Delamination of a strong film from a ductile substrate duringindentation unloading
A. Abdul-Baqi and E. Van der GiessenDelft University of Technology, Koiter Institute Delft, Mekelweg 2, 2628 CD Delft, The Netherlands
(Received 23 October 2000; accepted 23 February 2001)
In this work, a finite element method was performed to simulate the sphericalindentation of a ductile substrate coated by a strong thin film. Our objective was tostudy indentation-induced delamination of the film from the substrate. The film wasassumed to be linear elastic, the substrate was elastic–perfectly plastic, and the indenterwas rigid. The interface was modeled by means of a cohesive surface. The constitutivelaw of the cohesive surface included a coupled description of normal and tagentialfailure. Cracking of the coating itself was not included. During loading, it was foundthat delamination occurs in a tangential mode rather than a normal one and wasinitiated at two to three times the contact radius. Normal delamination occurred duringthe unloading stage, where a circular part of the coating, directly under the contactarea was lifted off from the substrate. Normal delamination was imprinted on the loadversus displacement curve as a hump. There was critical value of the interfacialstrength above which delamination was prevented for a given material system and agiven indentation depth. The energy consumption by the delamination process wascalculated and separated from the part dissipated by the substrate. The effect ofresidual stress in the film and waviness of the interface on delamination was discussed.
I. INTRODUCTIONIndustrial application of thin hard-film-coated systems
continuously progresses. Coatings are commonly used toenhance reliability, such as chemical resistance, wear re-sistance, corrosion resistance, and thermal barriers. Ad-hesion between the film and the substrate determines, toa great deal, the durability of that system. The enhance-ment gained by the coating may be accompanied by therisk of poor adhesion between the coating and the sub-strate. Failure of the interface between the coating andthe substrate may lead to premature failure of otherwiselong lasting systems. Indentation is one of the traditionalmethods to quantify the mechanical properties of mate-rials, and during the last decades it has also been advo-cated as a tool to characterize the properties of thin filmsor coatings. At the same time, for example for hard wear-resistant coatings, indentation can be viewed as an el-ementary step of concentrated loading. For these reasons,many experimental as well as theoretical studies havebeen devoted to indentation of coated systems duringrecent years.
Interfacial delamination is commonly observed inindentation experiments to be accompanied by otherfailure phenomena, such as coating cracking and subse-quent spalling.1,2 The corresponding load versus
displacement curves show a reduction in the stiffness oreven a sudden discontinuity which is usually attributed tothe coating cracking. Delamination without any accom-panying through-thickness cracks has been observed byLi and Bhushan2 in their nanoindentation experiments onsingle and multilayer coatings. There is no evidence inthe literature, to the authors’ knowledge, whether delami-nation can give rise to any characteristic fingerprint onthe load versus displacement curve.
Bagchi and Evans3 have reviewed the mechanics ofthin film decohesion motivated by residual stress. Theemphasis in their work is on the role of the interfacedebond energy and the methods of its quantitative meas-urement. They argue that most thin film adhesion tests donot measure the interface debond energy because thestrain energy release rate cannot be deconvoluted fromthe work done by the external load. Viable procedures toextract the interfacial energy from indentation experi-ments will depend strongly on the precise mechanismsinvolved. The relative contribution of each mechanism tothe overall observed behavior and failure mode dependson the material properties and loading conditions in acomplex manner. In the case of ductile films on ahard substrate, coating delamination is coupled to plas-tic expansion of the film with the driving force for
delamination being delivered via buckling of the film(see also Ref. 4). On the other hand, coatings on rela-tively ductile substrates often fail during indentation byradial and in some cases circumferential cracks throughthe film. The mechanics of delamination in such systemshas been analyzed by Drory and Hutchinson5 for deepindentation with depths that are 2 to 3 orders of magni-tude larger than the coating thickness. They have alsoreviewed briefly the commonly used test methods forevaluating adhesion.
Hainsworthet al.6 have suggested a simple model forestimating the work of interfacial debonding from themaximum indentation depth and the final delaminationradius. In this model, the elastic energy of the indentedcoating is approximated by the elastic energy of a cen-trally loaded disc. The idea has also been used in cross-sectional indentation by Sa´nchez et al.7 as a newtechnique to characterize interfacial adhesion. The pro-portionality between the delamination area and the filmlateral deflection predicted by the model was confirmedby the experimental results.
