Available online at www.sienediret.om Engineering Frature Mehanis 75 (008) 311 3130 www.elsevier.om/loate/engframeh Elasti anisotropy effet on indentation-indued thin film interfaial delamination B. Yang a, *, A.A. Volinsky b a Department of Mehanial and Aerospae Engineering, Florida Institute of Tehnology, Melbourne, FL 3901, United States b Department of Mehanial Engineering, University of South Florida, Tampa, FL 3360, United States Reeived 1 August 007; reeived in revised form 3 Deember 007; aepted 13 Deember 007 Available online 5 Deember 007 Abstrat Effet of elasti anisotropy on indentation-indued thin film interfaial delamination, espeially, at the initiation and early growth stage, is examined. The indentation load is modeled as a onstant pressure over an expanding semi-spherial avity. The delamination proess is approahed by a ohesive zone model. The rest of the problem is formulated within the general anisotropi elastiity theory, and solved numerially by the boundary element method employing a speial Green s funtion for multilayers. The material system of a Cu(0 0 1) film on a Si(0 0 1) substrate is studied as an example. The interfaial damage initiation and rak development under indentation are aptured in the simulation. By omparing the preditions with the materials being modeled as isotropi and as anisotropi (of the ubi symmetry as they are), it is shown that the elasti anisotropy of the opper film plays a signifiant role in determining the delamination pattern. In the isotropi model, the delamination rak fronts are irular refleting the problem axisymmetry. In ontrast, rak fronts are square with rounded orners in the anisotropi ase. This signifiant differene neessitates a three-dimensional anisotropi stress analysis of the indentation-indued delamination of strongly anisotropi films. Ó 007 Elsevier Ltd. All rights reserved. Keywords: Adhesion; Anisotropi elastiity; Boundary element method; Cohesive zone model; Delamination; Expanding avity model; Frature mehanis; Indentation; Thin films 1. Introdution Indentation test is one of the promising tehniques for measuring thin film adhesion energy [13,17,1,19,10]. Pressed into a film, an indenter indues large lateral stress by repelling surrounding material aside, whih in turn may suffie to drive the film to delaminate from the substrate. Suess of the indentation tehnique relies on aurate haraterization of the indentation load and the delamination rak size. If the film is transparent, the delamination rak size an be measured in situ under an optial mirosope [17,5]. Otherwise, it is measured ex situ after the film delaminates and bukles [14,19]. * Corresponding author. Tel.: +1 31 674 7713; fax: +1 31 674 8813. E-mail address: boyang@fit.edu (B. Yang). 0013-7944/$ - see front matter Ó 007 Elsevier Ltd. All rights reserved. doi:10.1016/j.engframeh.007.1.008
31 B. Yang, A.A. Volinsky / Engineering Frature Mehanis 75 (008) 311 3130 A full model of plastiity and rak growth is neessary to aurately predit the proess of film delamination under indentation. However, the orresponding omputational task is prohibitively elaborate given the urrent omputer tehnology. Therefore, simplified but elegant models have been developed to haraterize the indentation loading in ertain extreme ases. For the ase of a film undergoing extensive plasti deformation on a rigid substrate, Marshall and Evans model [13] has been developed and widely adopted. It assumes that the plasti zone under an indenter takes a ylindrial shape and that the indentation volume is entirely translated into lateral elasti displaement to load the delamination rak. In another ase of a small plasti zone under indentation (relative to the film thikness), the expanding avity model an be used [11]. The plasti zone shape is assumed to be semi-spherial, and a onstant pressure (a fration of the film yield strength) is applied at the plasti zone boundary to load the delamination rak. The plasti zone size is determined from the volume onservation, i.e., the volume swept by