仪器化压入Conventional Vickers and true instrumented indentation

Conventional Vickers and true instrumented indentation hardness determined by instrumented indentation tests Seung-Kyun Kang Department of Materials Science and Engineering, Seoul National University, Seoul 151-744, Korea Ju-Young Kima) Materials Science, California Institute of Technology, Pasadena, California 91106 Chan-Pyoung Park, Hyun-Uk Kim, and Dongil Kwon Department of Materials Science and Engineering, Seoul National University, Seoul 151-744, Korea (Received 16 September 2009; accepted 18 November 2009) We evaluate Vickers hardness and true instrumented indentation test (IIT) hardness of 24 metals over a wide range of mechanical properties using just IIT parameters by taking into account the real contact morphology beneath the Vickers indenter. Correlating the conventional Vickers hardness, indentation contact morphology, and IIT parameters for the 24 metals reveals relationships between contact depths and apparent material properties. We report the conventional Vickers and true IIT hardnesses measured only from IIT contact depths; these agree well with directly measured hardnesses within �6% for Vickers hardness and �10% for true IIT hardness. I. INTRODUCTION The fundamental advantage of instrumented indenta- tion testing (IIT) over conventional hardness testing is that mechanical properties such as elastic modulus,1–15 tensile properties,16–29 and hardness can be measured by analyzing the indentation force–depth curve and without observing the residual indentation marks. However, elas- toplastic deformation of materials around the indenter, i.e., plastic pileup or sink-in,30–37 makes it difficult to determine the true contact depth in the loaded state. A contact depth can be determined by taking into account the response of two major materials to an indentation, elastic deflection from the initial sample surface, and plastic pileup/sink-in around the indenter.30–37 The elas- tic deflection depth (hd) is given by the widely used Oliver and Pharr method3,12: hd ¼ ePmax S ; ð1Þ where Pmax and S are the maximum indentation force and initial unloading stiffness, respectively, and e is a geometrical constant (0.75 for a conical indenter). How- ever, the plastic pileup and sink-in cannot be expressed as an analytical equation because the plastic deformation underneath the indenter is far more complex than elastic deflection. Many studies30–37 have been performed to evaluate the contact depth, taking into account the plastic pileup/ sink-in of a sharp indenter. From extensive finite- element analysis (FEA) work on a wide range of elasto- plastic materials, Cheng and Cheng13,25,30 proposed to measure elastic modulus and hardness without determin- ing the contact area directly by correlating these proper- ties and the indentation work ratio. Alcala et al.,35 using FEA, suggested a relationship between strain-hardening exponent n and pileup/sink-in height. Cheng and Cheng also suggested a correction parameter f, the ratio of con- tact depth (hc) to maximum indentation depth (hmax), and showed that f is given by the product of a function of the strain-hardening exponent n and the ratio of yield strength to reduced elastic modulus (sys/Er). 31,32 Choi et al.36 proposed a modified correction parameter f in- cluding indenter tip bluntness (△hb) as: f ¼ hc hmax þ Dhb ¼ 1:2445 1� 0:6nð Þ 1� 7:2 sys Er � � : ð2Þ However, previous research on this topic has all or some of the following limitations that require further studies: (i) The data were obtained by FEA simulation. Even though FEA is a well-established tool, more real data are certain to be obtained by experimental results. (ii) In the FEA simulations, conical indenters were used to simulate sharp indenters such as Berkovich and Vickers indenters. While this difference does not matter for true IIT hardness, it is critical when conventional