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A. Abdul-Baqi et al.: Delamination of a strong film from a ductile substrate during indentation unloadingdelamination being delivered via buckling of the film II. PROBLEM FORMULATION(see also Ref. 4). On the other hand, coatings on rela- The interface between the coating and the substrate istively ductile substrates often fail during indentation by modeled by means of a cohesive surface, where a smallradial and in some cases circumferential cracks through displacement jump between the film and substrate is al-the film. The mechanics of delamination in such systems lowed, with normal and tangential components n andhas been analyzed by Drory and Hutchinson5 for deep t, respectively. The interfacial behavior is specified inindentation with depths that are 2 to 3 orders of magni- terms of a constitutive equation for the correspondingtude larger than the coating thickness. They have also traction components Tn and Tt at the same location.reviewed briefly the commonly used test methods for The constitutive law we adopt in this study is an elasticevaluating adhesion. one, so that any energy dissipation associated with sepa- Hainsworth et al.6 have suggested a simple model for ration is ignored. Thus, it can be specified through aestimating the work of interfacial debonding from the potential, i.e.,maximum indentation depth and the final delaminationradius. In this model, the elastic energy of the indentedcoating is approximated by the elastic energy of a cen- T =− = n, t . (1)trally loaded disc. The idea has also been used in cross-sectional indentation by Sa nchez et al. 7 as a new ´technique to characterize interfacial adhesion. The pro- The potential reflects the physics of the adhesion be-portionality between the delamination area and the film tween coating and substrate. Here, we use the potentiallateral deflection predicted by the model was confirmed that was given by Xu and Needleman9by the experimental results. The objective of the present paper is to offer an im- n n 1−qproved understanding of indentation-induced delamina- = + exp − 1−r+ n n n n r−1tion and to test the validity of the above-mentionedsimple estimates. For this purpose, we perform a numeri- r−q n t 2 − q+ exp − . (2)cal simulation of the process of indentation of thin elastic r−1 n t 2film on a relatively soft substrate with a small sphericalindenter. The complete cycle of the indentation process, with n and t the normal and tangential works ofboth loading and unloading, is simulated. The indenter is separation (q t/ n) and n and t two characteris-assumed to be rigid, the film is elastic and strong, and the tics lengths. The parameter r governs the couplingsubstrate is elastic–perfectly plastic. The interface is between normal and tangential responses. As shown inmodeled by a cohesive surface, which allows one to Fig. 2, both tractions are highly nonlinear functionsstudy initiation and propagation of delamination during of separation with a distinct maximum of the nor-the indentation process. Separate criteria for delamina- mal (tangential) traction of max ( max) which occurs attion growth are not needed in this way. The aim of this a separation of n n ( t t/√2). The normalstudy 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. FIG. 1. Geometry of the analyzed problem. J. Mater. Res., Vol. 16, No. 5, May 2001 1397
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A. Abdul-Baqi et al.: Delamination of a strong film from a ductile substrate during indentation unloading The coating is assumed to be a strong, perfectly elastic material with Young’s modulus Ec and Poisson’s ration c (subscript c for coating). The substrate is supposed to be a standard isotropic elastoplastic material with plastic flow being controlled by the von Mises stress. For numerical convenience, however, we adopt a rate-sensitive version of this model, expressed by 3 sij . p n . .p . e ij p = = y , (5) 2 e y . . .e for the plastic part of the strain rate, p ij ij ij. Here, sij are the deviator components of the Piola– . Kirchhoff stress ij and ij are the dual Lagran- gean strain-rate components. Furthermore, e √(3/2)sijsij is the von Mises stress, n is the rate sensitivity . exponent, and y is a reference strain rate. In the limit of n → , this constitutive model reduces to the rate- independent von Mises plasticity with yield stress y. Values of n on the order of 100 are frequently used for metals (see e.g., Ref. 10), so that the value of e at yield is within a few percent of y for the strain rates that are encountered in our analysis. The elastic part of the strain . rate, e , is given in terms of the Jaumann stress rate as ij ij .e Rijkl kl , (6) with the elastic modulus tensor Rijkl being determined by Young’s modulus Es and Poisson’s ration s (subscript s for substrate). The problem actually solved is illustrated in Fig. 1.FIG. 2. Normal and tangential responses according to the interfacial The indenter is assumed rigid and to have a spherical tippotential [Eq. (1)]: (a) normal response Tn( n ); (b) tangential response characterized by its radius R. The film is characterized byTt( t). Both are normalized by their respective