the elasti plasti boundary through its displaement is equated to the indentation volume. These two models are essentially idential, assuming that the entire indentation volume is translated into elasti loading, with the only differene in the shape of the elasti plasti boundary. While the former is appliable to strong interfaes on a rigid substrate, the latter is appliable to relatively weak interfaes or thik films. In previous studies, an axi-symmetri two-dimensional (D) model has been adopted [13,1]. It implies that the film and the substrate materials are both isotropi or at least transversely isotropi normal to the film. It also implies that the delamination front is irular. The present work aims to relax the axisymmetry assumption, and arry out a 3D simulation of the delamination proess in an anisotropi film and substrate material system. Based on the simulation, the effet of the elasti anisotropy on the delamination pattern is examined. The material system of a Cu(001) film on a Si(0 01) substrate is studied as an example. It is shown that the elasti anisotropy of opper makes a signifiant differene in the delamination pattern. In ontrast, the relatively weak anisotropy of silion makes little differene. The simulations have aptured the delamination damage initiation and further rak development proesses under indentation. The rest of the paper is organized as follows. In Setion, the expanding avity model of indentation is summarized [11], followed by the ohesive zone model of delamination damage and rak development [,7]. In Setion 3, the speial boundary element method for multilayers [,3], adopted to solve the problem, is summarized. Sine the Green s funtion for multilayers used satisfies the interfaial ontinuity and trationfree surfae onditions [5,30], only the delamination rak and the indentation semi-spherial elasti plasti boundary where pressure is applied need to be numerially disretized. This enabled us to effiiently and aurately ondut an extensive parametri study of the progressive delamination problem under indentation. In Setion 4, numerial results are disussed. In Setion 5, onlusions are drawn.. Problem formulation Consider an indentation of a thin film of thikness t on a semi-infinite substrate with a rigid indenter, shematially shown in Fig. 1a. When the indenter is pressed into the film with a fore P I, it auses a permanent impression of volume V I. The material originally residing in the impression volume is pushed away in all possible diretions, inluding a signifiant fration that indues elevated lateral stresses to load the film-substrate interfae. This loading due to finite plastiity is muh larger than the elasti portion. These lateral stresses may a P I V I b x 1 x Film r p plasti S p elasti Film p I x 3 S p t Substrate Delamination rak Delamination proess zone Substrate Delamination rak Delamination proess zone Fig. 1. (a) Shematis showing delamination damage and rak in a thin-film-on-substrate struture under indentation; (b) solid to be simulated that exludes the indentation plasti zone.
B. Yang, A.A. Volinsky / Engineering Frature Mehanis 75 (008) 311 3130 313 be high enough to ause the film to delaminate from the substrate. During this proess, there are two foal points of interest: the indentation plasti zone, and the delamination rak. Of our greater interest is the latter; thus, an approximate expanding avity model is adopted to haraterize the indentation loading. The delamination proess is approahed by the ohesive zone model in order to predit an arbitrary shape of the delamination front. With the ohesive zone model, the delamination proess is onveniently aptured as a natural debonding proess of the film from the substrate, desribed by a ohesive fore law. The apability of prediting an arbitrary delamination front is neessary in order to examine the effet of film and/or substrate elasti anisotropy on the delamination pattern of our interest in the present work. The expanding avity model, as shown in Fig. 1b, assumes a semi-spherial shape of