Finite element modeling of the stresses, fracture and delamination during the indentation of hard elastic films on elastic–plastic soft substrates

Ž .Thin Solid Films 392 2001 65�74 Finite element modeling of the stresses, fracture and delamination during the indentation of hard elastic films on elastic�plastic soft substrates� R.M. Souzaa,�, G.G.W. Mustoeb, J.J. Moorec aSurface Phenomena Laboratory, Department of Mechanical Engineering, Polytechnic School, Uni�ersity of Sao Paulo, A�. Prof. Mello˜ Moraes 2231, Sao Paulo, SP 05508-900, Brazil bDi�ision of Engineering, Colorado School of Mines, Golden, CO 80401, USA c ( )Ad�anced Coatings and Surface Engineering Laboratory ACSEL , Colorado School of Mines, Golden, CO 80401, USA Received 3 May 2000; received in revised form 22 March 2001; accepted 22 March 2001 Abstract In this work, the mechanical behavior of hard films on soft substrates was studied based on the finite element analysis of an indentation with normal forces. As an attempt to reproduce situations found in practice, defects were considered during the preparation of the finite element mesh, both in the film and at the interface. A sequence of steps was considered during the Ž . Ž .loading sequence applied in the models. Initially, the deposition intrinsic and thermal extrinsic stresses were introduced to account for all residual stresses present in thin films deposited by processes such as sputtering. Later, a normal load of 50 N was applied on the pre-stressed system. The effects of a crack that propagated along the film�substrate interface was studied directly, by calculating the normal and shear stresses that develop at the film surface and the film�substrate interface, and indirectly, by looking at the behavior of cracks located at the film surface and propagating perpendicular to the interface. The results indicated that the suppression of the constraint imposed by the interface resulted in a decrease in the stresses in the film. However, the crack at the interface apparently did not interact with the stresses responsible for the array of circular cracks usually observed at the contact edge of the indentation of coated systems with soft substrates. � 2001 Elsevier Science B.V. All rights reserved. Keywords: Adhesion; Stress; Triboblogy 1. Introduction The deposition of coatings is a widespread tech- nology for improving the behavior of parts subjected to wear. These methods are particularly important in the case of soft substrates, such as aluminum, which are � �known for their poor tribological properties 1 . In recent years, several processes were developed to deposit thin films on a substrate, each one associated with a series of process parameters that can signifi- � �cantly affect the characteristics of the film 2 . There- � Corresponding author. � This paper was presented at ICMCTF 2000, San Diego, Califor- nia. fore, the number of possibilities to be tested for a given application is not restricted to the choice of the coating and its dimensions, but involves the selection of other factors, such as the process and its parameters. Since this number of possibilities is elevated, methods were developed to give insights into the wear behavior of coated systems without the need for long-term tests. Some of these methods refer to the use of analytical � � � �3,4 or finite element analyses 5�8 to evaluate the contact stresses originated when the system is sub- jected to the normal and�or tangential loads character- istic of many situations involving wear. � �In previous works 9�11 , the authors presented a sequence of finite element analyses studying the con- tact stresses developed during the normal indentation of systems with an elastic�plastic aluminum substrate 0040-6090�01�$ - see front matter � 2001 Elsevier Science B.V. All rights reserved. Ž .PII: S 0 0 4 0 - 6 0 9 0 0 1 0 0 9 5 9 - 2 ( )R.M. Souza et al.�Thin Solid Films 392 2001 65�7466 coated with an elastic wear-resistant film. In those works, the authors improved the correlation between the simulations and the situations found in practice by Ž .considering that: i the film was pre-stressed with thermal residual stresses developed during the film � � Ž .processing 9 ; and ii the film contained superficial circular cracks that propagated perpendicular to the interface under radial stresses developed during the � �indentation 10 . Another common assumption found in the literature is the consideration that the interface between the film and the substrate is perfect. Once again, perfect inter- faces may not be observed during the indentation of coated systems. In fact, indentation techniques are used to evaluate the adhesion of thin films to sub- � �strates 12�15 . The main difficulty related to the analysis of cracks that