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A three-point method for evaluating the tilt status between the indenter axis and the sample surface
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Measurement Science and Technology Meas. Sci. Technol. 25 (2014) 017001 (4pp)
doi:10.1088/0957-0233/25/1/017001
Technical Design Note
A three-point method for evaluating the tilt status between the indenter axis and the sample surface Hu Huang, Hongwei Zhao 1 , Chengli Shi, Xiaoli Hu and Ye Tian College of Mechanical Science & Engineering, Jilin University, Renmin Street 5988, Changchun, Jilin 130025, People’s Republic of China E-mail:
[email protected] Received 11 September 2013, in final form 21 October 2013 Published 26 November 2013 Abstract
Non-perpendicularity nanoindentation experiments commonly exist in previous research work and affect nanoindentation results. In this design note, a new measuring method named the three-point method is proposed to evaluate the tilt status between the indenter axis and the sample surface. The measuring principle is addressed in detail. By the proposed measuring method, the tilt status between the indenter axis and the sample surface can be evaluated by the selected three points on the sample surface. A measuring example is given to verify the feasibility of the three-point method. This method is independent of other measuring equipment and it has the potential to be used in commercial nanoindentation instruments for evaluating the tilt status between the indenter axis and the sample surface. Keywords: tilt, three-point method, nanoindentation (Some figures may appear in colour only in the online journal)
There are many potential sources leading to the tilt, such as assembly errors of instruments, non-parallelism of the top and bottom surfaces of the sample, uneven bonding layer between the sample and the stage and uneven sample surfaces. When sample size is small, it is hard to obtain flat surfaces by polishing, especially near the edges, and the edge rounding effects near the edges will be obvious. When polishing binary materials such as two-phase cutting materials, inclusions make the polishing difficult to get flat locally. The effects of the tilt on nanoindentation results have been studied in detail by finite element simulations, theoretical analysis and experiments in the literature [9, 15–17]. As mentioned in [17], for indentation with a cone of semicone angle 70.3◦ with area function equivalent to the Berkovich indenter on a sample tilted 5◦ , the contact area is underestimated by 8%, leading to 8% overestimation in hardness and 4% overestimation in modulus. These errors change approximately with the square of the tilt angle. So, when the tilt angle is large (>1◦ ), measurement and corrections
1. Introduction The technique of nanoindentation has commonly been used to characterize the mechanical properties of materials in micro-/nano-scales [1, 2], but its testing results are very sensitive to many factors [3] coming from the instrument itself, the theoretical method and the sample preparation due to its high measuring resolution and small measuring scale. The most commonly used analysis methods—the Oliver–Pharr method [4] and the international standard ISO 14 577 [5]—present good solutions for some factors. However, some affecting factors [6–10] are still not fully considered in the Oliver–Pharr method or ISO 14 577. One of these factors is the tilt between the indenter axis and the sample surface, which usually exists in practical nanoindentation experiments [11–14]. 1
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Meas. Sci. Technol. 25 (2014) 017001
Technical Design Note
by tracking the displacement of the indenter in x–y planes. So, the coordinates of the three points on the sample surface, A (xA, yA, zA), B (xB, yB, zB) and C (xC, yC, zC), can be obtained based on the established Cartesian coordinate system. After the coordinates of the three points on the sample → −→ −→ − surface are obtained, three vectors, AB, BC and CA can be derived as −→ AB = (xB − xA , yB − yA , zB − zA ) − → BC = (xC − xB , yC − yB , zC − zB ) −→ CA = (xA − xC , yA − yC , zA − zC ).
Figure 1. Schematic diagram of the three-point method for
evaluating the tilt status between the indenter axis and the sample surface.
Then, the surface normal n of the sample surface, n = (xn , yn , zn ), can be calculated by the following equations
of the tilt angle are necessary. A correction method of the tilt for conical and pyramidal indentation has been proposed in [17]. However, the method to evaluate the tilt status between the indenter axis and the sample surface is rarely reported, which is the premise for practical correction of the tilt. In reference [18], we proposed a theoretical approach to measure the tilt angle between the triangular pyramidal indenter and the sample surface by the residual indent morphology. In this design note, another measuring method— a three-point method for evaluating the tilt status between the indenter axis and the sample surface is presented. The measuring principle and an example are given in detail.
