Determination of the mechanical properties of metallic thin lms and substrates from indentation tests

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1 PHILOSOPHICAL MAGAZINE A, 2002, VOL. 82, NO. 10, 2013±2029 Determination of the mechanical properties of metallic thin lms and substrates from indentation tests K. Tunvisut, E. P. Busso, N. P. O Dowdy Department of Mechanical Engineering, Imperial College, London SW7 2BX, UK and H. P. Brantner Montanuniversitaet Leoben, Leoben, Austria [Received 16 July 2001 and accepted in revised form 1 March 2002] Abstract A procedure to obtain the elasto-plastic mechanical properties of strainhardening materials from indentation tests, based on dimensional analysis and nite-element techniques, is proposed. The method is applicable to homogeneous materials and to coatings deposited on substrates of known mechanical properties. The Young s modulus of the material is extracted from the initial slope of the unloading indentation curve and the yield strength and strainhardening exponent are obtained from the maximum indentation load and the contact area after unloading. The method is used to obtain the properties of a high-alloy steel and Mo and AlSi coatings deposited on a steel substrate by plasma spraying. The sensitivity of the measurement to the depth of indentation is discussed. } 1. Introduction In recent years, indentation tests have been used to determine elasto-plastic properties such as Young s modulus, yield strength and strain-hardening exponent (for example Doerner and Nix (1986), Oliver and Pharr (1992) and Cheng and Cheng (1999)). For instance, Young s modulus may be inferred from the unloading indentation load±depth curve and the yield strength from the maximum indentation load. In addition, a method to extract the ow stress and the strain-hardening exponent using indentation data has been proposed by Giannakopoulos and Suresh (1999). However, a common limitation to these approaches when applied to coatings is that the indentation test must be performed at relatively shallow depths, where the in uence of the substrate is negligible. In this way, the coating can be treated as a homogeneous material and the e ect of the substrate is ignored. However, as illustrated here, the prediction of bulk coating properties from indentation tests carried out at very small indentation depths is di cult and may lead to an overestimation of the mechanical properties. More recently, an alternative method has been proposed by Tunvisut et al. (2000, 2001) to estimate the elasto-plastic behaviour of thin lms from indentation tests. y n.odowd@ic.ac.uk. Philosophical Magazine A ISSN 0141±8610 print/issn online # 2002 Taylor & Francis Ltd DOI: /

2 2014 K. Tunvisut et al. The method takes into account the e ect of the substrate on the indentation response and therefore can be used to analyse indentation tests on thin lms performed at both shallow and deep depths. Tunvisut et al. (2001) carried out dimensional analysis and nite element studies of indentation of coated substrates and showed that the elasto-plastic mechanical properties (Young s modulus, yield strength and strainhardening behaviour) could be uniquely identi ed from measurement of the peak load and unloading slope of the indentation curve and the contact area of indentation (the region where permanent deformation was observed after removal of the indenter). Tunvisut et al. (2000) showed that the method could also be applied to uncoated substrates and comparison was made with existing and proposed techniques for determining coating properties from indentation tests (for example Oliver and Pharr (1992) and Giannakopoulos and Suresh (1999)). It was shown that the use of the contact area avoided the problem of uniqueness which may arise if the indentation curve alone is used to predict the material properties. In this work, a procedure is outlined, on the basis of these numerical studies, to determine directly from indentation measurements the mechanical properties of coatings and homogeneous substrates. The use of the method is illustrated for the case of a high-chromium steel (AISI D2) and for plasma-sprayed Mo and AlSi coatings deposited on a steel substrate. The latter coatings have been used to improve the wear behaviour of synchronizers used in automotive gear boxes. In order to predict the in-service behaviour of such coatings, an accurate description of the elasto-plastic mechanical behaviour of the coating is required (for example Yan et al. (2000)). The sensitivity of the predicted elasto-plastic properties of the coatings to the indentation depth is also discussed. } 2. Indentation methods 2.1. Indentation of bulk or uncoated substrates (method A) In previous work (Tunvisut et al. 2000, 2001), the indentation of an elasto-plastic solid by an ideally sharp rigid frictionless, conical indenter of half-angle ˆ 70 was studied ( gure 1 (a)). The stress±strain behaviour of the material was assumed to be linear in the elastic regime with Young s modulus E, and in the plastic regime the stress±strain behaviour is described by a power-law relation 8 ¼ >< " ˆ ¼ y " y >: ¼=¼ y n for 8 >< ¼ 4 ¼ y ; >: ¼ > ¼ y ; where ¼ and " are the stress and (total) strain respectively, ¼ y is the yield strength, " y ˆ ¼ y =E is the yield strain and n is the strain-hardening exponent. The material is assumed to obey the von Mises yield criterion and, under multiaxial loading, the stress and strain in equation (2) are replaced by the equivalent von Mises stress and strain respectively. It was found that conical indentation of such a material is described by the following dimensionless relationships: (1) (2)

