Materials Research, Vol. 7, No. 3, 2004Vol. 7, No. 3, 483-491, 2004. The Use of a Vickers Indenter in Depth Sensing Indentation for Measuring Elastic Modulus and Vickers Hardness © 2004 483 The Use of a Vickers Indenter in Depth Sensing Indentation for Measuring Elastic Modulus and Vickers Hardness Adonias Ribeiro Franco Jr.a, Giuseppe Pintaúdeb, Amilton Sinatoraa, Carlos Eduardo Pinedoc, André Paulo Tschiptschina* a Dpto. de Eng. Metalúrgica e de Materiais and Dpto. de Eng. Mecânica, Escola Politécnica, USP, Av. Prof. Mello Moraes, 2463, 05508-900 São Paulo - SP, Brazil b Dpto. Acadêmico de Mecânica, Centro Federal de Educação Tecnológica do Paraná Av. Sete de Setembro, 3165, 80230-901 Curitiba - PR, Brazil c Núcleo de Pesquisas Tecnológicas, Universidade de Mogi das Cruzes Av. Dr. Cândido Xavier de A. Souza 200, 08780-210 Mogi das Cruzes - SP, Brazil Received: January 27, 2003; Revised: January 26, 2004 Depth sensing indentation is a powerful experimental technique for determining mechanical properties of materials. In this work a computational routine was developed based on OliverPharr method for measuring a more precise values of elastic modulus using a Fischerscope H100 - depth sensing indentation apparatus, with a Vickers indenter. This computational routine aims also to measure Vickers hardness, as the equipment does not have software for this purpose. From indentation data it was possible to determine initial unloading stiffness, contact depth, projected contact area, reduced modulus, elastic modulus and Vickers hardness of materials. The validity of the routine was verified analyzing two coatings and nine bulk specimens with different elastic-plastic behaviors. It was verified that the elastic moduli determined through the software of the equipment resulted in great discrepancies when low loads were applied. A good estimate of the elastic moduli of the tested materials is given by the developed routine. For several testing loads, the diagonals determined by means of analytical procedure were compared with the same diagonals measured by image analysis. A good estimate of the Vickers hardness of the above-mentioned materials is given by the developed routine using different testing loads. Keywords: depth sensing indentation, elastic modulus, Vickers hardness, Oliver-Pharr method, ISE effect 1. Introduction Load and displacement sensing indentation technique allows determining mechanical properties at penetration depths as low as 20 nm, avoiding the substrate effect on the measurements1. The possibility to carry out tests in so small scales makes this technique one of the tools chosen to characterize mechanical properties of thin films, coatings, second phase particles and magnetic hard disk recording media2-4. A well defined indenter geometry is required to get well defined indentation impressions. A perfect tip shape is difficult to achieve. Berkovich is a three-sided pyramid, and *e-mail: aptschip@usp.br Article presented at the XV CBECIMAT, Natal - RN, November/2002 provides a sharply pointed tip, compared to the Vickers indenter, which is a four-sided pyramid and has a slight offset (0.5- µm)5,6. This is the main reason why three-sided Berkovich indenters are used in depth sensing indentation machines. However, any indenter with a sharp tip suffers from a finite but an exceptionally difficult to measure tip bluntness. Experimental procedures have been developed to correct the tip shape, of both Vickers and Berkovich indenters1, 7-10. When tests are carried out with Berkovich indenters, the registered data can be analyzed using the method pro- 484 Franco et al. posed in 1992 by Oliver and Pharr7, which has its origins in an earlier treatment by Doerner and Nix11. Nowadays, both methods are accepted for the analysis of the indentation data by the ISO/FDIS 14577-1 standard12. The Oliver-Pharr method consists in a series of loading cycles to avoid thermal drift and plastic reversion, while the Doerner-Nix method uses only a single cycle to obtain the indentation data. The Fischerscope H100 - depth sensing indentation machine uses the latter method, which is less time consuming and simpler but takes into account only a few data points, leading to greater inaccuracy. Moreover, the Fischerscope H100 uses the constant hardness calibration method8,13 to correct the indenter tip shape, which is not adequate since work-hardening may happen during the test. The Oliver-Pharr method is widely used in depth sensing indentation machines with Berkovich