The objective of the present paper is to offer an im-proved understanding of indentation-induced delamina-tion and to test the validity of the above-mentionedsimple estimates. For this purpose, we perform a numeri-cal simulation of the process of indentation of thin elasticfilm on a relatively soft substrate with a small sphericalindenter. The complete cycle of the indentation process,both loading and unloading, is simulated. The indenter isassumed to be rigid, the film is elastic and strong, and thesubstrate is elastic–perfectly plastic. The interface ismodeled by a cohesive surface, which allows one tostudy initiation and propagation of delamination duringthe indentation process. Separate criteria for delamina-tion growth are not needed in this way. The aim of thisstudy is to investigate the possibility and the phenom-enology of interfacial delamination with emphasis on theunloading part of the indentation process and the asso-ciated normal delamination. The interfacial failure duringthe loading part has been studied by the authors in aprevious work.8 Delamination was found to occur ina tangential mode driven by the shear stress at the inter-face. It is initiated at a radial distance which is two orthree times the contact radius resulting in a ring-shapeddelaminated area and imprinted on the load–displacement curve as a kind.8 In this paper we will studythe characteristics of normal delamination, conditionsfor the occurrence/suppression this mode of failure, andits fingerprint on the load–displacement curve and pro-vide some quantitative measures about the interfacialstrength. The effect of residual stress in the film andwaviness of the interface on delamination will also beinvestigated. It is emphasized that the calculations as-sume that other failure events, mainly through-thicknesscoating cracks, do not occur.
II. PROBLEM FORMULATION
The interface between the coating and the substrate ismodeled by means of a cohesive surface, where a smalldisplacement jump between the film and substrate is al-lowed, with normal and tangential componentsDn andDt, respectively. The interfacial behavior is specified interms of a constitutive equation for the correspondingtraction componentsTn and Tt at the same location.The constitutive law we adopt in this study is an elasticone, so that any energy dissipation associated with sepa-ration is ignored. Thus, it can be specified through apotential, i.e.,
Ta = −f
Da
~a = n, t! . (1)
The potential reflects the physics of the adhesion be-tween coating and substrate. Here, we use the potentialfthat was given by Xu and Needleman9
f = fn + fn expS−Dn
dnDHF1 − r +
Dn
dnG 1 − q
r − 1
− Fq + Sr − q
r − 1DDn
dnG expS−
Dt2
dt2DJ . (2)
with fn and ft the normal and tangential works ofseparation (q 4 ft/fn) and dn and dt two characteris-tics lengths. The parameterr governs the couplingbetween normal and tangential responses. As shown inFig. 2, both tractions are highly nonlinear functionsof separation with a distinct maximum of the nor-mal (tangential) traction ofsmax (tmax) which occurs ata separation ofDn 4 dn (Dt 4 dt/√2). The normal
FIG. 1. Geometry of the analyzed problem.
A. Abdul-Baqi et al.: Delamination of a strong film from a ductile substrate during indentation unloading
J. Mater. Res., Vol. 16, No. 5, May 2001 1397
(tangential) work of separation,fn (ft ), can now be ex-pressed in terms of the corresponding strengthssmax
(tmax) as
fn = exp~1!smaxdn ft = Î1
2exp~1!tmaxdt . (3)
Using these along with the definitionq = ft /fn, we canrelate the normal and shear strengths through
smax =1
q=2 exp~1!
dt
dntmax . (4)
The coupling parametersr andq are chosen such that theshear peak traction decreases with positiveDn and in-creases with negativeDn [Fig. 2(b)]. More details aregiven in Ref. 8.
The coating is assumed to be a strong, perfectly elasticmaterial with Young’s modulusEc and Poisson’s rationnc (subscript c for coating).
The substrate is supposed to be a standard isotropicelastoplastic material with plastic flow being controlledby the von Mises stress. For numerical convenience,however, we adopt a rate-sensitive version of this model,expressed by
h.
ijp =
3
2
sij
see.p e
.p = e.y Sse
syDn
, (5)
for the plastic part of the strain rate,h. p
ij 4 h.
ij 4 h. e
ij .Here, sij are the deviator components of the Piola–Kirchhoff stresstij and h
ij is the von Mises stress,n is the rate sensitivityexponent, ande
.y is a reference strain rate. In the limit of
n → `, this constitutive model reduces to the rate-independent von Mises plasticity with yield stresssy.Values ofn on the order of 100 are frequently used formetals (see e.g., Ref. 10), so that the value ofse at yieldis within a few percent ofsy for the strain rates that areencountered in our analysis. The elastic part of the strainrate,h
. eij , is given in terms of the Jaumann stress rate as
t=ij 4 Rijklh
. ekl , (6)
with the elastic modulus tensorRijkl being determined byYoung’s modulusEs and Poisson’s rationns (subscript sfor substrate).
The problem actually solved is illustrated in Fig. 1.The indenter is assumed rigid and to have a spherical tipcharacterized by its radiusR.The film is characterized byits thicknesst and is bonded to a half-infinite substrateby an interface specified above. Assuming both coatingand substrate to be isotropic, the problem is axisymmet-ric, with radial coordinater and axial coordinatez in theindentation direction. The actual calculation is carriedout for a substrate of heightL − t and radiusL, but L istaken large enough so that the solution is independent ofL and thus approaches the half-infinite substrate solution.
The analysis is carried out numerically using a finitestrain, finite element method. It uses a total Lagrangianformulation in which equilibrium is expressed in terms ofthe principle of virtual work as
*vtijdhij dv + *
SiTadDa dS = *
vtidui ds . (7)
Here,v is the totalL × L region analyzed andv is itsboundary, both in the undeformed configuration. Withxi 4 (r, z, u) the coordinates in the undeformed configu-ration,ui and ti are the components of displacement and
FIG. 2. Normal and tangential responses according to the interfacialpotential [Eq. (1)]: (a) normal responseTn(Dn); (b) tangential responseTt(Dt). Both are normalized by their respective peak valuessmax andtmax.