propagate along bimaterial interfaces comes from the naturally complicated nature of this type of prob- lem. The near-tip stress fields are given by a complex � �function 16�18 , where tension and shear effects are usually inseparable. Another complication arises from Ž .the fact that the fracture energy of an interface �i depends on the way the load is applied to the system, usually presenting different values, depending on the relative amount of tensile and shear contributions � �18,19 . It would be ideal to provide the finite element mod- els with a criterion that, for a given system, would allow the calculation of the amount of propagation of an interfacial crack under a given indentation load. How- ever, to the knowledge of the authors, the difficulties mentioned above prevent such a procedure from being applied using two fracture criteria for crack propaga- � �tion available in the ABAQUS software 20 . On the other hand, a third criterion in ABAQUS allows the user to specify a crack length as a function of time behavior, which can be adopted to force the propagation of an interfacial crack during the finite element analysis. In this work, a normal load of 50 N was applied to the system and the amount of propagation of 15 cracks located entirely in the film was calculated using the � �procedure adopted previously 10 . In addition, two interfacial crack as a function of time behaviors were imposed to a crack located at the film�substrate inter- face to study how the stresses in the system vary when debonding occurs, and how the propagation of the Ž .superficial film cracks was affected by this debonding. 2. Model description The ABAQUS software was used to run the finite Ž .element method FEM models, using the mesh pre- sented in Fig. 1a. Schematics of the most refined por- tions of the mesh are presented in Fig. 1b,c. The indenter shown in Fig. 1a was assumed to be rigid and to apply only normal loads to the system. A Ž .diameter of 1.59 mm 1�16 inch was selected, which is the diameter of a Rockwell F indenter. A total of 8867 four-noded elements was selected to model the substrate. Since only normal loads were applied, there was symmetry with respect to the inden- tation axis, and axisymmetric elements could be used. In all models, the substrate was composed of a 6061 aluminum alloy, with elastic�plastic behavior. The elas- Žtic and thermal properties of the aluminum elastic modulus E�68.9 GPa, Poisson ratio ��0.33 and co- �6 �1.efficient of thermal expansion ��23.6�10 K � �were taken from the literature 21 , and the plastic behavior, including strain hardening, was selected based on tensile tests conducted on 6061 aluminum samples � �11 . The film layer of wear-resistant material was as- sumed to be elastic in all cases, having ��9.8�10�6 �1 � �K 22 , E�280 GPa and ��0.3. The additional number of elements specified for the film was 3150, the thickness was fixed at t�2.1 �m and two fracture toughness values were studied, K �1.5 and �2.5Ic MPa �m. As an attempt to better reproduce the situations found in practice, defects were considered in the preparation of the finite element mesh, both in the film and at the interface. For the film cracks, results ob- � �tained previously 9 indicated that the location of the maximum radial stresses at the film surface varied according to the film thickness. Thicker films tended to have a peak of tensile stresses close to the contact edge Ž .region ‘A’ in Fig. 1b , while higher radial stresses were Žfound inside the contact region for thinner films re- .gion ‘B’ in Fig. 1b . Since the intent was to evaluate the effect of super- ficial cracks, all the analyses conducted in this work considered the presence of 15 cracks distributed along � � Ž .the film surface 10 Fig. 1b . It must be noted that, due to the axisymmetric geometry, the initial cracks are in fact circular cracks that propagate mainly under Ž .radial � stresses. A more precise configuration wouldr Ž .also consider cracks propagating under hoop �� stresses, but such consideration is impossible in a two- dimensional axisymmetric geometry. Fortunately, it was � �already observed 23,24 that, in systems with wear- resistant films on soft substrates, indentation results in an array of parallel cracks centered at the indentation axis, which suggests that circular cracks are the domi- nant mechanism of film fracture in the situations stud- ied in this work. It is also important to mention that the axisymmetric geometry requires a crack to simultaneously propagate over its entire perimeter. Therefore, it was not possible to reproduce facts that may be observed in practice, such as a crack propagating only over a circumferential � �section 23 , or a crack that locally propagates through ( )R.M. Souza et al.