau forms in the load –displacement curve (see Fig.8b)[34,44–47],which can be used to produce a quantitative estimate of the coating fracture tough-ness.Li et al.(Fig.8a)[34]suggested that fracture proceeds as follows:(1)the first circumferential through-thickness crack forms around the indenter by high stress in the contact area;(2)delamination and buckling occur around the contact area at the coating/substrate inter-face due to high lateral pressure;(3)a second circumferential through-thickness crack forms,and spallation is generated by highFirst ring-likethrough-thickness crack formationDelamination and bucklingSecond ring-like through-thickness crack formationLateral cracking during unloadingPartial spalling formationbaFig.8.(a)Schematic illustration of the three stages in nanoindentation fracture for thin coatings;(b)schematic of a load –displacement curve,showing a step during the load-ing cycle and associated energy release [34].Wedge modelHalf-infinite plate model Crack growth= 10No coating 6-10-20 -30 -40 -50-60-70-80Indenterba20406080100-20-80-60-40-100Fig.7.(a)Schematic of indentation geometry;(b)crack growth front for different film fracture toughnesses K c [43].2379S.Zhang,X.Zhang /Thin Solid Films 520(2012)2375–2389bending stresses at the edges of the buckled thin coating.The third stage,circumferential through-thickness cracking and spallation of the coating,causes a sudden excursion of the indenter in displace-ment,which induces a step in the load–displacement curve(Fig.8b). Given area ABC in Fig.8b representing the energy U dissipated upon coating cracking,the fracture toughness is obtained asK C¼EU1−νÀÁA"#1=2;ð12Þin which A=2πC R t is the crack area,2πC R is the crack length in the coating plane,C R is the radius of circumferential through-thickness crack formed around the indenter,and t is the coating thickness.E andνare Young's modulus and Poisson's ratio of the coating,respec-tively.In Eq.(12),the coating fracture energy U is the irreversible work W irr of the indenter during the excursion from A to B.However,extraction of the irreversible work W irr became the focal point of much debate.denToonder et al.[44]suggested the lower and upper boundaries of the W irr as shown in Fig.9:the areas of OAB and ABFR,which correspond to the cases of full elastic deformation and full plastic deformation of the coated system,respectively.Chen and Bull [47]considered the area ABQE as representing the irreversible work W irr,with AE and BQ being the unloading curves at excursion start point and end point.Further,they provided a method to obtain the unloading curve AE through the linear relationship between the ratio of displacementδf/δ1and that of hardness over Young's modulus(H s/E s):δf δ1¼1−λH sE s;ð13Þwhere H s and E s are the hardness and Young's modulus of the substrate.λ=4.5for a Berkovich indenter.δf,andδ1are the residual displacement and full displacement of indentation as shown in Fig.9.Expression(13) is developed for bulk materials without fracture[48].It approximately applies to coated systems in which the substrate dominates the deforma-tion in deep indentations.In this way,the lower boundary ABE and the upper boundary ABFE of the irreversible work W irr are obtained.Michel et al.[46]realized that apart from the fracture energy U of the coating,the energy consumed in substrate deformation is also in-cluded in the work done by the indenter.He proposed ABH in Fig.10as the fracture energy U of the coating.In Fig.10,ABEF represents the total work of the indenter during circumferential cracking and spallation of the coating,and segment GB represents a partial loading curve of the silicon substrate.Malzbender et al.[44,49]studied the change of the irreversible en-ergy W irr versus load P through conducting a set of loading–unloading cycles before and after the spallation of a coating.They found that the curve of W irr vs.P was divided into several segments of straight lines representing different events during cracking:radial cracking,delam-ination,circumferential through-thickness cracking,and chipping (Fig.11a).Apparently,the irreversible work W irr was obtained from the energy difference U fr c just before and after the circumferential through-thickness cracking.Further,they found that W irr is coating-thickness dependent[50]:the thinner the coating is,the larger the ir-reversible energy dissipation W irr.The authors claimed that it is due to the fact that more substrate deformation is involved for a thinner coating during indentation.Accordingly,the fracture energy U is obtained through extrapolating W irr of the coating to infinite coating thickness(Fig.11b).Chen and Bull[45,51]extracted the fracture energy U of a coating from the curve of total work W t versus displacement