- Email: [email protected]
Determination of an effective zero-point and extraction of indentation stress–strain curves without the continuous stiffness measurement signal
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Scripta Materialia 60 (2009) 439–442 www.elsevier.com/locate/scriptamat
Determination of an effective zero-point and extraction of indentation stress–strain curves without the continuous stiffness measurement signal Siddhartha Pathak, Joshua Shaffer and Surya R. Kalidindi* Department of Materials Science and Engineering, Drexel University, Philadelphia 19104, USA Received 5 September 2008; revised 17 November 2008; accepted 17 November 2008 Available online 3 December 2008
A novel approach is presented for converting spherical nanoindentation load–displacement data into indentation stress–strain curves, without the need for continuous stiffness measurement (CSM) – an option available only on a limited number of machines. This new method entails novel approaches for estimating (i) the effective zero-point and (ii) the effective contact radius from the raw load–displacement data. The results from the new analysis methods are shown to be in good agreement with the results obtained using CSM. Ó 2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: B. Nanoindentation; A. Aluminum tungsten; B. Elastic behavior; B. Yield phenomena; B. Plastic deformation
In recent work [1], we outlined new data analysis methods for converting raw load–displacement–continuous stiffness measurement (CSM) data obtained from spherical nanoindentation into much more meaningful indentation stress–strain curves. When compared to other currently used data analysis methods [2,3], these indentation stress–strain curves have been demonstrated to be more successful in correlating the elastic moduli measured in loading and unloading segments [1,4,5], measuring accurately the changes in the indentation yield points in annealed and deformed samples [6], and identifying and explaining several of the surface preparation artifacts typically encountered in nanoindentation measurements [5]. Extraction of reliable indentation stress–strain curves from the measured load–displacement data is essentially a two-step process (see Ref. [1] for more details). The first step in this process is an accurate estimation of the point of effective initial contact in the given data set, i.e. a clear identification of a zero-point that makes the measurements in the initial elastic loading segment consistent with the predictions of Hertz’s theory [7,8]. In our previous work [1], for spherical nanoindentation we expressed this relationship as
* Corresponding author; e-mail: [email protected]
e P 3 P 3P S¼ ¼ ; 2he 2 ~he h
ð1Þ
where Pe , ~he and S are, respectively, the measured load signal, the measured displacement signal and the CSM signal in the initial elastic loading segment from the machine, and P* and h* denote the values of the load and displacement signals at the point of effective initial contact. A linear regression analysis was used to establish the point of effective initial contact (P* and h*) in the indentation experiment. In the second step of the data analysis method, Hertz’s theory [7,8] was recast for frictionless, elastic, spherical indentation as (e.g. [1,4]): rind ¼ Eeff eind ; eind ¼
rind ¼
4 he he ; 3p a 2:4a
P ; pa2
a¼
1 1 m2s 1 m2i ¼ þ ; Es Ei Eeff
S ; 2Eeff
1 1 1 ¼ þ ; Reff Ri Rs
ð2Þ
where rind and eind are the indentation stress and the indentation strain, a is the radius of the contact boundary at the indentation load P, he is the elastic indentation depth, S is the elastic stiffness described earlier, Reff and Eeff are the effective radius and the effective modulus of
1359-6462/$ - see front matter Ó 2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2008.11.028
440
S. Pathak et al. / Scripta Materialia 60 (2009) 439–442 1000
Load, mN
a
R ht hr
800
he
hc Complete unload Max load
600
3 a 4
400
Idealized primary zone of indentation
A
2.4a
200
-200
he= ht – hr he =kP2/3
hr
0
B
ht 0
200
400 600 800 1000 1200 Displacement, nm
Figure 1. (A) Schematic of a spherical indentation showing the primary zone of indentation. (B) Schematic of the procedure involved in establishing the contact radius, a, at different load levels. The unloading segment at each of these load levels is fitted to Eq. (4) to estimate hr and k. Reff and a can then be calculated using Eqs. (3) and (5).