3 Mechanical properties of thin lms and substrates 2015 Figure 1. (a) Schematic diagram of an indentation test using a conical indenter. (b) Typical indentation load±displacement curve. F m Eh 2 m A f h 2 m ˆ ^P b " y ; n; ; ; ˆ ^P g " y ; n; ; ; 3 4 where F m and h m are the maximum indentation load and depth respectively (see gure 1 (b)) and A f is the area of the indentation after unloading. A similar relationship to equation (3) has been provided by Cheng and Cheng (1999) for perfectly plastic (non-hardening) materials, and more recently for power-law-hardening materials by Cheng and Cheng (2000). The functional relations given in equations (3) and (4) have been determined for a range of material behaviours and typical results from parametric nite-element (FE) studies are shown in gure 2. Here, the maximum normalized indentation load and nal contact area are given in terms of the yield strain for di erent strain hardening exponents. The value of Poisson s ratio was taken to be 0.3 in all cases. Equations (3) and (4) in conjunction with data of the type illustrated in gure 2 can then be solved to evaluate the yield strain and strain-hardening exponent of the material. To allow the method to be applied directly, accurate ts to the data of the type shown in gure 2 have been obtained using the least-squares method. It has been found that F m Eh 2 m A f h 2 m ˆ 73" 0:82 y 87:3" 0:98 y 0:24 ln " y 0:36 ˆ 6 ln " y 178" 0:13 y 4:54 ln " y 5:86 n 0:26 ln " y 0:10 ; 5 n 0:1 ln " y 0:1 155:7: 6 The implicit dependence in equations (5) and (6) of the two unknown properties, namely the yield strain " y and the strain-hardening exponent n, on the known quantities requires that the equations be solved numerically using, for instance, an iterative Newton-type method. If Young s modulus E of the material is unknown, it was shown by Tunvisut et al. (2000) that this may be obtained directly from the initial unloading slope at the onset of unloading (i.e. at h ˆ h m ). This method generalizes the approach of Oliver and Pharr (1992), which may be inaccurate for soft materials, that is ¼ y =E < 0:005

4 2016 K. Tunvisut et al. Figure 2. Results of FE parametric studies for the uncoated substrate showing (a) the maximum normalized indentation load and (b) the normalized nal contact area in terms of the normalized yield strength for di erent strain-hardening exponents. (Bolshakov et al. 1997). The dependence of unloading slope on the normalized yield strength is illustrated in gure 3 (a). Although not shown, it has been found from FE studies that the initial unloading slope is almost independent of the hardening exponent n. For low-strength materials, ¼ y =E < 0.01, the normalized initial unloading slope depends strongly on ¼ y =E while, for high-strength materials, ¼ y =E 5 0:01, the dependence is weaker and the unloading slope can be assumed to be independent of ¼ y =E. For the latter case, we can then write 1 Eh m df dh hˆhm ˆ 6:8; for h f h m 4 0:875: 7 The range of applicability of equation (7) is given in terms of a directly measurable quantity, h f =h m, the ratio of the unloaded depth to the maximum depth, rather than Figure 3. Results of FE parametric studies for the uncoated substrate showing relationships between (a) the initial unloading slope, normalized using h m, and ¼ y =E and (b) the initial unloading slope, normalized using h f, and ¼ y =E.