peak values max and its thickness t and is bonded to a half-infinite substrate max. by an interface specified above. Assuming both coating and substrate to be isotropic, the problem is axisymmet-(tangential) work of separation, n ( t ), can now be ex- ric, with radial coordinate r and axial coordinate z in thepressed in terms of the corresponding strengths max indentation direction. The actual calculation is carried( max) as out for a substrate of height L − t and radius L, but L is taken large enough so that the solution is independent of 1 L and thus approaches the half-infinite substrate solution. n = exp 1 max n t = exp 1 max t . (3) The analysis is carried out numerically using a finite 2 strain, finite element method. It uses a total LagrangianUsing these along with the definition q = t / n, we can formulation in which equilibrium is expressed in terms ofrelate the normal and shear strengths through the principle of virtual work as 1 dv + dS = (7) t max = max . (4) ij ij T ti ui ds . v Si v q 2 exp 1 nThe coupling parameters r and q are chosen such that the Here, v is the total L × L region analyzed and v is itsshear peak traction decreases with positive n and in- boundary, both in the undeformed configuration. Withcreases with negative n [Fig. 2(b)]. More details are xi (r, z, ) the coordinates in the undeformed configu-given in Ref. 8. ration, ui and ti are the components of displacement and1398 J. Mater. Res., Vol. 16, No. 5, May 2001
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A. Abdul-Baqi et al.: Delamination of a strong film from a ductile substrate during indentation unloadingtraction vector, respectively. The virtual strains ij cor- The substrate is simply supported at the bottom, sorespond to the virtual displacement field ui via the strain that the remaining boundary conditions readdefinition, uz r, L = 0 for 0 r L 1 ur 0, z = 0 for 0 z L . (12) ij = ui, j + uj,i + uk,iuk, j (8) 2 However the size L will be chosen large enough thatwhere a comma denotes (covariant) differentiation with the solution is independent from the precise remoterespect to xi. The second term in the left-hand side of conditions.Eq. (7) is the contribution of the interface, which is heremeasured in the deformed configuration (Si {r|z = t}). III. MODEL PARAMETERSThe (true) traction transmitted across the interface hascomponents T , while the displacement jump is , with There are various material parameters that enter the being either the local normal direction ( n) or the problem, but the main ones are the interfacial normaltangential direction ( t) in the (r, z)-plane. Here, and strength max, the coating thickness t, the coatingin the remainder, the axisymmetry of the problem is ex- Young’s modulus Ec, the maximum indentation depthploited, so that u t i 0. hmax, and the substrate yield strength y. In the results to i The precise boundary conditions are also illustrated in be presented subsequently we focus mainly on the effectFig. 1. The indentation process is performed incremen- of the interfacial normal strength max, keeping the same . value of y 1.0 GPa (with a reference strain rate oftally with a constant indentation rate h . Outside the con- .tact area with radius a in the reference configuration, the y 0.1 s−1 and n 100). The elastic properties arefilm surface is traction free, taken to be Ec 500 GPa, c 0.33, Es 200 GPa, and s 0.33. tr(r, 0) t z(r, 0) 0 for a r L . (9) For the cohesive surface we have chosen the sameInside the contact area we assume perfect sliding condi- values for n and t, namely 0.1 m. As in the previoustions. The boundary conditions are specified with respect study,8 the coupling parameters r and q are both takento a rotated local frame of reference ( , , ) as shown in equal to 0.5 which give rise to qualitatively realistic cou- . pling between normal and tangential responses of theFig. 1. In the normal direction, the displacement rate uis controlled by the motion of the indenter, while in the interface. The values of max that have been investigatedtangential direction the traction t is set to zero; i.e., vary approximately between 0.5 and 2.0 GPa. These cor- respond to interfacial energies for normal failure rangingu (r, z)˙ ˙ hcos , t (r, z) 0 for 0 r a . (10) from 150 to 600 J/m2, which are realistic values for the interface toughnesses of well-adhering deposited films.11Numerical experiments using perfect sticking conditions Note that a constant value of q implies that the shearinstead have shown that the precise boundary condi- strength max always scales with the normal strengthtions only have a significant effect very close to the max according to Eq. (4).contact area and do not alter the results for delamination We have used an indenter of radius R 25 m andto be presented later. During the loading part, contact most of the results are for a film thickness t 2.5 m.nodes are identified by their spatial location with respect Indentation as well as retraction are performed at a con-to the indenter; simply, at a certain indentation depth h . stant rate h ±1 mm/s. The size L of the system ana-and displacement increment h, the node is considered to lyzed (Fig. 1) is taken to be 50t. This proved