Research Papers

Aspects of Experimental Errors and Data Reduction Schemes From Spherical Indentation of Isotropic Materials

[+] Author and Article Information
J. K. Phadikar

Department of Mechanical Engineering,
University of Delaware,
Newark, DE 19711
e-mail: jpha@udel.edu

T. A. Bogetti

U.S. Army Research Laboratory,
Aberdeen Proving Ground,
Aberdeen, MD 21001
e-mail: travis.a.bogetti.civ@mail.mil

A. M. Karlsson

Department of Mechanical Engineering,
University of Delaware,
Newark, DE 19711
Washkewicz College of Engineering,
Cleveland State University,
Cleveland, OH 44115
e-mail: a.karlsson@csuohio.edu

This is a similar concept as used by previous researchers [32,37,40], with 36 coefficients required to describe the functions. As noted there, we are not striving to develop a relationship where the parameters have physical significance, but we are finding “fitting parameters” to accurately capture the response.

Alternatively, one shape function from depth-to-radius ratio of (hm/R)1 and two shape functions from depth-to-radius ratio of (hm/R)2 can be used. These results are omitted for brevity, since these gave almost the same condition number (for example, selecting Su and We from (hm/R)1 = 10% and Wt from (hm/R)1 = 40% gave almost same results as selection of Wt from (hm/R)1 = 10% and Su and We from (hm/R)1 = 40%).

The maximum experimental error in measuring shape functions from indentation testing has been reported to be within 5.1% [39,46].

Depth-to-radius ratio of 100% can be achieved by using a spherical indenter of sufficiently small radius, R. If R is large, cracking may ensue [37], invalidating the indentation analysis. Similarly, depth-to-radius ratio of 1% can be achieved by keeping R large. If R is small, the indentation depth will be small which will result in surface roughness effect [3] and size effect [47], invalidating the protocol.

Note that the combination (hm/R)1 = 1%, (hm/R)2 = 100% with shape function combination (We2,Su1,We1) results in a condition number that is lower than the “best” dual conical indentation (shape function combination (We2,Su1,We1) with half-angles 50 deg and 80 deg) as determined by the mentioned previous work [12].

1Corresponding author.

Contributed by the Materials Division of ASME for publication in the JOURNAL OF ENGINEERING MATERIALS AND TECHNOLOGY. Manuscript received September 24, 2013; final manuscript received April 28, 2014; published online May 15, 2014. Assoc. Editor: Georges Cailletaud.

This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States. Approved for public release; distribution is unlimited.

J. Eng. Mater. Technol 136(3), 031005 (May 15, 2014) (8 pages) Paper No: MATS-13-1175; doi: 10.1115/1.4027549 History: Received September 24, 2013; Revised April 28, 2014

Sensitivity to experimental errors determines the reliability and usefulness of any experimental investigation. Thus, it is important to understand how various test techniques are affected by expected experimental errors. Here, a semi-analytical method based on the concept of condition number is explored for systematic investigation of the sensitivity of spherical indentation to experimental errors. The method is employed to investigate the reliability of various possible spherical indentation protocols, providing a ranking of the selected data reduction protocols from least to most sensitive to experimental errors. Explicit Monte Carlo sensitivity analysis is employed to provide further insight of selected protocol, supporting the ranking. The results suggest that the proposed method for estimating the sensitivity to experimental errors is a useful tool. Moreover, in the case of spherical indentation, the experimental errors must be very small to give reliable material properties.