�Thin Solid Films 392 2001 65�74 67 Ž . Ž .Fig. 1. Mesh used in the finite element analyses to calculate contact stresses during the indentation of coated systems: a overview; b schematic Ž .of the distribution of surface cracks; and c schematic of the initial location of the interfacial crack. the film and later propagates over the perimeter of a � �circle 18 . In terms of defect size, the actual initial value de- pends on the quality of the film, but for the uniformity of the analyses, an initial crack size c �0.2 �m was0 considered reasonable and adopted in all cases. For the interfacial cracks, the debonding pattern � �observed during indentation of coated systems 12 suggests that the propagation of this type of crack starts inside the contact region, at a point located at a certain radial distance from the indentation axis. Addi- tionally, a circular pattern was also observed during the � �propagation of such crack 12 . Thus, part of the analy- ses conducted in this work also considered the pres- ence of a crack at the film�substrate interface, which was initially located at an arbitrary position distant from the model axis. In all cases, this position was Ž .under superficial crack number 12 Fig. 1c . Once again, due to the axysimmetric geometry, this was also a circular crack. Three steps were used to load the system. Initially, a uniform biaxial stress was imposed on the film ele- ments to account for the intrinsic stresses that result from film processing. A compressive value of 1 GPa was selected, although it is recognized that larger val- Ž .ues were reported for physical vapor deposition PVD � �processes 25 . In the second loading step, it was as- sumed that the temperature reached during deposition was 498 K, a temperature that is in the range usually observed during sputtering processes. Thermal residual stresses were then calculated when the system was Ž . � �cooled to room temperature 298 K 9 . In the third step, a normal load of 50 N was gradually applied on a Ž .reference node on the indenter Fig. 1a , following the sequence presented in Fig. 2. The stresses during in- dentation were calculated, both at maximum load and after the system was unloaded. During the indentation part of all analyses, the su- Ž .perficial cracks Fig. 1b were allowed to propagate in pure mode I along a predetermined path perpendicular to the interface. Based on the assumption of elastic films, the most suitable ABAQUS criterion for crack ( )R.M. Souza et al.�Thin Solid Films 392 2001 65�7468 Fig. 2. Evolution of load and interfacial crack size through the finite element analyses used to calculate contact stresses during the inden- tation of coated systems. � �propagation was the maximum stress 20 , according to which, the film cracks were allowed to propagate when a critical stress value was reached at a certain position ahead of the crack tip. Further information regarding the model and the criterion for propagation of the � �superficial cracks can be obtained elsewhere 10,11 . As mentioned before, crack propagation as a func- tion of time behaviors were imposed to induce propa- gation of the interfacial crack. Three different situa- tions were studied. In the first, no crack was present at Ž .the interface and, in other two, the crack length L as a function of time behaviors were those presented in Fig. 2. These types of behavior were selected based on � �the work by Jindal et al. 12 , who measured the amount of debonding as a function of the load applied by a spherical indenter. Once crack propagation occurred, a frictionless con- dition was specified for the contact of cracked surfaces, both for cracks located entirely in the film and for the crack at the film�substrate interface. This frictionless condition was also adopted for the contact between the indenter and the film, since the friction coefficient is usually small during the contact between a TiN film � �and a part made of steel 1 . It is important to mention that, as in the previous � �studies 3�11 , the microstructural features of the wear-resistant film were considered to affect the film mechanical properties, and thus indirectly affected the results. In addition to computational limitations, the direct effect of these microstructural features was ne- glected, based on the dimensional characteristics of the model. As will be presented in the results, the indenter radius and the applied load resulted in stresses dis- tributed in areas that are approximately 30-fold larger than the film thickness. Therefore, independent of the film microstructure, the regions under high radial Ž .stresses including the peaks would necessarily occur in regions with film defects, and the propagation of a film crack, or the sliding or splitting of the columns of � �films with a columnar structure 24 , would probably occur at the same position, independent of the film � �structure 23 . In other words, with the dimensions studied, the array of circular cracks would be similar for films with similar mechanical properties and initial crack sizes. 