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Relation between the ratio of elastic work to the total work of indentation and the ratio of hardness to Young’s modulus for a perfect conical tip

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Abstract

A linear relationship between the ratio of elastic work to the total indentation work and hardness to reduced modulus, i.e., We/Wt = λ H/Er, has been derived analytically and numerically in a number of studies and has been widely accepted. However, the scaling relationship between We/Wt and H/Er has recently been questioned, and it was found that λ is actually not a constant but is related to material properties. In this study, a new relationship between We/Wt and H/Er has been derived, which shows excellent agreement with numerical simulation and experimental results. We also propose a method for obtaining the elastic modulus and hardness of a material without invoking the commonly used Oliver and Pharr method. Furthermore, it is demonstrated that this method is less sensitive to tip imperfections than the Oliver and Pharr approach is.

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References

  1. N.A. Stilwell and D. Tabor: Elastic recovery of conical indentations. Proc. Phys. Soc. London 78, 169 (1961).

    Article  Google Scholar 

  2. Y-T. Cheng, Z. Li, and C-M. Cheng: Scaling relationships for indentation measurements. Philos. Mag. A 82, 1821 (2002).

    Article  CAS  Google Scholar 

  3. Y-T. Cheng and C-M. Cheng: Relationships between hardness, elastic modulus, and the work of indentation. Appl. Phys. Lett. 73, 614 (1998).

    Article  CAS  Google Scholar 

  4. A.E. Giannakopoulos and S. Suresh: Determination of elastoplastic properties by instrumented sharp indentation. Scr. Mater. 40, 1191 (1999).

    Article  CAS  Google Scholar 

  5. D. Ma, W. Chung, J. Liu, and J. He: Determination of Young’s modulus by nanoindentation. Sci. China, Ser. E 47, 398 (2004).

    Article  CAS  Google Scholar 

  6. J. Malzbender: Comment on the determination of mechanical properties from the energy dissipated during indentation. J. Mater. Res. 20, 1090 (2005).

    Article  CAS  Google Scholar 

  7. W.C. Oliver: Alternative technique for analyzing instrumented indentation data. J. Mater. Res. 16, 3202 (2001).

    Article  CAS  Google Scholar 

  8. S.V. Hainsworth, H.W. Chandler, and T.F. Page: Analysis of nanoindentation load-displacement loading curves. J. Mater. Res. 11, 1987 (1996).

    Article  CAS  Google Scholar 

  9. W.C. Oliver and G.M. Pharr: An improved technique for determining hardness and elastic-modulus using load and displacement sensing indentation experiments. J. Mater. Res. 7, 1564 (1992).

    Article  CAS  Google Scholar 

  10. G.M. Pharr and A. Bolshakov: Understanding nanoindentation unloading curves. J. Mater. Res. 17, 2660 (2002).

    Article  CAS  Google Scholar 

  11. Y-T. Cheng and C-M. Cheng: Scaling, dimensional analysis, and indentation measurements. Mater. Sci. Eng., R 44, 91 (2004).

    Article  Google Scholar 

  12. I.N. Sneddon: The relation between load and penetration in the axisymmetric Boussinesq problem for a punch of arbitrary profile. Int. J. Eng. Sci. 3, 47 (1965).

    Article  Google Scholar 

  13. J.C. Hay, A. Bolshakov, and G.M. Pharr: A critical examination of the fundamental relations used in the analysis of nanoindentation. J. Mater. Res. 14, 2296 (1999).

    Article  CAS  Google Scholar 

  14. J. Alkorta, J.M. Martínez-Esnaola, and J. Gil Sevillano: Comments on “Comment on the determination of mechanical properties from the energy dissipated during indentation” by J. Malzbender [J. Mater. Res. 20, 1090 (2005)]. J. Mater. Res. 21, 302 (2006).

    Article  Google Scholar 

  15. Y. Choi, H.S. Lee, and D. Kwon: Analysis of sharp-tip-indentation load-displacement curve for contact area determination taking into account pile-up and sink-in effects. J. Mater. Res. 19, 3307 (2004).

    Article  CAS  Google Scholar 

  16. M. Sakai: Simultaneous estimate of elastic/plastic parameters in depth-sensing indentation tests. Scr. Mater. 51, 391 (2004).

    Article  CAS  Google Scholar 

  17. V. Marx and H. Balke: A critical investigation of the unloading behavior of sharp indentation. Acta Mater. 45, 3791 (1997).

    Article  CAS  Google Scholar 

  18. J. Malzbender and G. de With: Indentation load-displacement curve, plastic deformation and energy. J. Mater. Res. 17, 502 (2002).

    Article  CAS  Google Scholar 

  19. J. Chen and S.J. Bull: A critical examination of the relationship between plastic deformation zone size and Young’s modulus to hardness ratio in indentation testing. J. Mater. Res. 21, 2617 (2006).

    Article  CAS  Google Scholar 

  20. M. Li, W. Chen, N. Liang, and L. Wang: A numerical study of indentation using indenters of different geometry. J. Mater. Res. 19, 73 (2004).

    Article  Google Scholar 

  21. S. Soare: Design of a rotating sensor for stress measurement in metallization. Ph.D. Thesis, University of Newcastle upon Tyne, UK, 2004.

    Google Scholar 

  22. S. Jayaraman, G.T. Hahn, W.C. Oliver, C.A. Rubin, and P.C. Bastias: Determination of monotonic stress-strain curve of hard materials from ultra-low-load indentation tests. Int. J. Solids Struct. 35, 365 (1998).

    Article  Google Scholar 

  23. T.A. Venkatesh, K.J. Van Vleit, A.E. Ginnakopoulos, and S. Suresh: Determination of elasto-plastic properties by instrumented sharp indentation: Guidelines for property extraction. Scr. Mater. 42, 833 (2000).

    Article  CAS  Google Scholar 

  24. J.F. Palacio: Mechanical properties of high performance fuller-ene-like CNx coatings assessed by nanoindentation. Ph.D. Thesis, University of Newcastle upon Tyne, UK, 2006.

    Google Scholar 

  25. Y.T. Cheng and C.M. Cheng: Further analysis of indentation loading curves: Effects of tip rounding on mechanical property measurements. J. Mater. Res. 13, 1059 (1998).

    Article  CAS  Google Scholar 

  26. J. Chen and S.J. Bull: On the relationship between plastic zone radius and maximum depth during nanoindentation. Surf. Coat. Technol. 201, 4289 (2006).

    Article  CAS  Google Scholar 

  27. H. Bei, E.P. George, J.L. Hay, and G.M. Pharr: Influence of indenter tip geometry on elastic deformation during nanoindentation. Phys. Rev. Lett. 95(045501), 1 (2005).

    Google Scholar 

  28. J. Chen and S.J. Bull: Multicycling nanoindentation study on thin optical coatings on glass. J. Phys. D: Appl. Phys. 41, 074009 (2008).

    Article  Google Scholar 

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  1. School of Chemical Engineering and Advanced Materials, Newcastle University, Merz Court, Newcastle upon Tyne NE1 7RU, United Kingdom

    J. Chen & S. J. Bull

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  1. J. Chen
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  2. S. J. Bull
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Correspondence to S. J. Bull.

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Chen, J., Bull, S.J. Relation between the ratio of elastic work to the total work of indentation and the ratio of hardness to Young’s modulus for a perfect conical tip. Journal of Materials Research 24, 590–598 (2009). https://doi.org/10.1557/jmr.2009.0086

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  • DOI: https://doi.org/10.1557/jmr.2009.0086

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