A. Abdul-Baqi et al.: Delamination of a strong film from a ductile substrate during indentation unloading
J. Mater. Res., Vol. 16, No. 5, May 20011398
traction vector, respectively. The virtual strainsdhij cor-respond to the virtual displacement fielddui via the straindefinition,
hij =1
2~ui, j + uj,i + uk,iuk, j! (8)
where a comma denotes (covariant) differentiation withrespect toxi. The second term in the left-hand side ofEq. (7) is the contribution of the interface, which is heremeasured in the deformed configuration (Si 4 { r|z = t}).The (true) traction transmitted across the interface hascomponentsTa, while the displacement jump isDa, witha being either the local normal direction (a 4 n) or thetangential direction (a 4 t) in the (r, z)-plane. Here, andin the remainder, the axisymmetry of the problem is ex-ploited, so thatuu 4 tu 4 tiu 4 hiu 4 0.
The precise boundary conditions are also illustrated inFig. 1. The indentation process is performed incremen-tally with a constant indentation rateh
.. Outside the con-
tact area with radiusa in the reference configuration, thefilm surface is traction free,
tr(r, 0) 4 tz(r, 0) 4 0 for a ø r ø L . (9)
Inside the contact area we assume perfect sliding condi-tions. The boundary conditions are specified with respectto a rotated local frame of reference (r, z, u) as shown inFig. 1. In the normal direction, the displacement rate u
.z
is controlled by the motion of the indenter, while in thetangential direction the tractiontr is set to zero; i.e.,
uz(r, z) 4 hcosf, tr(r, z) 4 0 for 0 ø r ø a . (10)
Numerical experiments using perfect sticking conditionsinstead have shown that the precise boundary condi-tions only have a significant effect very close to thecontact area and do not alter the results for delaminationto be presented later. During the loading part, contactnodes are identified by their spatial location with respectto the indenter; simply, at a certain indentation depthhand displacement incrementDh, the node is considered tobe in contact if the vertical distance between the node andthe indenter is not greater thanDh. During the unloadingpart, a node is released from contact on the basis of bothits spatial location and the force it exerts on the indenter;if the normal component of the nodal force is smallerthan a critical value, and the vertical distance between thenode and the indenter is positive, the node is releasedfrom contact. The critical value for the nodal force istaken to be 1% of the average current nodal force. Itshould be noted that using a value 1 order of magnitudesmaller did not significantly affect the results. The in-dentation forceF is computed from the tractions in thecontact region,
F = *0
atz~r, 0!2pr dr . (11)
The substrate is simply supported at the bottom, sothat the remaining boundary conditions read
uz~r, L! = 0 for 0 ø r ø Lur ~0, z! = 0 for 0 ø z ø L . (12)
However the sizeL will be chosen large enough thatthe solution is independent from the precise remoteconditions.
III. MODEL PARAMETERS
There are various material parameters that enter theproblem, but the main ones are the interfacial normalstrength smax, the coating thicknesst, the coatingYoung’s modulusEc, the maximum indentation depthhmax, and the substrate yield strengthsy. In the results tobe presented subsequently we focus mainly on the effectof the interfacial normal strengthsmax, keeping the samevalue of sy 4 1.0 GPa (with a reference strain rate ofe.y 4 0.1 s−1 and n 4 100). The elastic properties are
taken to beEc 4 500 GPa,nc 4 0.33,Es 4 200 GPa,andns 4 0.33.
For the cohesive surface we have chosen the samevalues fordn anddt, namely 0.1mm. As in the previousstudy,8 the coupling parametersr and q are both takenequal to 0.5 which give rise to qualitatively realistic cou-pling between normal and tangential responses of theinterface. The values ofsmax that have been investigatedvary approximately between 0.5 and 2.0 GPa. These cor-respond to interfacial energies for normal failure rangingfrom 150 to 600 J/m2, which are realistic values for theinterface toughnesses of well-adhering deposited films.11
Note that a constant value ofq implies that the shearstrengthtmax always scales with the normal strengthsmax according to Eq. (4).
We have used an indenter of radiusR 4 25 mm andmost of the results are for a film thicknesst 4 2.5mm.Indentation as well as retraction are performed at a con-stant rate h
.4 ±1 mm/s. The sizeL of the system ana-
lyzed (Fig. 1) is taken to be 50t. This proved to be largeenough that the results are independent ofL and thereforeidentical to those for a coated half-infinite medium. Themesh is an arrangement of 12,000 quadrilateral elementsand 12,342 nodes. The elements are built up of four lin-ear strain triangles in a cross arrangement to minimizenumerical problems due to plastic incompressibility. Toresolve properly the high stress gradients under theindenter and for an accurate detection of the contactnodes, the mesh is made very fine locally near the con-tact area with an element size oft/10.