to be largebe in contact if the vertical distance between the node and enough that the results are independent of L and thereforethe indenter is not greater than h. During the unloading identical to those for a coated half-infinite medium. Thepart, a node is released from contact on the basis of both mesh is an arrangement of 12,000 quadrilateral elementsits spatial location and the force it exerts on the indenter; and 12,342 nodes. The elements are built up of four lin-if the normal component of the nodal force is smaller ear strain triangles in a cross arrangement to minimizethan a critical value, and the vertical distance between the numerical problems due to plastic incompressibility. Tonode and the indenter is positive, the node is released resolve properly the high stress gradients under thefrom contact. The critical value for the nodal force is indenter and for an accurate detection of the contacttaken to be 1% of the average current nodal force. It nodes, the mesh is made very fine locally near the con-should be noted that using a value 1 order of magnitude tact area with an element size of t/10.smaller did not significantly affect the results. The in- Consistent with the type of elements in the coatingdentation force F is computed from the tractions in the and the substrate, linear two-noded elements are usedcontact region, along the interface. Integration of the cohesive surface a contribution in Eq. (7) is carried out using two-point F= t z r, 0 2 r dr . (11) 0 Gauss integration. Failure, or delamination, of the J. Mater. Res., Vol. 16, No. 5, May 2001 1399
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A. Abdul-Baqi et al.: Delamination of a strong film from a ductile substrate during indentation unloadinginterface at any location develops when exceeds . A is a result of the resistance of the substrate to the filmpractical definition of when a complete crack has formed bending in this region. It was demonstrated by the au-is 2 .12 thors8 that the normal displacement induced by this stress The maximum indentation depth applied in all calcu- will reduce the interfacial shear strength [Fig. 2(b)],lations is hmax 2t. Further indentation can be done but which in turn may lead to shear delamination.was not considered relevant since real coatings will have As the indenter is withdrawn, at the same rate as dur-cracked by then and the present model is no longer ing loading, the elastically bent coating tends to seek itsapplicable. original flat shape. For the material parameters here this peeling tendency induces reverse plastic flow in the sub- strate under the indenter. As this proceeds, the initiallyIV. RESULTS AND DISCUSSION compressive stress evolves into a tensile stress in theA. Perfect interface interface directly under the initial contact region (Fig. 3). For the purpose of reference, we first consider a sys- The figure also shows that the tensile area increasestem with a perfect interface; i.e., its strength is suffi- slowly in size during the process of unloading, and itsciently higher than the stresses induced by the particular final size is roughly the same as the maximum contactloading. This can be achieved by rigidly connecting the radius amax.coating to the substrate, which corresponds to taking To study the evolution of the tensile normal stress at the interface, its maximum value max is recorded to- max/ y → . Of particular relevance here, is the devel- nopment of the stress distribution along the interface dur- gether with its position r along the interface, as shown ining the unloading stage and, in particular, the component Fig. 4. In the initial stages of unloading, tension is foundnormal to the interface n. From this, we can already get only in the ring outside the contact area (Fig. 3). Uponqualitative insight into when and where delamination continued unloading, the peeling effect causes interfacialmay occur. tension to develop rapidly, Fig. 4(a), with the location of Figure 3 shows the normal stress at the interface at the maximum closely following the instantaneous con-different instants between maximum indentation depth tact radius a [Fig. 4(b)]. The largest value of max nand complete retraction of the indenter, as specified 2.7 GPa obtained in this particular case is reached at thethrough the load F relative to the maximum indenter end of the unloading and located at the symmetry axis.load. At the maximum indentation depth, the interface On the basis of these results, interfacial failure leadingstress is of course compressive and almost uniform over to normal delamination may be expected during the un-the current contact area due to plastic flow in the sub- loading stage when the interfacial strength max is lowerstrate. The compressive stress attains a peak value of than the maximum tensile stress max reached at any napproximately 4 GPa just outside the contact region moment. In the present case, normal delamination isof radius amax. Relatively low tensile normal stresses are avoided on the other hand if the interfacial strength maxfound beyond the compressive region, at r ≈ 3amax. This 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 of indentation experiments. Its importance stems from the fact that it is a signature of the indented material system. Several techniques have been reported in the literature to extract the mechanical properties of both homogeneous and 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 in more detail and provide some qualitative measures of the interfacial strength. B. Finite-strength interface In this section, and throughout the rest of this paper, we will study interfaces with finite strengths to allow for interfacial delamination to develop. To demonstrate theFIG. 