Skip to main content
SpringerLink
Log in

Weightings on the Propagation of Errors in the Vickers Hardness Parameters

  • Condensed Matter
  • Published:
Brazilian Journal of Physics Aims and scope Submit manuscript

Abstract

It is known that the value of an experimental quantity is considered to be adequately determined when there is an indication of its respective error (or uncertainty); otherwise, it is not possible to know at which level of precision it is being working. The most common method to evaluate the measurement uncertainty is the Guide to the expression of Uncertainty in Measurement (GUM), based on the Law of Propagation of Uncertainty (LPU), which in physics sciences is known as Law of Propagation of Errors (LPE). This work presents a theoretical evaluation of the parameters that influence the Vickers hardness (HV) error obtained by the error propagation method. Vickers indentations were performed on soda–lime–silica glasses to compare the results of descriptive statistics with the error by propagating errors. In addition, Monte Carlo Simulations (MCS) were performed and validated by the experimental data. It can be said that the error model proved to be more accurate than the conventional approach; in other words, the descriptive statistic overestimates the uncertainty value of the Vickers hardness. Comparing the model results for the error propagation in HV with the MCS, it can be said that the MCS method was validated by the experimental results and, therefore, it is recommended to use the error propagation to evaluate the uncertainty of Vickers hardness.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. R.L. Smith, G.E. Sandland, An accurate method of determining the hardness of metals, with particular reference to those of a high degree of hardness. Proc. Inst. Mech. Eng. B J. Eng. 102(1), 623–641 (1922). https://doi.org/10.1243/PIME_PROC_1922_102_033_02

    Article  Google Scholar 

  2. A. Yonezu, M. Arino, T. Kondo, H. Hirakata, K. Minoshima, On hydrogen-induced vickers indentation cracking in high-strength steel. Mech. Res. Commun. 37(2), 230–234 (2010). https://doi.org/10.1016/j.mechrescom.2010.01.001

    Article  Google Scholar 

  3. Y. Liu, M. Wang, W. Wang, Electric induced curing of graphene/cement-based composites for structural strength formation in deep-freeze low temperature. Mater. Des. 160, 783–793 (2018). https://doi.org/10.1016/j.matdes.2018.10.008

    Article  Google Scholar 

  4. T.D.L. Gontarski, R.M. Casali, A. Mikowski, Vickers hardness – definition, standardization and research perspectives: a review. Braz. J. Dev. 7(2), 15736–15754 (2021). https://doi.org/10.34117/bjdv7n2-274

  5. S. Amiri, N. Lecis, A. Manes, M. Giglio, A study of a micro-indentation technique for estimating the fracture toughness of al6061-t6. Mech. Res. Commun. 58, 10–16 (2014). https://doi.org/10.1016/j.mechrescom.2013.10.013

    Article  Google Scholar 

  6. S.M. Domínguez-Nicolas, A.L. Herrera-May, L. García-González, L. Zamora-Peredo, J. Hernández-Torres, J. Martínez-Castillo, E.A. Morales-González, C.A. Cerón-Álvarez, A. Escobar-Pérez, Algorithm for automatic detection and measurement of vickers indentation hardness using image processing. Meas. Sci. Technol. 32(1), 015407 (2020). https://doi.org/10.1088/1361-6501/abaa66

  7. N.H. Faisal, R.L. Reuben, R. Ahmed, An improved measurement of vickers indentation behaviour through enhanced instrumentation. Meas. Sci. Technol. 22(1), 015703 (2010). https://doi.org/10.1088/0957-0233/22/1/015703

  8. L.R. Pendrill, Optimised measurement uncertainty and decision-making when sampling by variables or by attribute. Measurement: Journal of the International Measurement Confederation 39(9), 829–840 (2006). https://doi.org/10.1016/j.measurement.2006.04.014

    Article  Google Scholar 

  9. JCGM: JCGM 100 - Evaluation of measurement data — Guide to the expression of uncertainty in measurement. Joint Committee for Guides in Metrology, Sèvres (2008). Joint Committee for Guides in Metrology

  10. P.D.S. Hack, C.S.T. Caten, Measurement uncertainty: literature review and research trends. IEEE Trans. Instrum. Meas. 61(8), 2116–2124 (2012). https://doi.org/10.1109/TIM.2012.2193694