the indenter and the specimen system, m and E are the Poisson’s ratio and the Young’s modulus, and the subscripts s and i refer to the specimen and the indenter, respectively (see also the schematic in Fig. 1). Note that both the estimation of the zero point (Eq. (1)) and the extraction of the indentation stress–strain curves (Eq. (2)) require the use of the CSM signal (S ¼ dP =dhe ; also referred to as the CSM mode in the MTS XPÒ nanoindentation machine). However, the CSM module is an optional accessory on the MTS/Agilent1 nanoindenter machine, and hence is available only on some machines. Moreover, other nanoindentation machines (manufactured by companies other than MTS/Agilent) do not presently offer CSM as an option. Thus, there is a critical need to develop data analysis methods for extracting indentation stress–strain curves without CSM. In this paper, we formulate and validate new data analysis methods for the extraction of the indentation stress–strain curves without using CSM. This is achieved in a two-step process: (i) establishing the effective zeropoint and (ii) estimating the contact radius, a, which is required for the computation of indentation stress and strain. The indentation stress–strain curves obtained using this non-CSM approach is then compared to our previously reported [1,5,6] indentation stress–strain curves obtained using CSM. As mentioned earlier, determining of the point of initial contact constitutes the first step in the analyses of the raw data obtained from spherical nanoindentation. This problem, discussed in detail in our previous work [1,4,5] and in other literature [9–13], assumes great importance in computing the values of indentation stress and indentation strain. The error in the zero-point arising from the default procedures of the nanoindentation machines have been reported to be anywhere from ±2 nm [3] to ±30 nm [10]. Since the initial elastic segment corresponds to only a few (10–20 nm) nanometers of indentation depth, errors of this magnitude pose a major hurdle in extracting reliable values of elastic and yield properties from this data.
1
At the time of writing of this manuscript Agilent Technologies had acquired the Nano Instruments business unit of MTS Systems Corporation.
In this work, we have established the zero-load and the zero-displacement by fitting the recorded initial elastic load ð Pe Þ and displacement ð~he Þ signals to the predictions of Hertz theory. According to Hertz’s theory [7,8], the load and displacement during elastic loading in a spherical nanoindentation experiment should be related by: " # 2=3 3 1 1 ~ e pffiffiffiffiffiffiffiffi : he h ¼ k P P ; k¼ ð3Þ 4 Eeff Reff Note that k in Eq. (3) is a constant for the entire initial elastic loading segment. The values of P* and h* that yield the lowest residual error in the least-squares fit of the initial elastic loading segment to Eq. (3) were chosen to correspond to the effective zero-point. This approach ensures that the corrected data set would be highly consistent with Hertz’s theory. Note that, just as in Eq. (1), Eq. (3) also identifies an ‘‘effective” point of initial contact which allows us to de-emphasize any artifacts created at the initial contact by the unavoidable surface conditions (e.g. surface roughness or the presence of an oxide layer) [5]. For this study, samples of electropolished aluminum and tungsten were indented with two differently sized indenter tips (1.4 and 20 lm radii) in a MTS XPÒ machine. The low degree of elastic anisotropy in these two materials as well as the strong variation in their respective mechanical properties (aluminum exhibits a low modulus and a low yield strength while tungsten exhibits a high modulus and a high yield strength) makes them an ideal choice for validating the new data analysis procedures presented in this paper. The sample surfaces were prepared by electropolishing in order to avoid the undesirable effects caused by other preparation techniques (such as mechanical hardening on the sample surface due to polishing and grinding). Figure 2 shows an example of a comparison between the zero-points (values of P* and h*) for tungsten as determined by the two methods: (i) Eq. (1) using CSM, and (ii) Eq. (3) without using the CSM signal. As seen from Figure 2, both the CSM and the non-CSM data analysis methods for tungsten yield nearly identical values of P* and h*. This point is further emphasized in Table 1, where the values of P* and h* are given for three tests each in both aluminum and tungsten for each indenter size. It is clearly seen from these results that the zeropoints established by both methods are in very good agreement with each other. The Young’s moduli for the tungsten and aluminum samples as tested were determined to be 418 ± 4 and 71 ± 2 GPa (average and standard deviation from analyses of the six measurements shown in Table 1), respectively, using the non-CSM data analysis procedures described above. These values are in good agreement with the values reported in literature from other testing methods [2,14] and also agree very well with our previously reported values on the same materials using CSM (414 ± 6 GPa for tungsten and 67 ± 2 GPa for aluminum [1]). It is worth reiterating here that we are able to analyze the relatively small initial elastic loading segments with remarkable accuracy because the use of Eq. (1) or (3) to establish the effective zero-point does not warrant prior knowledge of the values of Reff and Eeff. This is
S. Pathak et al. / Scripta Materialia 60 (2009) 439–442
log (residual error)
W
m ,n h*
P*, mN
Zero poihnt without using CSM, Eq. (3) Zero point using the CSM [1]
Figure 2. Comparison between the effective zero-points identified by the CSM technique [1] and the non-CSM technique (Eq. (3)) for tungsten using a 1.4 lm spherical indenter (corresponding to Test #2 in Table 1). The residual error is obtained from regression analyses as measure of the degree of fit to Eq. (3).
