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substrate systems
Surface and Coatings Technology 179 (2004) 324–332
Effect of work hardening on the critical indentation limit in spherical nano-indentation of thin filmysubstrate systems Yo-Han Yooa, Woong Leeb, Hyunho Shina,* a First R&D Centre, Agency for Defence Development, P.O. Box 35-1, Yuseong, Daejeon, South Korea Department of Materials Science and Engineering, Yonsei University, 134 Shinchon-dong, Seoul 120-749, South Korea
b
Received 12 March 2003; accepted 3 June 2003
Abstract A finite element analysis has been carried out to investigate the effect of work hardening of materials on the critical indentation limit (the ratio of indentation depth to the film thickness) to ensure the determination of the ‘film-only’ properties in spherical nano-indentation of thin filmysubstrate layered systems. The work hardening of the film decreased the critical limit whilst that of substrate did not show any appreciable influence. The film hardening effect was more apparent when a smaller indenter was used. For the case when the film is harder than substrate, the influence of film hardening was more apparent especially when the filmy substrate yield stress ratio was low. However, for soft filmyhard substrate system, the influence of work hardening was more or less similar regardless of the yield stress ratio. Thus, it is suggested that care has to be taken to include the influence of work hardening when drawing out a critical indentation limit. 䊚 2003 Elsevier B.V. All rights reserved. Keywords: Work hardening; Critical indentation limit; Spherical indentation; Numerical analysis
1. Introduction In indentation of thin filmysubstrate layered systems, substrate is usually deformed together with the thin film and thus care has to be taken in characterising ‘filmonly’ properties from the indentation experiment. In order to ensure a reliable measurement of the ‘filmonly’ properties, indentation depth has to be less than a critical limit (ratio of the indentation depth to the film thickness), beyond which the measured apparent film properties are different from the real film material because the influence of the substrate is included. Thus, there has been great interest for the determination of the critical indentation limit. For this purpose, numerical analysis using finite element method is very useful for such purpose w1–9x. In this analysis, the numerically determined result from the layered system is compared with that from a hypothetical monolithic film material that has exactly same properties as the thin film. Then *Corresponding author. Tel.: q82-17-675-7753; fax: q82-42-8212221. E-mail address: [email protected] (H. Shin).
the critical indentation limit is determined by investigating that how much of the indentation limit ensures the same test results between the two cases, i.e. layered system and the hypothetical monolith. In order to guarantee the ‘film-only’ properties in an indentation experiment, in general, elastic or plastic deformation induced by the indenter has to be limited within the film. In case the deformed zone extends beyond the filmysubstrate interface toward substrate, the measured properties include the influence of the substrate. Thus, it is important to understand the evolution of the strain field with indentation depth to establish the critical indentation depth criterion. In He and Veprek w9x, the critical indentation limit was correlated with the initiation of plastic deformation in the substrate which is influenced by the mechanical properties of materials such as the ratio of Young’s modulus, Poisson’s ratio, yield stress and work hardening, of film to those of substrate. The influence of the former three properties on the evolution of the plastic deformation, which in turn, governs the critical indentation depth, have been studied w8x whilst the influence
0257-8972/04/$ - see front matter 䊚 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0257-8972(03)00814-4
Y.-H. Yoo et al. / Surface and Coatings Technology 179 (2004) 324–332
Fig. 1. Geometry of the model for finite element analysis when radius of indenter is 10 mm.
of work hardening is less studied although it has been known that film shows a considerable work hardening characteristics during indentation w10x. Some works dealt with the influence of work hardening of indented materials on the critical indentation depth based on the load discrepancy between the layered system of interest and bulk film material during the indentation process w2–4x. However, these were performed for a given material system, i.e. at constant material properties including work hardening. From a practical viewpoint, it is necessary to investigate whether the influence of work hardening is equally important for all material systems with different material properties or it has to be considered more seriously for a certain systems. It is also interest to investigate the significance of work hardening depending on the geometry of indenter, e.g. radius of indenter, for instance, in spherical indentation. Therefore, in the present study, it has been aimed to systematically investigate the influence of work hardening on critical depth for a number of materials systems with different film to substrate yield stress ratios especially for spherical indentations with different radius of indenter. The critical indentation depth was determined by numerically investigating the evolution of the elasto-plastic deformation as well as load discrepancy between the layered system and hypothetical bulk film material during the indentation process.