Vickers hardness is measured by IIT, since it depends only on contact depth at corners. a)Address all correspondence to this author. e-mail: juyoung1@snu.ac.kr DOI: 10.1557/JMR.2010.0045 J. Mater. Res., Vol. 25, No. 2, Feb 2010 © 2010 Materials Research Society 337 (iii) To use the proposed equations, tensile properties such as strain-hardening exponent and yield strength are required. Since the proposed equations cannot be applied only with IIT for unknown materials, their applications are limited. To explain the three different hardness values and corre- sponding contact depths in IIT using a Vickers indenter, we show in Fig. 1 a schematic of contact morphology for pileup around a Vickers indenter. First, conventional IIT hardness (HO-P) is given by themaximum indentation force divided by the projected contact area corresponding to the contact depth (hc,O-P) in the Oliver–Pharr method 3,12,38: HO-P ¼ Pmax 24:5� hc;O-P � �2 ; ð3Þ which subtracts the elastic deflection depth hd from the maximum indentation depth hmax. This hardness is measured only with IIT parameters such as maximum applied force and unloading stiffness, as shown in Eq. (1), and thus is widely used although it does not take into account plastic pileup/sink-in. Second, true IIT hardness (Htrue) is given by the maximum force divided by the true projected contact area hproc � � : Htrue ¼ Pmax 24:5� hprocð Þ2 ; ð4Þ considering plastic pileup/sink-in. Since plastic pileup/ sink-in is more constrained at the corners of the Vickers indenter, the projected contact depth representing the projected contact area may lie between the contact depths at the center and at a corner. Third, the conven- tional Vickers hardness is given by the maximum force divided by the four-sided pyramidal contact area, which is evaluated from the diagonals of the residual indenta- tion marks after unloading. If the recovery in the in- plane direction during unloading is negligible,35 the diagonals evaluated from the contact depth at corners in the loaded state hVc � � should be the same as those of the residual indentation marks. Vickers hardness is repre- sented by contact depth at corners as HV ¼ Pmax 26:43� hVc � �2 : ð5Þ Vickers hardness, the result of conventional hardness testing using a self-similar indenter, is widely used and has an extensive number of databases. Nevertheless, the evaluation algorithms for Vickers hardness using IIT are insufficient for its industrial uses. The problem arises from the different definitions of contact area in Vickers hardness and IIT hardness. Vickers hardness calculates the contact area from the corner-to-corner diagonal, ig- noring the difference in pileup/sink-in around the side of indenter, which must be contained to derive the IIT hardness. Thus, algorithms for evaluating not only true IIT hardness but also Vickers hardness that take into account the real contact depth through IIT will be simple and useful techniques in hardness testing. In this study, we propose methods for measuring Vickers hardness and true IIT hardness that use IIT alone by taking into account the pileup height at the corner and the representative pileup height, respectively. To this end, we correlate the pileup heights with experimentally measured IIT parameters for 24 metals over a wide range of mechanical properties. We show that the true IIT hard- ness and conventional Vickers hardness measured by the proposed methods for the 24 metals agree well with the true values within �6% for Vickers hardness and �10% for true IIT hardness. II. EXPERIMENTS Twenty-four metal samples with a wide range of me- chanical properties—Al alloys, Mg alloys, Cu alloys, Ti alloys, Ni alloys, carbon steels, API X-grade steels, fer- rite-based stainless steels, and austenite-based stainless steels—were prepared for indentation tests and uniaxial tensile tests. For the indentation tests, one side of the samples was finely polished with 1 mm alumina powder. IITs were conducted using the AIS 3000 instrumented indentation system (Frontics Inc., Seoul, Korea) with force resolution of 55 mN and displacement resolution of 100 nm and a Vickers indenter. The IITs wereFIG. 