Consistent with the type of elements in the coatingand the substrate, linear two-noded elements are usedalong the interface. Integration of the cohesive surfacecontribution in Eq. (7) is carried out using two-pointGauss integration. Failure, or delamination, of the
A. Abdul-Baqi et al.: Delamination of a strong film from a ductile substrate during indentation unloading
J. Mater. Res., Vol. 16, No. 5, May 2001 1399
interface at any location develops whenDa exceedsda. Apractical definition of when a complete crack has formedis Da 4 2da.12
The maximum indentation depth applied in all calcu-lations ishmax ø 2t. Further indentation can be done butwas not considered relevant since real coatings will havecracked by then and the present model is no longerapplicable.
IV. RESULTS AND DISCUSSION
A. Perfect interface
For the purpose of reference, we first consider a sys-tem with a perfect interface; i.e., its strength is suffi-ciently higher than the stresses induced by the particularloading. This can be achieved by rigidly connecting thecoating to the substrate, which corresponds to takingsmax/sy → `. Of particular relevance here, is the devel-opment of the stress distribution along the interface dur-ing the unloading stage and, in particular, the componentnormal to the interfacesn. From this, we can already getqualitative insight into when and where delaminationmay occur.
Figure 3 shows the normal stress at the interface atdifferent instants between maximum indentation depthand complete retraction of the indenter, as specifiedthrough the loadF relative to the maximum indenterload. At the maximum indentation depth, the interfacestress is of course compressive and almost uniform overthe current contact area due to plastic flow in the sub-strate. The compressive stress attains a peak value ofapproximately 4 GPa just outside the contact regionof radius amax. Relatively low tensile normal stresses arefound beyond the compressive region, atr ≈ 3amax. This
is a result of the resistance of the substrate to the filmbending in this region. It was demonstrated by the au-thors8 that the normal displacement induced by this stresswill reduce the interfacial shear strength [Fig. 2(b)],which in turn may lead to shear delamination.
As the indenter is withdrawn, at the same rate as dur-ing loading, the elastically bent coating tends to seek itsoriginal flat shape. For the material parameters here thispeeling tendency induces reverse plastic flow in the sub-strate under the indenter. As this proceeds, the initiallycompressive stress evolves into a tensile stress in theinterface directly under the initial contact region (Fig. 3).The figure also shows that the tensile area increasesslowly in size during the process of unloading, and itsfinal size is roughly the same as the maximum contactradius amax.
To study the evolution of the tensile normal stress atthe interface, its maximum valuesn
max is recorded to-gether with its positionr along the interface, as shown inFig. 4. In the initial stages of unloading, tension is foundonly in the ring outside the contact area (Fig. 3). Uponcontinued unloading, the peeling effect causes interfacialtension to develop rapidly, Fig. 4(a), with the location ofthe maximum closely following the instantaneous con-tact radius a [Fig. 4(b)]. The largest value ofsn
max 42.7 GPa obtained in this particular case is reached at theend of the unloading and located at the symmetry axis.
On the basis of these results, interfacial failure leadingto normal delamination may be expected during the un-loading stage when the interfacial strengthsmax is lowerthan the maximum tensile stresssn
max reached at anymoment. In the present case, normal delamination isavoided on the other hand if the interfacial strengthsmax
exceeds 2.7 GPa.Figure 5, curve (e), shows the indentation load versus
displacement curve for this case of a perfect interface.Such a curve is one of the most common outputs ofindentation experiments. Its importance stems from thefact that it is a signature of the indented material system.Several techniques have been reported in the literature toextract the mechanical properties of both homogeneousand composite or coated materials from indentation ex-periments (e.g., Refs. 13–17). In the forthcoming section,we will therefore study the interfacial failure process inmore detail and provide some qualitative measures of theinterfacial strength.
B. Finite-strength interface
In this section, and throughout the rest of this paper,we will study interfaces with finite strengths to allow forinterfacial delamination to develop. To demonstrate theeffect of the interfacial failure on the load–displacementdata, Fig. 5 shows the predicted curves for different val-ues of interfacial strengthsmax. The rest of the material
FIG. 3. Normal stress variations along a perfect interface at the be-ginning of unloading (F = Fmax) until complete retrieval of the in-denter (F 4 0).
A. Abdul-Baqi et al.: Delamination of a strong film from a ductile substrate during indentation unloading
J. Mater. Res., Vol. 16, No. 5, May 20011400
and geometrical parameters are the same as before. In-terfacial delamination during unloading was found in allcases shown in Fig. 5 (except case e). Compared to theperfect interface case (curve e), the initiation of delami-nation is seen to result in a rather sudden reduction of theunloading stiffness at sufficiently smallF. For higherinterfacial strengths, delamination is imprinted on theload versus displacement curve as a hump wherethe stiffness becomes negative. This phenomena will beexplained in more detail later in this section. Anothercharacteristic of delamination that can be observed in theload–displacement curve is the negligible residual inden-tation depth at the end of the unloading. In the absence ofdelamination (case e), the residual indentation depth ismore than half the maximum indentation depth. Curve a,which corresponds to the lowestsmax, shows a little de-crease in the stiffness at the end of the loading stage. This
reduction is due to shear delamination at that stage, asdiscussed in detail in Ref. 8. In all other cases shown, theinterface strength was large enough to prevent shear de-lamination but not normal delamination.