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when the radius of indenter is 18 mm. The radius of the filmysubstrate system was 10 R in order to effectively eliminate the interaction of indentation-induced deformation with the free boundary at radial end w13x. Thickness of the substrate along axial direction was also 10 R. The modelled thickness of the film was constant (3 mm) without regard to the radius of indenter so that the resultant ratios of indenter radius to film thickness, Ryt, have been 0.6, 2, 6 and 60. The 4-node isoparametric linear axi-symmetric element has been employed and the total number of element has been 6310 in the model shown in Fig. 1, for instance. The elasto-plastic von Mises model has been utilised as the constitutive model for the film and substrate materials with appropriate consideration of work hardening as schematically shown in Fig. 2. Elastic and plastic properties of the materials used in the numerical analysis are shown in Table 1 by referring to Murakami and Yuan w14x who numerically simulated indentation of monolithic materials with various hardening moduli. In their work on the analysis of work hardening characteristics depending on the elastic-linear-hardening material model, most of the metallic materials showed work hardening modulus (Et) values of 0.5 sy –2.0 sy and thus Etssy has been assumed in the present study both for film and substrate. The behaviour of the indenter, which is usually high-hardness and high-yield stress material such as diamond, has been assumed to be rigid since such an assumption was shown to yield no appreciable difference from the calculation based on the assumption of elastically deformable object w15x. The modelled contact condition between the indenter surface and upper surface of the film has been frictionless since the friction coefficient during indentation was shown to be very small and it did not have appreciable influence to the computation result in a previous work w16x. For numerical analysis an explicit finite element code ABAQUSyExplicit was utilised. In explicit time integration method, time step is very small due to the requirement of stability condition and thus the simulation of a
2. Numerical analysis Finite element analysis has been performed for the spherical nano-indentation process using two-dimensional axi-symmetric model, in which a thin film is assumed to be completely adhered to substrate as shown in Fig. 1. In order to investigate the effect of the radius of spherical indenter (R), various radii have been modelled, i.e. 1.8, 6, 18 and 180 mm, considering the practical size of the indenter w11,12x. Fig. 1 shows the model
Fig. 2. Schematic illustration of the bilinear constitutive relation.
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quasi-static event takes enormous computational time. When the loading speed is less than approximately 1% of the wave velocity in materials, inertia effect is negligible and thus a reasonably acceptable solution can be achieved even when the loading rate is artificially increased. Thus, the indenter speed up technique has been employed in the present work to reduce the computational time. This technique has been commonly used in simulating forming processes of sheet or bulk metals w17,18x. 3. Results and discussion 3.1. Hard filmysoft substrate system We first investigate the influence of substrate in nanoindentation process by monitoring load-displacement relation. Fig. 3 shows the load–displacement diagram for hypothetical bulk film material and hard filmysoft substrate layered system for the case when no strain hardening is present and radius of indenter is 18 mm for a hard filmysoft substrate system. As seen in the figure, the load–displacement curve of the layered system is similar to that of bulk film material in a small indentation depth regime whilst the layered system shows increasingly low load values as the indentation depth increases. This is because the effect of the soft substrate is getting increased with displacement, requiring less level of load for the displacement. In the present work, the deviation due to the influence of substrate has been quantified by taking the difference of the load at a given displacement divided by the load of bulk material. The right vertical axis in Fig. 3 indicates the influence of substrate in percent scale. As seen in Fig. 3, the load discrepancy seems to start from the indentation depth less than approximately 40 nm. In general, when the critical indentation limit is determined based on the load deviation curve such as
Fig. 3. Load–displacement relations for hypothetical bulk film material and a hard filmysoft substrate system (syfysyss6, Etfs0, Etss 0) by an indenter with Rs18 mm.
Fig. 3, it is determined at the load deviation of 2 or 10% depending on how much rigorous indentation limit is required. The critical indentation limit determined at 2% load deviation is considered a rigorous criterion whilst that at 10% is a practical limit. In the present work, the establishment of the critical indentation limit, based on some physically meaningful parameter that is associated with the load deviation, was also pursued and used in addition to the conventionally used load based critical indentation limit. For this purpose, the development of equivalent plastic strain (EQPS, scalar magnitude of the plastic strain tensor) field in the layered system during the nano-indentation process (shown in Fig. 3) was studied especially around the indentation depth where the load discrepancy takes place and the result is plotted in Fig. 4. This figure shows the contour diagrams of EQPS as the indentation depth increases for the hard filmysoft substrate system (Fig. 3). The con-
Table 1 Elastic and plastic properties of film and substrate materials used in the finite element analysis System
Component
Young’s modulus, E (GPa)
Poisson’s ratio, n
Yield stress sy (MPa)
Hardening modulus, Et (MPa)
Hard film on soft substrate
Film
200
0.3
200
0 200 0 600 0 100
Soft film on hard substrate
600 Substrate
150
0.3
100
Film
150
0.3
100
Substrate
200
0.3
200 600
0 100 0 200 0 600
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Fig. 4. Evolution of equivalent plastic strain during indentation of a hard filmysoft substrate system (syfysyss6, Etfs0, Etss0) by an indenter with Rs18 mm. Monitored indentation depths are (a) 20 nm, (b) 30 nm, (c) 50 nm and (d) 60 nm.