5 Mechanical properties of thin lms and substrates 2017 the ratio ¼ y =E, which is not known a priori. For the case where h f =h m > (i.e. a low-strength material), a di erent normalizing parameter was used; the initial unloading slope is divided by h f E rather than by h m E, where h f is the nal depth after unloading (see gure 1 (b)). Figure 3 (b) shows that, when this normalization is used, the dependence on ¼ y =E is weak for ¼ y =E < 0:01, leading to the following expression: 1 Eh f df dh hˆhm ˆ 7:8; for h f h m > 0:875: Equations (7) and (8) allow direct identi cation of Young s modulus of the material from the initial unloading slope of the indentation curve for high- and low-strength materials respectively. It is worth noting that the applicability of the method depends strongly on the accuracy associated with the experimental indentation data. For instance, a 5% error in the unloading slope may result in a maximum error of 5%, 30% and 50% in the predicted values of E, ¼ y and n respectively (Tunvisut et al. 2000) Indentation of a coated substrate (method B) Consider a system similar to that discussed in 2.1 except that the indented solid is now replaced with a coating of thickness h 0, Young s modulus E c, yield strength ¼ yc, Poisson s ratio c and strain-hardening exponent n c, deposited on an elastic substrate of known Young s modulus E s and Poisson s ratio s. The relevant functional dependences of the geometrical and material parameters for the thin lm± substrate bimaterial system have been presented by Tunvisut et al. (2001) and are given as 1 h 0 E s df dh hˆhm ˆ ^P d h m h 0 ; E c E s ; c; s; ; 9 F m h 2 0 E ˆ ^P a s A f h 2 0 ˆ ^P l h m h 0 ; E c E s ; ¼ yc E s ; c; s; n c ; ; 10 h m h 0 ; E c E s ; ¼ yc E s ; c; s; n c ; : 11 Parametric FE analyses of the indentation of a range of coating±substrate systems for a maximum indentation depth h m ˆ 0:33h 0, were performed to calibrate the functions given in equations (9)±(11). Typical results are shown in gure 4. Figure 4 (a) can be used to determine Young s modulus of the coating from the initial unloading slope, and the yield strength and strain-hardening exponent can be obtained from gures 4 (b) and (c). An accurate calibration of the relationship de ned in equation (9) from the type of FE results shown in gure 4 gives 8 1 df < 2:1E 0:86 ; E 4 1; 12 h 0 E s dh hˆhmˆ : 7E 0:32 27:1E 0:01 13:9E 0:002 22:3; E > 1; 13 where E=E c =E s. It proves convenient to separate each of the dimensionless functions de ned by equations (10) and (11) into a term that is independent of the strain-hardening

6 2018 K. Tunvisut et al. Figure 4. Typical results from the FE parametric study of indentation of the coating/ substrate system showing (a) (df=dh hˆhm versus E c =E s, (b) F m =h 2 0E s versus ¼ yc =E s and (c) A f =h 2 0 versus ¼ yc =E s.