D(see example Fig.12).First,the initial W t vs.D curve is extrapolated from the crack-ing start point A to the cracking end point C.Then W t vs.D curve after cracking is extrapolated back to the cracking start point.Vertical W t differences AB and CD are thus obtained.AB represents the work dif-ference consumed in the elastic–plastic deformation of the coating/ substrate system before and after the coating fracture,and CD is the total work done during the cracking.The difference between CD and AB is thus deemed the fracture energy U.Displacement-controlled nanoindentation of thin ceramic coat-ings with a sharp indenter(cube corner tip with radius of40nm when new)was extensively investigated by Chen and Bull[45,51]. Displacement-controlled indentation was supposed to be more sensi-tive to the coating cracking because the load drop at coating fracture is unambiguously related to the loss of the contact of the indenter with the coating/substrate system.On the contrary,in addition to the indent-er movement due to the loss of contact,there is an additional movement of the indenter due to the deformation of the coating/substrate system at the fracture load.Fig.13a shows a typical load–displacement curve of a displacement-controlled nanoindentation conducted on a400nm TiO x N y coating on a glass substrate,in which the load jump between B and C is associated with the radial through-thickness cracking of the coating[45].The W t vs.D method was used to obtain the fracture energy U and the fracture toughness was obtained according to Eq.(12).The fracture behavior of a thin hard coating in nanoindentation as described by Li et al.(Fig.8a)[34]and Malzbender et al.[49]isthe Fig.10.Schematic diagram representing the fracture energy U of a coating at the pla-teau in a nanoindentation load–displacement curve:the segment(GB)represents a partial loading curve of the Si substrate;the dotted area(ABEF)represents the total work under the step(done by the indenter);the gray area1(ABH)represents the en-ergy released on circumferential cracking and spallation,while the area2(BEF)repre-sents the energy of Si deformation[46].Fig.9.Schematic illustration of the boundaries of the irreversible work W irr at the pla-teau in a nanoindentation load–displacement curve[47].2380S.Zhang,X.Zhang/Thin Solid Films520(2012)2375–2389basis for the energy-based nanoindentation methodology.After radial cracking,delamination and buckling,it is the circumferential through-thickness cracking and spallation that results in a step in the load –displacement curve which is used to extract the fracture energy U of the coating.However,not only the circumferential cracking and spall-ation,but also radial cracking of the coating [52–54],delamination of the interface [55–56],cracking or spallation of the brittle substrate[52,57],and even the dislocation nucleation and phase transformation of the substrate material [58]induce a step in the loading curve of a nanoindentation.These steps may overlap in some cases.For example,the catastrophic delamination of a compressively stressed coating after buckling overlaps with the circumferential cracking and spall-ation that follows.This can be explained by the following one-dimensional blister model of a coating.The driving force (energy release rate of the interface)G i of the in-terfacial delamination after buckling is [59–61]G i 0¼m 1−σc σr2 !;ð14Þwhere G 0=(1−υ)t σr 2/E and m =[1+0.9021(1−υ)]−1;σr is the re-sidual stress of the coating,andσc ¼1:2235E 1−υtb 2ð15Þis the buckling stress;t is the coating thickness,and b is the radius of the delaminated zone.25002000150010005002000150010005000.51 1.52100200300400Depth (nm)L o a d (µN )baFig.13.Displacement-controlled nanoindentation of a 400-nm-thick TiO x N y coating on a glass substrate.In (a),position A is where plastic deformation of the softer substrate starts;points B and C are the start point and end point of through-thickness cracking;D and E are the start point and end point of interfacial fracture.The circle in (b)marks an area of uplift associated with interfacial fracture [45].00.51 1.52 2.53 3.554321Fig.12.Schematic illustration extrapolating the total work vs.displacement curve be-fore and after cracking to determine the fracture dissipated energy CD –AB.Here,thepoints A and D are the start and end points of the excursion in the measured work vs.displacement curve [45].300250200150100500cracking Radial Delamination Chipping30025020015010050000.10.10.20.20.30.30.40.50.6baFig.11.(a)Energy irreversibly dissipated during indentation as a function of the peakload applied during the indentation [49];(b)the energy irreversibly dissipated during indentation as a function of the inverse coating thickness t [50].2381S.Zhang,X.Zhang /Thin Solid Films 520(2012)2375–2389Eq.