3. Testing the model Two different procedures were applied to test the meshes developed for the FEM. Initially, models with an aluminum substrate and no film were developed to test the substrate mesh. The size of the smallest ele- Ž .ments in region ‘C’ Fig. 1b was gradually reduced Ž .until minimal variation �1% in the results was ob- tained. The selected mesh was then used in the calcula- Ž .tion of stresses during the indentation 5 N of an elastic aluminum substrate, and the results were com- Ž .pared with analytical Hertzian values. A further test of the substrate mesh was obtained by comparing the calculated depth profile formed during the indentation Ž .600 N of an elastic�plastic aluminum substrate with Ž .that of a Rockwell F 600 N test conducted on a 6061 aluminum alloy. Good agreement was found when comparing FEM data with analytical and experimental � �results 11 . A similar procedure was conducted to establish the size and number of film elements that are on both sides of the superficial crack paths. The radial size of the elements in regions ‘A’ and ‘B’ was gradually decreased and the number of elements through film thickness was gradually increased until minimal variation was observed in terms of the calculated stresses, and also in the amount that each of the 15 cracks propagated � �during the entire analysis 10,11 . 4. Results The results were obtained based on the effect of the propagation of the interfacial cracks on the amount of Ž .propagation of the superficial film cracks and on the contact stresses developed during the indentation. Fig. 3 presents the amount of propagation of the 15 superficial cracks for all the six analyses conducted in this work. The results indicate a significant difference, depending on the values of the film fracture toughness Ž .K . The amounts of crack propagation were similarIc Ž .when K was set at 2.5 MPa �m Fig. 3d�f . In mostIc cases, only cracks 10�15 were activated and only mini- mal crack propagation was observed after unloading. When the film fracture toughness was reduced to 1.5 Ž .MPa �m Fig. 3a�c , more superficial cracks were acti- vated and significant differences were observed, de- pending on the presence or absence of interfacial crack propagation. The propagation of cracks 7�11 proceeded during the unloading portion of the analysis without Ž .the interfacial crack Fig. 3a and remained arrested in ( )R.M. Souza et al.�Thin Solid Films 392 2001 65�74 69 Ž .Fig. 3. Length of circular superficial film cracks calculated during the 50-N normal indentation of systems for a film with E�280 GPa and Ž . Ž .t�2.1 �m deposited on an elastic�plastic aluminum substrate: a no interfacial crack and K �1.5 MPa �m; b short interfacial crack andIc Ž . Ž . Ž .K �1.5 MPa �m; c long interfacial crack and K �1.5 MPa �m; d no interfacial crack and K �2.5 MPa �m; e short interfacial crack andIc Ic Ic Ž .K �2.5 MPa �m; and f long interfacial crack and K �2.5 MPa �mIc Ic the situations considering crack propagation along the Ž .interface Fig. 3b,c . In terms of the contact stresses developed during the indentation, Fig. 4 presents the distribution of radial Ž . Ž .Fig. 4a and axial Fig. 4b stresses obtained after unloading, for one of the analysis considering the pres- ence of the interfacial crack. In Fig. 4, the fracture toughness of the film was 2.5 MPa �m and the interfa- cial crack length as a function of time behavior was Ž .that given by curve 1 in Fig. 2 short interfacial crack . The contour plot shown in Fig. 4b is reproduced in Fig. 5b. Fig. 5 presents a comparison of the distribution of the radial stresses obtained after unloading, for the three cases where the fracture toughness of the film was 2.5 MPa �m. Fig. 5a presents the case without the interfacial crack and Fig. 5b,c the cases where the interfacial crack propagated according to the short and Ž .long behaviors Fig. 2 , respectively. More localized information regarding the variation of the contact stresses is presented in Figs. 6 and 7. Fig. Ž .6a shows a comparison at maximum load of the radial Ž . Ž .� stresses along the film surface z�0 , with andr ( )R.M. Souza et al.�Thin Solid Films 392 2001 65�7470 Fig. 4. Contour plots of stresses calculated during the 50-N normal indentation of systems for a film with E�280 GPa, t�2.1 �m and K �2.5Ic MPa �m film deposited on an elastic�plastic aluminum substrate. Short interfacial crack and values obtained after the load was entirely released: Ž . Ž .a radial stresses, � and b axial stresses, � .r z without crack propagation along the interface. The same results are presented in Fig. 6b for the variation Ž .of the hoop � stresses along the film surface; in Fig.