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Van Vliet, K. J., Prchlik, L., and Smith, J. F., 2004, “Direct Measurement of Indentation Frame Compliance,” J. Mater. Res., 19(1), pp. 325–331. [CrossRef]
Field, J. S., and Swain, M. V., 1995, “Determining the Mechanical Properties of Small Volumes of Material From Submicrometer Spherical Indentations,” J. Mater. Res., 10(1), pp. 101–112. [CrossRef]
Kim, J.-Y., Kang, S.-K., Lee, J.-J., Jang, J.-I., Lee, Y.-H., and Kwon, D., 2007, “Influence of Surface-Roughness on Indentation Size Effect,” Acta Mater., 55(10), pp. 3555–3562. [CrossRef]
Huang, Y., Zhang, F., Hwang, K. C., Nix, W. D., Pharr, G. M., and Feng, G., 2006, “A Model of Size Effects in Nano-Indentation,” J. Mech. Phys. Solids, 54(8), pp. 1668–1686. [CrossRef]
Higuchi, R., Mochizuki, M., and Toyoda, M., 2010, “A Method for Evaluating the Stress–Strain Relationship of Materials in the Microstructural Region Using a Triangular Pyramidal Indenter,” Weld. Int., 24(8), pp. 611–619. [CrossRef]
Feng, G., and Ngan, A. H. W., 2002, “Effects of Creep and Thermal Drift on Modulus Measurement Using Depth-Sensing Indentation,” J. Mater. Res., 17(3), pp. 660–668. [CrossRef]
Ahn, J.-H., and Kwon, D., 2001, “Derivation of Plastic Stress–Strain Relationship From Ball Indentations: Examination of Strain Definition and Pileup Effect,” J. Mater. Res., 16(11), pp. 3170–3178. [CrossRef]
Jiang, P., Zhang, T., Feng, Y., Yang, R., and Liang, N., 2009, “Determination of Plastic Properties by Instrumented Spherical Indentation: Expanding Cavity Model and Similarity Solution Approach,” J. Mater. Res., 24(3), pp. 1045–1053. [CrossRef]
Lan, H., and Venkatesh, T. A., 2007, “Determination of the Elastic and Plastic Properties of Materials Through Instrumented Indentation With Reduced Sensitivity,” Acta Mater., 55(6), pp. 2025–2041. [CrossRef]
Lan, H., and Venkatesh, T. A., 2007, “On the Sensitivity Characteristics in the Determination of the Elastic and Plastic Properties of Materials Through Multiple Indentation,” J. Mater. Res., 22(4), pp. 1043–1063. [CrossRef]
Lan, H., and Venkatesh, T. A., 2007, “On the Uniqueness and Sensitivity Issues in Determining the Elastic and Plastic Properties of Power-Law Hardening Materials Through Sharp and Spherical Indentation,” Philos. Mag., 87(30), pp. 4671–4729. [CrossRef]
Phadikar, J. K., Bogetti, T. A., and Karlsson, A. M., 2013, “On the Uniqueness and Sensitivity of Indentation Testing of Isotropic Materials,” Int. J. Solids Struct., 50(20–21), pp. 3242–3253. [CrossRef]
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(6), pp. 1564–1583. [CrossRef]
Phadikar, J. K., Bogetti, T. A., Kaliakin, V. N., and Karlsson, A. M., 2013, “Conical Indentation of a Viscoelastic Sphere,” ASME J. Eng. Mater. Technol., 135(4), p. 041001. [CrossRef]
Cheng, Y.-T., and Cheng, C.-M., 2004, “Scaling, Dimensional Analysis, and Indentation Measurements,” Mater. Sci. Eng.: R: Reports, 44(4–5), pp. 91–149. [CrossRef]
Phadikar, J. K., Bogetti, T. A., and Karlsson, A. M., 2012, “On Establishing Elastic-Plastic Properties of a Sphere by Indentation Testing,” Int. J. Solids Struct., 49(14), pp. 1961–1972. [CrossRef]
Dieter, G. E., and Bacon, D., 1986, Mechanical Metallurgy, McGraw-Hill, New York.