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tours in Fig. 4 are illustrated in equal steps (0.01%) up to 0.1% whilst the region with plastic strain higher than 0.1% is in grey colour (electronic publication only).1 It is noted in Fig. 4 that the plastic deformation zone appears only at the upper portions of the film up to indentation depth of 20 nm. At indentation depth of approximately 30 nm, a new plastically deformed zone appears at the upper region of the substrate before the plastically deformed zone in film reaches the substrate. Such feature is possible when the stress field around the bottom of the film region is insufficient for yielding whilst it is greater than the yield yield stress of the substrate material. The plastic zone growing in the film reaches substrate at approximately 60 nm of the indentation depth. Comparing the load discrepancy in Fig. 3 with the development of strain field in Fig. 4, it is suggested that the onset point of the equivalent plastic strain at upper region of the substrate (appears at around the indentation depth of 30 nm in Fig. 4) would serve as a more sensitive and an earlier warning criterion for the influence of the substrate than the criterion based on the load discrepancy (at approx. 40 nm indentation depth in Fig. 3). Therefore, the indentation depth at which equivalent plastic deformation sets off at the upper region of substrate (evolution of 0.01% EQPS) has been utilised to establish the critical indentation depth to ensure a reliable measurement of the ‘film-only’ properties for hard filmysoft substrate system in this work. Such method to establish the critical indentation depth criteria was also used by He and Veprek w9x. Having set up the way to define critical indentation depth as the onset point of the plastic strain in upper region of substrate, we have investigated the development of EQPS in substrate with indentation depth for various hard filmysoft substrate systems with different work hardening behaviours and the result is shown in Fig. 5. As seen in the figure, the critical indentation limit (the onset point of EQPS) as well as the slope after the onset is much influenced by the radius of indenter (R), the yield stress ratio of film to substrate (syf y sys), and work hardening of film (presence of Etf), whilst work hardening of substrate (presence of Ets) does not result in any appreciable difference. For a clear investigation of the influence of work hardening, the critical indentation depths (onset points of EQPS) for different hard filmysoft substrate combinations shown in Fig. 5 are compared in Fig. 6 as a function of indenter radius. In Fig. 6, the negligible influence of strain hardening of substrate is not shown for clarity. It is first noted in Fig. 6 for the EQPS based solid lines that the cases when Etfssyf (marked as j and m) are located below the cases when Etfs0 (d 1 The cutting of the 0.1% contour line is chosen arbitrarily for a clear presentation purpose.
Fig. 5. Development of equivalent plastic strain at the upper region of substrate with indentation depth for various hard filmysoft substrate material combinations for the case when indenter radii are (a) 6 mm and (b) 18 mm.
and %, respectively), indicating that the work hardening of film decreases the critical indentation limit at a given yield stress ratio (syf y sys) and indenter radius. Such influence of work hardening is more enhanced when a smaller spherical indenter is used. Second, the work hardening-induced decrease in critical limit, i.e. the difference between the lines for syf y syss2 (d and j), are larger than the gap between the lines for syf y syss 6 (% and m), indicating that the influence of work hardening is more pronounced when syf y sys ratio is low, i.e. when film is slightly stronger than the substrate. Second, the solid lines for syf y syss2 (d and j) are located above the lines for syf y syss6 (% and m), indicating that the critical indentation limit is low when the syf y sys ratio is high, i.e. as film is getting harder.
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Fig. 6. Change in critical indentation limit dcryt as a function of indenter radius R (normalised to film thickness, t) for various hard filmysoft substrate systems.
For the case when the radius of indenter is 1.8 mm (Ry ts0.6), the effect of film work hardening decreases the critical indentation depth from 4.7 to 3.8% for a system with yield stress ratio of 6 and from 14.1 to 8.5% for yield stress ratio of 2.2 Therefore, it is suggested that the influence of film work hardening has to be taken care of when drawing out a guideline for critical indentation depth especially for an indenter with small radius and a system with small yield stress ratio because unless otherwise the guideline may overestimate the indentation depth criteria. Fig. 6 also compares conventionally used 2% load– deviation-based critical indentation criteria (dotted lines: the indentation depth at which the influence of substrate, i.e. the load deviation is 2%). The trend of the influence of work hardening mentioned in the previous paragraph is the same also in the light of the load-based criteria whilst it is confirmed from Fig. 6 that the EQPS based criteria provides an earlier warning for the influence of substrate compared with the load-based criteria for the hard filmysoft substrate systems.
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(Etsssys) and when the radius of indenter is 6 mm. As seen in the figure, once load deviation occurs, the layered system shows increasingly higher load values as the indentation depth increases, which trend is reversed from the case of hard filmysoft substrate shown in Fig. 3. This is because the hard substrate influences to resist the deformation of soft film thereby it increases the load of indentation as compared with the bulk film material. Fig. 8 shows the development of EQPS as indentation depth increases for the indentation event shown in Fig. 7. As seen in Fig. 8, EQPS in substrate does not appear until the plastic strain in film reaches the filmysubstrate boundary and even after it grows fairly in radial direction (indentation depth of 400 nm). The plastic strain in substrate appears at the indentation depth of approximately 500 nm and this condition is much sensitive compared with the load deviation point in Fig. 7, i.e. after approximately 1000 nm. Thus, the development of EQPS in the hard substrate with indentation depth was also investigated for various soft filmyhard substrate systems with different work hardening behaviours similar to the way for the hard filmysoft substrate system and the result for the case when Rs6 mm is shown in Fig. 9. As seen in the figure, yield stress ratio of substrate to film and work hardening of film is associated with the onset condition of the plastic strain at substrate whilst the work hardening of the substrate does not show any appreciable influence to the onset condition, all of which are qualitatively similar to the hard filmysoft substrate system. We now compare the onset point of critical indentation depths in Fig. 9 for various soft filmyhard substrate systems and the results are shown in Fig. 10 as a function of the radius of indenter. In Fig. 10, the critical
3.2. Soft filmyhard substrate system The load–displacement relation and development of plastic strain in soft filmyhard substrate system has also been investigated as a counterpart for the hard filmysoft substrate system studied in the previous section and the results are shown in Figs. 7 and 8. Fig. 7 shows the load–displacement relation for the case when work hardening is present at both film (Etfssyf) and substrate 2
Based on the 2% load-error-based criteria in Fig. 6, the work hardening decreases the critical depth from 11.0 to 10.0% for yield stress ratio of 6 and from 18.3 to 14.7% for yield stress ratio of 2.