indenter. However, it is equally applicable to the case depth sensing indentation using a Vickers indenter, with good results as mentioned in literature8,14,15. The depth sensing indentation machine used in this work analyzes indentation data using a software based on the Doerner-Nix method, and uses an incorrect area function to describe the indenter tip shape, leading to overestimated elastic modulus and hardness values due to ISE effect - indentation size effect16. Additionally, the Fischerscope apparatus software does not give Vickers numbers, which are useful to compare with wellknown data of phases and micro-constituents given in literature. The present work aims to develop a computational routine based on the Oliver-Pharr method for measuring more precise values of elastic modulus and to obtain Vickers hardness numbers, using a Fischerscope H100 - depth sensing indentation apparatus, equipped with a Vickers diamond indenter. 2. Theoretical Aspects Depth sensing indentation technique consists of printing an impression on the material surface by applying a known load with an indenter of known geometry and subsequently analyzing the load vs. displacement data. Equations from the elastic punch theory can be used to determine the elastic modulus, E, and hardness, H, provided that the following conditions during the initial withdrawal of the indenter are ensured: • the material’s recovery follows an elastic behavior; • the contact area between the indenter and the specimen remains constant. In this case, the Sneddon’s solutions17,18 for the case of the indentation of an elastic half-space for a cylindrical punch approach to the elastic behavior. One of the Sneddon’s solutions leads to a simple relation between the load, P, and the penetration depth, h, of the form: Materials Research (1) where a is the radius of the cylinder; µ is the shear modulus; and ν is Poisson’s ratio. Knowing that the area of the contact circle projected onto the surface, Ac, is equal to πa2 and that the shear modulus is related to the elastic modulus in the following way: (2) and substituting (2) in (1) and differentiating the obtained expression with respect to h: (3) one can obtain the contact stiffness S = dP/dh. The elastic modulus, E, can be taken directly from the initial unloading slope, S, when Poisson’ ratio, ν, and contact area, Ac, are given. The latter can be measured independently as a function of contact depth, hc. As the elastic modulus of the indenter is not infinite, Eq. 3 should be written in terms of combined elastic modulus specimen/indenter (Er), which is, according to Hertz Equation: (4) where E, Ei, ν and ν i are the elastic moduli and Poisson’s ratios of the specimen and indenter, respectively. Therefore, for the indentation of a plane surface of a semi-infinite elastic solid by a rigid punch, Eq. 3 can be rewritten: or (5) The above Equation shows that, for axisymmetric indenters, the relationship between unloading stiffness, S, and contact area, Ac, does not depend upon indenter geometry. Pharr, Oliver, and Brotzen19 have shown experimentally that the analysis used for determining elastic moduli and contact areas from contact stiffness S is not limited to punch geometry. Using finite elements method, King20 has introduced to Eq. 5 a correction factor for non-axisymmetric indenters: (6) Vol. 7, No. 3, 2004 The Use of a Vickers Indenter in Depth Sensing Indentation for Measuring Elastic Modulus and Vickers Hardness where b corresponds to a correction factor related to the lack of symmetry of the indenter, which is equal to 1.0124 for Vickers indenters, and Ac is the projected contact area. Figures 1 to 3 show the main parameters used in analyzing indentation data. In Figs 1 and 2, hmax corresponds to the maximum depth, a to the half-diagonal projected on the surface, hf to the residual depth, hc to the contact depth, and hs to the deflection depth. In Fig. 3, S corresponds to the unloading stiffness for h = hmax. As the unloading from hmax to hf is elastic, one of the Sneddon’s solutions 8, for a conical punch (Figs. 1 and 2), shows that the deflection of the surface at the contact is: (7) Another Sneddon’s solution shows that, for h = hmax, the load is related to the elastic depth: (8) Substituting (7) in (8) and noting that the contact area of interest is that at peak load, P = Pmax, it follows: or 485 (10) Therefore: (11) Considering that usually the indenters are not conical, but square or triangular base pyramids (Vickers or Berkovich indenters) it