7 Mechanical properties of thin lms and substrates 2019 exponent n c and another term that depends on n c, that is we obtain, for a xed value of h m =h 0, F m h 2 0 E s A f h 2 0 ˆ ^P a E; ¼ ^P b E; ¼; n c ; 14 ˆ ^P c E; ¼ ^P d E; ¼; n c ; 15 where ¼=¼ yc =E s is the normalized coating yield stress. In the above, ^P a and ^P c provide the solution for a non-hardening material (n c ˆ 1) and the functions ^P b and ^P d provide the corrections to ^P a and ^P c respectively to account for strain hardening. The dimensionless functions ^P a, ^P b, ^P c and ^P d used in equations (14) and (15) obtained by tting to FE data are given in appendix A. Thus equations (14) and (15) constitute a set of simultaneous nonlinear equations from which the two unknown coating parameters, namely ¼ yc and n c, can be obtained. The implicit nature of the equations require that equations (14) and (15) must be solved numerically for ¼ and n c using for example an iterative Newton-type scheme. As discussed in the previous section, the method proposed in this work relies strongly on accurate measurements of the maximum indentation load, initial unloading slope and nal contact area. A sensitivity analysis showed that a 5% error in the measurement of the unloading slope could result in errors of up to 40% in the predicted value of ¼ yc. The method for uncoated materials discussed in 2.1, method A, can also be used to determine the properties of a thin lm provided that the indentation depth does not exceed a certain fraction of its thickness to ensure that the presence of the underlying substrate does not a ect the indentation curve. The indentation depth under which the e ect of the substrate is negligible depends on the mismatch in the mechanical properties of the coating and the substrate. Appendix B provides the conditions under which method A may be used to predict coating behaviour for a wide range of coating/substrate systems. Further discussion of these analyses has been provided by Tunvisut (2002). } 3. Experimental work Indentation tests have been carried out on an alloy steel (AISI D2) and on AlSi and Mo coatings deposited on a case hardened steel (with E s =194 GPa) by plasma spraying. The test on the alloy steel was used as a validation of the procedure as the mechanical properties of the steel were also measured in uniaxial tensile tests. Prior to performing the tests, all samples were polished and the coating thicknesses were measured. The mean thickness of the AlSi and Mo coatings were found to be 295 and 270 mm respectively. Indentation tests were carried out using Vickers and Berkovich indenters up to di erent depths, the details of which will be given below and at least two tests were performed for the same maximum indentation depth on each material. Details of the experimental data obtained for each material are presented in appendix C. The results presented in 2, which can be used to extract the mechanical properties of a material from an indentation test, are for a conical indenter. It has been shown in numerical studies (for example Cheng and Cheng (1998)) that indentation curves for a Berkovich indenter, a Vickers indenter and a 70 conical indenter are

8 2020 K. Tunvisut et al. almost identical (they di er by less than 5% for the wide range of cases examined). Furthermore, based on the approach of Oliver and Pharr (1992), it can be inferred that the contact areas for the 70 conical indenter, the Berkovich indenter and the Vickers indenter at the same indentation depth di er by less than 5% (i.e. the geometrical constant used to obtain the contact area from the slope of the unloading curve di ers by less than 5% for these indenters). Thus the relations presented in 2 may be used to analyse indentation tests using Berkovich and Vickers indenters as well as conical indenters Shallow indentations (h m < 0:5 m) Indentation tests up to a maximum depth of less than 0.5 mm were carried out using a nanoindenter instrument with a diamond Berkovich indenter. A typical indentation curve measured in the AlSi coating system is shown in gure 5 (a). The image of the indentation after unloading was obtained using a nanoscope. The shape of a typical impression is shown in gure 5 (b). The nal contact area A f, after unloading for a perfectly sharp indenter can be calculated directly from the image, assuming a triangular contact region: A f ˆ d 2 3 1=2 sin ; 16 where is the angle between the central line and the face of the indenter; here ˆ 65:3, and d ˆ d1 d 2 d 3 : 17 3 The dimensions d 1 ; d 2 and d 3 in equation (17) are the diagonals of the indentation area after unloading (see gure 5 (b)). In practice, owing to the e ect of indenter tip rounding, equation (16) will not precisely describe the relationship between indentation depth and contact area for an actual indenter. To determine this relationship an experi- Figure 5. Typical results from nanoindentation tests on the AlSi coating: (a) indentation curve; (b) image of indentation after unloading.