Lubliner, J., 1990, Plasticity Theory, Macmillan, New York.
Cao, Y. P., Qian, X. Q., Lu, J., and Yao, Z. H., 2005, “An Energy-Based Method to Extract Plastic Properties of Metal Materials From Conical Indentation Tests,” J. Mater. Res., 20(5), pp. 1194–1206. [CrossRef]
Dao, M., Lim, C., and Suresh, S., 2003, “Mechanics of the Human Red Blood Cell Deformed by Optical Tweezers,” J. Mech. Phys. Solids, 51(11–12), pp. 2259–2280. [CrossRef]
Ogasawara, N., Chiba, N., and Chen, X., 2009, “A Simple Framework of Spherical Indentation for Measuring Elastoplastic Properties,” Mech. Mater., 41(9), pp. 1025–1033. [CrossRef]
Yan, J., Chen, X., and Karlsson, A., 2007, “Determining Equi-Biaxial Residual Stress and Mechanical Properties From the Force-Displacement Curves of Conical Microindentation,” ASME J. Eng. Mater. Technol., 129(2), pp. 200–206. [CrossRef]
Yan, J., Karlsson, A., and Chen, X., 2007, “Determining Plastic Properties of a Material With Residual Stress by Using Conical Indentation,” Int. J. Solids Struct., 44(11–12), pp. 3720–3737. [CrossRef]
Buckingham, E., 1914, “On Physically Similar Systems; Illustrations of the Use of Dimensional Equations,” Phys. Rev., 4(4), pp. 345–376. [CrossRef]
Alkorta, J., Martínez-Esnaola, J. M., and Sevillano, J. G., 2005, “Absence of One-to-One Correspondence Between Elastoplastic Properties and Sharp-Indentation Load–Penetration Data,” J. Mater. Res., 20(2), pp. 432–437. [CrossRef]
Capehart, T. W., and Cheng, Y.-T., 2003, “Determining Constitutive Models From Conical Indentation: Sensitivity Analysis,” J. Mater. Res., 18(4), pp. 827–832. [CrossRef]
Cheng, Y.-T., and Cheng, C.-M., 1999, “Can Stress–Strain Relationships be Obtained From Indentation Curves Using Conical and Pyramidal Indenters?,” J. Mater. Res., 14(9), pp. 3493–3496. [CrossRef]
Liu, L., Ogasawara, N., Chiba, N., and Chen, X., 2009, “Can Indentation Technique Measure Unique Elastoplastic Properties?,” J. Mater. Res, 24(3), pp. 784–800. [CrossRef]
Tho, K. K., Swaddiwudhipong, S., Liu, Z. S., Zeng, K., and Hua, J., 2004, “Uniqueness of Reverse Analysis From Conical Indentation Tests,” J. Mater. Res., 19(8), pp. 2498–2502. [CrossRef]
Cao, Y., Qian, X., and Huber, N., 2007, “Spherical Indentation Into Elastoplastic Materials: Indentation-Response Based Definitions of the Representative Strain,” Mater. Sci. Eng.: A, 454–455, pp. 1–13. [CrossRef]
Zhao, M., Ogasawara, N., Chiba, N., and Chen, X., 2006, “A New Approach to Measure the Elastic-Plastic Properties of Bulk Materials Using Spherical Indentation,” Acta Mater., 54(1), pp. 23–32. [CrossRef]
Cao, Y. P., and Lu, J., 2004, “A New Method to Extract the Plastic Properties of Metal Materials From an Instrumented Spherical Indentation Loading Curve,” Acta Mater., 52(13), pp. 4023–4032. [CrossRef]
Xu, B., and Chen, X., 2010, “Determining Engineering Stress-Strain Curve Directly From the Load-Depth Curve of Spherical Indentation Test,” J. Mater. Res., 25(12), pp. 2297–2307. [CrossRef]
Kang, B. S. J., Yao, Z., and Barbero, E. J., 2006, “Post-Yielding Stress-Strain Determination Using Spherical Indentation,” Mech. Adv. Mater. Struct., 13(2), pp. 129–138. [CrossRef]