must take into account that for any revolution paraboloyd (including Vickers indenters), ε is about 0.75 19. As one can see in Fig. 4a, the contact area, Ac, can be expressed as a function of the diagonal d 21: (12) Substituting this expression in Eq. 6, one obtains the diagonal as a function of the indentation parameters: (13) Eventually, the Vickers hardness numbers can be determined by the average diagonal, d, estimated from such parameters: (9) (14) To determine the contact depth from experimental data, one can note that in Fig. 1: Figure 1. Profile of the surface before and after indentation. Figure 2. Main parameters used in analyzing unloading vs. indenter depth curves. Figure 3. Schematic representation of load-displacement data for a depth sensing indentation experiment. 486 Franco et al. Materials Research Table 1 shows the main testing conditions. In all cases the indentation depth did not surpass 10% of the specimen thickness (film or substrate), avoiding any contribution from the substrate22. 3.2. Materials The materials used in this study were chosen in order to include a wide range of hardness and elastic modulus values: PVD TiN and HVOF WC-12%Co coatings, alumina, AISI D2 and AISI H13 tool steels, AISI 316 stainless steel, aluminum, gold, soda-lime glass, Au-12%Pt and Co-25%Cr odontological alloys. Table 2 presents elastic moduli for these materials reported in the literature. Bulk specimens were mechanically polished and finished until 1 µm diamond paste. The AISI H13 tool steel specimens coated with TiN and with WC - 12%Co were carefully cleaned using an ultra-sound apparatus in alcohol bath and subsequently dried in warm air. The WC-12%Co coating thickness is about 300 µm, while the TiN coating thickness is about 5 µm. (a) 3.3. Analytical procedure used by Fischerscope equipment (b) Figure 4. a) Vickers indenter: Ac(d) = d2/2; β [≡ (180 - 2α)/2]= 22°; and d = 2a. b) Under loading, the apex for Vickers indenter, 2Ψ, is equal to 148°. (15) Fischerscope procedure for measuring hardness and elastic modulus consists in estimating the initial slope from unloading data based on the Doerner-Nix analytical treatment and subsequently calculating the correct depth through a calibration curve, previously established for tip-shape indenter correction, frequently called hardness constant method8,13. Figure 5 shows the load-displacement data for alumina. Considering the unloading curve, the initial unloading slope, defined as contact stiffness, dP/dh = S, is determined taking the upper 1/3 of the unloading data and fitting it by a linear relation of the form: (16) 3. Experimental Details where 3.1. Equipment and indentation procedure A straight line is fitted to the unloading curve down to its first 1/3 region and then extrapolated to zero, determining the depth hs and stiffness S. Fitting the unloading data by a linear relation, yields S = 0.4719 mN/nm and hs = 84 nm. Because the Vickers indenter tip is not perfect, hmax is replaced by a corrected depth, hcorr, in a depth function of the form: A Fischerscope H100 - depth sensing indentation machine equipped with a Vickers diamond indenter, manufactured by Helmut Fischer GmbH, was used. Such equipment allows applying loads from 1 mN to 1000 mN and registering penetration depths as a function of applied loads. Other important testing characteristics of the equipment are: • control of loading and unloading rates (dP/dt and d √ P/dt); • long or short delay times at maximum load; • minimum step among indentations equal to 10 mm; • definition of the number and position of the indentations (mapping). . (17) where hcorr corresponds to equivalent depths from an ideal Vickers indenter, and k and n are empirical parameters. The surface area of the indenter is determined through the expression: Vol. 7, No. 3, 2004 The Use of a Vickers Indenter in Depth Sensing Indentation for Measuring Elastic Modulus and Vickers Hardness (18) In the above expression, the factor 26.43 is used instead of 24.5 because the universal hardness Hu, taken for tip shape calibration, is based on surface area of an ideal Vickers indenter. The surface area, A, together with the initial stiffness, S, is then substituted in Eq. 6, yielding a value of 325.4 GPa for reduced modulus, Er. Using Eq. 4 leads to E = 440 GPa for alumina. 