9 Mechanical properties of thin lms and substrates 2021 mental calibration is carried out closely following the procedure of Oliver and Pharr (1992). The contact area shape function has been calibrated in the following form: A f ˆ d 2 3 1=2 sin C 1 d C 2 d 1=2 C 3 d 1=4 C 8 d 1=128 ; 18 where C 1, C 2, C 3, C 4, C 5, C 6, C 7 and C 8 are constants, here found to be 4.16, 10.38, 15.9, 17.5, 17.7, 15.6, 10.7 and 8.7 respectively. The lead term in equation (18) describes a perfect Berkovich indenter; the others describe deviations from this idealized geometry (e.g. tip rounding). The indenter geometry is precisely de ned through the use of equation (18). However, for certain materials, the shape of the contact region may deviate considerably from a triangle depending on the material properties and indentation depth, invalidating the use of equation (18). However, for the range of the materials and indentation depths studied in the current work, insigni cant deviation has been observed and the assumption of a triangular impression is acceptable Deep indentations (h m > 0:5 m) For 0:5 mm < h m 4 10 mm, tests were performed using a Fisherscope dynamic hardness tester with a Vickers indenter. The maximum load available was 1 N; thus, for larger indentations, a macrohardness tester was used. A typical macroindentation curve for the AlSi system is shown in gure 6 (a) together with a typical indentation impression in gure 6 (b). For the macrohardness measurements, only the indentation load and the image of the indented region could be recorded. Thus for these measurements the indentation depth h in gure 6 (a) was estimated from the mean value of the two impression diagonals d 1 and d 2 in gure 6 (b). Then h d 7 ˆ 1 7 d 1 d 2 : 19 2 Figure 6. Typical results from macroindentation tests on the AlSi coating: (a) indentation curve; (b) image of indentation after unloading.

10 2022 K. Tunvisut et al. Figure 7. Ratio of the impression diagonal to the indentation depth for a Vickers indenter for di erent materials. Note that the value of h calculated via equation (19) is the contact depth, which will include the e ects of piling-up and sinking-in and thus may overestimate or underestimate the actual indentation depth. Figure 7 provides values of d=h, obtained from FE calculations, for a range of materials, that is a range of ¼ y =E. When d=h 7 the e ect of piling-up or sinking-in is negligible, and equation (19) is appropriate in the determination of the indentation depth from the impression diagonal. For hard materials (i.e. high ¼ y =E), sinking-in is observed, d=h < 7, and equation (19) overestimates the indentation depth. On the other hand, when piling-up takes place (materials with low ¼ y =E), equation (19) underestimates the indentation depth (d=h > 7). However, for the range of materials being studied in this paper, 0:004 < ¼ y =E < 0:01; it may be seen in gure 7 that d=h 7 and thus the indentation depth estimated from equation (19) is acceptable; equation (19) was found to overestimate h by approximately 3% for the Mo coating and to underestimate h by <8% for the AlSi coating, which will result in less than a 15% error in the material yield strength using the current method (see also Tunvisut (2002)). The contact area A f for the Vickers macroindentation can be obtained directly from the indentation impression by assuming that the indenter is perfectly sharpðan acceptable approximation for deep indentations. The contact area can then be approximated from A f ˆ d 2 1:85 : 20 Again, the assumption inherent in equation (20) is that the indentation impression is of diamond shape. We have observed this shape of impression for all the indentation tests carried out (see, for example gure 6 (b)).