especially beneficial in establishing a reliable value of Ri (note that in the initial elastic loading segment Reff = Ri). Using measurements on samples of known Young’s moduli (e.g. Si standards), we estimated that the indenter radii for the two indenters used in this study to be 1.4 and 20 lm respectively, even though the manufacturer had claimed otherwise. Both these estimates were subsequently confirmed by scanning electron microscopy. The second step in the extraction of indentation stress–strain curves is an accurate estimation of the contact radius, a, which evolves continuously during the indentation experiment. Majority of the methods used for estimation of a in the literature [11,15–17] are motivated by the spherical geometry of the indenter, where a is calculated directly from the indentation depth and the radius of the indenter, Ri. However, the contact radius is defined through the contact height, hc, and not ht, as shown in Figure 1. Although a number of different approaches (including area functions [2,3]) have been explored for relating hc and ht, they have not yet produced consistent and reliable results for a broad range of samples [18]. Table 1. Comparison between the effective zero-point as identified by the CSM technique [1] and the non-CSM technique (Eq. (3)) for tungsten and aluminum using two different indenter radii (1.4 and 20 lm).
1.4 lm indenter
A1
W
20 lm indenter
A1
W
Using CSM h* (nm) P* (mN)
Without CSM h* (nm) P* (mN)
Test Test Test Test Test Test
#1 #2 #3 #1 #2 #3
10.29 5.25 10.21 15.77 27.195 10.29
0.039 0.007 0.041 0.146 0.059 0.10
10.17 3.75 9.36 14.95 27.63 7.71
0.041 0.003 0.037 0.139 0.084 0.063
Test Test Test Test Test Test
#1 #2 #3 #1 #2 #3
4.35 6.08 4.34 10.95 9.29 4.07
0.028 0.027 0.027 0.135 0.295 0.209
4.19 5.91 4.65 10.73 8.31 3.77
0.028 0.029 0.029 0.141 0.248 0.181
441
Alternatively, one can impose an elastic unloading segment2 at any point of interest and analyze it using Hertz’s theory in order to estimate the contact radius. Indeed, this is exactly what is done in estimating the contact radius using CSM [2,3,19]. In the analyses of an unloading segment after some imposed plastic deformation in the loading segment, Reff can no longer be assumed to be equal to Ri. As we shall see later, Reff changes quite dramatically with any imposed plastic deformation by the indenter. In this work, we have established the value of Reff by imposing unloading segments at several selected locations on the measured load–displacement data, and analyzing them carefully using Hertz’s theory, as illustrated in Figure 1. Each unloading segment is fit to the expected Hertz’s relationship between the total indentation depth, ht, and the indentation load, P, which may be conveniently expressed as he ¼ ht hr ¼ kP 2=3 :
ð4Þ
In Eq. (4) he is the elastic indentation depth, hr is the residual (or plastic) indentation depth, and k is a factor containing Eeff and Reff as defined in Eq. (3). Once the value of Eeff is established from the initial loading curve (or if the Young’s modulus of the sample material is already known), a regression analysis on the unloading segment can determine both hr and Reff. The value of the contact radius, a, at any point in the unloading segment can then be determined from: pffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a ¼ Reff he ¼ Reff ðht hr Þ: ð5Þ It should be noted that this relationship between Reff and a is implicit in Hertz’s theory (for the quadratic contacting surfaces). Applying this equation to the data point just before the initiation of the unloading segment provides the value of the contact radius at that point in the original loading segment. It is important to note here that each unloading curve will produce only one point on the indentation stress– strain curve. This necessitates a large number of loading–unloading segments on the sample in order to be able to obtain a complete description of the indentation stress–strain curve for a given sample. Figure 3 shows the values of Reff estimated at different indentation depths, using the data analysis procedure described earlier, in the measurements obtained using a 20 lm spherical indenter on samples of electropolished aluminum and tungsten. As seen from this figure, the values for Reff change dramatically with imposed plastic deformation by the indenter. In fact, the changes in the effective radius are most dramatic in the initial stages of plastic deformation under the indenter. It is also seen that Reff takes on much higher values for the softer aluminum samples compared to the harder tungsten samples (at the same depth of indentation). Indeed, it was observed that Reff is not just a function of indentation depth (or load) alone, but varies substantially with the details of the elastic 2
The unloading segment of an indentation experiment is generally assumed to be elastic because reloading at the same spot often reproduces this portion of the load–displacement curve.