⎧ −→ ⎫ ⎪ ⎨n · AB = 0⎪ ⎬ − → (1) n · BC = 0 . ⎪ ⎪ ⎩ −→ ⎭ n · CA = 0 Giving unit vectors along the three coordinate axes as nx = (1, 0, 0), ny = (0, 1, 0) and nz = (0, 0, 1), the angles between the three coordinate axes and the surface normal n of the sample surface can be expressed as
n · ni n, ni = arc cos |n| · |ni | ⎞ ⎛ x x + y y + z z n ni n ni n ni ⎠ = arc cos ⎝ 2 2 2 2 xn + yn + zn · xni + y2ni + z2ni
2. A three-point method
i = x, y, z.
Before presenting the three-point method, an assumption is given that the indenter axis is vertical, and sources of the tilt mainly come from sample preparation. Figure 1 is a schematic diagram of the three-point method. The basic principle is that three points define a plane. If the coordinates of the three points on the sample surface are obtained, the surface normal n of the sample surface used to evaluate the tilt status between the indenter axis and the sample surface can be determined. So, the key of the three-point method is to obtain the coordinates of three points on the sample surface. In order to solve this problem, the Cartesian coordinate system is established at the tip of the indenter as shown in figure 1. The point O just above the first position point A is the origin, which is located at the tip of the indenter. The z axis is along the indentation direction. Initially, there is a gap between the tip of the indenter and the sample surface. Then, the indentation experiment is carried out at the position point A. During the indentation experiment, displacement of the indenter, before and after it contacts the sample surface, is measured. Here, use the change in load to detect the contact between the indenter and the sample surface. The displacement of the indenter from the origin to the contact point is just the z coordinate zA of the position point A. Similarly, z coordinates zB and zC of the other two position points B and C can be obtained. In particular, when positions of the indenter are adjusted only in the x–y plane, the x and y coordinates of the points A, B and C can be obtained easily
(2)
The angles between the three coordinate axes and the surface normal n of the sample surface can illustrate the tilt status more directly, and they can also be used for correction of the tilt during indentation testing by the conical indenter according to the analysis results presented in [17]. 3. A measuring example In order to verify the feasibility of the three-point method, a measuring example is given in this section. A selfmade nanoindentation instrument whose structures and performances have been presented in [19] was used, which has the function to measure and track displacement of the indenter before and after it contacts the sample surface. A 5 × 5 × 1 mm3 bulk metallic glass was polished and then adhered on the sample stage by paraffin wax. Three position points with the x and y coordinates (0, 0), (50, 0) and (50, 50) are selected to carry out indentation experiments. All the coordinate position values are given in micrometres. The experimental results around the contact points are illustrated in figure 2. The fluctuation of the load before the indenter contacts the sample surface is about 30 μN, and the critical value of 0.1 mN was selected to determine the contact between the indenter and the sample surface. From figure 2, the z coordinates zA, zB and zC of the selected three points are obtained, and they are 2.783, 4.428 and 4.866, respectively. The coordinates 2
Meas. Sci. Technol. 25 (2014) 017001
Technical Design Note
(a)
(b)
(c)
Figure 2. Contact detection around the selected three position points, by which the z coordinates, zA, zB and zC, are obtained.
of the selected three position points, A, B and C are (0, 0, 2.783), (50, 0, 4.428) and (50, 50, 4.866), respectively. The critical value of 0.1 mN for contact detection is a little large for nanoindentation testing but it has a small effect on measuring of the z coordinate. → −→ −→ − So, three vectors, AB, BC and CA can be derived as −→ AB = (50, 0, 1.645)
Acknowledgments This research is funded by the National Natural Science Foundation of China (grant nos. 50905073, 51275198), Special Projects for Development of National Major Scientific Instruments and Equipments (grant no. 2012YQ030075), National Hi-tech Research and Development Program of China (863 Program) (grant no. 2012AA041206), Key Projects of Science and Technology Development Plan of Jilin Province (grant no. 20110307) and Program for New Century Excellent Talents in University of Ministry of Education of China (grant no. NCET-12-0238). Also, we thank the reviewer for the professional review and valuable suggestions.
− → BC = (0, 50, 0.438) −→ CA = ( − 50, −50, −2.083). Then, the surface normal n of the sample surface can be obtained by equation (1), and it is n = (3.7557, 1.0000, − 114.1552). By equation (2), angles between the surface normal n of the sample surface and the x axis, the y axis and the z axis are 88.1152◦ , 89.4960◦ and 178.0466◦ , respectively. According to equation (7) in [17], for indentation with a cone of semi-cone angle 70.3◦ with area function equivalent to the Berkovich indenter, this tilt angle of 1.9534◦ between the surface normal n of the sample surface and the z axis will result in approximately 1.37% underestimation of the projected area.