Fig. 7. Load–displacement relations for hypothetical bulk film material and a soft filmyhard substrate system (syfysyss0.5, Etfssyf, Etsssys) by an indenter with Rs6 mm.
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Fig. 8. Evolution of equivalent plastic strain during indentation of a soft filmyhard substrate system (syfysys s0.5, Etf ssyf , Etsssys) by an indenter with Rs6 mm. Monitored indentation depths are (a) 400 nm, (b) 500 nm, (c) 1000 nm and (d) 2000 nm.
indentation limit criteria based on 2% load deviation (««) are also presented for comparison with the EQPS based criteria (——). As seen in Fig. 10, for the case when yield stress ratio syf y sys is 0.5 (marked as j and d), the EQPS criteria provides an earlier warning for the influence of substrate as compared to the load based criteria (h and s), consistent with the hard filmysoft substrate system where the yield stress ratios were 2 and 6 (Fig. 6). However, for the case when the yield stress ratio syf y sys is very low, e.g. 0.17 in Fig. 10, the load based criteria (n and ) provide an earlier warning. This is because the substrate material is too hard to allow an appreciable plastic deformation (onset of EQPS is retarded) even after the film deformation is prohibited much by the presence of the hard substrate (after significant load deviation). Thus, the EQPS based criteria for the system with yield stress ratio syf y sys of 0.17 (m and %) would predict an erroneous result. It is noted via further investigation of the EQPS based criteria
∑
that the work hardening-induced decrease in critical limit (gap between % and m) seems to be more apparent when the indenter radius is large and this contradicts to the result based on the load-based criteria («« in Fig. 10) as well as the other systems with different yield stress ratios (Fig. 6). Therefore, the use of EQPS based criteria is inappropriate for the layered system with a very low yield stress ratio. In such a materials system, the interaction of elastic field with substrate is responsible for the load deviation. In case the detailed knowledge of the evolution of elastic field in such a system3 is unavailable, it is suggested to use the load-based criteria for such a system. As shown in the load-based criteria Fig. 10 (««), the influence of work hardening is more apparent as the radius of indenter decreases, consistent with the case of 3 The research results on this issue by the current group will be published elsewhere.
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is 6 mm, the work hardening of film decreases the loadbased critical indentation depth from 53 to 50% for the system with yield stress ratio of 0.5 (3% increase) and from 60 to 57% for yield stress ratio of 0.17 (3% increase). 4. Summary and conclusions
Fig. 9. Development of equivalent plastic strain at the upper region of substrate with indentation depth for various soft filmyhard substrate material combinations for the case when indenter radius is 6 mm.
hard filmysoft substrate system (Fig. 6). Therefore, it is suggested that the influence of film work hardening has to be taken care of also for the soft filmyhard substrate system especially for a small indenter for more rigorous critical indentation depth criteria. However, the work hardening-induced decrease in critical limits (gap and n; between s and h) are more or less between similar at a given indenter radius regardless of the studied yield stress ratios: when the radius of indenter
∑
A finite element analysis has been carried out to investigate the effect of work hardening of materials on the critical indentation limit in spherical nano-indentation of thin filmysubstrate layered systems. The onset condition for the development of equivalent plastic strain (EQPS) in substrate provided an earlier warning for the influence of substrate compared with the load-based criteria and thus the onset of EQPS could be utilised to establish a more rigorous critical indentation depth. However, vice versa was the case when filmysubstrate yield stress ratio was too low, e.g. 0.17. The work hardening of film decreased the critical indentation depth whilst that of substrate did not show any appreciable influence. The influence of film hardening was more apparent when a smaller indenter was used. For the case when film was harder than substrate, the influence of work hardening was more apparent when the filmysubstrate yield stress ratio was low. However, for soft filmyhard substrate system, the influence of work hardening was more or less similar regardless of the ratio. It is thus suggested that care for considering the influence of work hardening has to be
Fig. 10. Change in critical indentation limit dcryt as a function of indenter radius R (normalised to film thickness, t) for various soft filmyhard substrate systems.
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taken when drawing out a critical indentation limit especially when the radius of indenter is small. References w1x A.K. Bhattacharya, W.D. Nix, Int. J. Solids Struct. 27 (1991) 1047. w2x T.A. Laursen, J.C. Simo, J. Mater. Res. 7 (1992) 618. w3x X. Cai, H. Bangert, Thin Solid Films 264 (1995) 59. w4x J.A. Knapp, D.M. Follstaedt, J.C. Barbour, S.M. Myers, Nucl. Instr. Methods Phys. Res. B 127–128 (1997) 935. w5x C.-J. Lu, D.B. Bogy, Int. J. Solids Struct. 32 (1995) 1759. w6x Y. Sun, T. Bell, S. Zheng, Thin Solid Films 258 (1995) 198. w7x M. Lichinchi, C. Lenardi, J. Haupt, R. Vitali, Thin Solid Films 312 (1998) 240.