11 Mechanical properties of thin lms and substrates 2023 } 4. Results and discussion 4.1. Uncoated substrate The methodology proposed here will rst be used to determine the mechanical properties of the alloy steel. Method A is used and the resulting predictions for the material s yield stress, Young s modulus, hardening coe cient and 0.2% yield strength ¼ 0:2 using two di erent indenter types and three maximum indentation depths are presented in table 1. As seen in table C1 in appendix C, di erent results are obtained from the nanoindentation measurements (h m < 0:35 mm) depending on whether the indentation is located in the carbide or martensite phases. To obtain the macroscopic elastic±plastic properties, the values for the individual phases were averaged. The use of an unweighted average to calculate the material properties may be open to question, but the result is included in table 1 for completeness. In addition to the indentation tests, uniaxial tensile tests were carried out and these data are also given. The measured stress±strain uniaxial response of the AISI D2 steel is shown in gure 8, together with the predicted responses calculated using the material parameters obtained from indentation at di erent indentation depths. The result for the lowest indentation depth is not included. It may be seen that the results are similar within the experimental scatter; however, on the basis of the mean result there is a small depth dependence of the result, with strength increasing somewhat with decreasing indentation depth. Overall there is good agreement with the tensile data particularly at the deepest indentation depth Mo and AlSi coatings The results from indentation tests on the Mo and AlSi coatings have been analysed using the proposed methods. For the tests with maximum indentation depths less than 0.5 mm, which is relatively shallow compared with the coating thickness, method A can be used. For AlSi and Mo coatings indented to the depth of 97 and 89 mm respectively (h m =h 0 =0.33%) the substrate has a strong e ect on the indentation test results and therefore method B is used. For the latter data, Young s modulus cannot be determined because the unloading curve is not available. Therefore the average values of Young s modulus obtained from the shallow indentation tests were used to interpret these data. The calculated Young s modulus, yield strength and hardening exponent of the coatings at di erent depths are presented in tables 2 and 3. It can be seen that, as for the steel substrate, there is again good agreement in the predicted Young s moduli, but in this case there is a strong dependence of the yield strength values on depth. It should be pointed out that this depth dependence is also evident in the raw experimental data; for example the Table 1. Predicted and measured elasto-plastic properties for the AISI D2 steel. h m Indenter E ¼ y ¼ 0:2 Method (mm) type (GPa) (MPa) n (MPa) Uniaxial test ± ± ± A <0.35 Berkovich A Vickers A Vickers

12 2024 K. Tunvisut et al. Figure 8. Stress±strain curve for AISI D2 steel. Table 2. Predicted elasto-plastic properties of Mo coating using indentation test data. Indenter h m E ¼ y ¼ 0:2 Method type (mm) (GPa) (MPa) n (MPa) A Berkovich A Berkovich A Vickers B Vickers 89 ± Table 3. Predicted elasto-plastic properties of AlSi coating using indentation test data. Indenter h m E ¼ y ¼ 0:2 Method type (mm) (GPa) (MPa) n (MPa) A Berkovich A Berkovich A Vickers B Vickers 97 ± normalized maximum load, as speci ed by equation (3) is depth dependent for the lowest depths with the same indenter type, even taking the experimental scatter into consideration (see tables C 2 and C 3 in appendix C) Indentation size e ects Good agreement between the measured properties and that determined from the tensile curve has been obtained for the steel substrate with a small depth dependence of the elastic±plastic properties. However, our results suggest that for the AlSi and

13 Mechanical properties of thin lms and substrates 2025 Mo coatings the properties depend on the depth of indentation. Similar observations have been reported for example by Iost and Bigot (1996) and Rother et al. (1998). This `size e ect is also seen here in the raw data, as discussed in the previous section, and so cannot be associated with uncertainties in the proposed method. The observed depth dependence may be due to a number of factors, for example the e ect of inhomogeneities in the coatings (there is known to be porosity in the Mo coatings and some unmelted particles in the AlSi coating which may have a larger e ect at small indentation depth) or the e ect of tip rounding which is more important at smaller indentation depths (for example Cheng and Cheng (1998)) or experimental uncertainties at shallow depths (for example Blau (1983)). Alternatively, such size e ects may be due to the nature of the plastic deformation at small size scales. Plastic deformation under small indentation loads leads to large local strain gradients which can cause enhanced strain hardening due to the additional contribution from geometrically necessary dislocations (for example De Guzman et al. (1993), Fleck and Hutchinson (1993) and Busso et al. (2000)). The incorporation of these e ects will lead naturally to an indentation size e ect (hardness and apparent yield strength increasing with decreasing depth). Based on this concept, Nix and Gao (1998) studied indentation size e ects for ductile materials. By taking into account the e ect of the geometrically necessary dislocations generated near the indenter surface and using Taylor s relation for the ow stress (for example Hull and Bacon (1984)), they obtained a simple relationship for the depth dependence of the hardness H (indentation force divided by contact area): H H 0 ˆ 1 h* h 1=2 : 21 In equation (21), h is the indentation depth, H 0 the hardness in the limit of in nite depth and h* is a characteristic length that depends on the shape of the indenter, the shear modulus and H 0. It has been shown by Tunvisut (2002) using similar arguments that the expression for the depth dependence of the yield strength may be written as ¼ y ¼ y0 2ˆ k 1 h* h ; 22 where ¼ y is the yield strength obtained for a given depth h of indentation, ¼ y0 represents the value obtained in the limit of in nite depth and k is a constant which depends only on material properties, that is Young s modulus, yield strength, hardening behaviour and indentation depth. The relationship obtained from equation (22) is plotted in gure 9 for the AlSi and Mo coatings together with the measured data. It may be seen that the data seem to follow the trend predicted by equation (22). Thus there is some support for the use of strain gradient approaches to account for depth dependence of predicted material properties from indentation tests. However, further studies are needed to validate the approach fully. In any case, the observed results highlight the fact that care needs to be taken if material properties are obtained from indentations at very small length scales, using for example nanoindentation techniques.