3. Normal stress variations along a perfect interface at the be- effect of the interfacial failure on the load–displacementginning of unloading (F = Fmax) until complete retrieval of the in- data, Fig. 5 shows the predicted curves for different val-denter (F 0). ues of interfacial strength max. The rest of the material1400 J. Mater. Res., Vol. 16, No. 5, May 2001
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A. Abdul-Baqi et al.: Delamination of a strong film from a ductile substrate during indentation unloading FIG. 5. Load versus displacement curves for several values of inter- facial strength max: (a) 0.55; ( b) 1.1; (c) 1.5; (d) 2.2 GPa. Curve (e) is for a perfect interface. reduction is due to shear delamination at that stage, as discussed in detail in Ref. 8. In all other cases shown, the interface strength was large enough to prevent shear de- lamination but not normal delamination. The interfacial strength above which delamination is prevented is found to be max 2.21 GPa (curve d in Fig. 5). From the results discussed above for a perfect interface, however, we expected delamination at even higher strengths, up to 2.7 GPa. The difference must be attributed to the fact that the cohesive surface description for the finite-strength interface provides additional com- pliance to the system even before failure. This additionalFIG. 4. (a) Evolution of the maximum normal stress max with inden- compliance results from the limited normal opening at ntation depth during unloading. (b) Corresponding location at the in- the interface ( n < n), whereas a perfect interface, byterface at which the stress is maximum. definition, does not allow such opening. Although the energy consumed at the interface in this state is ex- tremely small, the extra compliance does give rise to aand geometrical parameters are the same as before. In- small redistribution of the normal stress over the inter-terfacial delamination during unloading was found in all face and a reduction of the maximum normal stress max. ncases shown in Fig. 5 (except case e). Compared to the Figure 6(a) shows a contour plot of the von Misesperfect interface case (curve e), the initiation of delami- effective stress at the end of the loading stage (F = Fmax)nation is seen to result in a rather sudden reduction of the for the case (c) in Fig. 5 with max 1.5 GPa. The sizeunloading stiffness at sufficiently small F. For higher of the plastic zone at this depth of h 2t is about 5 timesinterfacial strengths, delamination is imprinted on the the maximum contact radius. To illustrate the delamina-load versus displacement curve as a hump where tion process, Fig. 6(b) shows a contour plot of the verti-the stiffness becomes negative. This phenomena will be cal stress component zz at the end of the unloadingexplained in more detail later in this section. Another process (F 0). The first thing to observe is that thecharacteristic of delamination that can be observed in the radius of the delaminated zone, rd, is about 50% largerload–displacement curve is the negligible residual inden- than the maximum contact radius amax reached duringtation depth at the end of the unloading. In the absence of indentation. Second, we observe a region with compres-delamination (case e), the residual indentation depth is sive normal stress in front of the delamination tip. Thismore than half the maximum indentation depth. Curve a, region is the remainder of the compressive region gen-which corresponds to the lowest max, shows a little de- erated during the loading stage, which has apparentlycrease in the stiffness at the end of the loading stage. This hardly changed during unloading. It thus seemed that J. Mater. Res., Vol. 16, No. 5, May 2001 1401
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A. Abdul-Baqi et al.: Delamination of a strong film from a ductile substrate during indentation unloading FIG. 7. Evolution of the delamination radius during unloading for hmax 5 m and several values of max (or equivalently n). at a relatively high initial propagation velocity compared . to the indentation rate h and then reaches a lower ve- . locity on the order of h . The crack is stopped when it reaches the region with sufficiently high compressive stress (Fig. 6). The final delamination radius is about 1.5 times the maximum contact area for all values of max. It is clear in the figure that for lower interfacial strengths, delamination starts earlier in the unloading process. On the other hand, the lower the interfacial strength, the lower the residual indentation depth hr (per- manent indentation depth left at the end of the unload- ing). Figure 7 reveals that residual