442
S. Pathak et al. / Scripta Materialia 60 (2009) 439–442 120 100 80 60 40 20 0
2000
Reff, µm
1500
Tungsten
0
500
1000
1500
1000 Aluminum Tungsten
500
0 0
500
1000
1500
Displacement, nm
Figure 3. Values of Reff computed at different indentation depths which differ significantly between the softer aluminum and the harder tungsten sample. The values for Reff are also significantly larger than (Ri = 20 lm) for both samples. Inset: expanded view for the tungsten sample.
and plastic properties of the sample. Also, the values of Reff measured in Figure 3 are substantially larger than the indenter radius (Ri = 20 lm) indicating that the approximation Reff Ri [15,16,20–22] is not valid in the post-elastic regions of the samples. Figure 4A and B shows comparisons between the indentation stress–strain curves obtained using both the CSM method described in our earlier work [1] and the non-CSM method described in this work on aluminum and tungsten samples for both 1.4 and 20 lm radii indenter sizes. It is seen that the indentation stress–strain curves from the CSM and the non-CSM methods agree well with each other for both indenters. The indentation stress–strain curve produced using the non-CSM method is able to capture all the major features of the stress–strain curves, including the linear elastic regime, the plastic yield point and the post-yield strain-hardening. As expected, the indentation stress–strain curve
A
B
Figure 4. Comparison between the indentation stress–strain curves obtained using the CSM method [1] and the non-CSM method on (A) aluminum and (B) tungsten samples with the 1.4 and 20 lm radii indenters. The insets show expanded views of the indentation stress– strain curves for the larger 20 lm indenter.
for tungsten shows more strain hardening compared to that for the aluminum sample. These results also demonstrate the feasibility of capturing the details of the popin phenomenon [5,23] occurring in the smaller 1.4 lm indenter with the non-CSM method. The good agreement between indentation stress–strain curves measured by these two very different indenter sizes for two very different materials provides additional validation for the data analyses procedures used in this work. These findings indicate that in spite of the non-continuous nature of the non-CSM calculations, this method can be successfully used to characterize the mechanical response of the material during spherical nanoindentation. This work was partially supported by the National Science Foundation Grant No. CMMI-0654179. The MTS XPÒ System used in this study is maintained and operated by the Centralized Research Facilities in the College of Engineering at Drexel University. S.P. also wishes to acknowledge the support from the Sigma Xi Grants-inAid of Research (GIAR) program for this work. [1] S.R. Kalidindi, S. Pathak, Acta Materialia 56 (2008) 3523. [2] W.C. Oliver, G.M. Pharr, Journal of Materials Research 7 (1992) 1564. [3] W.C. Oliver, G.M. Pharr, Journal of Materials Research 19 (2004) 3. [4] S. Pathak, S.R. Kalidindi, C. Klemenz, N. Orlovskaya, Journal of the European Ceramic Society 28 (2008) 2213. [5] S. Pathak, D. Stojakovic, R. Doherty, S.R. Kalidindi, Journal of Materials Research, 2009, in press. [6] S. Pathak, D. Stojakovic, S. R. Kalidindi, 2008, unpublished work. [7] H. Hertz, Miscellaneous Papers, Macmillan, New York, 1896. [8] A.E.H. Love, Journal of Mathematics 10 (1939) 161. [9] T. Chudoba, N. Schwarzer, F. Richter, Surface and Coatings Technology 127 (2000) 9. [10] J. Deuschle, S. Enders, E. Arzt, Journal of Materials Research 22 (2007) 3107. [11] A.C. Fischer-Cripps, Surface and Coatings Technology 200 (2006) 4153. [12] J. Mencik, M.V. Swain, Journal of Materials Research 10 (1995) 1491. [13] C. Ullner, Measurement 27 (2000) 43. [14] G. Simmons, H. Wang, Single Crystal Elastic Constants and Calculated Aggregate Properties, second ed., MIT Press, Cambridge, MA, 1971. [15] S. Basu, A. Moseson, M.W. Barsoum, Journal of Materials Research 21 (2006) 2628. [16] J.S. Field, M.V. Swain, Journal of Materials Research 8 (1993) 297. [17] E.G. Herbert, G.M. Pharr, W.C. Oliver, B.N. Lucas, J.L. Hay, Thin Solid Films 398–399 (2001) 331. [18] M. Radovic, E. Lara-Curzio, L. Riester, Materials Science and Engineering A A368 (2004) 56. [19] X.D. Li, B. Bhushan, Materials Characterization 48 (2002) 11. [20] J.S. Field, M.V. Swain, Journal of Materials Research 10 (1995) 101. [21] A. Murugaiah, M.W. Barsoum, S.R. Kalidindi, T. Zhen, Journal of Materials Research 19 (2004) 1139. [22] M.V. Swain, Materials Science and Engineering A 253 (1998) 160. [23] C.A. Schuh, Materials Today 9 (2006) 32.