an elasti plasti boundary underneath an indenter, and a uniform pressure transmitted through the elasti plasti boundary to load a delamination rak. The size of the plasti zone is determined by equating the volume swept by the elasti plasti boundary through its displaement to the indentation volume. It is implied that the model an be inaurate if signifiant pile-up ours around the indenter. Another possible soure of inauray is the assumption of uniform pressure along the elasti plasti boundary. It is lear that the pressure is two thirds of the film yield strength away from the free surfae. Near the free surfae, the pressure is in ontrast equal to the film yield strength [11]. Sine the delamination rak is loaded by the pressure aligned in the lateral diretions, it is taken to be equal to the film yield strength. The model appliation is limited to a small plasti zone size, before it reahes the interfae in order to avoid further ompliation/inauray. The onfiguration exluding the indentation plasti zone that will be numerially modeled later is shown in Fig. 1b. The total indentation fore P I and the indentation volume V I an be derived from the stress and displaement fields along the elasti plasti boundary S p : P I ¼ p I n 3 ds; ð1þ S p V I ¼ u n ds; ðþ S p where p I is the pressure, n 3 is the third omponent of the outward normal n, and u n is the normal omponent of the displaement, respetively, over S p. The Cartesian oordinate system with the third axis perpendiular to the film surfae is established. The ubi root of V I has linear dependene on the indentation depth, and the proportionality oeffiient depends on the indenter tip geometry. For the details of the stress field in the plasti zone, one may refer to the textbook of Johnson [11] in the ase of elasti-perfetly plasti materials and to more reent publiations of Gao [9] and Gao et al. [8] in the ase of appreiable hardening and loal effets. The ohesive zone model is well suited for simulating arbitrary rak front development during the indentation proess. Originally developed for modeling plasti deformation at a rak tip (i.e., a strip yield model) [7] and atomi deohesion [], the ohesive zone model has been advaned to take into aount loading rate dependene [3,1], unloading [16,6], fatigue loading [4,15], et. An exellent review of the ohesive zone model and its appliations an be found in Yang and Cox [9]. In the present appliation, the delamination rak is entirely losed, and loaded only in the shear mode. Sine the trations on opposite rak faes are equal in magnitude and are opposite in diretion, either one an be used to represent the loading of the ohesive zone material, and is related to the orresponding deformation, i.e., the rak displaement jump [u]. The ohesive zone thikness effet is negleted. The tration p on one side of the rak is related to [u] as p ¼ k½uš; ð3þ where k is the ohesive zone stiffness, assumed to be a salar, as in a line-spring model. The stiffness k softens with a damage parameter w d, taken to be the maximum magnitude of the displaement jump that the ohesive zone material has ever experiened: k ¼ p d ; ð4þ w d where p d is a funtion of w d, representing the damage lous. The damage lous is taken to be a onstant, trunated at a ritial damage parameter w f, as shown in Fig.. The damage proess is irreversible. When the ohesive zone material is loaded monotonially, it will eventually reah the damage lous and subsequently