14 2026 K. Tunvisut et al. Figure 9. Illustration of indentation depth dependence of the yield strength for Mo and AlSi coating. } 5. Conclusions Methods to determine the mechanical properties of materials from indentation tests have been described and these methods have been used to extract Young s modulus, yield strength and hardening exponent of an AISI D2 steel and of AlSi and Mo coatings. The values of Young s modulus determined from the indentation tests are relatively independent of indentation depth and for the steel substrate the data are consistent with uniaxial tensile data. Quite a strong depth dependence of the estimated values of yield strength has been found for the AlSi and Mo coating, that is increasing yield strength with decreasing indentation depth. This dependence may be due to the indentation size e ect which can be signi cant when the indentation depth is small relative to intrinsic material length scales. Such size e ects should be taken into account if nanoindentation measurements are to be used to predict bulk mechanical properties. ACKNOWLEDGEMENTS Support for this work has been provided by the government of Thailand and by the European Union through Brite EUram project BE The authors are grateful to Dr N. Renevier, from Teer Coatings Ltd, UK, for providing us with the microindentation test results on the AISI D2 steel. Helpful comments from one of the reviewers are acknowledged. APPEN DIX A D i m e n s i o n l e s s f u nc t i o n s f o r m e t h o d B The dimensionless functions which appear in equations (14) and (15) are given below. In the equations, E ˆ E c =E s and ¼ ˆ ¼ yc =E s.

15 For 0.01< E 4 0.1, Mechanical properties of thin lms and substrates 2027 ^P a ˆ 0:6E 2 0:77E 0:003 ¼ 17E2 3:8E 0:01 ; A 1 ^P b ˆ 0:014 ln ¼E 0:84 0:1 ln ¼ 0:09 E0:3 n c ; A 2 ^P c ˆ 0:7E 0:42 ¼ 0:1E 0:586 ; A 3 For 0.1< E 4 1, ^P d ˆ 0:27 ln ¼E 0:5 0:7E 0:44 n 1:3E0:1 ¼ 0:14E0:2 c : A 4 ^P a ˆ 0:32E 2 0:91E 0:003 ¼ 0:33E2 0:57E 0:2 ; A 5 ^P b ˆ 0:014 ln ¼E 0:84 0:1 ln ¼ 0:09 E0:3 n c ; A 6 ^P c ˆ 0:7E 0:42 ¼ 0:1E 0:59 ; A 7 For 1< E 4 10, ^P d ˆ 0:27 ln ¼E 0:5 0:7E 0:4 n 1:3E0:1 ¼ 0:14E0:2 c : A 8 ^P a ˆ 0:02E 2 0:26E 0:38 ¼ 0:004E2 0:05E 0:412 ; A 9 ^P b ˆ 0:014 ln ¼E 0:84 0:1 ln ¼ 0:09 E0:3 n c ; A 10 ^P c ˆ 0:8E 0:3 ¼ 0:25E 0:1 ; A 11 ^P d ˆ 0:3 ln ¼E 0:5 0:7E 0:4 n 1:3E0:1 ¼ 0:14E0:2 c : A 12 APPEN DIX B S el ect i o n o f i n d e n tat i o n m e t h o d Tables B 1±B 3 can be used as guidelines in selecting an appropriate indentation method to be used in determining the mechanical properties of coatings deposited on a substrate from indentation tests carried out at di erent thicknesses. Table B 1. Indentation depth of one tenth of the coating thickness. Method for following values of ¼ yc =E s E c =E s A A A A 0.1 A A B B 1 A A A A 2 A A A A 10 A A A A