w8x D.E. Vlachos, Y.P. Markopoulos, V. Kostopoulos, Comput. Mech. B 27 (2001) 138. w9x J.L. He, S. Veprek, Surf. Coat. Technol. 163–164 (2003) 374. w10x K.-D. Bouzakis, N. Michailidis, G. Erkens, Surf. Coat. Technol. 142–144 (2001) 102. w11x J.S. Field, M.V. Swain, J. Mater. Res. 10 (1995) 101. w12x J.S. Field, M.V. Swain, J. Mater. Res. 8 (1993) 297. w13x S.Dj. Mesarovic, N.A. Fleck, Proc. R. Soc. Lond. A 455 (1999) 2707. w14x Y. Murakami, L.P. Yuan, J. Test. Eval. 20 (1992) 15. w15x K.C. Tang, A. Faulkner, N. Schwarzer, R.D. Arnell, F. Richter, Thin Solid Films 300 (1997) 177. w16x K. Komvopoulos, ASME J. Trib. 111 (1989) 430. w17x L.M. Kutt, A.B. Pifko, J.A. Nardiello, J.M. Papazian, Comput. Struct. 66 (1998) 1. w18x J. Rojek, E. Onate, E. Postek, J. Mater. Process Technol. 80– 81 (1998) 620.
Effect of work hardening on the critical indentation limit in spherical nano-indentation of thin filmysubstrate systems Yo-Han Yooa, Woong Leeb, Hyunho Shina,* a First R&D Centre, Agency for Defence Development, P.O. Box 35-1, Yuseong, Daejeon, South Korea Department of Materials Science and Engineering, Yonsei University, 134 Shinchon-dong, Seoul 120-749, South Korea
b
Received 12 March 2003; accepted 3 June 2003
Abstract A finite element analysis has been carried out to investigate the effect of work hardening of materials on the critical indentation limit (the ratio of indentation depth to the film thickness) to ensure the determination of the ‘film-only’ properties in spherical nano-indentation of thin filmysubstrate layered systems. The work hardening of the film decreased the critical limit whilst that of substrate did not show any appreciable influence. The film hardening effect was more apparent when a smaller indenter was used. For the case when the film is harder than substrate, the influence of film hardening was more apparent especially when the filmy substrate yield stress ratio was low. However, for soft filmyhard substrate system, the influence of work hardening was more or less similar regardless of the yield stress ratio. Thus, it is suggested that care has to be taken to include the influence of work hardening when drawing out a critical indentation limit. 䊚 2003 Elsevier B.V. All rights reserved. Keywords: Work hardening; Critical indentation limit; Spherical indentation; Numerical analysis
1. Introduction In indentation of thin filmysubstrate layered systems, substrate is usually deformed together with the thin film and thus care has to be taken in characterising ‘filmonly’ properties from the indentation experiment. In order to ensure a reliable measurement of the ‘filmonly’ properties, indentation depth has to be less than a critical limit (ratio of the indentation depth to the film thickness), beyond which the measured apparent film properties are different from the real film material because the influence of the substrate is included. Thus, there has been great interest for the determination of the critical indentation limit. For this purpose, numerical analysis using finite element method is very useful for such purpose w1–9x. In this analysis, the numerically determined result from the layered system is compared with that from a hypothetical monolithic film material that has exactly same properties as the thin film. Then *Corresponding author. Tel.: q82-17-675-7753; fax: q82-42-8212221. E-mail address: [email protected] (H. Shin).
the critical indentation limit is determined by investigating that how much of the indentation limit ensures the same test results between the two cases, i.e. layered system and the hypothetical monolith. In order to guarantee the ‘film-only’ properties in an indentation experiment, in general, elastic or plastic deformation induced by the indenter has to be limited within the film. In case the deformed zone extends beyond the filmysubstrate interface toward substrate, the measured properties include the influence of the substrate. Thus, it is important to understand the evolution of the strain field with indentation depth to establish the critical indentation depth criterion. In He and Veprek w9x, the critical indentation limit was correlated with the initiation of plastic deformation in the substrate which is influenced by the mechanical properties of materials such as the ratio of Young’s modulus, Poisson’s ratio, yield stress and work hardening, of film to those of substrate. The influence of the former three properties on the evolution of the plastic deformation, which in turn, governs the critical indentation depth, have been studied w8x whilst the influence
0257-8972/04/$ - see front matter 䊚 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0257-8972(03)00814-4
Y.-H. Yoo et al. / Surface and Coatings Technology 179 (2004) 324–332
Fig. 1. Geometry of the model for finite element analysis when radius of indenter is 10 mm.
of work hardening is less studied although it has been known that film shows a considerable work hardening characteristics during indentation w10x. Some works dealt with the influence of work hardening of indented materials on the critical indentation depth based on the load discrepancy between the layered system of interest and bulk film material during the indentation process w2–4x. However, these were performed for a given material system, i.e. at constant material properties including work hardening. From a practical viewpoint, it is necessary to investigate whether the influence of work hardening is equally important for all material systems with different material properties or it has to be considered more seriously for a certain systems. It is also interest to investigate the significance of work hardening depending on the geometry of indenter, e.g. radius of indenter, for instance, in spherical indentation. Therefore, in the present study, it has been aimed to systematically investigate the influence of work hardening on critical depth for a number of materials systems with different film to substrate yield stress ratios especially for spherical indentations with different radius of indenter. The critical indentation depth was determined by numerically investigating the evolution of the elasto-plastic deformation as well as load discrepancy between the layered system and hypothetical bulk film material during the indentation process.