indentation depth hr for several values of max 0.55 GPa. Lower interfacial strengths even lead to small negative residual indentation depths, where the coating bulges upwards at the end of the unloading. The observations indicate that delamination is the out- come of a complex interaction between various mecha-FIG. 6. (a) Contour plot of the von Mises stress at the end of loading nisms. To get further insight into this competition,(F = Fmax). (b) Contour plot of the stress component zz at the end ofthe unloading (F 0) for max 1.5 GPa (curve c in Fig. 5). The plot Fig. 8(a) shows the decomposition of the total energy ofalso shows the delaminated region. the system into interfacial energy Uin, elastic energy Uel (in the film and substrate), and dissipated, plastic energydelamination was initiated under the retrieving indenter, Upl for the case of max 1.5 GPa (curve c in Fig. 5).expanded in the radial direction and was arrested in this Other values of interfacial strength show the same quali-compressive interfacial stress region. tative behavior. In this particular case, delamination ini- The progressive development of delamination with tiated at h 1.5t 3.75 m. It is clear in the figure thatcontinued unloading is shown in Fig. 7 for several values the plastic energy is constant at the initial stage of theof max. It should be noted that, except for max unloading, i.e., the initial stage for the unloading is al-2.2 GPa, delamination starts at a distance from the sym- most purely elastic. This is in agreement with what ismetry axis. For these cases rd represents the location of commonly observed in indentation experiments Ref. 13.the delamination tip which is traveling away from the Limited reverse plasticity is seen to have contributed to asymmetry axis. Since the other tip reaches the symmetry little increase (less than 10%) in the plastic energy. At theaxis almost immediately, rd can be considered to a good onset of delamination, the plastic energy reaches a con-approximation as the radius of the delaminated circular stant value. The contribution of the film and the substratearea. In all cases shown in Fig. 7 delamination starts to the elastic energy is demonstrated in Fig. 8(b). The1402 J. Mater. Res., Vol. 16, No. 5, May 2001
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A. Abdul-Baqi et al.: Delamination of a strong film from a ductile substrate during indentation unloading stiffness, shown in Fig. 7, is now readily attributed to the spontaneous opening of the interface at the initial stage of delamination (Fig. 7). As explained in the previous para- graph and shown in Fig. 8, the processes that control the system during delamination are the unflexing of the coat- ing and the interfacial delamination. The coating evi- dently provides a positive contribution to the overall stiffness, whereas the energy release from the interface gives a negative contribution. This can be seen in Fig. 8, where the stiffness provided by each energy source is the curvature of the corresponding curve. For relatively strong interfaces, the energy release from the interface dominates during the first stage of delamination when the . rate of propagation, relative to the indentation rate h , is high. During the second stage, the process is governed by the unflexing of the coating, thus giving rise to a positive overall stiffness (note that the coating response is con- strained by the indenter which is withdrawn at a given rate). It is this complex interplay between these two terms 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 delamination areas, is hindered by the rather complicated interplay between the film elastic energy and the interfacial en- ergy. A simple estimate for the work of interfacial debonding from final delamination results has been given by Hainsworth et al.6 This estimate is based on an energy balance involving the interfacial energy and the elasticFIG. 8. (a) Decomposition of total energy into interfacial energy (Uin), energy in the coating (the elastic energy in the substrateelastic energy (Uel), and plastic energy (Upl ). (b) Contribution of thefilm and substrate to the elastic energy. In (a) and (b) max 1.5 GPa, is neglected). The latter is approximated by the elasticthe normalization constant is Umax ∫hmaxF dh, and the vertical dashed 0 energy of a centrally loaded disc of radius rd withlines identify the initiation of delamination. clamped edges. On the basis of this model, the interfacial work of separation is estimated byelastic energy of the substrate is seen to decrease morerapidly compared to the elastic energy of the film at the 2Ect 3 hmax2 − hr2initial stage of the unloading. This is in agreement with est n = 2 , (13) 31 c rd4what is reported in the literature that the initial stiffnessof the unloading is predominantly controlled by the sub- in terms of directly measurable quantities.strate for indentation depths larger than the film thick- As the model shows a strong dependence on the coat-ness.13–17 At the onset of delamination, the substrate ing thickness t and the maximum indentation depth hmax,elastic energy reaches a constant value, whereas the we have chosen to vary these two parameters over afilm