314 B. Yang, A.A. Volinsky / Engineering Frature Mehanis 75 (008) 311 3130 p Damage lous p d (w d ) p y k w d w f [u] Fig.. Cohesive damage lous of a onstant damage-flow stress p y, trunated at a ritial failure parameter w f. follow it with further damage progression. When unloaded at a ertain point of the damage lous, the damage parameter w d is held onstant, and the state linearly reedes by the orresponding slope k, as shown in Fig.. When reloaded, it follows the same path of slope k bak to the damage lous and subsequently follows the damage lous until omplete failure at w f. The onsideration of unloading behavior is neessary in order to simulate an arbitrary proess of film interfaial delamination in three dimensions. After exluding the indentation plasti zone and applying the ohesive zone model to the delamination proess, the rest of the problem is to model the film and the substrate deformation applying the anisotropi elastiity theory [18]. The equilibrium ondition requires that r ij;j ¼ 0; ð5þ where r is the stress tensor, the prime in the subsript indiates the partial differentiation with respet to the axis that follows, and the repeated subsript indies imply the Einstein onvention of summation over their ranges. The onstitutive law is given by r ij ¼ C ijkl u k;l ; ð6þ where C is the elasti stiffness matrix. The film surfae is tration-free. Pressure equal to the yield strength of the film is imposed at the boundary of the indentation plasti zone. The film-substrate interfae is initially perfetly bonded, and subjeted to progressive delamination damage and raking under indentation by the above ohesive damage law. The remote radiation ondition, i.e., zero displaement at infinite distane, is enfored. 3. Boundary element method for multilayers Applying Green s funtion for multilayers [5,30], the displaement field in the solid as desribed in the previous setion is expressed as ij u j ðxþ ¼ ðu ij ðx; xþp jðxþ p ij ðx; xþu jðxþþds þ p ij ðx; xþ½u jðxþšds; ð7þ S p S where S p is the semi-spherial loading boundary, i.e., the boundary of indentation plasti zone, S is one side of the rak, and is a onstant, depending on the loation of X. IfX is loated inside the solid, is given by the identity matrix (i.e., the Kroneker delta funtion d ij ). If X is loated on a smooth boundary, it is given by one half of the identity matrix. Green s funtions u * and p *, respetively, represent the displaement and the tration at x indued by a unit point fore applied at X in the system of perfetly bonded film and substrate and a free surfae. Note that the above integral equation involves only the altered boundary and the rak. Taking derivatives of both sides of Eq. (7) with respet to X and applying Eq. (6) (i.e., the onstitutive law), the integral equation of stress an be derived. If X is a boundary point, multiplying both sides of the derived integral equation of stress by the outward normal n at X yields the integral equation of tration. It is given by p i ðxþ ¼ ðu ij ðx; xþp jðxþ P ij ðx; xþu jðxþþds þ P ij ðx; xþ½u jðxþšds; ð8þ S p S where U * and P * are ombinations of u * and p * derivatives, respetively.
B. Yang, A.A. Volinsky / Engineering Frature Mehanis 75 (008) 311 3130 315 Eqs. (7) and (8) an be applied to develop an effiient and aurate boundary element method to solve the above formulated problem. Eq. (7) is applied to the boundary where the indentation pressure is applied, whilst Eq. (8) is applied to the rak. The loading boundary and the rak are disretized into onstant elements. Substituting the approximated fields into Eqs. (7) and (8) yields a system of algebrai equations: u m i ¼ X ns p ðg mn ij pn j þ hmn ij un j ÞþX ns h mn ij ½un j Š; ð9þ p m i ¼ X ns p ðg mn ij pn j þ H mn ij un j ÞþX ns H mn ij ½un j Š; ð10þ where the oeffiients, g, h, G and H are integrals of Green s funtions over an element, and the supersript indiates a node. The overall omputational effiieny of the boundary element method is largely determined by the evaluation effiieny of these oeffiients. A semi-analytial method has been reently developed to ompute these integrals effiiently and aurately [8], whih is adopted in the present study. Given the pressure over the indentation loading boundary S p and the ohesive fore law governing the delamination proess, the above system of algebrai equations an be solved numerially to obtain the displaement along the loading boundary and the displaement jump aross the rak. The iterative sheme of suessive over-relaxation, whih has been suessfully applied to solve similar problems [7,,3], is adopted to solve the present problem. For the sake of