The interfacial strength above which delamination isprevented is found to besmax 4 2.21 GPa (curve d inFig. 5). From the results discussed above for a perfectinterface, however, we expected delamination at evenhigher strengths, up to 2.7 GPa. The difference must beattributed to the fact that the cohesive surface descriptionfor the finite-strength interface provides additional com-pliance to the system even before failure. This additionalcompliance results from the limited normal opening atthe interface (Dn < dn), whereas a perfect interface, bydefinition, does not allow such opening. Although theenergy consumed at the interface in this state is ex-tremely small, the extra compliance does give rise to asmall redistribution of the normal stress over the inter-face and a reduction of the maximum normal stresssn
max.Figure 6(a) shows a contour plot of the von Mises
effective stress at the end of the loading stage (F = Fmax)for the case (c) in Fig. 5 withsmax 4 1.5 GPa. The sizeof the plastic zone at this depth ofh 4 2t is about 5 timesthe maximum contact radius. To illustrate the delamina-tion process, Fig. 6(b) shows a contour plot of the verti-cal stress componentszz at the end of the unloadingprocess (F 4 0). The first thing to observe is that theradius of the delaminated zone,rd, is about 50% largerthan the maximum contact radius amax reached duringindentation. Second, we observe a region with compres-sive normal stress in front of the delamination tip. Thisregion is the remainder of the compressive region gen-erated during the loading stage, which has apparentlyhardly changed during unloading. It thus seemed that
FIG. 4. (a) Evolution of the maximum normal stresssnmax with inden-
tation depth during unloading. (b) Corresponding location at the in-terface at which the stress is maximum.
FIG. 5. Load versus displacement curves for several values of inter-facial strengthsmax: (a) 0.55; (b) 1.1; (c) 1.5; (d) 2.2 GPa. Curve (e)is for a perfect interface.
A. Abdul-Baqi et al.: Delamination of a strong film from a ductile substrate during indentation unloading
J. Mater. Res., Vol. 16, No. 5, May 2001 1401
delamination was initiated under the retrieving indenter,expanded in the radial direction and was arrested in thiscompressive interfacial stress region.
The progressive development of delamination withcontinued unloading is shown in Fig. 7 for several valuesof smax. It should be noted that, except forsmax 42.2 GPa, delamination starts at a distance from the sym-metry axis. For these casesrd represents the location ofthe delamination tip which is traveling away from thesymmetry axis. Since the other tip reaches the symmetryaxis almost immediately,rd can be considered to a goodapproximation as the radius of the delaminated circulararea. In all cases shown in Fig. 7 delamination starts
at a relatively high initial propagation velocity comparedto the indentation rateh
.and then reaches a lower ve-
locity on the order ofh.. The crack is stopped when it
reaches the region with sufficiently high compressivestress (Fig. 6). The final delamination radius is about1.5 times the maximum contact area for all values ofsmax. It is clear in the figure that for lower interfacialstrengths, delamination starts earlier in the unloadingprocess. On the other hand, the lower the interfacialstrength, the lower the residual indentation depthhr (per-manent indentation depth left at the end of the unload-ing). Figure 7 reveals that residual indentation depthhr
for several values ofsmax4 0.55 GPa. Lower interfacialstrengths even lead to small negative residual indentationdepths, where the coating bulges upwards at the end ofthe unloading.
The observations indicate that delamination is the out-come of a complex interaction between various mecha-nisms. To get further insight into this competition,Fig. 8(a) shows the decomposition of the total energy ofthe system into interfacial energyUin, elastic energyUel
(in the film and substrate), and dissipated, plastic energyUpl for the case ofsmax 4 1.5 GPa (curve c in Fig. 5).Other values of interfacial strength show the same quali-tative behavior. In this particular case, delamination ini-tiated ath 4 1.5t 4 3.75mm. It is clear in the figure thatthe plastic energy is constant at the initial stage of theunloading, i.e., the initial stage for the unloading is al-most purely elastic. This is in agreement with what iscommonly observed in indentation experiments Ref. 13.Limited reverse plasticity is seen to have contributed to alittle increase (less than 10%) in the plastic energy. At theonset of delamination, the plastic energy reaches a con-stant value. The contribution of the film and the substrateto the elastic energy is demonstrated in Fig. 8(b). The
FIG. 6. (a) Contour plot of the von Mises stress at the end of loading(F = Fmax). (b) Contour plot of the stress componentszzat the end ofthe unloading (F 4 0) for smax4 1.5 GPa (curve c in Fig. 5). The plotalso shows the delaminated region.
FIG. 7. Evolution of the delamination radius during unloading forhmax 4 5 mm and several values ofsmax (or equivalentlyfn).