325
when the radius of indenter is 18 mm. The radius of the filmysubstrate system was 10 R in order to effectively eliminate the interaction of indentation-induced deformation with the free boundary at radial end w13x. Thickness of the substrate along axial direction was also 10 R. The modelled thickness of the film was constant (3 mm) without regard to the radius of indenter so that the resultant ratios of indenter radius to film thickness, Ryt, have been 0.6, 2, 6 and 60. The 4-node isoparametric linear axi-symmetric element has been employed and the total number of element has been 6310 in the model shown in Fig. 1, for instance. The elasto-plastic von Mises model has been utilised as the constitutive model for the film and substrate materials with appropriate consideration of work hardening as schematically shown in Fig. 2. Elastic and plastic properties of the materials used in the numerical analysis are shown in Table 1 by referring to Murakami and Yuan w14x who numerically simulated indentation of monolithic materials with various hardening moduli. In their work on the analysis of work hardening characteristics depending on the elastic-linear-hardening material model, most of the metallic materials showed work hardening modulus (Et) values of 0.5 sy –2.0 sy and thus Etssy has been assumed in the present study both for film and substrate. The behaviour of the indenter, which is usually high-hardness and high-yield stress material such as diamond, has been assumed to be rigid since such an assumption was shown to yield no appreciable difference from the calculation based on the assumption of elastically deformable object w15x. The modelled contact condition between the indenter surface and upper surface of the film has been frictionless since the friction coefficient during indentation was shown to be very small and it did not have appreciable influence to the computation result in a previous work w16x. For numerical analysis an explicit finite element code ABAQUSyExplicit was utilised. In explicit time integration method, time step is very small due to the requirement of stability condition and thus the simulation of a
2. Numerical analysis Finite element analysis has been performed for the spherical nano-indentation process using two-dimensional axi-symmetric model, in which a thin film is assumed to be completely adhered to substrate as shown in Fig. 1. In order to investigate the effect of the radius of spherical indenter (R), various radii have been modelled, i.e. 1.8, 6, 18 and 180 mm, considering the practical size of the indenter w11,12x. Fig. 1 shows the model
Fig. 2. Schematic illustration of the bilinear constitutive relation.
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quasi-static event takes enormous computational time. When the loading speed is less than approximately 1% of the wave velocity in materials, inertia effect is negligible and thus a reasonably acceptable solution can be achieved even when the loading rate is artificially increased. Thus, the indenter speed up technique has been employed in the present work to reduce the computational time. This technique has been commonly used in simulating forming processes of sheet or bulk metals w17,18x. 3. Results and discussion 3.1. Hard filmysoft substrate system We first investigate the influence of substrate in nanoindentation process by monitoring load-displacement relation. Fig. 3 shows the load–displacement diagram for hypothetical bulk film material and hard filmysoft substrate layered system for the case when no strain hardening is present and radius of indenter is 18 mm for a hard filmysoft substrate system. As seen in the figure, the load–displacement curve of the layered system is similar to that of bulk film material in a small indentation depth regime whilst the layered system shows increasingly low load values as the indentation depth increases. This is because the effect of the soft substrate is getting increased with displacement, requiring less level of load for the displacement. In the present work, the deviation due to the influence of substrate has been quantified by taking the difference of the load at a given displacement divided by the load of bulk material. The right vertical axis in Fig. 3 indicates the influence of substrate in percent scale. As seen in Fig. 3, the load discrepancy seems to start from the indentation depth less than approximately 40 nm. In general, when the critical indentation limit is determined based on the load deviation curve such as
Fig. 3. Load–displacement relations for hypothetical bulk film material and a hard filmysoft substrate system (syfysyss6, Etfs0, Etss 0) by an indenter with Rs18 mm.
Fig. 3, it is determined at the load deviation of 2 or 10% depending on how much rigorous indentation limit is required. The critical indentation limit determined at 2% load deviation is considered a rigorous criterion whilst that at 10% is a practical limit. In the present work, the establishment of the critical indentation limit, based on some physically meaningful parameter that is associated with the load deviation, was also pursued and used in addition to the conventionally used load based critical indentation limit. For this purpose, the development of equivalent plastic strain (EQPS, scalar magnitude of the plastic strain tensor) field in the layered system during the nano-indentation process (shown in Fig. 3) was studied especially around the indentation depth where the load discrepancy takes place and the result is plotted in Fig. 4. This figure shows the contour diagrams of EQPS as the indentation depth increases for the hard filmysoft substrate system (Fig. 3). The con-
Table 1 Elastic and plastic properties of film and substrate materials used in the finite element analysis System
Component
Young’s modulus, E (GPa)
Poisson’s ratio, n
Yield stress sy (MPa)
Hardening modulus, Et (MPa)
Hard film on soft substrate
Film
200
0.3
200
0 200 0 600 0 100
Soft film on hard substrate
600 Substrate
150
0.3
100
Film
150
0.3
100
Substrate
200
0.3
200 600
0 100 0 200 0 600
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Fig. 4. Evolution of equivalent plastic strain during indentation of a hard filmysoft substrate system (syfysyss6, Etfs0, Etss0) by an indenter with Rs18 mm. Monitored indentation depths are (a) 20 nm, (b) 30 nm, (c) 50 nm and (d) 60 nm.