brevity, it is not desribed here. 4. Results and analysis The analyzed material system onsists of a Cu(1 0 0) film on a Si(0 0 1) substrate. In order to demonstrate the effet of elasti anisotropy on the indentation-indued interfaial delamination, simulations are run with the materials being modeled first as isotropi and then as anisotropi (of the ubi symmetry as they are). The elasti onstants of the two models are given in Tables 1 and, respetively. The anisotropi elasti onstants of the opper and the silion were taken from Cleri and Rosato [4] and Levinshtein et al. [1], respetively. Meanwhile, the isotropi elasti onstants of both materials were taken from Dolbow and Gosz [6]. ener s anisotropy index is defined as a =(C 66 )/(C 11 C 1 ). It is equal to one if the material is fully isotropi. Deviation from one indiates anisotropy, where greater deviation means stronger anisotropy. The anisotropy index is 3. for opper, and 1.56 for silion. Therefore, opper exhibits stronger anisotropy than silion. The pressure imposed upon the elasti plasti boundary is taken to be the yield strength of the opper film, equal to 100 MPa [6,0]. The delamination damage-flow stress p y is taken to be 4 MPa, and the failure rak displaement jump w f is taken to be 10 4 t, where t is the film thikness. Thus, the film adhesion energy in the units of Joule is 400 t, where t is in meters. For example, if the film is 1 mm thik, the adhesion energy is 0.4 J. In the rest of text, inluding figures, all of the quantities of the length dimension are normalized by t, and all of the stress quantities are normalized by E 0 = 1 GPa. Note that the interfaial properties are hosen suh that Table 1 Elasti onstants for opper and silion in the isotropi model Young s modulus (GPa) Poisson ratio Copper 18 0.36 Silion 165 0. Table Elasti onstants for opper and silion in the anisotropi (ubi) model C 11 (GPa) C 1 (GPa) C 44 (GPa) Copper 176 15 8 Silion 165 63.9 79.6
316 B. Yang, A.A. Volinsky / Engineering Frature Mehanis 75 (008) 311 3130 the ohesive zone size an be resolved aurately with the urrently available omputer power. Consequently, the following alulation may be inappliable to a very thin film of thikness below one miron. For the isotropi ase, a set of simulations with various radii r p of the indentation plasti zone S p was arried out. Sine the loading over this semi-spherial boundary is axi-symmetri about the x 3 -axis, the indued stress field and hene the delamination pattern along the film-substrate interfae is expeted to be axi-symmetri about the normal diretion. During the simulations, the total indentation fore P I and the indentation volume V I are omputed using Eqs. (1) and (), respetively. Variations of the normalized indentation fore P I / (E 0 t ffiffiffiffiffi ) and the ubi root of the indentation volume 3p p V I/t with respet to rp /t are plotted in Fig. 3. Four snapshots of the tangential tration ðs ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 1 þ p Þ ontour over the interfae are plotted in Fig. 4. The normal tration omponent, i.e., pressure normal to the interfae, does not ontribute to the delamination proess. A spider-web mesh of seven radial divisions per unit length and forty eight irumferential divisions was used as the potential zone of interfaial delamination damage and raking. This mesh and the mesh used for the 0.3 P I /(E 0 t ) and (V I ) (1/3) /t 0. 0.1 P I /(E 0 t )=0.3099(r p /t) 1.9989 3 y = 0.1788x 0.9975 V I /t=0.1788(r p /t) 0.99751 0 0 0. 0.4 0.6 0.8 1 r p /t Fig. 3. Variation of normalized indentation fore P I /(E 0 t ffiffiffiffiffi ) and normalized ubi root of indentation volume 3p V I/t with indentation plasti zone radius r p /t. Also shown are the power-law urve fitting equations for the two data sets in the ase of isotropi material model. Fig. 4. Contour plots of tangential tration (GPa) along the interfae at various values of r p /t: (a) 0.3; (b) 0.4; () 0.5; (d) 0.8, in the ase of isotropi material model. All the figures share the same sale bar as given in (d).