A. Abdul-Baqi et al.: Delamination of a strong film from a ductile substrate during indentation unloading
J. Mater. Res., Vol. 16, No. 5, May 20011402
elastic energy of the substrate is seen to decrease morerapidly compared to the elastic energy of the film at theinitial stage of the unloading. This is in agreement withwhat is reported in the literature that the initial stiffnessof the unloading is predominantly controlled by the sub-strate for indentation depths larger than the film thick-ness.13–17 At the onset of delamination, the substrateelastic energy reaches a constant value, whereas thefilm elastic energy decreases as the film unbends. Thisindicates that the main contribution to the energy release,and hence the advance of delamination, come from thefilm. It is also interesting to notice that, at the end ofthe unloading, there still exists some elastic energy in thesystem. This energy is small compared to the dissipatedenergy (plastic energy), but when compared with the in-terfacial energy,Uin, it seems to have a significant value.
On the basis of the above observations, the unstablepart of the load–displacement curves, with negative
stiffness, shown in Fig. 7, is now readily attributed to thespontaneous opening of the interface at the initial stage ofdelamination (Fig. 7). As explained in the previous para-graph and shown in Fig. 8, the processes that control thesystem during delamination are the unflexing of the coat-ing and the interfacial delamination. The coating evi-dently provides a positive contribution to the overallstiffness, whereas the energy release from the interfacegives a negative contribution. This can be seen in Fig. 8,where the stiffness provided by each energy source is thecurvature of the corresponding curve. For relativelystrong interfaces, the energy release from the interfacedominates during the first stage of delamination when therate of propagation, relative to the indentation rate h
., is
high. During the second stage, the process is governed bythe unflexing of the coating, thus giving rise to a positiveoverall stiffness (note that the coating response is con-strained by the indenter which is withdrawn at a givenrate). It is this complex interplay between these twoterms which shapes the overall behavior of the system,including the load–displacement curve.
C. Comparison with a simple estimate
Deduction of quantitative information about the inter-facial strength from indentation experiments, in particu-lar from load–displacement curves and delaminationareas, is hindered by the rather complicated interplaybetween the film elastic energy and the interfacial en-ergy. A simple estimate for the work of interfacialdebonding from final delamination results has been givenby Hainsworthet al.6 This estimate is based on an energybalance involving the interfacial energy and the elasticenergy in the coating (the elastic energy in the substrateis neglected). The latter is approximated by the elasticenergy of a centrally loaded disc of radiusrd withclamped edges. On the basis of this model, the interfacialwork of separation is estimated by
fnest =
2Ect3 ~hmax
2 − hr2!
3~1 1 nc2!rd
4 , (13)
in terms of directly measurable quantities.As the model shows a strong dependence on the coat-
ing thicknesst and the maximum indentation depthhmax,we have chosen to vary these two parameters over acertain range and compare the model predictions withour FEM findings. A set of calculations using a conicalindenter with a 68° semiangle is also performed to ex-amine the model’s sensitivity to the indenter’s geometrywhich is not captured by Eq. (13).
Despite its very approximate nature, Eq. (13) doescapture some of the qualitative trends, as shown in Fig. 9.For instance, one expects from (13) thatrd
2 ~ hmax for agiven interfacial strength (or energy) and coating prop-erties (and neglecting the residual indentation depth).
FIG. 8. (a) Decomposition of total energy into interfacial energy (Uin),elastic energy (Uel), and plastic energy (Upl ). (b) Contribution of thefilm and substrate to the elastic energy. In (a) and (b)smax4 1.5 GPa,the normalization constant isUmax4 ∫0hmaxF dh, and the vertical dashedlines identify the initiation of delamination.
A. Abdul-Baqi et al.: Delamination of a strong film from a ductile substrate during indentation unloading
J. Mater. Res., Vol. 16, No. 5, May 2001 1403
The results of a series computations for two differentstrengths are summarized in Fig. 9(a) and are seen to beconsistent with this scaling. The conical indenter resultspresented in the figure show the same trend. Sa´nchezet al.7 have used Eq. (13) and a modified version of it ontheir cross-sectional indentation data, and they have alsoconfirmed the linear relation between the delaminationarea and the maximum deflection of the coating. Second,according to (13),rd
4/3 is proportional tot, with all otherquantities being the same. Our results, shown in Fig. 9(b)are consistent with this as well. Finally, over the range ofEc 4 350–600 GPa, the proportionality betweenrd
4
andEc is also found to be consistent with the predictionof Eq. (13).
However, not all trends are correct. For example,Eq. (13) predicts a lower slope for the delamination areaversus hmax curve for higher values of interfacialstrength, whereas the FEM results presented in Fig.9(a)show the opposite tendency.
The more serious limitation of Eq. (13) is that the in-terfacial energy estimated from the numerical results donot agree quantitatively with the actual energies. As dem-onstrated in Tables I–III, the interfacial energies are se-verely overestimated. In Table I we notice that the higherthe maximum indentation depth, the better the estimate.This can be understood by recalling that the model isbased on the expression for the deflection of a clampeddisc loaded at the center,18 where the deformation isassumed to be pure bending. The contribution of thestretching is ignored; this is reasonable when the radiusof the disc is large compared to its thickness. In thecase of indentation, this condition is analogous to contactradius (or maximum indentation depth) being larger thanthe film thickness. This explains the better estimation atlarger maximum indentation depths. This trend is alsoobserved for the conical indenter in Table II, but thequality of the estimate here is even worse. The reason isthat the cone produces more stretching of the film thanthe sphere, resulting in less accuracy of the model. InTable III, the smaller the coating thickness, the better theestimate according to Eq. (13). The same explanation
FIG. 9. (a) Delamination areaprd2 versus the maximum indentation
depthhmax. (b) rd4/3 versus coating thicknesst.