(14)is schematically represented by Fig.14,whereΓi(ψ)is the fracture toughness of the coating/substrate interface.Given an initial-ly delaminated region of width2b i,the coating will buckle atσr=σc. The blister(buckled coating)would then spread dynamically whenσr reachesσ*(where G i=Γi(ψ)),and would be arrested at b=b*(where G i=Γi(ψ).The buckling also favors cracking and spallation of the coating because a tensile stress in the coating on the inner side near the edge is induced after bending.When the driving force G f for cracking of the coating satisfies the equationG f G i NΓfΓiψðÞ;ð16ÞwhereΓf is the fracture toughness of the coating,the coating cracks and spalls away from substrate at a particular angle[62].Both the dy-namic processes,i.e.,the interfacial delamination and the cracking and spallation of the coating,blend together.More fundamentally,the fracture toughness should be obtained from the energy release rate(or stress intensity factor)as catastrophic fracture starts near equilibrium.(With reference to Fig.14,this would be at the intersection of the vertical line at b i and the horizontal line at “1”).In Fig.14,let A be the area under the curve and above line“1”,the fracture toughness obtained from the spallation process would then be equivalent to A/(b*−b i);obviously this grossly overestimates the energy.A more recent paper[63]scrutinized,from extensive published data,the steps and the coating thicknesses,and concluded that the steps are formed due to loss of contact of the indenter with the sample. Upon catastrophic fracture of the coating,the indenter undergoes freefall of a distance approximately equal to the thickness of the coat-ing.The size of such a step has no logical relationship with the energy dissipation that fractures the coating.3.3.Toughness evaluation from channel crackingChannel cracking is“through-thickness cracking”in which the coating is cracked all the way through to the substrate as the crack propagates.The“through-thickness”characteristic is maintained dur-ing crack propagation,forming a channel-like crack(Fig.15).In tensile loading,as the crack length reaches approximately three times the coating thickness,the channel crack propagates at steady state until complete fracture[64].Taking into consideration the substrate constraint on coating cracking,the stress intensity factor of the coating can be expressed as[61,64–68]K I¼σ1−ν22πtg!1=2;ð17Þwhereg¼gα;β;σsy;n s!;in whichσsy and n s are the yielding stress and strain hardening expo-nents of the substrate;andαandβare Dunders parameters describ-ing the elastic mismatch between the coating and the substrate:α¼E−E sþsandβ¼1μ1−2νsðÞ−μs1−2νðÞμ1−νsðÞþμs;where E s andνs are the elastic constants of the substrate,respectively,μ=E/2(1+ν)denotes the shear modulus,and E¼E=1−ν2is aplane strain tensile modulus.In Eq.(17),the substrate effect is contained in g.Studies[64,66]ofg indicate that a ductile substrate promotes channel cracking(i.e.,atlarger g,as can also seen in the ratioσσsyin the definition of g,ductilematerials have much smaller yielding stress than brittle materials)and thus requires less stress to reach K I in Eq.(17).For a thin ceramic coating on a ductile substrate,bending of the duc-tile substrate causes channel cracking of the coating.Thus multi-strainflexure tests[69–70]and sphere indentation tests[67–68]have beenproposed for the fracture toughness of this type of coating/substratesystem.A multi-strainflexure test[69–70]is illustrated in Fig.16.A ce-ramic coating is deposited on a ductile(metallic)substrate and thecoating patterned into strips.The sample is then placed underflexuresuch that the ceramic coating strips are on the side surface of thebending beam,and the strips are aligned along the beam axis.Duringbending,a linear strain gradient is induced in the beam from the bot-tom to the top:tensile strain on the top,compressive strain at thebottom,and zero strain along the neutral plane.Thus,coating stripsat different positions are subjected to different strains.Coating stripswith strains larger than a critical value fracture.As such,the criticalstrain can be indentified and the critical stress is thus obtained fromthe stress–strain relationship(Hooke's law,assuming the ceramiccoating fractures in purely elastic deformation).Inserting the criticalstress in Eq.(17)yields the fracture toughness of the coating.In sphere indentation,a spherical indenter of large radius isindented into the coating to cause circular cracking(Fig.17)[67–68].Advancingcrack frontxytFig.15.Three-dimensional channelingof a crack across a thin bonded coating[66]. Fig.14.Schematic illustration of instability analysis of a one-dimensional blister[61].2382S.Zhang,X.Zhang/Thin Solid Films520(2012)2375–2389。