B. Yang, A.A. Volinsky / Engineering Frature Mehanis 75 (008) 311 3130 317 Normalized damage, rak and ohesive zone sizes 5 4 3 1 damage and rak rak ( A damage d ( A + A ) /( πt ) d + A ) /( π t ) A /( π t ) A /( π t ) 0 0 0.1 0. 0.3 P I /(E 0 t ) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi Fig. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5. Variationp of normalized damage radius ða d þ A Þ=pt, rak radius A =pt, and ohesive ða d þ A Þ=pt ffiffiffiffiffiffiffiffiffiffiffiffiffi A =pt with normalized indentation fore P I /(E 0 t ) in the ase of isotropi material model. zone depth semi-spherial loading boundary S p were both heked to be adequately dense to provide aurate solution to the problem. The red 1 (dark) olor indiates the delamination proess zone. The white (bright) olor indiates the loation of trivial tangential tration. These figures learly show the proess of delamination damage initiation, rak initiation, and their further extension in both inward and outward radial diretions. Further- ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the pohesive ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi damage and prak ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi areas A d and p A are alulated. Their normalized variants, pmore, A =ðpt Þ, ða d þ A Þ=ðpt Þ, and ða d þ A Þ=ðpt Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A =ðpt Þ, indiating the average damage and rak radii and the average ohesive zone depth, respetively, are plotted as funtions of normalized indentation fore P I /(E 0 t )infig. 6. A power-law urve fitting to the numerial data in Fig. 3 reveals that the ubi root of indentation volume V I is proportional to r p and that the total indentation fore P I is a quadrati funtion of r p, whih is onsistent with the analytial solution of Johnson [11]. It is evident that ffiffiffiffiffi the present formulation is valid and that the mesh used is adequate. Note that the linear dependene of 3p V I and quadrati dependene of PI on r p are identified over the entire proess of no raking in the beginning to later signifiant delamination damage. This is expeted beause the delamination rak is always losed (before bukling of delaminated film) and beause the indentation behavior is mainly dependent on the deformation in the normal diretion to the film surfae. It an be seen from Fig. 4 that the delamination damage initiates at a ertain distane from the middle symmetry point where the interfaial shear stress is the maximum and eventually reahes the interfaial strength, p y. When the indentation progresses with inreasing P I, the damage extends in both inward and outward radial diretions. Upon the preeding damage saturation, a rak is initiated (roughly at the same loation of damage initiation), and extends following the damage fronts. The inner ligament eventually diminishes, meanwhile the outer rak and damage fronts ontinue to advane. This proess of damage and rak development is quantitatively pffiffiffiffiffiffiffiffiffiffi shown pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi through the evolution of rak and damage areas in Fig. 5. After the ligament pdiminishes, ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A p=p and ða d þ A Þ=p indiate the radii of the rak and damage fronts, and ða d þ A Þ=p ffiffiffiffiffiffiffiffiffiffi A =p indiates the ohesive zone size. It an be seen that the damage evolution inluding its initiation and propagation is a gradual proess. In ontrast, the rak initiation is a jump-like event. Quikly after the initiation, the rak extends in steady state, indiated by a onstant ohesive zone size. Here, we did not onsider film bulking effets, so the rak length is assumed to be always smaller than the ritial bulking length. Another set of simulations was arried out, with the materials modeled as anisotropi. The numerial parameters were all the same as above exept the material model. Thus, any differene in the preditions must be attributed to the effet of elasti anisotropy of the film and the substrate. Similar sets of results as above were obtained. Sine the overall indentation behavior in the ase of anisotropi material model is found to be virtually the same as above (although the onstants of these two isotropi and anisotropi material models 1 For interpretation of olor in Figs. 4 and 6, the reader is referred to the web version of this artile.