  11. D.W. Preston, E.R. Dietz, The art of experimental physics, 1st edn. (John Wiley & Sons Ltd., New York 1991), p. 448

  12. J.R. Taylor, Introduction to error analysis, 2nd edn. (University Science Books, New York, 1997), p. 327

  13. H.H. Ku, Notes on the use of propagation of error formulas. Journal of Research of the National Bureau of Standards - C. Engineering and Instrumentation 70(4), 263–273 (1966)

  14. A. Mikowski, R.M. Casali, P. Soares, W.B. da Silva, B.S. Barra, Methodology for error propagation analysis of the complex stiffness modulus of asphalt mixes. Constr. Build. Mater. 290, 123156 (2021). https://doi.org/10.1016/j.conbuildmat.2021.123156

  15. J. Palma, R. Rivero, I. Lira, M. François, Measurement of the residual stress tensor on the surface of a specimen by layer removal and interferometry: uncertainty analysis. Meas. Sci. Technol. 20(11), 115302 (2009). https://doi.org/10.1088/0957-0233/20/11/115302

  16. M. Matus, Uncertainty of the variation in length of gauge blocks by mechanical comparison: a worked example. Meas. Sci. Technol. 23(9), 094003 (2012). https://doi.org/10.1088/0957-0233/23/9/094003

  17. S.R. Díaz, On the propagation of methodological uncertainties in Depth Sensing Indentation data analysis: a brief and critical review. Mech. Res. Commun. 105 (2020). https://doi.org/10.1016/j.mechrescom.2020.103516

  18. A. Mikowski, F.C. Serbena, C.E. Foerster, A.R. Jurelo, C.M. Lepienski, A method to measure fracture toughness using indentation in \(reba_{2}cu_{3}o_{7}\)-\(\delta\) superconductor single crystals. J. Appl. Phys. 110(10), 103504 (2011). https://doi.org/10.1063/1.3662121

  19. D.A. Lucca, K. Herrmann, M.J. Klopfstein, Nanoindentation: Measuring methods and applications. CIRP Ann. Manuf. Technol. 59(2), 803–819 (2010). https://doi.org/10.1016/j.cirp.2010.05.009

    Article  Google Scholar 

  20. R. Cagliero, G. Barbato, G. Maizza, G. Genta, Measurement of elastic modulus by instrumented indentation in the macro-range: Uncertainty evaluation. Int. J. Mech. Sci. 101–102, 161–169 (2015). https://doi.org/10.1016/j.ijmecsci.2015.07.030

    Article  Google Scholar 

  21. M.I. Mohamed, G.A. Aggag, Uncertainty evaluation of shore hardness testers. Measurement: Journal of the International Measurement Confederation 33(3), 251–257 (2003). https://doi.org/10.1016/S0263-2241(02)00087-8

    Article  Google Scholar 

  22. H. Kumar, G. Moona, P.K. Arora, A. Haleem, J. Singh, R. Kumar, A. Kumar, Monte Carlo method for evaluation of uncertainty of measurement in Brinell hardness scale. Indian J. Pure Appl. Phys. 55(6), 445–453 (2017)

    Article  ADS  Google Scholar 

  23. G.M. Mahmoud, R.S. Hegazy, Comparison of GUM and Monte Carlo methods for the uncertainty estimation in hardness measurements. Int.J. Metrol. Qual. Eng. 8, 1–9 (2017). https://doi.org/10.1051/ijmqe/2017014

    Article  Google Scholar 

  24. I. Elizabeth, R. Kumar, N. Garg, M. Asif, R.M. Manikandan, Girish, S.S.K. Titus, Measurement uncertainty evaluation in Vickers hardness scale using law of propagation of uncertainty and Monte Carlo simulation. Mapan - Journal of Metrology Society of India 34(3), 317–323 (2019). https://doi.org/10.1007/s12647-019-00341-9

  25. R. Kessel, R. Kacker, M. Berglund, Coefficient of contribution to the combined standard uncertainty. Metrologia 43(4), 189–195 (2006). https://doi.org/10.1088/0026-1394/43/4/s04

  26. G. Wübbeler, M. Krystek, C. Elster, Evaluation of measurement uncertainty and its numerical calculation by a Monte Carlo method. Meas. Sci. Technol. 19(8), 084009 (2008). https://doi.org/10.1088/0957-0233/19/8/084009