elastic energy decreases as the film unbends. This certain range and compare the model predictions withindicates that the main contribution to the energy release, our FEM findings. A set of calculations using a conicaland hence the advance of delamination, come from the indenter with a 68° semiangle is also performed to ex-film. It is also interesting to notice that, at the end of amine the model’s sensitivity to the indenter’s geometrythe unloading, there still exists some elastic energy in the which is not captured by Eq. (13).system. This energy is small compared to the dissipated Despite its very approximate nature, Eq. (13) doesenergy (plastic energy), but when compared with the in- capture some of the qualitative trends, as shown in Fig. 9.terfacial energy, Uin, it seems to have a significant value. For instance, one expects from (13) that rd2 hmax for a On the basis of the above observations, the unstable given interfacial strength (or energy) and coating prop-part of the load–displacement curves, with negative erties (and neglecting the residual indentation depth). J. Mater. Res., Vol. 16, No. 5, May 2001 1403
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A. Abdul-Baqi et al.: Delamination of a strong film from a ductile substrate during indentation unloadingThe results of a series computations for two different The more serious limitation of Eq. (13) is that the in-strengths are summarized in Fig. 9(a) and are seen to be terfacial energy estimated from the numerical results doconsistent with this scaling. The conical indenter results not agree quantitatively with the actual energies. As dem-presented in the figure show the same trend. Sanchez ´ onstrated in Tables I–III, the interfacial energies are se-et al.7 have used Eq. (13) and a modified version of it on verely overestimated. In Table I we notice that the highertheir cross-sectional indentation data, and they have also the maximum indentation depth, the better the estimate.confirmed the linear relation between the delamination This can be understood by recalling that the model isarea and the maximum deflection of the coating. Second, based on the expression for the deflection of a clampedaccording to (13), rd4/3 is proportional to t, with all other disc loaded at the center,18 where the deformation isquantities being the same. Our results, shown in Fig. 9(b) assumed to be pure bending. The contribution of theare consistent with this as well. Finally, over the range of stretching is ignored; this is reasonable when the radiusEc 350–600 GPa, the proportionality between rd4 of the disc is large compared to its thickness. In theand Ec is also found to be consistent with the prediction case of indentation, this condition is analogous to contactof Eq. (13). radius (or maximum indentation depth) being larger than However, not all trends are correct. For example, the film thickness. This explains the better estimation atEq. (13) predicts a lower slope for the delamination area larger maximum indentation depths. This trend is alsoversus h max curve for higher values of interfacial observed for the conical indenter in Table II, but thestrength, whereas the FEM results presented in Fig.9(a) quality of the estimate here is even worse. The reason isshow the opposite tendency. that the cone produces more stretching of the film than the sphere, resulting in less accuracy of the model. In Table III, the smaller the coating thickness, the better the estimate according to Eq. (13). The same explanation TABLE I. Estimates for n from Eq. (13) on the basis of the com- puted values hr and rd for t 2.5 m and several values of hmax. The actual value is n 500 J/m2. est hmax ( m) hr ( m) rd ( m) n / n 2.5 0.79 8.14 15.02 3.0 0.67 10.62 7.86 3.5 0.64 12.25 6.15 4.0 0.64 13.61 5.32 4.5 0.63 14.82 4.82 5.0 0.63 15.94 4.46 TABLE II. Same as in Table I but for a conical indenter. est hmax ( m) hr ( m) rd ( m) n / n 2.5 0.43 4.75 139.34 3.0 0.42 6.49 58.24 3.5 0.43 7.81 37.92 4.0 0.45 9.03 27.84 4.5 0.47 10.24 21.32 5.0 0.49 11.45 16.86 TABLE III. Estimates for n from E8. (13) on the basis of the com- puted values of hr and rd for hmax 5 m and several values of t. The actual value is n 500 J/m2. est t ( m) hr ( m) rd ( m) n / n 2.5 0.63 15.94 4.46 3.0 0.58 16.59 6.58 3.5 0.56 17.23 9.00 4.0 0.54 17.84 11.69 4.5 0.54 18.43 14.62FIG. 9. (a) Delamination area rd2 versus the maximum indentation 5.0 0.54 19.00 17.75depth hmax. (b) rd4/3 versus coating thickness t.1404 J. Mater. Res., Vol. 16, No. 5, May 2001