3.4. Oliver-Pharr Analytical Procedure Referring to the unloading curve obtained experimentally for alumina (Fig. 5), one can note that it presents a non-linear behavior. Pharr, Oliver, and Brotzen19 showed that even metals present unloading curves with non-linear behavior and that these curves are better described by nonlinear relationships. In the Oliver-Pharr method7, the data taken from the upper portion of the unloading curve (Fig. 5) are fitted by a power-law relation of the form: 487 was verified that the contact stiffness, S, was underestimated. Probably, this effect is associated with the limitation of the Fischerscope equipment, which does not allow the inclusion of multiple cycles for minimizing the effects of thermal drift and plastic reversion. Fitting the unloading data of Fig. 5 by a power law relation, it is obtained m equal to 1.386620 and A equal to 0.053065, as one can see in Fig. 6. Rearranging Eq. 19, the residual depth can be predicted: ⇒ (20) When P = Pmax = 50 mN, the maximum penetration depth, h =hmax, is 270 nm. Substituting the values of m, A, P and h in Eq. 20, a value of 130 nm is obtained for hf. The contact stiffness, S, is obtained by means of Eq. 19 differentiating P with respect to h: (21) (19) where P is the indenter load, m and A are empirical constants determined after unloading data fitting, hf is the residual depth, and h is the elastic displacement. Although fittings corresponding to 80% of the unloading curve are recommended in the Oliver-Pharr method, the fitting used in the present work was only 52%. When trying to fit more than 52% of the data of the unloading curve, it Table 1. Indentation conditions. Number of measurements Loading time, loading rate Dwell time at maximum load (“creep”) Unloading time, unloading rate 20 12 s, d√ P/dt 20 s 20s, d√ P/dt Substituting the values of m, A and hf in Eq. 21, the contact stiffness S = 0.4966 mN/nm is obtained. The indenter-specimen contact depth, hc, can be obtained using one of Sneddon’s solutions given in Eq. 11. Substituting the values of ε, Pmax and S in the referred expression, a value of 195 nm is obtained for hc. The projected contact area, Ac, can be determined through the indenter area function, shown in Fig. 7. This calibration curve for tip shape correction was previously established by means of an iterative procedure, called the constant elastic modulus method, described by OliverPharr7. Fine fittings of the empirical constants were performed using an alternative method32 consisting of comparing the indentation diagonals measured by image analysis with that predicted through Eq. 13. Table 2. Elastic moduli reported in literature for the used materials. Material Literature modulus (GPa) WC-12%Co 495 TiN 417 (111) Al2O3 (99.8%) 375, 393 Co-25%Cr 211 H13 tool steel 210 D2 tool steel 207 316 stainless steel 192, 195 Au-12%Pt 77 Au (99.98%) 77 Aluminum 70,4 Soda-lime glass 69,9 Reference 23 24 25, 26 27 28 28 29, 30 31 31 7 7 Figure 5. Load-displacement curve obtained experimentally for an alumina (Al2O3) specimen. 488 Franco et al. Then, substituting hc in the area function of the Vickers indenter: Ac = 2.197912 µm2 Then, after substituting this value together with that for contact stiffness in Eq. 6, a value of 293.2 GPa for reduced modulus Er is determined. Hence alumina modulus can be determined by using the Hertz expression (Eq. 4): E = 380.3 GPa 4. Discussion 4.1. Elastic Modulus Materials Research yielded by the software of the equipment, together with values reported in the literature. With respect to the values determined using the proposed routine, a good agreement is observed when compared with that reported in literature. On the other hand, Table 3 shows that elastic moduli measured by the equipment are overestimated for materials with high elastic moduli. One can see in Table 3 that elastic moduli determined for alumina, through the Fischerscope routine, are greater overestimated the less is the applied load. This effect is known as indentation size effect - “ISE effect”16, indicating that the analytical procedure used by the equipment introduces inaccuracies when low loads are applied. On the other hand, the proposed routine based on Table 3 compares the results of elastic moduli measured using the developed computational routine with that Figure 6. Unloading curve fitted by a power law relation of the form P = A (h - hf)m. Figure 7. Area function, which takes into account roundness of the Vickers indenter tip, determined previously by the constant elastic modulus method and image analysis. Figure 8. Vickers impressions taken beneath the nitrided layer of a H13 tool steel specimen. Load = 50 mN. Figure 9. Vickers impression on top of