  27. JCGM: JCGM 101 - Evaluation of measurement data — Supplement 1 to the “Guide to the expression of uncertainty in measurement” propagation of distributions using a Monte Carlo method. Joint Committee for Guides in Metrology, Sèvres (2008). Joint Committee for Guides in Metrology

  28. A. Mikowski, P. Soares, S.E. Braun, S. Buchner, E. Neves, C.M. Lepienski, Statistical analysis of indentation fracture toughness of high voltage insulator glass. IV International Symposium on Non-Crystalline Solids, VII Brazilian Symposium on Glass and Related Materials, 1 (2007)

  29. A.L. Yurkov, N.V. Jhuravleva, E.S. Lukin, Kinetic microhardness measurements of sialon-based ceramics. J. Mater. Sci. 29, 6551–6560 (1994). https://doi.org/10.1007/BF00354021

    Article  ADS  Google Scholar 

  30. I.Y. Yanchev, E.P. Trifonova, C. Karakotsou, A.N. Anagnostopoulos, G.L. Bleris, Analysis of microhardness data in tlxin1-xse. J. Mater. Sci. 30 (1995). https://doi.org/10.1007/BF00356689

  31. F. Antonio Dorini, G. Pintaude, R. Sampaio, Maximum entropy approach for modeling hardness uncertainties in Rabinowicz’s abrasive wear equation. J. Tribol. 136(2) (2014). https://doi.org/10.1115/1.4026421.021607

  32. J.-M. Schneider, M. Bigerelle, A. Iost, Statistical analysis of the Vickers hardness. Mater. Sci. Eng. A 262(1), 256–263 (1999). https://doi.org/10.1016/S0921-5093(98)01000-4

    Article  Google Scholar 

  33. J.W. Eaton, D. Bateman, S. Hauberg, R. Wehbring, GNU Octave, 5th edn. (Boston, 2020), p. 1094 . https://octave.org/octave.pdf

  34. R.F. Cook, G.M. Pharr, Direct observation and analysis of indentation cracking in glasses and ceramics. J. Am. Ceram. Soc. 73(4), 787–817 (1990). https://doi.org/10.1111/j.1151-2916.1990.tb05119.x

    Article  Google Scholar 

  35. J.L. Wang, D.J. Ma, L. Sun, The influence of crack forms on indentation hardness test results for ceramic materials. J. Mater. Sci. 7 (2015). https://doi.org/10.1007/s10853-015-9162-2

  36. C.M. Lepienski, M.D. Michel, P.J.G. Araújo, C.A. Achete, Indentation fracture of a-C:H thin films from chemical vapour deposition. Philos. Mag. 86(33–35), 5397–5406 (2006). https://doi.org/10.1080/14786430600732085

    Article  ADS  Google Scholar 

  37. M.D. Michel, L.V. Muhlen, C.A. Achete, C.M. Lepienski, Fracture toughness, hardness and elastic modulus of hydrogenated amorphous carbon films deposited by chemical vapor deposition. Thin Solid Films 496(2), 481–488 (2006). https://doi.org/10.1016/j.tsf.2005.08.342

    Article  ADS  Google Scholar 

  38. ISO: ISO 6507-2:2018: Metallic materials — Vickers hardness test — Part 2: Verification and calibration of testing machines. International Organization for Standardization, Geneva (2018). International Organization for Standardization

  39. M.D. Michel, A. Mikowski, C.M. Lepienski, C.E. Foerster, F.C. Serbena, High temperature microhardness of soda-lime glass. J. Non-Cryst. Solids 348, 131–138 (2004). https://doi.org/10.1016/j.jnoncrysol.2004.08.138. Proceedings of the 6th Brazilian Symposium of Glases and Related Materials and 2nd International Symposium on Non-Crystalline Solids in Brazil

  40. N. Maharjan, W. Zhou, N. Wu, Direct laser hardening of AISI 1020 steel under controlled gas atmosphere. Surf. Coat. Technol. 385, 125399 (2020). https://doi.org/10.1016/j.surfcoat.2020.125399

  41. J. Suwanprateeb, A comparison of different methods in determining load- and time-dependence of vickers hardness in polymers. Polym. Testing 17(7), 495–506 (1998). https://doi.org/10.1016/S0142-9418(97)00040-8

    Article  Google Scholar 

Download references

Acknowledgements

This study was financed in part by the Coordenação de Aperfeiçamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001. The fourth and last author received grants from Capes during the period of experimental data acquisition. The second author is currently funded by a Capes grant.