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A. Abdul-Baqi et al.: Delamination of a strong film from a ductile substrate during indentation unloadingholds 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 for est nin Eq. 13 must be multiplied by (1 + c)/(3 + c).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 delaminationradius rd on the substrate properties Es and y. Varyingthe substrate Young’s modulus Es from 100 to 500 GPa,the resulting delamination radius increases with Es by25%. On the other hand, an increase of the yield stress yfrom 0.72 to 2.0 GPa gives values of rd 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 the FIG. 10. Load– displacement curves for several values of hmax, for aindenter; the latter is what is controlling the delamination coating strength of max 1.85 GPa.radius. However, it should be noted, as will be discussedin the next section, that the yield stress plays a major For values of hmax less than the coating thickness (trole in determining whether delamination will take 2.5 m), c max shows a relatively rapid increase,place at all. Fig. 11(a). This increase is attributed to the increase inD. Critical value of interfacial strength the bending moment in the coating. The bending momentfor delamination is proportional to the curvature of the coating which in- creases rapidly with the indentation depth until the coat- Whether or not delamination takes place depends on ing takes the shape of the indenter. After that point, thethe tensile normal stress that can be generated at the curvature does not change much but the bent regioninterface during the unloading process. The ultimate propagates outward, and this corresponds to the slowervalue of this stress relative to the interface strength max increase in c max for higher indentation depths.depends on almost all parameters involved in the bound- Figure 11(b) shows also an initial rapid increase inary value problem in a rather complex way. We have the critical strength with the coating thickness due to theperformed a parameter study involving the coating elas- increase of the bending moment with t3. For thicker coat-tic modulus, the substrate yield stress, the maximum in- ings, the critical strength decreases due to the decrease indentation depth, and the coating thickness. For each the coating curvature because the substrate becomes rela-parameter combination, delamination is suppressed if the tively softer. Figure 11(c) shows an almost linear in-interfacial strength is higher than a critical value of c . max crease of the critical strength with the coating Young’s As an example, Fig. 10 shows load–displacement modulus. The increase of the critical strength with thecurves for different values of maximum indentation substrate yield stress y is shown in Fig. 11(d). This in-depths. Delamination is seen to occur if hmax is above a crease is caused by the reverse plasticity that takes placecertain critical value, and it is recognized by the hump prior to delamination (Fig. 8). The higher the yield stress,left on the curve and the negligible residual indentation the higher the stresses which can be reached at the sub-depth. Lower indentation depths do not create normal strate. Since the normal stress is continuous across thestresses that exceed the interfacial strength max and, interface, higher tensile normal stress can be reachedtherefore, do not lead to delamination. with increasing y, thus making it possible to delaminate Figure 11 shows the variation of the critical strength c stronger interfaces. 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 coating E. Residual stresses and interfacial wavinessYoung’s modulus Ec, the coating thickness t 3, or the Coated systems generally contain residual stresses.maximum indentation depth hmax lead to delamination of These are due to the deposition process itself, to thestronger interfaces. These are explained by the fact that thermal expansion mismatch between the coating andthe driving force for delamination is the unbending of the the substrate, or a combination of the two. To study thecoating. Despite the limitations of the circular disc model influence of residual stresses on delamination, wepointed out before, these trends are roughly consistent have introduced uniform in-plane stress in the film priorwith Eq. (13) but not when looked at in more detail. to indentation. This has been achieved, for numerical J. Mater. Res., Vol. 16, No. 5, May 2001 1405
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A. Abdul-Baqi et al.: Delamination of a strong film from a ductile substrate during indentation unloading c FIG. 11. Critical value of the interfacial strength max versus (a) hmax, (b) t, (c) Ec, and (d) y.convenience, by assigning different thermal expansion valleys and crests where the normal stress component hascoefficients to coating and substrate and by subjecting a local maximum. Neighboring delaminated areas link upthe system to various temperature changes to generate before the delamination front propagates to the nextstresses ranging from −10 GPa (compressive) to 10 GPa crest/valley. Even though the precise evolution of de-(tensile). Subsequently, we perform the indentation cal- lamination depends on the waviness of the interface, forculations 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 of t 2.5 m with a interfacialstrength of max 1.84 GPa was found earlier to de-laminate after indentation to h max 5 m [seeFig. 11(a)], but delamination is prevented under a re-sidual stress of −10 GPa. The delamination radius rd isrelatively insensitive to the residual stress: over a rangeof −7.5 to 10 GPa, rd varies between 14.4 and 16.7 mcompared to rd 15.94 m 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 of FIG. 12. Example of normal delamination for a case with a roughroughness on delamination, a wave of an amplitude up to interface, modeled as a sinusoidal wave with an amplitude of 0.12t and0.2t and a wavelength up to 2t were introduced along the a wavelength equal to t. In this case, hmax 5 m and maxinterface; see Fig. 12. Delamination is found to start at 1.85 GPa.1406 J. Mater. Res., Vol. 16, No. 5, May 2001