TABLE I. Estimates forfn from Eq. (13) on the basis of the com-puted valueshr andrd for t 4 2.5mm and several values ofhmax. Theactual value isfn 4 500 J/m2.
TABLE III. Estimates forfn from E8. (13) on the basis of the com-puted values ofhr andrd for hmax4 5 mm and several values oft. Theactual value isfn 4 500 J/m2.
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J. Mater. Res., Vol. 16, No. 5, May 20011404
holds here too. Evidently, the assumption that the disc isclamped at its boundary in questionable. If it is assumedthat the disc is simply supported, the expression forfn
est
in Eq. 13 must be multiplied by (1 +nc)/(3 + nc).This will give better estimates, but large errors are stillpossible.
Note that Eq. 13 does not incorporate the influence ofthe substrate. To see the accuracy of this approximation,we have investigated the dependence of the delaminationradiusrd on the substrate propertiesEs andsy. Varyingthe substrate Young’s modulusEs from 100 to 500 GPa,the resulting delamination radius increases withEs by25%. On the other hand, an increase of the yield stresssy
from 0.72 to 2.0 GPa gives values ofrd that decrease byonly 6%. The reason for this is that the yield stress de-termines the size of the plastic zone in the substrate butnot the permanent deformation immediately below theindenter; the latter is what is controlling the delaminationradius. However, it should be noted, as will be discussedin the next section, that the yield stress plays a majorrole in determining whether delamination will takeplace at all.
D. Critical value of interfacial strengthfor delamination
Whether or not delamination takes place depends onthe tensile normal stress that can be generated at theinterface during the unloading process. The ultimatevalue of this stress relative to the interface strengthsmax
depends on almost all parameters involved in the bound-ary value problem in a rather complex way. We haveperformed a parameter study involving the coating elas-tic modulus, the substrate yield stress, the maximum in-dentation depth, and the coating thickness. For eachparameter combination, delamination is suppressed if theinterfacial strength is higher than a critical value ofsc
max.As an example, Fig. 10 shows load–displacement
curves for different values of maximum indentationdepths. Delamination is seen to occur ifhmax is above acertain critical value, and it is recognized by the humpleft on the curve and the negligible residual indentationdepth. Lower indentation depths do not create normalstresses that exceed the interfacial strengthsmax and,therefore, do not lead to delamination.
Figure 11 shows the variation of the critical strengthsc
max with (a) the maximum indentation depth, (b) thecoating thickness, (c) the coating Young’s modulus, and(d) the substrate yield stress. Higher values of the coatingYoung’s modulusEc, the coating thicknesst 3, or themaximum indentation depthhmax lead to delamination ofstronger interfaces. These are explained by the fact thatthe driving force for delamination is the unbending of thecoating. Despite the limitations of the circular disc modelpointed out before, these trends are roughly consistentwith Eq. (13) but not when looked at in more detail.
For values ofhmax less than the coating thickness (t 42.5 mm), sc
max shows a relatively rapid increase,Fig. 11(a). This increase is attributed to the increase inthe bending moment in the coating. The bending momentis proportional to the curvature of the coating which in-creases rapidly with the indentation depth until the coat-ing takes the shape of the indenter. After that point, thecurvature does not change much but the bent regionpropagates outward, and this corresponds to the slowerincrease insc
max for higher indentation depths.Figure 11(b) shows also an initial rapid increase inthe critical strength with the coating thickness due to theincrease of the bending moment witht3. For thicker coat-ings, the critical strength decreases due to the decrease inthe coating curvature because the substrate becomes rela-tively softer. Figure 11(c) shows an almost linear in-crease of the critical strength with the coating Young’smodulus. The increase of the critical strength with thesubstrate yield stresssy is shown in Fig. 11(d). This in-crease is caused by the reverse plasticity that takes placeprior to delamination (Fig. 8). The higher the yield stress,the higher the stresses which can be reached at the sub-strate. Since the normal stress is continuous across theinterface, higher tensile normal stress can be reachedwith increasingsy, thus making it possible to delaminatestronger interfaces.
E. Residual stresses and interfacial waviness
Coated systems generally contain residual stresses.These are due to the deposition process itself, to thethermal expansion mismatch between the coating andthe substrate, or a combination of the two. To study theinfluence of residual stresses on delamination, wehave introduced uniform in-plane stress in the film priorto indentation. This has been achieved, for numerical
FIG. 10. Load–displacement curves for several values ofhmax, for acoating strength ofsmax 4 1.85 GPa.
A. Abdul-Baqi et al.: Delamination of a strong film from a ductile substrate during indentation unloading
J. Mater. Res., Vol. 16, No. 5, May 2001 1405
convenience, by assigning different thermal expansioncoefficients to coating and substrate and by subjectingthe system to various temperature changes to generatestresses ranging from −10 GPa (compressive) to 10 GPa(tensile). Subsequently, we perform the indentation cal-culations as before.