1. Diagonal area and projected area at pileup and sink-in. S-K. Kang et al.: Conventional Vickers and true instrumented indentation hardness determined by instrumented indentation tests J. Mater. Res., Vol. 25, No. 2, Feb 2010338 performed at constant displacement rate 0.3 mm/min with maximum indentation depth of 80 mm. After inden- tation, the residual indentation marks were observed by optical microscopy to evaluate the conventional Vickers hardness. The relationship between the diagonal and con- tact depth at the corner of the residual indent is given by hVc ¼ d 2 ffiffiffi 2 p tan yV ; ð6Þ where yV is the half angle of the Vickers indenter, 68�. The ratio of contact depth at the corner to maximum indentation depth is defined as fV ¼ h V c hmax : ð7Þ Because the elastic recovery during unloading hap- pens mainly along loading direction and negligible in-plane direction,35 the projected contact area at the maximum indentation depth can be measured directly from the area of the residual indentation mark. The pro- jected area Aproc was measured using an image analyzer (NIS Elements, Nikon, Tokyo, Japan), from which the projected contact depth hproc was calculated by hproc ¼ ffiffiffiffiffiffiffiffi Aproc p 2 tan yV ; ð8Þ and the ratio of the projected contact depth to maximum indentation depth is defined as fpro ¼ h pro c hmax : ð9Þ Figure 1 shows the contact morphology of Vickers indentation for material pileup around the indenter. The contact depth at the corner hVc determines conventional Vickers hardness, while the projected contact depth hproc is the parameter determining the true IIT hardness. Uniaxial tensile tests were carried out using an Instron 5582 (Instron Inc., Grove City, PA) at cross-head speed 1 mm/min; the gauge length and diameter of the cylin- drical specimens were 25 and 6 mm, respectively, in accordance with the ASTM standard.39 Elastic moduli of the samples were measured by an ultrasonic pulse- echo technique using a two-channel digital real-time oscilloscope. III. RESULTS AND DISCUSSION To explore the relationship between plastic pileup underneath the Vickers indenter and material plastic properties, we show in Fig. 2 the relationship of the contact depth function fV versus the strain-hardening exponent n and the ratio of elastic modulus to yield strength, E/sys, corresponding to the inverse of the yield strain, both of which have been widely used to describe plastic indentation pileup.31,32,35,36 For the metal sam- ples, fV shows good linearity with E/sys, while it has much less relation with n. If we assume a Tabor relation- ship between yield strength and hardness (H = Csys, where C is plastic constraint factor of 3),9 fV can also be related to E/H, as shown in Fig. 3. It is notable that, in Fig. 3, hardness H and elastic modulus E are determined by optically measured projected area of residual impres- sion and ultrasonic pulse-echo technique, respectively, not by the E/sy data in Fig. 2. The ratio of hardness FIG. 2. Relation between contact depth function and tensile proper- ties. (a) Hardening exponent and (b) ratio of elastic modulus and yield strength. FIG. 3. Relation between contact depth function and ratio of elastic modulus and hardness. S-K. Kang et al.: Conventional Vickers and true instrumented indentation hardness determined by instrumented indentation tests J. Mater. Res., Vol. 25, No. 2, Feb 2010 339 and elastic modulus can be expressed as the ratio of irreversible work (Wtotal � We) to total work (Wtotal) during indentation: 1� k H E ¼ Wtotal �We Wtotal ; ð10Þ where We is the work recovered during unloading and k is a constant commonly taken as 5.36 By using linear