Scripta Materialia 60 (2009) 439–442 www.elsevier.com/locate/scriptamat
Determination of an effective zero-point and extraction of indentation stress–strain curves without the continuous stiffness measurement signal Siddhartha Pathak, Joshua Shaffer and Surya R. Kalidindi* Department of Materials Science and Engineering, Drexel University, Philadelphia 19104, USA Received 5 September 2008; revised 17 November 2008; accepted 17 November 2008 Available online 3 December 2008
A novel approach is presented for converting spherical nanoindentation load–displacement data into indentation stress–strain curves, without the need for continuous stiffness measurement (CSM) – an option available only on a limited number of machines. This new method entails novel approaches for estimating (i) the effective zero-point and (ii) the effective contact radius from the raw load–displacement data. The results from the new analysis methods are shown to be in good agreement with the results obtained using CSM. Ó 2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: B. Nanoindentation; A. Aluminum tungsten; B. Elastic behavior; B. Yield phenomena; B. Plastic deformation
In recent work [1], we outlined new data analysis methods for converting raw load–displacement–continuous stiffness measurement (CSM) data obtained from spherical nanoindentation into much more meaningful indentation stress–strain curves. When compared to other currently used data analysis methods [2,3], these indentation stress–strain curves have been demonstrated to be more successful in correlating the elastic moduli measured in loading and unloading segments [1,4,5], measuring accurately the changes in the indentation yield points in annealed and deformed samples [6], and identifying and explaining several of the surface preparation artifacts typically encountered in nanoindentation measurements [5]. Extraction of reliable indentation stress–strain curves from the measured load–displacement data is essentially a two-step process (see Ref. [1] for more details). The first step in this process is an accurate estimation of the point of effective initial contact in the given data set, i.e. a clear identification of a zero-point that makes the measurements in the initial elastic loading segment consistent with the predictions of Hertz’s theory [7,8]. In our previous work [1], for spherical nanoindentation we expressed this relationship as
* Corresponding author; e-mail: [email protected]
e P 3 P 3P S¼ ¼ ; 2he 2 ~he h
ð1Þ
where Pe , ~he and S are, respectively, the measured load signal, the measured displacement signal and the CSM signal in the initial elastic loading segment from the machine, and P* and h* denote the values of the load and displacement signals at the point of effective initial contact. A linear regression analysis was used to establish the point of effective initial contact (P* and h*) in the indentation experiment. In the second step of the data analysis method, Hertz’s theory [7,8] was recast for frictionless, elastic, spherical indentation as (e.g. [1,4]): rind ¼ Eeff eind ; eind ¼
rind ¼
4 he he ; 3p a 2:4a
P ; pa2
a¼
1 1 m2s 1 m2i ¼ þ ; Es Ei Eeff
S ; 2Eeff
1 1 1 ¼ þ ; Reff Ri Rs
ð2Þ
where rind and eind are the indentation stress and the indentation strain, a is the radius of the contact boundary at the indentation load P, he is the elastic indentation depth, S is the elastic stiffness described earlier, Reff and Eeff are the effective radius and the effective modulus of
1359-6462/$ - see front matter Ó 2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2008.11.028
440
S. Pathak et al. / Scripta Materialia 60 (2009) 439–442 1000
Load, mN
a
R ht hr
800
he
hc Complete unload Max load
600
3 a 4
400
Idealized primary zone of indentation
A
2.4a
200
-200
he= ht – hr he =kP2/3
hr
0
B
ht 0
200
400 600 800 1000 1200 Displacement, nm
Figure 1. (A) Schematic of a spherical indentation showing the primary zone of indentation. (B) Schematic of the procedure involved in establishing the contact radius, a, at different load levels. The unloading segment at each of these load levels is fitted to Eq. (4) to estimate hr and k. Reff and a can then be calculated using Eqs. (3) and (5).