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tours in Fig. 4 are illustrated in equal steps (0.01%) up to 0.1% whilst the region with plastic strain higher than 0.1% is in grey colour (electronic publication only).1 It is noted in Fig. 4 that the plastic deformation zone appears only at the upper portions of the film up to indentation depth of 20 nm. At indentation depth of approximately 30 nm, a new plastically deformed zone appears at the upper region of the substrate before the plastically deformed zone in film reaches the substrate. Such feature is possible when the stress field around the bottom of the film region is insufficient for yielding whilst it is greater than the yield yield stress of the substrate material. The plastic zone growing in the film reaches substrate at approximately 60 nm of the indentation depth. Comparing the load discrepancy in Fig. 3 with the development of strain field in Fig. 4, it is suggested that the onset point of the equivalent plastic strain at upper region of the substrate (appears at around the indentation depth of 30 nm in Fig. 4) would serve as a more sensitive and an earlier warning criterion for the influence of the substrate than the criterion based on the load discrepancy (at approx. 40 nm indentation depth in Fig. 3). Therefore, the indentation depth at which equivalent plastic deformation sets off at the upper region of substrate (evolution of 0.01% EQPS) has been utilised to establish the critical indentation depth to ensure a reliable measurement of the ‘film-only’ properties for hard filmysoft substrate system in this work. Such method to establish the critical indentation depth criteria was also used by He and Veprek w9x. Having set up the way to define critical indentation depth as the onset point of the plastic strain in upper region of substrate, we have investigated the development of EQPS in substrate with indentation depth for various hard filmysoft substrate systems with different work hardening behaviours and the result is shown in Fig. 5. As seen in the figure, the critical indentation limit (the onset point of EQPS) as well as the slope after the onset is much influenced by the radius of indenter (R), the yield stress ratio of film to substrate (syf y sys), and work hardening of film (presence of Etf), whilst work hardening of substrate (presence of Ets) does not result in any appreciable difference. For a clear investigation of the influence of work hardening, the critical indentation depths (onset points of EQPS) for different hard filmysoft substrate combinations shown in Fig. 5 are compared in Fig. 6 as a function of indenter radius. In Fig. 6, the negligible influence of strain hardening of substrate is not shown for clarity. It is first noted in Fig. 6 for the EQPS based solid lines that the cases when Etfssyf (marked as j and m) are located below the cases when Etfs0 (d 1 The cutting of the 0.1% contour line is chosen arbitrarily for a clear presentation purpose.
Fig. 5. Development of equivalent plastic strain at the upper region of substrate with indentation depth for various hard filmysoft substrate material combinations for the case when indenter radii are (a) 6 mm and (b) 18 mm.
and %, respectively), indicating that the work hardening of film decreases the critical indentation limit at a given yield stress ratio (syf y sys) and indenter radius. Such influence of work hardening is more enhanced when a smaller spherical indenter is used. Second, the work hardening-induced decrease in critical limit, i.e. the difference between the lines for syf y syss2 (d and j), are larger than the gap between the lines for syf y syss 6 (% and m), indicating that the influence of work hardening is more pronounced when syf y sys ratio is low, i.e. when film is slightly stronger than the substrate. Second, the solid lines for syf y syss2 (d and j) are located above the lines for syf y syss6 (% and m), indicating that the critical indentation limit is low when the syf y sys ratio is high, i.e. as film is getting harder.
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Fig. 6. Change in critical indentation limit dcryt as a function of indenter radius R (normalised to film thickness, t) for various hard filmysoft substrate systems.
For the case when the radius of indenter is 1.8 mm (Ry ts0.6), the effect of film work hardening decreases the critical indentation depth from 4.7 to 3.8% for a system with yield stress ratio of 6 and from 14.1 to 8.5% for yield stress ratio of 2.2 Therefore, it is suggested that the influence of film work hardening has to be taken care of when drawing out a guideline for critical indentation depth especially for an indenter with small radius and a system with small yield stress ratio because unless otherwise the guideline may overestimate the indentation depth criteria. Fig. 6 also compares conventionally used 2% load– deviation-based critical indentation criteria (dotted lines: the indentation depth at which the influence of substrate, i.e. the load deviation is 2%). The trend of the influence of work hardening mentioned in the previous paragraph is the same also in the light of the load-based criteria whilst it is confirmed from Fig. 6 that the EQPS based criteria provides an earlier warning for the influence of substrate compared with the load-based criteria for the hard filmysoft substrate systems.