318 B. Yang, A.A. Volinsky / Engineering Frature Mehanis 75 (008) 311 3130 Fig. 6. Contour plots of normalized tangential tration (GPa) on the interfae at various values of r p /t: (a) 0.3; (b) 0.4; () 0.48; (d) 0.5; (e) 0.7; (f) 0.9, in the ase of anisotropi material model. All the figures share the same sale bar as given in (f). Normalized damage, rak and ohesive zone sizes 5 4 3 1 damage and rak rak ( A A ) /( t ) damage d + π ( A + A ) /( π t ) d A /( π t ) A /( π t ) 0 0 0.1 0. 0.3 P I /(E 0 t ) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Fig. pffiffiffiffiffiffiffiffiffiffiffiffiffi 7. Variation of normalized damage radius ða d þ A Þ=pt, rak radius A =pt, and ohesive zone depth ða d þ A Þ=pt A =pt with normalized indentation fore P I /(E 0 t ) in the ase of anisotropi material model. in Tables 1 and were taken from different referenes), the plot similar to Fig. 3 is omitted. The omparable results to those in Figs. 4 and 5 are plotted in Figs. 6 and 7. Almost the same qualitative disussion an be made about the loading and the delamination proesses in the ase of anisotropi material model as in the previous ase of isotropi material model. However, a striking
differene is observed in the shape of the delamination damage and rak fronts being square with rounded orners ompared to the irular shape attaining to the axi-symmetry in the previous isotropi ase. The only possible explanation to this result is the material elasti anisotropy that has been taken into aount. This strong deviation of the delamination front from the irular shape needs to be aounted for when interpreting experimental results in suh material systems. Besides the signifiant effet of materials elasti anisotropy on delamination pattern, it is also seen that both damage and rak initiations are gradual, unlike disontinuous in the isotropi ase. Finally, further simulations were run with the opper film and the silion substrate modeled as either isotropi or anisotropi. When the opper film is modeled as isotropi and the silion substrate as anisotropi, the delamination front is irular. However, when the opper film is modeled as anisotropi and the silion substrate as isotropi, it is square with rounded orners. This suggests that it is not the elasti anisotropy of the silion substrate but that of the opper film that transforms the delamination pattern. Reall that the anisotropy index of the opper film is 3., ompared to 1.56 for the silion substrate. 5. Conlusions B. Yang, A.A. Volinsky / Engineering Frature Mehanis 75 (008) 311 3130 319 A omparative omputational study of the indentation-indued delamination proess of thin films with and without elasti anisotropy has been arried out. The indentation load was treated by the expanding avity model. The delamination proess was treated by the ohesive zone model so that an arbitrary delamination front an be onveniently predited. The main onlusion of this study is that in the ase of a Cu(0 01) film on a Si(0 0 1) substrate, the elasti anisotropy plays a signifiant role in determining the delamination pattern. This is evidened by very different preditions of a square delamination front with rounded orners in the anisotropi material model and a irular delamination front in the isotropi material model. The whole proess of delamination damage and rak development at a film-substrate interfae under indentation was aptured. This study employing the semi-spherial avity model is limited to the ases of relatively small indentation plasti zone ompared to the film thikness, thus relatively weakly bonded interfaes. The present extensive parametri study was arried out by applying an effiient and aurate boundary element method employing the speial Green s funtion for multilayers. Beause Green s funtion satisfies the interfaial ontinuity and tration-free surfae ondition, the numerial disretization was only performed along the semispherial loading boundary and one side of the delamination rak. Aknowledgement This work is supported by the National Siene Foundation (CMMI-060066 and CMMI-060031), the Division of Civil, Mehanial and Manufaturing Innovation. BY also aknowledges summer support from the National Institute of Standards and Tehnology that initiated this study. AV aknowledges the NSF IREE supplement support for the CMMI-060031 Grant. Referenes [1] Abdul-Baqi A, Van der Giessen E. Indentation-indued interfae delamination of a strong film on a dutile substrate. Thin Solid Films 001;381:143 54. [] Barenblatt GI. The mathematial theory of equilibrium raks in brittle frature. Adv Appl Meh, vol. VII. Aademi Press; 196. p. 55 19. [3] Bazant P, Li Y-N. Cohesive rak model with rate-dependent opening and visoelastiity: I. Mathematial model and saling. Int J Frat 1997;86:47 65. [4] Cleri F, Rosato V. Tight-binding potentials for transition metals and alloys. Phys Rev B 1993;48: 33. [5] Cook RF, Pharr GM. Diret observation and analysis of indentation raking in glasses and eramis. J Am Ceram So 1990;73:787 817. [6] Dolbow J, Gosz M. Effet of out-of-plane properties of a polyimide film on the stress fields in miroeletroni strutures. Meh Mater 1996;3:311 1. [7] Dugdale DS. Yielding in steel sheets ontaining slits. J Meh Phys Solids 1960;8:100 4. [8] Gao X-L, Jing XN, Subhash G. Two new expanding avity models for indentation deformations of elasti strain-hardening materials. Int J Solids Strut 006;43:193 08.
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