the indenter and the specimen system, m and E are the Poisson’s ratio and the Young’s modulus, and the subscripts s and i refer to the specimen and the indenter, respectively (see also the schematic in Fig. 1). Note that both the estimation of the zero point (Eq. (1)) and the extraction of the indentation stress–strain curves (Eq. (2)) require the use of the CSM signal (S ¼ dP =dhe ; also referred to as the CSM mode in the MTS XPÒ nanoindentation machine). However, the CSM module is an optional accessory on the MTS/Agilent1 nanoindenter machine, and hence is available only on some machines. Moreover, other nanoindentation machines (manufactured by companies other than MTS/Agilent) do not presently offer CSM as an option. Thus, there is a critical need to develop data analysis methods for extracting indentation stress–strain curves without CSM. In this paper, we formulate and validate new data analysis methods for the extraction of the indentation stress–strain curves without using CSM. This is achieved in a two-step process: (i) establishing the effective zeropoint and (ii) estimating the contact radius, a, which is required for the computation of indentation stress and strain. The indentation stress–strain curves obtained using this non-CSM approach is then compared to our previously reported [1,5,6] indentation stress–strain curves obtained using CSM. As mentioned earlier, determining of the point of initial contact constitutes the first step in the analyses of the raw data obtained from spherical nanoindentation. This problem, discussed in detail in our previous work [1,4,5] and in other literature [9–13], assumes great importance in computing the values of indentation stress and indentation strain. The error in the zero-point arising from the default procedures of the nanoindentation machines have been reported to be anywhere from ±2 nm [3] to ±30 nm [10]. Since the initial elastic segment corresponds to only a few (10–20 nm) nanometers of indentation depth, errors of this magnitude pose a major hurdle in extracting reliable values of elastic and yield properties from this data.
1
At the time of writing of this manuscript Agilent Technologies had acquired the Nano Instruments business unit of MTS Systems Corporation.
In this work, we have established the zero-load and the zero-displacement by fitting the recorded initial elastic load ð Pe Þ and displacement ð~he Þ signals to the predictions of Hertz theory. According to Hertz’s theory [7,8], the load and displacement during elastic loading in a spherical nanoindentation experiment should be related by: " # 2=3 3 1 1 ~ e pffiffiffiffiffiffiffiffi : he h ¼ k P P ; k¼ ð3Þ 4 Eeff Reff Note that k in Eq. (3) is a constant for the entire initial elastic loading segment. The values of P* and h* that yield the lowest residual error in the least-squares fit of the initial elastic loading segment to Eq. (3) were chosen to correspond to the effective zero-point. This approach ensures that the corrected data set would be highly consistent with Hertz’s theory. Note that, just as in Eq. (1), Eq. (3) also identifies an ‘‘effective” point of initial contact which allows us to de-emphasize any artifacts created at the initial contact by the unavoidable surface conditions (e.g. surface roughness or the presence of an oxide layer) [5]. For this study, samples of electropolished aluminum and tungsten were indented with two differently sized indenter tips (1.4 and 20 lm radii) in a MTS XPÒ machine. The low degree of elastic anisotropy in these two materials as well as the strong variation in their respective mechanical properties (aluminum exhibits a low modulus and a low yield strength while tungsten exhibits a high modulus and a high yield strength) makes them an ideal choice for validating the new data analysis procedures presented in this paper. The sample surfaces were prepared by electropolishing in order to avoid the undesirable effects caused by other preparation techniques (such as mechanical hardening on the sample surface due to polishing and grinding). Figure 2 shows an example of a comparison between the zero-points (values of P* and h*) for tungsten as determined by the two methods: (i) Eq. (1) using CSM, and (ii) Eq. (3) without using the CSM signal. As seen from Figure 2, both the CSM and the non-CSM data analysis methods for tungsten yield nearly identical values of P* and h*. This point is further emphasized in Table 1, where the values of P* and h* are given for three tests each in both aluminum and tungsten for each indenter size. It is clearly seen from these results that the zeropoints established by both methods are in very good agreement with each other. The Young’s moduli for the tungsten and aluminum samples as tested were determined to be 418 ± 4 and 71 ± 2 GPa (average and standard deviation from analyses of the six measurements shown in Table 1), respectively, using the non-CSM data analysis procedures described above. These values are in good agreement with the values reported in literature from other testing methods [2,14] and also agree very well with our previously reported values on the same materials using CSM (414 ± 6 GPa for tungsten and 67 ± 2 GPa for aluminum [1]). It is worth reiterating here that we are able to analyze the relatively small initial elastic loading segments with remarkable accuracy because the use of Eq. (1) or (3) to establish the effective zero-point does not warrant prior knowledge of the values of Reff and Eeff. This is
S. Pathak et al. / Scripta Materialia 60 (2009) 439–442
log (residual error)
W
m ,n h*
P*, mN
Zero poihnt without using CSM, Eq. (3) Zero point using the CSM [1]
Figure 2. Comparison between the effective zero-points identified by the CSM technique [1] and the non-CSM technique (Eq. (3)) for tungsten using a 1.4 lm spherical indenter (corresponding to Test #2 in Table 1). The residual error is obtained from regression analyses as measure of the degree of fit to Eq. (3).