Compressive stress in the coating is found to delay thedelamination process, or to even prevent delamination,whereas the opposite happens with tensile stresses. Thisis explained by the fact that residual stress will have anout-of-plane component after the deformation of thecoating. In the case of tensile stress, this component willtend to enhance the unbending of coating during the un-loading and, thus, will assist delamination. As a conse-quence, the critical strength to prevent delamination willincrease with residual tension in the coating. Compres-sive stress has the opposite effect. For example, a coatingof the default thickness oft 4 2.5mm with a interfacialstrength ofsmax 4 1.84 GPa was found earlier to de-laminate after indentation tohmax 4 5 mm [seeFig. 11(a)], but delamination is prevented under a re-sidual stress of −10 GPa. The delamination radiusrd isrelatively insensitive to the residual stress: over a rangeof −7.5 to 10 GPa,rd varies between 14.4 and 16.7mmcompared tord 4 15.94mm for the stress-free coating(cf. Table I).
Roughness of the interface is commonly simplified bya sinusoidal wave (e.g., Ref. 19). To study the effect ofroughness on delamination, a wave of an amplitude up to0.2t and a wavelength up to 2t were introduced along theinterface; see Fig. 12. Delamination is found to start at
valleys and crests where the normal stress component hasa local maximum. Neighboring delaminated areas link upbefore the delamination front propagates to the nextcrest/valley. Even though the precise evolution of de-lamination depends on the waviness of the interface, for
FIG. 11. Critical value of the interfacial strengthscmax versus (a)hmax, (b) t, (c) Ec, and (d)sy.
FIG. 12. Example of normal delamination for a case with a roughinterface, modeled as a sinusoidal wave with an amplitude of 0.12t anda wavelength equal tot. In this case,hmax 4 5 mm and smax 41.85 GPa.
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J. Mater. Res., Vol. 16, No. 5, May 20011406
all cases considered here we did not find a significanteffect on the critical indentation depth at which delami-nation starts nor on the final delamination radius.
V. CONCLUSIONS
For the purpose of studying interfacial delamination,numerical simulations have been carried out of the in-dentation process of a coated material by a sphericalindenter. To describe interfacial failure, the interface be-tween the film and the substrate was modeled by meansof a cohesive surface, with a coupled constitutive law forthe normal and the tangential response. Failure of theinterface by normal or tangential separation, or a combi-nation, is embedded in the constitutive model and doesnot require any additional criteria.
Normal delamination occurs during the unloadingstage of the indentation process. A circular part of thecoating, directly under the contact area, is lifted off fromthe substrate, driven by the bending moment in the coat-ing. Normal delamination is recognized by the imprintleft on the load versus displacement curve and the neg-ligible residual indentation depth. For any given inden-tation depth, the normal stress that can be attained at theinterface is larger for thicker coatings, for coatings witha higher Young’s modulus, or for substrates with a higheryield strength. To prevent delamination of such coatings,stronger interfaces are necessary.
It should be noted that shear delamination can occurduring indentation, before normal delamination takesplace. Compared to normal delamination, shear delami-nation can occur for relatively low interfacial strength.Conversely, if the interface strength is high enough toprevent normal delamination, shear delamination willalso be avoided.
The energy consumed by the delamination process hasbeen explicitly calculated and separated from the partdissipated by plastic deformation in the substrate. Asmall amount of elastic energy, but still comparable withthe total interfacial energy, is left in the system afterunloading. Delamination is driven by the coating energyas it unflexes to retain its initial configuration. Deductionof quantitative information about the interfacial work ofseparation or strength is hindered by the complex inter-play between the coating elastic energy and the interfa-cial energy. However, the present model does allow foran inverse approach by which the work of separation canbe derived iteratively.
The disc model estimate6 has been compared with ournumerical findings for a range of parameters. It doescapture some of the qualitative aspects of delamination.But, it tends to strongly overestimate the interfacialstrength or energy of separation.
Critical values of the interfacial strength were calcu-lated for several parameter combinations. The generaltrends of the variation of these critical values with theinvolved parameters are easily interpreted, whereasthe details of this variation are governed by the nonlinearnature of the problem.
Compressive residual stress in the film delays delami-nation, and if high enough, it might even preventdelamination, whereas tensile residual stress has an op-posite effect. Waviness of the interface was not found tohave a significant effect on delamination. Both conclu-sions, however, are intimately tied to the assumption thatthe coating remains intact during indentation.
REFERENCES
1. M.D. Kriese and W.W. Gerberich, J. Mater. Res.14,3019 (1999).2. X. Li and B. Bhushan, Thin Solid Films315,214 (1998).3. A. Bagchi and A.G. Evans, Interface Sci.3, 169 (1996).4. B.D. Marshall and A.G. Evans, J. Appl. Phys.56, 2632 (1984).5. M.D. Drory and J.W. Hutchinson, Proc. R. Soc. London, Ser. A