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A. Abdul-Baqi et al.: Delamination of a strong film from a ductile substrate during indentation unloadingall cases considered here we did not find a significant The disc model estimate6 has been compared with oureffect on the critical indentation depth at which delami- numerical findings for a range of parameters. It doesnation starts nor on the final delamination radius. capture some of the qualitative aspects of delamination. But, it tends to strongly overestimate the interfacial strength or energy of separation.V. CONCLUSIONS Critical values of the interfacial strength were calcu- For the purpose of studying interfacial delamination, lated for several parameter combinations. The generalnumerical simulations have been carried out of the in- trends of the variation of these critical values with thedentation process of a coated material by a spherical involved parameters are easily interpreted, whereasindenter. To describe interfacial failure, the interface be- the details of this variation are governed by the nonlineartween the film and the substrate was modeled by means nature of the problem.of a cohesive surface, with a coupled constitutive law for Compressive residual stress in the film delays delami-the normal and the tangential response. Failure of the nation, and if high enough, it might even preventinterface by normal or tangential separation, or a combi- delamination, whereas tensile residual stress has an op-nation, is embedded in the constitutive model and does posite effect. Waviness of the interface was not found tonot require any additional criteria. have a significant effect on delamination. Both conclu- Normal delamination occurs during the unloading sions, however, are intimately tied to the assumption thatstage of the indentation process. A circular part of the the coating remains intact during indentation.coating, 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 imprint REFERENCESleft on the load versus displacement curve and the neg- 1. M.D. Kriese and W.W. Gerberich, J. Mater. Res. 14, 3019 (1999).ligible residual indentation depth. For any given inden- 2. X. Li and B. Bhushan, Thin Solid Films 315, 214 (1998).tation depth, the normal stress that can be attained at the 3. A. Bagchi and A.G. Evans, Interface Sci. 3, 169 (1996).interface is larger for thicker coatings, for coatings with 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. Aa higher Young’s modulus, or for substrates with a higher 452, 2319 (1996).yield strength. To prevent delamination of such coatings, 6. S.V. Hainsworth, M.R. McGurk, and T.F. Page, Surf. Coat. Tech-stronger interfaces are necessary. nol. 102, 97 (1998). It should be noted that shear delamination can occur 7. J.M. Sa nchez, S. El-Mansy, B. Sun, T. Scherban, N. Fang, ´during indentation, before normal delamination takes D. Pantsuo, W. Ford, M.R. Elizalde, J.M. Martınez-Esnaola, ´ A. Martın-Meizoso, J. Gil-Sevillano, M. Fuentes, and J. Maiz, ´place. Compared to normal delamination, shear delami- Acta Mater. 47, 4405 (1999).nation can occur for relatively low interfacial strength. 8. A. Abdul-Baqi and E. Van der Giessen, Thin Solid Films 381, 143Conversely, if the interface strength is high enough to (2001).prevent normal delamination, shear delamination will 9. X-P. Xu and A. Needleman, Model. Simul. Mater. Sci. Eng. 1,also be avoided. 111 (1993). 10. R. Becker, A. Needleman, O. Richmond, and V. Tvergaard, The energy consumed by the delamination process has J. Mech. Phys. Solids 36, 317 (1988).been explicitly calculated and separated from the part 11. Y. Wei and J.W. Hutchinson, Int. J. Fract. 10, (1999).dissipated by plastic deformation in the substrate. A 12. X-P. Xu and A. Needleman, J. Mech. Phys. Solids 42, 1397small amount of elastic energy, but still comparable with (1994).the total interfacial energy, is left in the system after 13. M. Doerner and W. Nix, J. Mater. Res. 4, 601 (1986). 14. A.K. Bhattacharya and W.D. Nix, Int. J. Solids Struct. 24, 1287unloading. Delamination is driven by the coating energy (1988).as it unflexes to retain its initial configuration. Deduction 15. H. Gao, C-H. Chiu, and J. Lee, Int. J. Solids Struct. 29, 2471of quantitative information about the interfacial work of (1992).separation or strength is hindered by the complex inter- 16. R.B. King, Int. J. Solids Struct. 23, 1657 (1987).play between the coating elastic energy and the interfa- 17. Y.Y. Lim, M.M. Chaudhri, and Y. Enomoto, J. Mater. Res. 14, 2314 (1999).cial energy. However, the present model does allow for 18. S. Timoshinko and S. Woinowsky-Krieger, Theory of Plates andan inverse approach by which the work of separation can Shells, 2nd ed. (McGraw-Hill, New York, 1959).be derived iteratively. 19. D.R. Clarke and W. Pompe, Acta Mater. 47, 1749 (1999). J. Mater. Res., Vol. 16, No. 5, May 2001 1407
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