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(Etsssys) and when the radius of indenter is 6 mm. As seen in the figure, once load deviation occurs, the layered system shows increasingly higher load values as the indentation depth increases, which trend is reversed from the case of hard filmysoft substrate shown in Fig. 3. This is because the hard substrate influences to resist the deformation of soft film thereby it increases the load of indentation as compared with the bulk film material. Fig. 8 shows the development of EQPS as indentation depth increases for the indentation event shown in Fig. 7. As seen in Fig. 8, EQPS in substrate does not appear until the plastic strain in film reaches the filmysubstrate boundary and even after it grows fairly in radial direction (indentation depth of 400 nm). The plastic strain in substrate appears at the indentation depth of approximately 500 nm and this condition is much sensitive compared with the load deviation point in Fig. 7, i.e. after approximately 1000 nm. Thus, the development of EQPS in the hard substrate with indentation depth was also investigated for various soft filmyhard substrate systems with different work hardening behaviours similar to the way for the hard filmysoft substrate system and the result for the case when Rs6 mm is shown in Fig. 9. As seen in the figure, yield stress ratio of substrate to film and work hardening of film is associated with the onset condition of the plastic strain at substrate whilst the work hardening of the substrate does not show any appreciable influence to the onset condition, all of which are qualitatively similar to the hard filmysoft substrate system. We now compare the onset point of critical indentation depths in Fig. 9 for various soft filmyhard substrate systems and the results are shown in Fig. 10 as a function of the radius of indenter. In Fig. 10, the critical
3.2. Soft filmyhard substrate system The load–displacement relation and development of plastic strain in soft filmyhard substrate system has also been investigated as a counterpart for the hard filmysoft substrate system studied in the previous section and the results are shown in Figs. 7 and 8. Fig. 7 shows the load–displacement relation for the case when work hardening is present at both film (Etfssyf) and substrate 2
Based on the 2% load-error-based criteria in Fig. 6, the work hardening decreases the critical depth from 11.0 to 10.0% for yield stress ratio of 6 and from 18.3 to 14.7% for yield stress ratio of 2.
Fig. 7. Load–displacement relations for hypothetical bulk film material and a soft filmyhard substrate system (syfysyss0.5, Etfssyf, Etsssys) by an indenter with Rs6 mm.
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Fig. 8. Evolution of equivalent plastic strain during indentation of a soft filmyhard substrate system (syfysys s0.5, Etf ssyf , Etsssys) by an indenter with Rs6 mm. Monitored indentation depths are (a) 400 nm, (b) 500 nm, (c) 1000 nm and (d) 2000 nm.
indentation limit criteria based on 2% load deviation (««) are also presented for comparison with the EQPS based criteria (——). As seen in Fig. 10, for the case when yield stress ratio syf y sys is 0.5 (marked as j and d), the EQPS criteria provides an earlier warning for the influence of substrate as compared to the load based criteria (h and s), consistent with the hard filmysoft substrate system where the yield stress ratios were 2 and 6 (Fig. 6). However, for the case when the yield stress ratio syf y sys is very low, e.g. 0.17 in Fig. 10, the load based criteria (n and ) provide an earlier warning. This is because the substrate material is too hard to allow an appreciable plastic deformation (onset of EQPS is retarded) even after the film deformation is prohibited much by the presence of the hard substrate (after significant load deviation). Thus, the EQPS based criteria for the system with yield stress ratio syf y sys of 0.17 (m and %) would predict an erroneous result. It is noted via further investigation of the EQPS based criteria
∑
that the work hardening-induced decrease in critical limit (gap between % and m) seems to be more apparent when the indenter radius is large and this contradicts to the result based on the load-based criteria («« in Fig. 10) as well as the other systems with different yield stress ratios (Fig. 6). Therefore, the use of EQPS based criteria is inappropriate for the layered system with a very low yield stress ratio. In such a materials system, the interaction of elastic field with substrate is responsible for the load deviation. In case the detailed knowledge of the evolution of elastic field in such a system3 is unavailable, it is suggested to use the load-based criteria for such a system. As shown in the load-based criteria Fig. 10 (««), the influence of work hardening is more apparent as the radius of indenter decreases, consistent with the case of 3 The research results on this issue by the current group will be published elsewhere.
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is 6 mm, the work hardening of film decreases the loadbased critical indentation depth from 53 to 50% for the system with yield stress ratio of 0.5 (3% increase) and from 60 to 57% for yield stress ratio of 0.17 (3% increase). 4. Summary and conclusions
Fig. 9. Development of equivalent plastic strain at the upper region of substrate with indentation depth for various soft filmyhard substrate material combinations for the case when indenter radius is 6 mm.
hard filmysoft substrate system (Fig. 6). Therefore, it is suggested that the influence of film work hardening has to be taken care of also for the soft filmyhard substrate system especially for a small indenter for more rigorous critical indentation depth criteria. However, the work hardening-induced decrease in critical limits (gap and n; between s and h) are more or less between similar at a given indenter radius regardless of the studied yield stress ratios: when the radius of indenter
∑
A finite element analysis has been carried out to investigate the effect of work hardening of materials on the critical indentation limit in spherical nano-indentation of thin filmysubstrate layered systems. The onset condition for the development of equivalent plastic strain (EQPS) in substrate provided an earlier warning for the influence of substrate compared with the load-based criteria and thus the onset of EQPS could be utilised to establish a more rigorous critical indentation depth. However, vice versa was the case when filmysubstrate yield stress ratio was too low, e.g. 0.17. The work hardening of film decreased the critical indentation depth whilst that of substrate did not show any appreciable influence. The influence of film hardening was more apparent when a smaller indenter was used. For the case when film was harder than substrate, the influence of work hardening was more apparent when the filmysubstrate yield stress ratio was low. However, for soft filmyhard substrate system, the influence of work hardening was more or less similar regardless of the ratio. It is thus suggested that care for considering the influence of work hardening has to be
Fig. 10. Change in critical indentation limit dcryt as a function of indenter radius R (normalised to film thickness, t) for various soft filmyhard substrate systems.
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