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An energy-based method to extract plastic properties of metal materials from conical indentation tests

  • Yan Ping Cao (a1), Xiu Qing Qian (a2), Jian Lu (a3) and Zhen Han Yao (a2)


Based on dimensional analysis and finite element computations, an energy-based representative strain for conical indentation in elastoplastic materials has been proposed to establish an explicitly one-to-one relationship between the representative stress σr, the indentation loading curvature C, and the ratio of reversible work We to total work Wt performed by the indenter, i.e., σr/C = F0(We/Wt), where σr is the flow stress corresponding to the representative strain. The relationship provides a very simple method to evaluate the representative stress σr from the three directly measurable quantities We, Wt, and C. Numerical examples and further theoretical analysis reveal that a unique, stable solution can be obtained from the present method for a wide range of material properties, including both highly plastic materials (e.g., Ni for which Ey = 1070) and highly elastic materials (e.g., materials for which Ey = 25 and n = 0.5), using indenters with different tip apex angles. Based on the representative strains and stresses given by two indenters with different tip apex angles, e.g., (σr,80, ϵr,80) and (σr,65, ϵr,65), the plastic properties of materials, i.e., the yield strength σy and strain hardening exponent n can be further determined.


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1.Tabor, D.: Indentation hardness: Fifty years on—A personal view. Philos. Mag. A 74, 1207 (1996).
2.Doener, M.F. and Nix, W.D.: A method for interpreting the data from depth-sensing indentation instruments. J. Mater. Res. 1, 601 (1986).
3.Oliver, W.C. and Pharr, G.M.: An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments. J. Mater. Res. 7, 1564 (1992).
4.Giannakopoulos, A.E. and Suresh, S.: Determination of elastoplastic properties by instrumented sharp indentation. Scripta Mater. 40, 1191 (1999).
5.Dao, M., Chollacoop, N., Van Vliet, K.J., Venkatesh, T.A. and Suresh, S.: Computational modelling of the forward and reverse problems in instrumented sharp indentation. Acta Mater. 49, 3899 (2001).
6.Cheng, Y.T. and Cheng, C.M.: Can stress–strain relationships be obtained from indentation curves using conical and pyramidal indenters? J. Mater. Res. 14, 3493 (1999).
7.Capehart, T.W. and Cheng, Y.T.: Determining constitutive models from conical indentation: Sensitivity analysis. J. Mater. Res. 18, 827 (2003).
8.Tho, K.K., Swaddiwudhipong, S., Liu, Z.S., Zeng, K. and Hua, J.: Uniqueness of reverse analysis from conical indentation tests. J. Mater. Res. 19, 2498 (2004).
9.Tunvisut, K., O’Dowd, N.P. and Busso, E.P.: Use of scaling functions to determine mechanical properties of thin coatings from microindentation tests. Int. J. Solids Struct. 38, 335 (2001).
10.Mata, M. and Alcalá, J.: Mechanical property evaluation through sharp indentations in elastoplastic and fully plastic contact regimes. J. Mater. Res. 18, 1705 (2003).
11.Mata, M. and Alcalá, J.: The role of friction on sharp indentation. J. Mech. Phys. Solids. 52, 145 (2004).
12.Futakawa, M., Wakui, T., Tanabe, Y.J. and Ioka, I.: Identification of the constitutive equation by the indentation technique using plural indenters with different apex angles. J. Mater. Res. 16, 2283 (2001).
13.DiCarlo, A. and Yang, H.T.Y.: Semi-inverse method for predicting stress–strain relationship from cone indentations. J. Mater. Res. 18, 2068 (2003).
14.Bucaille, J.L., Stauss, S., Felder, E. and Michler, J.: Determination of plastic properties of metals by instrumented indentation using different sharp indenters. Acta Mater. 51, 1663 (2003).
15.Chollacoop, N., Dao, M. and Suresh, S.: Depth-sensing instrumented indentation with dual sharp indenters. Acta Mater. 51, 3713 (2003).
16.Swaddiwudhipong, S., Tho, K.K., Liu, Z.S. and Zeng, K.: Material characterization based on dual indenters. Int. J. Solids Struct. 42, 69 (2005).
17.Cao, Y.P. and Lu, J.: Depth-sensing instrumented indentation with dual indenters: Stability analysis and corresponding regularization schemes. Acta Mater. 52, 1143 (2004).