especially beneficial in establishing a reliable value of Ri (note that in the initial elastic loading segment Reff = Ri). Using measurements on samples of known Young’s moduli (e.g. Si standards), we estimated that the indenter radii for the two indenters used in this study to be 1.4 and 20 lm respectively, even though the manufacturer had claimed otherwise. Both these estimates were subsequently confirmed by scanning electron microscopy. The second step in the extraction of indentation stress–strain curves is an accurate estimation of the contact radius, a, which evolves continuously during the indentation experiment. Majority of the methods used for estimation of a in the literature [11,15–17] are motivated by the spherical geometry of the indenter, where a is calculated directly from the indentation depth and the radius of the indenter, Ri. However, the contact radius is defined through the contact height, hc, and not ht, as shown in Figure 1. Although a number of different approaches (including area functions [2,3]) have been explored for relating hc and ht, they have not yet produced consistent and reliable results for a broad range of samples [18]. Table 1. Comparison between the effective zero-point as identified by the CSM technique [1] and the non-CSM technique (Eq. (3)) for tungsten and aluminum using two different indenter radii (1.4 and 20 lm).
1.4 lm indenter
A1
W
20 lm indenter
A1
W
Using CSM h* (nm) P* (mN)
Without CSM h* (nm) P* (mN)
Test Test Test Test Test Test
#1 #2 #3 #1 #2 #3
10.29 5.25 10.21 15.77 27.195 10.29
0.039 0.007 0.041 0.146 0.059 0.10
10.17 3.75 9.36 14.95 27.63 7.71
0.041 0.003 0.037 0.139 0.084 0.063
Test Test Test Test Test Test
#1 #2 #3 #1 #2 #3
4.35 6.08 4.34 10.95 9.29 4.07
0.028 0.027 0.027 0.135 0.295 0.209
4.19 5.91 4.65 10.73 8.31 3.77
0.028 0.029 0.029 0.141 0.248 0.181
441
Alternatively, one can impose an elastic unloading segment2 at any point of interest and analyze it using Hertz’s theory in order to estimate the contact radius. Indeed, this is exactly what is done in estimating the contact radius using CSM [2,3,19]. In the analyses of an unloading segment after some imposed plastic deformation in the loading segment, Reff can no longer be assumed to be equal to Ri. As we shall see later, Reff changes quite dramatically with any imposed plastic deformation by the indenter. In this work, we have established the value of Reff by imposing unloading segments at several selected locations on the measured load–displacement data, and analyzing them carefully using Hertz’s theory, as illustrated in Figure 1. Each unloading segment is fit to the expected Hertz’s relationship between the total indentation depth, ht, and the indentation load, P, which may be conveniently expressed as he ¼ ht hr ¼ kP 2=3 :
ð4Þ
In Eq. (4) he is the elastic indentation depth, hr is the residual (or plastic) indentation depth, and k is a factor containing Eeff and Reff as defined in Eq. (3). Once the value of Eeff is established from the initial loading curve (or if the Young’s modulus of the sample material is already known), a regression analysis on the unloading segment can determine both hr and Reff. The value of the contact radius, a, at any point in the unloading segment can then be determined from: pffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a ¼ Reff he ¼ Reff ðht hr Þ: ð5Þ It should be noted that this relationship between Reff and a is implicit in Hertz’s theory (for the quadratic contacting surfaces). Applying this equation to the data point just before the initiation of the unloading segment provides the value of the contact radius at that point in the original loading segment. It is important to note here that each unloading curve will produce only one point on the indentation stress– strain curve. This necessitates a large number of loading–unloading segments on the sample in order to be able to obtain a complete description of the indentation stress–strain curve for a given sample. Figure 3 shows the values of Reff estimated at different indentation depths, using the data analysis procedure described earlier, in the measurements obtained using a 20 lm spherical indenter on samples of electropolished aluminum and tungsten. As seen from this figure, the values for Reff change dramatically with imposed plastic deformation by the indenter. In fact, the changes in the effective radius are most dramatic in the initial stages of plastic deformation under the indenter. It is also seen that Reff takes on much higher values for the softer aluminum samples compared to the harder tungsten samples (at the same depth of indentation). Indeed, it was observed that Reff is not just a function of indentation depth (or load) alone, but varies substantially with the details of the elastic 2
The unloading segment of an indentation experiment is generally assumed to be elastic because reloading at the same spot often reproduces this portion of the load–displacement curve.