� Ž .6c for the variation of the radial � stresses along ther film side of the interface; in Fig. 6d for the variation of Ž .the axial � stresses along the film side of the inter-z Ž .face and on Fig. 6e for the shear � stresses along therz film side of the interface. The same plots of Fig. 6 are shown on Fig. 7, but with stresses calculated after the load was entirely released. On Figs. 6 and 7, the frac- ture toughness of the film was 2.5 MPa �m and all the Ž .results were normalized by the contact area a andos Ž .the maximum contact pressure p obtained when aos rigid spherical indenter applies a normal load of 50 N � �on an elastic aluminum substrate 9�11 . 5. Discussion Different phenomena are experimentally observed during the spherical indentation of coated systems with soft substrates. Usually, an array of circular cracks is Žobserved close to the indentation contact edge region . � �‘A’ in Fig. 1b 11,23,24 . In many cases, these cracks nucleate at defects located close to the film surface, where the stresses are tensile due to the presence of Ž . � �film bending Fig. 6 8�11 . Weppelmann and Swain � �26 studied the mode I and II stress intensity factors associated with the circular cracks developed during spherical indentation, specifically for situations where Ž . Žthe crack size�thickness c�t ratio was small less .than 1�3 . Their conclusion was that mode I contribu- tions are predominant in the early stages of superficial crack propagation, especially in the cases where the Ž .ratio R�t of indenter radius�film thickness was large Ž .�2 . ŽIn this work, the ratio R�t was much larger R�t� .379 than those studied by Weppelmann and Swain � �26 , and it is possible to assume that the predominance of mode I stress intensity factors would extend to larger Ž .crack size�thickness c�t ratios. Therefore, even with ( )R.M. Souza et al.�Thin Solid Films 392 2001 65�74 71 Fig. 5. Contour plots of radial stresses calculated during the 50-N normal indentation of systems for a film with E�280 GPa, t�2.1 �m and Ž .K �2.5 MPa �m deposited on an elastic�plastic aluminum substrate. Values obtained after the load was entirely released: a no interfacialIc Ž . Ž .crack; b short interfacial crack; and c long interfacial crack. a pure mode I fracture criterion for the film cracks, it is possible to conclude that the model is efficient in predicting the cracks which would propagate in the analysis, and errors with respect to the situations found Ž .in practice effect of mode II would only occur after the film crack had already propagated a significant amount towards the interface. Based on these points, it is possible to assume that the model has correctly calculated the similarity in the number of film cracks that propagated with or without Ž .the propagation of an interfacial crack Fig. 3 . An explanation to this phenomenon can be given based on the fact that bending stresses are the most important contribution to the peak in radial stresses observed at the film surface close to the contact edge. Since bend- ing is associated with the deformation of the system, it is mainly affected by the characteristics of the sub- strate, and minimal differences in bending, and conse- quently in the number of cracks that propagated, should be observed for films with the same fracture toughness in situations with and without an interfacial crack. The morphology of the cracks may be different in Žareas close to the indentation axis region ‘B’ in Fig. .1b . Cracks may occur not only in circles, but also in � �the radial direction 11 , which indicates that the pre- Ž . Ž .dominant stresses are radial � and hoop � stressesr � Ž . � �Fig. 6a,b . Begley et al. 8 explained that high � andr � close to the model axis are observed when the� Ž .friction coefficient � between the indenter and the film is low. In cases where � is low, there is little restriction preventing the film from slipping under the indenter, which results in biaxial tensile stretching stresses that overcome the bending contribution. Ž .The effects of the unloading process Fig. 7 have � �already been presented by Montmitonet et al. 7 and � �previously discussed by the authors 10,11 . In those cases, it was verified that unloading results in an in- Ž .crease of � and � close to the model axis Fig. 7 ,r � Žwhich was assumed to be a result of the plastic perma- .nent deformation of the substrate. Substrate plastic deformation would prevent the elastic film from return- ing to its original position and would locally increase the stresses. In this work, the results obtained in cases with crack ( )R.M. Souza et al.