18.Cheng, Y.T. and Cheng, C.M.: Relationships between hardness, elastic modulus, and the work of indentation. Appl. Phys. Lett. 73, 614 (1998).
19.Cheng, Y.T. and Cheng, C.M.: Scaling approach to conical indentation in elastic-plastic solids with work hardening. J. App. Phys. 84(3), 1284 (1998).
20.Cheng, Y.T. and Cheng, C.M.: Scaling relationships in conical indentation of elastic-perfectly plastic solids. Int. J. Solids Struct. 36, 1231 (1999).
21.Cheng, Y.T. and Cheng, C.M.: Scaling, dimensional analysis, and indentation measurements. Mater. Sci. Eng. R 44, 91 (2004).
22.ABAQUS, : Theory Manual Version 6.4 (Hibbitt, Karlsson and Sorensen Inc, Pawtucket, RI, 2004).
23.Tabor, D.: Hardness of Metals (Clarendon Press, Oxford, U.K., 1951).
24.Johnson, K.L.: The correlation of indentation experiments. J. Mech. Phys. Solids 18, 115 (1970).
25.Larsson, P.L.: Investigation of sharp contact at rigid–plastic conditions. Int. J. Mech. Sci. 43, 895 (2001).
26.Bucaille, J.L. and Felder, E.: Finite-element analysis of deformation during indentation and scratch tests on elastic-perfectly plastic materials. Philos. Mag. A 82, 2003 (2002).
27.Mata, M., Anglada, M. and Alcalá, J.: Contact deformation regimes around sharp indentations and the concept of the characteristic strain. J. Mater. Res. 17, 964 (2002).
28.Cheng, Y.T. and Cheng, C.M.: Further analysis of indentation loading curves: Effects of tip rounding on mechanical property measurements. J. Mater. Res. 13, 1059 (1998).
29.Bucaille, J.L., Stauss, S., Schwaller, P. and Micher, J.: A new technique to determine the elastoplastic properties of thin metallic films using sharp indenters. Thin Solid Films 447, 239 (2004).
30.Sun, S., Zheng, S., Bell, T. and Smith, J.: Indenter tip radius and load frame compliance calibration using nanoindentation loading curves. Philos. Mag. Lett. 79, 649 (1999).
31.Thurn, J., Morris, D.J. and Cook, R.F.: Depth-sensing indentation at macroscopic dimensions. J. Mater. Res. 17, 2679 (2002).
32.Van, K.J., Prchlik, V.L. and Smith, J.F.: Direct measurement of indentation frame compliance. J. Mater. Res. 19, 325 (2004).
33.Oliver, W.C. and Pharr, G.M.: Measurement of hardness and elastic modulus by instrumented indentation: Advances in understanding and refinement to methodology. J. Mater. Res. 19, 3 (2004).
34.Cao, Y.P. and Lu, J.: A new method to extract the plastic properties of metal materials from an instrumented spherical indentation loading curve. Acta Mater. 52, 4023 (2004).
35.Suresh, S.: Graded materials for resistance to contact deformation and damage. Science 292, 2447 (2001).
36.Giannakopoulos, A.E.: Indentation of plastically graded substrates by sharp indenters. Int. J. Solids Struct. 39, 2495 (2002).
37.De Guzman, M.S., Neubauer, G., Flinn, P., and Nix, W.D.: The role of indentation depth on the measured hardness of materials, in Thin Films: Stresses and Mechanical Properties IV, edited by Townsend, P.H., Weihs, T.P., Sanchez, J.E. Jr., and Borgesen, P. (Mater. Res. Soc. Symp. Proc. 308, Pittsburgh, PA, 1993), pp. 613.
38.Stelmashenko, N.A., Walls, M.G., Brown, L.M. and Milman, Y.V.: Micro-indentation on W and Mo oriented single crystals: An STM study. Acta Metall. Mater. 41, 2855 (1993).
39.Nix, W.D. and Gao, H.: Indentation size effects in crystalline materials: A law for strain gradient plasticity. J. Mech. Phys. Solids 46, 411 (1998).
40.Arzt, E.: Size effect in materials due to microstructural and dimensional constraints: A comparative review. Acta Mater. 46, 5611 (1998).
41.Cao, Y.P. and Lu, J.: A new scheme for computational modelling of conical indentation in plastically-graded materials. J. Mater. Res. 19, 1703 (2004).
42.Cao, Y.P. and Lu, J.: Size-dependent sharp indentation I: A closed-form expression of the indentation loading curve. J. Mech. Phys. Solids 53, 33 (2005).
43.Cao, Y.P. and Lu, J.: Size-dependent sharp indentation II: A reverse algorithm to identify plastic properties of metallic materials. J. Mech. Phys. Solids 53, 49 (2005).


An energy-based method to extract plastic properties of metal materials from conical indentation tests

  • Yan Ping Cao (a1), Xiu Qing Qian (a2), Jian Lu (a3) and Zhen Han Yao (a2)


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