indentation loading and unloading curves, Wtotal and We are measured approximately by k H E ¼ We Wtotal � hmax � hf hmax : ð11Þ The ratio of elastic modulus to yield strength can thus be represented using Eqs. (10) and (11) by the maximum indentation depth hmax and final indentation depth hf. 12,36 Figure 4 shows the relationship between the contact depth function (fV) and contact depth ratio hmax/ (hmax � hf), which shows good linearity; we obtain the linear relation fV ¼ 9:90� 10�3 hmax hmax � hf þ 1:00 ; ð12Þ which can be written as hVc � hmax hmax ¼ 9:90� 10�3 hmax � hf hmax � ��1 ; ð13Þ where (hVc � hmax)/hmax is the normalized pileup from the reference plane, and (hmax � hf)/hmax is the normalized recovery depth. For a geometrically self- similar sharp indenter such as the Vickers indenter, the representative total strain is determined regardless of indentation depth,9,13 so the plastic pileup is most likely to be related to the ratio of elastically recovered depth to maximum indentation depth. Table I presents FIG. 4. Linear relation of contact depth function at corner and nor- malized recovery depth. TABLE I. Evaluation of contact depth at the corner and Vickers hardness using the contact depth function. Materials E/sys n hVc (mm) hVc Eq. (12) (mm) Error (%) Vickers hardness (HV) Vickers hardness Eq. (12) (HV) Error (%) Al alloy Al6061 323.86 0.063 84.95 87.81 3.4 117.38 109.85 �6.4 Al7075 136.19 0.080 84.77 83.76 �1.2 173.47 177.71 2.4 Mg alloy AZ61 272.73 0.300 85.52 86.53 1.2 44.94 43.90 �2.3 AZ910 229.59 0.396 83.49 85.48 2.4 58.44 55.75 �4.6 Cu alloy C1010 392.46 0.029 88.26 91.24 3.4 80.30 75.14 �6.4 C5101 551.17 0.328 86.52 88.85 2.7 85.66 81.22 �5.2 C62400 307.01 0.259 83.61 83.48 �0.2 212.68 213.36 0.3 Ti alloy Ti–10V– 2Fe–3Al 95.36 0.096 79.85 82.15 2.9 360.48 340.65 �5.5 Ti–7Al–4Mo 128.18 0.055 80.32 82.70 3.0 341.86 322.40 �5.7 Ni alloy Inconel 600 443.53 0.265 83.54 85.86 2.8 228.10 215.96 �5.3 Carbon steel S45C 584.15 0.258 90.71 88.67 �2.2 181.61 190.06 4.7 SCM21 714.93 0.222 90.03 89.40 �0.7 160.11 162.36 1.4 SCM4 324.59 0.130 83.78 83.81 0.0 285.95 285.78 �0.1 SKD61 574.92 0.241 89.62 87.55 �2.3 189.86 198.96 4.8 SKS3 481.92 0.218 89.78 88.38 �1.6 182.22 188.03 3.2 SUJ2 536.61 0.240 88.71 87.43 �1.4 195.36 201.12 2.9 API steel X100 350.84 0.141 87.85 85.30 �2.9 240.19 254.77 6.1 X70 348.16 0.117 87.37 86.18 �1.4 216.69 222.68 2.8 Ferrite base stainless steel SUS303F 560.65 0.334 87.40 87.37 0.0 175.06 175.18 0.1 SUS310S 720.29 0.294 93.86 93.89 0.0 129.46 129.36 �0.1 SUS316L 651.07 0.309 89.77 91.18 1.6 158.28 153.40 �3.1 Austenite base stainless steel SUS403 628.59 0.212 90.91 91.38 0.5 165.73 164.02 �1.0 SUS410 585.36 0.179 90.26 90.31 0.1 166.53 166.37 �0.1 SUS420J2 530.18 0.207 88.61 87.29 �1.5 204.08 210.26 3.0 S-K. Kang et al.: Conventional Vickers and true instrumented indentation hardness determined by instrumented indentation tests J. Mater. Res., Vol. 25, No. 2, Feb 2010340 the corner contact depths and Vickers hardnesses directly measured from the profile of residual indentation marks and evaluated only from IIT parameters using Eqs. (6) and (12), respectively. The errors in contact depth and hardness are below �4% and �7%, respectively. True IIT hardness is given by the maximum applied load divided by the projected area in the loaded state, which can be work for elastoplastic deformation per unit volume and mean pressure. We directly compared the contact depth function for projected area fpro to hmax/(hmax � hf) and found a linear relation for the contact depth function at the corner; Fig. 5 shows the relation as fpro ¼ 1:06� 10�2 hmax hmax � hf þ 1:00 : ð14Þ Table II presents true IIT hardnesses directly measured from the profiles of residual indentation marks and evaluated only from IIT parameters using Eq. (14). These results imply that the true projected contact depth in the loaded state and true IIT hardness