�Thin Solid Films 392 2001 65�7472 Fig. 6. Stresses calculated during the 50-N normal indentation of systems for a film with E�280 GPa, t�2.1 �m and K �2.5 MPa �m filmIc Ž . Ž .deposited on an elastic�plastic aluminum substrate. Values calculated at maximum load: a radial stresses at the film surface; b hoop stresses at Ž . Ž . Ž .the film surface; c radial stresses at the film side of the interface; d shear stresses at the film side of the interface; and e axial stresses at the film side of the interface. propagation along the interface provided further evi- dence of the stress contribution from the plastic defor- mation of the substrate. Fig. 3 shows that the crack Ž .propagation during unloading Fig. 3a was suppressed when there was crack propagation along the interface Ž .Fig. 3b,c . Therefore, this fact confirms that the in- crease in stresses during unloading was associated with some reaction imposed by the substrate, which was eliminated when the film�substrate interface was no longer perfect. In terms of the effect of the interfacial crack on the contact stresses, it was again observed that the propa- gation of cracks resulted in an overall and localized � �decrease in the stresses in the film 10 . In Figs. 4 and 5, this stress reduction is evident in areas immediately above the interfacial crack. The only exception is with respect to the areas immediately ahead of the crack Ž .tips interfacial and superficial , where the stress inten- sity factor associated with the crack resulted in an Ž .increase in the stresses in these areas Fig. 4 . The observation of a localized and overall decrease in the stresses due to the propagation of a crack along the interface is also evident when the stresses were calculated along the surface or along the film side of Ž .the interface Figs. 6 and 7 . In this case, it is possible to observe that the stress reduction is not restricted to ( )R.M. Souza et al.�Thin Solid Films 392 2001 65�74 73 Fig. 7. Stresses calculated during the 50-N normal indentation of systems for a film with E�280 GPa, t�2.1 �m and K �2.5 MPa �mIc Ž .deposited on an elastic�plastic aluminum substrate. Values calculated after the load was entirely released: a radial stresses at the film surface; Ž . Ž . Ž . Ž .b hoop stresses at the film surface; c radial stresses at the film side of the interface; d shear stresses at the film side of the interface; and e axial stresses at the film side of the interface. areas above the interfacial crack, but also occurs close Ž .to the model axis r�0 . Examples of this behavior can Ž . Ž .be found in the case of the radial � and hoop �r � stresses shown in Fig. 6a,b and Fig. 7a,b. Considering the morphology of cracks observed close to r�0, the results presented indicate that situations with interfa- cial crack would probably result in a reduction in the amount of circular and radial cracks observed in areas � �close to the indentation axis 11 . 6. Conclusions In this work, the finite element method was applied to study the effect of a crack propagating along the film�substrate interface on the contact stresses devel- oped during the indentation of coated systems with an elastic�plastic soft substrate. The effect of the interfa- cial crack was observed not only directly on the calcu- lated contact stresses, but also indirectly. The indirect analysis was conducted by looking at the behavior of cracks located entirely in the film and propagating in a direction perpendicular to the interface. The results indicated that, for the conditions studied, the interfa- Ž .cial cracks: i did not interact with the bending stresses responsible for the circular cracks observed close to the contact edge of indentations of coated systems with ( )R.M. Souza et al.�Thin Solid Films 392 2001 65�7474 Ž .soft substrates; ii reduced the constraints imposed by the substrate on the film, which may be responsible for the propagation of film cracks during the unloading Ž .portion of the indentation; and iii resulted in an overall and localized reduction in the film stresses in most portions of the model, which is important in areas close to the indentation axis, where a reduction in the amount of circular and radial cracks would be ex- pected. Acknowledgements The authors would like to thank Mr Argemiro L.A. Costa, Mr Ricardo M. Dias, Mr Eduardo G. Pinheiro and Pirelli Pneus S.A. in Brazil for their help in using the ABAQUS software. References � �1 S. Ramalingam, Y. Shimazaki, W.O. Winer, Thin Solid Films 80 Ž .1981 297. � �2 D.L. Smith, Thin-Film Deposition Principles & Practice, McGraw Hill, New York, 1995. � �3 P.K. Gupta, J.A. Walowit, Trans. ASME J. Lubr. Technol. 96 Ž .1974 250. � � Ž .4 L. Zheng, S. Ramalingam, Surf. Coat. 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