Haušild, P., Materna, A., and Nohava, J., 2012, “On the Identification of Stress–Strain Relation by Instrumented Indentation With Spherical Indenter,” Mater. Des., 37, pp. 373–378. [CrossRef]
Kang, S.-K., Kim, Y.-C., Kim, K.-H., Kim, J.-Y., and Kwon, D., 2013, “Extended Expanding Cavity Model for Measurement of Flow Properties Using Instrumented Spherical Indentation,” Int. J. Plasticity, 49, pp. 1–15. [CrossRef]
Chen, X., Ogasawara, N., Zhao, M., and Chiba, N., 2007, “On the Uniqueness of Measuring Elastoplastic Properties From Indentation: The Indistinguishable Mystical Materials,” J. Mech. Phys. Solids, 55(8), pp. 1618–1660. [CrossRef]
Cao, Y. P., and Lu, J., 2004, “Depth-Sensing Instrumented Indentation With Dual Sharp Indenters: Stability Analysis and Corresponding Regularization Schemes,” Acta Mater., 52(5), pp. 1143–1153. [CrossRef]
Chollacoop, N., Dao, M., and Suresh, S., 2003, “Depth-Sensing Instrumented Indentation With Dual Sharp Indenters,” Acta Mater., 51(13), pp. 3713–3729. [CrossRef]
Hyun, H. C., Kim, M., Lee, J. H., and Lee, H., 2011, “A Dual Conical Indentation Technique Based on FEA Solutions for Property Evaluation,” Mech. Mater., 43(6), pp. 313–331. [CrossRef]
Le, M.-Q., 2008, “A Computational Study on the Instrumented Sharp Indentations With Dual Indenters,” Int. J. Solids Struct., 45(10), pp. 2818–2835. [CrossRef]
Swaddiwudhipong, S., Tho, K. K., Liu, Z. S., and Zeng, K., 2005, “Material Characterization Based on Dual Indenters,” Int. J. Solids Struct., 42(1), pp. 69–83. [CrossRef]
3DS, 2009, abaqus, Version 6.9-2, Dassault Systèmes, Waltham, MA.
Bowden, F. P., and Tabor, D., 2001, “Appendix: Some Typical Values of Friction,” The Friction and Lubrication of Solids, Oxford University Press, New York.
Datta, B. N., 2010, Numerical Linear Algebra and Applications, Society for Industrial and Applied Mathematics, Philadelphia, PA, Chap. 4.
Wang, L., Ganor, M., and Rokhlin, S. I., 2005, “Inverse Scaling Functions in Nanoindentation With Sharp Indenters: Determination of Material Properties,” J. Mater. Res., 20(4), pp. 987–1001. [CrossRef]
Xu, Z.-H., and Li, X., 2006, “Sample Size Effect on Nanoindentation of Micro-/Nanostructures,” Acta Mater., 54(6), pp. 1699–1703. [CrossRef]
Moussa, C., Hernot, X., Bartier, O., Delattre, G., and Mauvoisin, G., 2014, “Evaluation of the Tensile Properties of a Material Through Spherical Indentation: Definition of an Average Representative Strain and a Confidence Domain,” J. Mater. Sci., 49(2), pp. 592–603. [CrossRef]
Phadikar, J. K., Bogetti, T. A., and Karlsson, A. M., “On the Construction of Confidence Interval for Indentation Testing” (in press).
Higham, N. J., 1996, Accuracy and Stability of Numerical Algorithms, Society for Industrial and Applied Mathematics, Philadelphia, PA, Chap. 7.
Rheinboldt, W. C., 1976, “On Measures of Ill-Conditioning for Nonlinear Equations,” Math. Comput., 30(133), pp. 104–111. [CrossRef]


Grahic Jump Location
Fig. 2

Stress–strain relationship of a linear-elastic, power-law hardening plastic material

Grahic Jump Location
Fig. 1

Schematic diagram of (a) spherical indentation on a half-space and (b) the typical force–displacement response obtained during loading and unloading




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