References [1] Schuh C A 2006 Nanoindentation studies of materials Mater. Today 9 32–40 [2] Oliver W C and Pharr G M 2010 Nanoindentation in materials research: past, present, and future MRS Bull. 35 897–907 [3] Fischer-Cripps A C 2011 Nanoindentation 3rd edn (New York: Springer) [4] Oliver W C and Pharr G M 1992 An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments J. Mater. Res. 7 1564–83 [5] International Standard 2002 Metallic materials—instrumented indentation test for hardness and materials parameters ISO 14577 (Geneva: ISO) [6] Garrido Maneiro M A and Rodr´ıguez J 2005 Pile-up effect on nanoindentation tests with spherical–conical tips Scr. Mater. 52 593–8 [7] Huang H, Zhao H W, Zhang Z Y, Yang Z J and Ma Z C 2012 Influences of sample preparation on nanoindentation behavior of a Zr-based bulk metallic glass Materials 5 1033–9 [8] Berke P, Houdaigui F E and Massart T J 2010 Coupled friction and roughness surface effects in shallow spherical nanoindentation Wear 268 223–32 [9] Xu Z H and Li X 2007 Effect of sample tilt on nanoindentation behaviour of materials Phil. Mag. 87 2299–312 [10] Mata M and Alcal´a J 2004 The role of friction on sharp indentation J. Mech. Phys. Solids 52 145–65 [11] Nowak S, Ochin P, Pasko A, Maciejak O, Aubert P and Champion Y 2009 Nanoindentation analysis of the mechanical behavior of Zr-based metallic glasses with Sn, Ta and W additions J. Alloys Compounds 483 139–42 [12] Bhattacharyya D, Mara N A, Dickerson P, Hoagland R G and Misra A 2010 A transmission electron microscopy study of the deformation behavior underneath nanoindents in nanoscale Al–TiN multilayered composites Phil. Mag. 90 1711–24
4. Conclusions In this design note, we focus on the measuring method of the tilt between the indenter axis and the sample surface. A Cartesian coordinate system was established at the tip of the indenter, based on which a three-point measuring method was proposed to evaluate the tilt status between the indenter axis and the sample surface. This method is used to obtain the coordinates of three selected position points on the sample surface by means of the nanoindentation instrument itself and then realizes the measurement of the tilt by simple mathematical calculation processes. A measuring example is given to verify the feasibility of the three-point method. Compared with the previous measuring method in reference [18], this new measuring method is independent of other equipment such as microscopes. Moreover, it is an in situ measuring method which can reduce measuring errors resulting from re-installation of the sample during microscope observation and indentation experiments existing in previous measuring methods. It has the potential to be used in commercial nanoindentation instruments for evaluating the tilt status between the indenter axis and the sample surface. 3
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[13] Wheeler J M, Raghavan R and Michler J 2011 In situ SEM indentation of a Zr-based bulk metallic glass at elevated temperatures Mater. Sci. Eng. A 528 8750–6 [14] Zhu X Y, Luo J T, Zeng F and Pan F 2011 Microstructure and ultrahigh strength of nanoscale Cu/Nb multilayers Thin Solid Films 520 818–23 [15] Kashani M S and Madhavan V 2007 The effect of surface tilt on nanoindentation results IMECE: Proc. ASME Int. Mech. Eng. Congress and Exposition (Seattle, USA) pp 1067–71 [16] Ellis J D, Smith S T and Hocken R J 2008 Alignment uncertainties in ideal indentation styli Precis. Eng. 32 207–14
[17] Madhavan V 2011 Analysis and correction of the effect of sample tilt on results of nanoindentation Acta Mater. 59 883–95 [18] Huang H, Zhao H W, Shi C L and Zhang L 2013 Using residual indent morphology to measure the tilt between the triangular pyramid indenter and the sample surface Meas. Sci. Technol. 24 105602 [19] Huang H, Zhao H W, Shi C L, Zhang L, Wan S G and Geng C Y 2013 Randomness and statistical laws of indentation-induced pop-out in single crystal silicon Materials 6 1496–505
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