aluminum. Load = 255 mN. Vol. 7, No. 3, 2004 The Use of a Vickers Indenter in Depth Sensing Indentation for Measuring Elastic Modulus and Vickers Hardness 489 Table 3. Comparison of elastic moduli measured by means of the analysis procedure based on Oliver Pharr method with those given by Fischerscope machine. Each measurement corresponds to the average of twenty curves. Material Load, mN WC-12%Co 50 100 30 50 500 1000 100 500 50 500 100 700 500 750 40 80 255 750 100 10 5 10 50 500 1000 TiN Al2O3 (99.8%) Co-25%Cr H13 tool steel D2 tool steel AISI 316 stainless steel Aluminum Au-12%Pt Au (99.98%) Soda-lime glass Experimental modulus (new routine), GPa 477 ± 49 481 ± 42 411 ± 20 380 ± 18 378 ± 06 379 ± 05 209 ± 10 208 ± 10 208 ± 07 207 ± 07 206 ± 07 207 ± 05 196 ± 08 197 ± 06 70 ± 02 69 ± 02 70 ± 02 69 ± 03 81 ± 04 77 ± 06 68 ± 02 69 ± 01 70 ± 01 70 71 the Oliver and Pharr method gives elastic moduli practical independent of the applied load, as one can see in Table 3. 4.2. Vickers Hardness Figure 8 shows a set of measurements undertaken 5, 15, 25 and 35 µm beneath the nitrided surface of a H13 tool steel specimen and Table 4 compares the diagonals measured by image analysis with those determined by means of Eq. 13, based on the calculated S and Er values. It can be seen that the values given by both methods are very close. Even in the case of large indentations (Fig. 9), the measured hardness values are in good agreement with those determined using Eq. 13, as can be seen in Table 5. Finally, Table 6 shows the Vickers hardness numbers determined through the proposed routine for different materials presenting different elastic-plastic behaviors and in some cases obtained with different applied loads. The hardness values shown in Table 6 agree fairly well with the well-known Vickers hardness numbers of the different listed materials. Therefore, artifacts in the measurements of the Vickers Experimental modulus (Fischer Software), GPa 526± 61 509± 40 510 ± 30 442 ± 19 410 ± 06 405 ± 05 230 ±10 210 ±10 220 ± 07 210 ± 05 230 ± 05 215 ± 07 182 ± 08 189 ± 6 69 ± 02 68 ± 02 68 ± 02 65 ± 03 80 ± 02 74 ± 02 79 ± 02 78 ± 01 80 ± 01 80 80 Literature modulus, GPa 490 417 (111) 375, 393 211 210 207 192, 195 70.4 77 77 69.9 hardness owing to ISE effect alone can be ruled out. The new routine adapted to Fischerscope - depth sensing indentation apparatus allows measuring Vickers hardness of tribological coatings using very low loads without the necessity to derive expressions that relate coating hardness to substrate hardness, and to deal with the composite response of film and substrate. 5. Conclusion • The proposed routine gives a better estimate of elastic moduli of materials, when compared to the values given by the equipment software. • The lesser the load, the greater are the differences between the two methods. • Very low loads can be used for determining elastic moduli of very thin layers, as the ISE effect was minimized. • The elastic contact theory equations and the OliverPharr approach offer an optimum prediction of the indentation diagonals and, consequently, of the Vickers hardness numbers of the tested materials. 490 Franco et al. Materials Research Table 4. Indentation diagonals beneath the surface in a nitrided specimen, determined by means of Eq. 13 and measured by image analysis. Maximum load: 50 mN. Distance from the nitrided surface (µm) ~5 ~15 ~25 ~35 Average diagonal using Eq. 13 (µm) 3.19 ± 0.07 3.80 ± 0.08 4.02 ± 0.06 4.04 ± 0.07 Table 5. Comparison of Vickers hardness number determined using Eq. 11 with that measured by image analysis for an aluminum specimen. Load = 255 mN. Average diagonal, d Area, AVickers Vickers number, HV (µm) (µm2) (kgf/mm2) 35.14 (Eq. 13) 665.96 38 34.65 (Image analysis) 647.46 39 Table 6. Hardness Vickers determined experimentally using different loading, for materials of different elastic-plastic behaviors. Material Hardness Vickers, kgf/mm2 5.0 µm TiN coating 2421 ± 119 Al2O3 (99.8%) 2005 ± 50 1947 ± 81 2028 ± 135 300 µm WC-12%Co coating 1399 ± 167 1455 ± 90 AISI D2 tool steel 650 ± 59 Soda-lime glass 550 ± 0 2 AISI H13 tool steel 485 ± 17 481 ± 9 Co-25%Cr 398 ± 35 403 ± 35 316 stainless steel 228 ± 7 Au (99.98%) 217 ± 11 Au-12%Pt 194 ± 14 Aluminum 35 ± 4 Load, mN 30 1000 500 50 50 100 250 50 50 500 100 500 750 10 100 750 Acknowledgments The financial support of the FAPESP, Fundação de Amparo à Pesquisa do Estado de São Paulo, is gratefully acknowledged. References 1. 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