16 2028 K. Tunvisut et al. Table B 2. Indentation depth of one sixth of the coating thickness. Method for following values of ¼ yc =E s E c =E s B B B B 0.1 A B B B 1 A A A A 2 A A A B 10 A A B B Table B 3. Indentation depth of one third of the coating thickness. Method for following values of ¼ yc =E s E c =E s B B B B 0.1 B B B B 1 A A A A 2 B B B B 10 B B B B APPEN DIX C I n d e n tat i o n t e s t m e a s u r e m e n t s Experimental data obtained from indentation tests for the high-cr steel (AISI D2), and plasma-sprayed Mo and AlSi coatings deposited on a steel substrate are reported in tables C 1±C 3. Table C 1. Measurements for indentation tests on AISI D2 steel. h m h f df=dh hˆhm A f F m (mm) (mm) (mn/mm) (mm 2 ) mn (carbides) mn (martensite) mn mn Table C 2. Measurements for indentation tests on the Mo coating. h m h f df=dh hˆhm A f F m =h 2 m F m (mm) (mm) (mn/mm) (mm 2 ) (N/mm 2 ) mn mn mn N 89 8 ± ±

17 Mechanical properties of thin lms and substrates 2029 Table C 3. Measurements for indentation tests on the AlSi coating. h m h f df=dh hˆhm A f F m =h 2 m F m (mm) (mm) (mn/mm) (mm 2 ) (N/mm 2 ) mn mn mn N 97 8 ± ± References Blau, P. J., 1983, Metallography, 16, 1. Bolshakov, A., Oliver, W. C., and Pharr, G. M., 1997, Mater. Res. Soc. Symp. Proc., 436, 141. Busso, E. P., Meissonnier, F., and O Dowd, N. P., 2000, J. Mech. Phys. Solids, 48, Cheng, Y. T., and Cheng, C. M., 1998, J. Mater. Res., 13, 1059; 1999, Int. J. Solids Structs, 36, 1231; 2000, Surf. Coating Technol., 133±134, 417. De Gusman, M. S., Neubauer, G., Flinn, P., and Nix, W. D., 1993, Mater. Res. Symp. Proc., 308, 613. Doerner, M. F., and Nix, W. D., 1986, J. Mater. Res., 1, 601. Fleck, N. A., and Hutchinson, J. W., 1993, Adv. appl. Mech., 33, 295. Giannakopoulos, A. E., and Suresh, S., 1999, Scripta mater., 40, Hull, D., and Bacon, D. J., 1984, Introduction to Dislocations (Oxford: Butterworth- Heinemann). Iost, A., and Bigot, R., 1996, J. Mater. Sci., 31, Nix, W. D., and Gao, H., 1998, J. Mech. Phys. solids, 46, 411. Oliver, W. C., and Pharr, G. M., 1992, J. Mater. Res., 7, Rother, B., Steiner, A., Dietrich, D. A., Jehn, H. A., Haupt, J., and Gissler, W., 1998, J. Mater. Res., 13, Tunvisut, K., 2002, PhD Thesis, University of London. Tunvisut, K., O Dowd, N. P., and Busso, E. P., 2000, Proceedings of the Fourth International Conference on Modern Practice in Stress and Vibration Analysis, Nottingham, 2000, edited by A. A. Becker (EMAS Limited), p. 89; 2001, Int. J. Solids. Structs, 38, 335. Yan, W., Busso, E. P., and O Dowd, N. P., 2000, Proc. R. Soc. A, 456, 2387.