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S. Pathak et al. / Scripta Materialia 60 (2009) 439–442 120 100 80 60 40 20 0
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Figure 3. Values of Reff computed at different indentation depths which differ significantly between the softer aluminum and the harder tungsten sample. The values for Reff are also significantly larger than (Ri = 20 lm) for both samples. Inset: expanded view for the tungsten sample.
and plastic properties of the sample. Also, the values of Reff measured in Figure 3 are substantially larger than the indenter radius (Ri = 20 lm) indicating that the approximation Reff Ri [15,16,20–22] is not valid in the post-elastic regions of the samples. Figure 4A and B shows comparisons between the indentation stress–strain curves obtained using both the CSM method described in our earlier work [1] and the non-CSM method described in this work on aluminum and tungsten samples for both 1.4 and 20 lm radii indenter sizes. It is seen that the indentation stress–strain curves from the CSM and the non-CSM methods agree well with each other for both indenters. The indentation stress–strain curve produced using the non-CSM method is able to capture all the major features of the stress–strain curves, including the linear elastic regime, the plastic yield point and the post-yield strain-hardening. As expected, the indentation stress–strain curve
A
B
Figure 4. Comparison between the indentation stress–strain curves obtained using the CSM method [1] and the non-CSM method on (A) aluminum and (B) tungsten samples with the 1.4 and 20 lm radii indenters. The insets show expanded views of the indentation stress– strain curves for the larger 20 lm indenter.
for tungsten shows more strain hardening compared to that for the aluminum sample. These results also demonstrate the feasibility of capturing the details of the popin phenomenon [5,23] occurring in the smaller 1.4 lm indenter with the non-CSM method. The good agreement between indentation stress–strain curves measured by these two very different indenter sizes for two very different materials provides additional validation for the data analyses procedures used in this work. These findings indicate that in spite of the non-continuous nature of the non-CSM calculations, this method can be successfully used to characterize the mechanical response of the material during spherical nanoindentation. This work was partially supported by the National Science Foundation Grant No. CMMI-0654179. The MTS XPÒ System used in this study is maintained and operated by the Centralized Research Facilities in the College of Engineering at Drexel University. S.P. also wishes to acknowledge the support from the Sigma Xi Grants-inAid of Research (GIAR) program for this work. [1] S.R. Kalidindi, S. Pathak, Acta Materialia 56 (2008) 3523. [2] W.C. Oliver, G.M. Pharr, Journal of Materials Research 7 (1992) 1564. [3] W.C. Oliver, G.M. Pharr, Journal of Materials Research 19 (2004) 3. [4] S. Pathak, S.R. Kalidindi, C. Klemenz, N. Orlovskaya, Journal of the European Ceramic Society 28 (2008) 2213. [5] S. Pathak, D. Stojakovic, R. Doherty, S.R. Kalidindi, Journal of Materials Research, 2009, in press. [6] S. Pathak, D. Stojakovic, S. R. Kalidindi, 2008, unpublished work. [7] H. Hertz, Miscellaneous Papers, Macmillan, New York, 1896. [8] A.E.H. Love, Journal of Mathematics 10 (1939) 161. [9] T. Chudoba, N. Schwarzer, F. Richter, Surface and Coatings Technology 127 (2000) 9. [10] J. Deuschle, S. Enders, E. Arzt, Journal of Materials Research 22 (2007) 3107. [11] A.C. Fischer-Cripps, Surface and Coatings Technology 200 (2006) 4153. [12] J. Mencik, M.V. Swain, Journal of Materials Research 10 (1995) 1491. [13] C. Ullner, Measurement 27 (2000) 43. [14] G. Simmons, H. Wang, Single Crystal Elastic Constants and Calculated Aggregate Properties, second ed., MIT Press, Cambridge, MA, 1971. [15] S. Basu, A. Moseson, M.W. Barsoum, Journal of Materials Research 21 (2006) 2628. [16] J.S. Field, M.V. Swain, Journal of Materials Research 8 (1993) 297. [17] E.G. Herbert, G.M. Pharr, W.C. Oliver, B.N. Lucas, J.L. Hay, Thin Solid Films 398–399 (2001) 331. [18] M. Radovic, E. Lara-Curzio, L. Riester, Materials Science and Engineering A A368 (2004) 56. [19] X.D. Li, B. Bhushan, Materials Characterization 48 (2002) 11. [20] J.S. Field, M.V. Swain, Journal of Materials Research 10 (1995) 101. [21] A. Murugaiah, M.W. Barsoum, S.R. Kalidindi, T. Zhen, Journal of Materials Research 19 (2004) 1139. [22] M.V. Swain, Materials Science and Engineering A 253 (1998) 160. [23] C.A. Schuh, Materials Today 9 (2006) 32.