can be measured without additional observation of the residual impression and material properties. Figure 6 shows the relationship between two different contact areas: diagonal contact area (Adia) and projected contact area A pro c � � (see Fig. 7). Aproc is �2% greater than Adiac : Aproc ¼ 1:02Adiac : ð15Þ Thus, when considering pileup dependent on indenter geometry as in Fig. 1, i.e., more pileup in the center and minimum pileup at each Vickers indenter corner, this result implies that the true contact area, which is the projected area at contact depth taking into account aver- age pileup along the contact morphology, is slightly great than projected area at corner contact depth, hVc . FIG. 5. Linear relation of contact depth function for projected area and normalized recovery depth. TABLE II. Evaluation of projected contact depth and true IIT hardness using the contact depth function. Materials hproc (mm) hproc Eq. (14) (mm) Error (%) True IIT hardness (GPa) True IIT hardness Eq. (14) (GPa) Error (%) Al alloy Al6061 88.63 88.40 �0.3 1.14 1.15 0.5 Al7075 86.75 84.08 �3.1 1.75 1.86 6.4 Mg alloy AZ61 83.49 87.01 4.2 0.50 0.46 �7.9 AZ910 81.82 85.89 5.0 0.64 0.58 �9.3 Cu alloy C1010 91.45 92.06 0.7 0.79 0.78 �1.3 C5101 84.78 89.50 5.6 0.94 0.85 �10.3 C62400 82.37 83.80 1.7 2.32 2.24 �3.4 Ti alloy Ti–10V–2Fe–3Al 79.86 82.40 3.2 3.81 3.58 �6.1 Ti–7Al–4Mo 80.43 82.99 3.2 3.60 3.38 �6.1 Ni alloy Inconel600 84.10 86.36 2.7 2.38 2.26 �5.2 Carbon steel S45C 91.49 89.34 �2.3 1.89 1.98 4.9 SCM21 90.53 90.11 �0.5 1.67 1.69 0.9 SCM4 85.21 84.18 �1.2 2.92 2.99 2.5 SKD61 90.02 88.14 �2.1 1.99 2.08 4.3 SKS3 90.02 89.03 �1.1 1.92 1.96 2.2 SUJ2 88.57 88.02 �0.6 2.07 2.10 1.3 API steel X100 90.02 85.76 �4.7 2.42 2.66 10.2 X70 90.52 86.69 �4.2 2.13 2.33 9.0 Ferrite base stainless steel SUS303F 87.82 87.93 0.1 1.83 1.83 �0.3 SUS310S 95.21 94.91 �0.3 1.33 1.34 0.6 SUS316L 90.36 92.01 1.8 1.65 1.59 �3.5 Austenite base stainless steel SUS403 92.57 92.23 �0.4 1.69 1.70 0.7 SUS410 91.98 91.08 �1.0 1.70 1.73 2.0 SUS420J2 89.50 87.87 �1.8 2.11 2.19 3.7 S-K. Kang et al.: Conventional Vickers and true instrumented indentation hardness determined by instrumented indentation tests J. Mater. Res., Vol. 25, No. 2, Feb 2010 341 IV. CONCLUSIONS On the basis of experiments on the 24 metals, we found relations between the contact depth functions and indentation parameters that let us determine the Vickers hardness and true IIT hardness only with IIT. Using Eqs. (12) and (14) with only IIT parameters enabled us to obtain Vickers hardness and true IIT hardness within �6% and �10% error, respectively, of the directly measured values. Notable findings of this study are: (1) The contact depth function correcting the pileup at corner of Vickers indenter is directly related to the ratio of yield strength to elastic modulus, rather than to the strain-hardening exponent. (2) The ratio of yield strength to elastic modulus is represented by the ratio of recovered depth (hmax � hf) to maximum indentation depth (hmax), which is propor- tional to the inverse of the normalized pileup depth. (3) Experimentally determined relationship between the contact depth functions ( fV, fpro) and the (hmax � hf)/ hmax made it possible to evaluate the contact area and hardness using only IIT parameters. (4) The projected area Apro has a linear relationship with the diagonal area Adia as in Eq. (15), suggesting that the average pileup height is slightly greater than that at the corners of the Vickers indenter. ACKNOWLEDGMENTS This research was supported in part by the Seoul R&BD Program (Grant No. TR080564) of the Seoul Development Institute, Korea, and in part by the Improvement of Standardization Technology Program (Grant No. B0010740) of the Ministry of Knowledge Economy, Korea. REFERENCES 1. S.I. Bulychev, V.P. Alekhin, M.K. Shorshorov, A.P. Ternovskii, and G.D. 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