Author information

Author notes

  1. Aline Peres Leal, Rafael Machado Casali, Sandro Elias Braun, Paulo Soares, André Luís Condino Fujarra, and Alexandre Mikowski contributed equally to this work.

Authors and Affiliations

  1. Department of Mobility Engineering, Federal University of Santa Catarina, Technological Center of Joinville, Graduate Program in Mechanical Engineering and Sciences, R. Dona Francisca, 8300 - Bloco U, Joinville, 89.219-600, SC, Brazil

    Thiago de Lima Gontarski, Rafael Machado Casali, André Luís Condino Fujarra & Alexandre Mikowski

  2. Polytechnic School of the University of São Paulo, Graduate Program in Naval and Ocean Engineering, Av. Prof. Luciano Gualberto, 380 - Butantã, São Paulo, 05.508-010, SP, Brazil

    Aline Peres Leal & André Luís Condino Fujarra

  3. State Secretary for Education of Rio Grande do Sul, Av. Borges de Medeiros, 1501, Porto Alegre, 90.119-900, RS, Brazil

    Sandro Elias Braun

  4. Department of Mechanical Engineering, Pontifical Catholic University of Paraná, Polytechnic School, R. Imaculada Conceição, 1155, Curitiba, 80.215-901, PR, Brazil

    Paulo Soares

Authors
  1. Thiago de Lima Gontarski
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Aline Peres Leal
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Rafael Machado Casali
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Sandro Elias Braun
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Paulo Soares
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. André Luís Condino Fujarra
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Alexandre Mikowski
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Thiago de Lima Gontarski.

Ethics declarations

Competing Interests

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary Information

Below is the link to the electronic supplementary material.

Supplementary file1 (PDF 314 KB)

Appendix 1

Appendix 1

Distribution of HV proposed by Schneider et al. [32].

1.1 Mean and Standard Deviation of HV

The values for mean (\(\mu _{HV}\)) and standard deviation (\(\sigma _{HV}\)) of the Vickers hardness are described by the following equations, respectively, which were approximated by a Taylor expansion [32].

$$\begin{aligned} \mu _{HV} = \frac{kP}{\mu _{d_{m}}^{2}} \left[ 1 + 3 \left( \frac{\sigma _{d_{m}}}{\mu _{d_{m}}} \right) ^{2} + 15\left( \frac{\sigma _{d_{m}}}{\mu _{d_{m}}} \right) ^{4} + ... \right] \end{aligned}$$
$$\begin{aligned} \sigma _{HV} = \left[ \left( \frac{kP}{\mu _{d_{m}}^{2}}\right) ^{2} \left[ 4 \left( \frac{\sigma _{d_{m}}}{\mu _{d_{m}}} \right) ^{2} + 57 \left( \frac{\sigma _{d_{m}}}{\mu _{d_{m}}} \right) ^{4} + ... \right] \right] ^{\frac{1}{2}} \end{aligned}$$

1.2 Probability Distribution for Vickers Hardness

The distribution proposed to represent Vickers hardness data is described by the following equation, probability density function (PDF) [32].

$$\begin{aligned} f(HV) = \frac{\sqrt{kP}}{2\sigma _{d}\sqrt{2\pi }}HV^{-\frac{3}{2}} \mathrm {exp}\left[ - \frac{1}{2} \left( \frac{\sqrt{\frac{kP}{HV}} - \mu _{d}}{\sigma _{d}} \right) ^{2} \right] \end{aligned}$$

where f(HV) is the PDF of Vickers hardness, k is the Vickers indenter constant, P is the test load [N], d is the mean diagonal of one indentation [\(\mu\)m], and \(\mu _{d}\) and \(\sigma _{d}\) are the mean and standard deviation of d [\(\mu\)m], respectively, for the entire data set.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gontarski, T.d.L., Leal, A.P., Casali, R.M. et al. Weightings on the Propagation of Errors in the Vickers Hardness Parameters. Braz J Phys 52, 107 (2022). https://doi.org/10.1007/s13538-022-01110-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s13538-022-01110-x

Keywords

Search

Navigation