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Nonparametric Method for Estimating the Distribution of Time to Failure of Engineering Materials

  • Antonio Meneses
  • Salvador NayaEmail author
  • Ignacio López-de-Ullibarri
  • Javier Tarrío-Saavedra
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 175)

Abstract

The aim of this work is to develop and assess a new method to estimate lifetime distribution in materials subjected to mechanical fatigue efforts. This problem is addressed from a statistical semiparametric and nonparametric perspective. Taking into account that fatigue failures in materials are due to crack formation and the subsequently induced crack growth, linear mixed effects regression models with smoothing splines (based on the linearized Paris-Erdogan model) are applied to estimate crack length as a function of the number of fatigue cycles. This model allows to simultaneously estimate the dependence between crack length and number of cycles in a sample of specimens. Knowing the crack length that induces material failure, the lifetime of each specimen is the crossing point of the crack length limit and the model crack length estimate. The authors propose to estimate the lifetime distribution function by applying nonparametric kernel techniques. In order to assess the influence of factors such as material type, material heterogeneity, and also that of the parameters of the estimation procedure, a simulation study consisting of different scenarios is performed. The results are compared with those of a procedure proposed by Meeker and Escobar (Statistical Methods for Reliability Data, Wiley, 1998, [16]) based on nonlinear mixed effects regression. Functional data analysis techniques are applied to perform this task. The proposed methodology estimates lifetime distribution of materials under fatigue more accurately in a wide range of scenarios.

Keywords

Fatigue crack growth Paris-Erdogan model Nonparametric kernel distribution function estimation Linear mixed effects Statistical learning Nonlinear mixed effects 

Notes

Acknowledgments

This research has been supported by the Spanish Ministry of Economy and Competitiveness, grant MTM2014-52876-R (ERDF included), and by the Secretariat for Higher Education, Science, Technology and Innovation of Ecuador (SENESCYT).

References

  1. 1.
    Atluri, S.N., Han, Z.D., Rajendran, A.M.: A new implementation of the meshless finite volume method, through the MLPG "mixed" approach. Comput. Model. Eng. Sci. 6, 491–514 (2004)Google Scholar
  2. 2.
    Box, G.E., Hunter, J.H., Hunter, W.G.: Statistics for Experimenters: Design, Innovation, and Discovery. Wiley (2005)Google Scholar
  3. 3.
    Braz, M.H.P.: Propriedades de Fadiga de Soldas de Alta Resistência e Baixa Liga com Diferentes Composições Microestruturais. Tese de M. Sc, USP, So Carlos, SP, Brasil (1999)Google Scholar
  4. 4.
    Callister, W., Rethwisch, D.G.: Materials Science and Engineering: an Introduction, 9th edn. Wiley (2013)Google Scholar
  5. 5.
    Campbell, F.C.: Elements of Metallurgy and Engineering Alloys. ASM International (2008)Google Scholar
  6. 6.
    Cuevas, A., Febrero, M., Fraiman, R.: On the use of the bootstrap for estimating functions with functional data. Comput. Stat. Data Anal. 51, 1063–1074 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Donahue, R.J., Clark, H.M., Atanmo, P., Kumble, R., McEvily, A.J.: Crack opening displacement and the rate of fatigue crack growth. Int. J. Fract.Mech. 8, 209–219 (1972)Google Scholar
  8. 8.
    Dong, L., Haynes, R., Atluri, S.N.: On improving the celebrated Paris’ power law for fatigue, by using moving least squares. CMC-Comput. Mater. Con. 45, 1–15 (2015)Google Scholar
  9. 9.
    Dowling, N.E.: Mechanical Behavior of Materials, Engineering Methods for Deformation, Fracture and Fatigue, Pearson Prentice Hall (2012)Google Scholar
  10. 10.
    Faraway, J.J.: Extending the Linear Model with R. Generalized Linear, Mixed Effects and Nonparametric Regression Models. Chapman & Hall/CRC (2005)Google Scholar
  11. 11.
    Febrero-Bande, M., Oviedo de la Fuente, M.: Statistical computing in functional data analysis: the R package fda.usc. J. Stat. Soft. 51, 1–28 (2012)CrossRefGoogle Scholar
  12. 12.
    FKM-Guideline: Fracture Mechanics Proof of Strength for Engineering Components (2004)Google Scholar
  13. 13.
    Forman, R.G., Kearney, V.E., Engle, R.M.: Numerical analysis of crack propagation in cyclic-loaded structures. J. Basic Eng. 89, 459–463 (1967)Google Scholar
  14. 14.
    Jones, R.: Fatigue crack growth and damage tolerance. Fatigue Fract. Eng. Mater. Struct. 37, 463–483 (2014)CrossRefGoogle Scholar
  15. 15.
    Meggiolaro, M.A., De Castro, J.T.P.: Equacionamento da curva de propagação de trincas por fadiga. III Seminário de Mecânica da Fratura/Integridade Estrutural (1997)Google Scholar
  16. 16.
    Meeker, W., Escobar, L.: Statistical Methods for Reliability Data. Wiley (1998)Google Scholar
  17. 17.
    Montgomery, D.: Design and Analysis of Experiments, 8th edn. Wiley (2013)Google Scholar
  18. 18.
    Montes, J., Cuevas, F., Cintas, J.: Ciencia e Ingeniería de los Materiales. Ediciones Paraninfo (2014)Google Scholar
  19. 19.
    Paris, P.C., Erdogan, F.: A critical analysis of crack propagation laws. J. Basic Eng. 85, 528 (1963)CrossRefGoogle Scholar
  20. 20.
    Pinheiro, J.C., Bates, D.M.: Approximations to the loglikelihood function in the nonlinear mixed-effects model. J. Comput. Graph. Stat. 4, 12–35 (1995)Google Scholar
  21. 21.
    Pinheiro, J.C., Bates, D.M.: Mixed-effects Models in S and S-plus. Springer (2000)Google Scholar
  22. 22.
    Pinheiro, J., Bates, D., DebRoy, S., Sarkar, D., R Core Team.: nlme: linear and nonlinear mixed effects Models (2014). R package version 3.1-116. http://CRAN.R-project.org/package=nlme
  23. 23.
    Polansky, A.M., Baker, E.R.: Multistage plug-in bandwidth selection for kernel distribution function estimates. J. Stat. Comput. Simul. 65, 63–80 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Quintela-del-Rio, A., Estevez-Perez, G.: Nonparametric kernel distribution function estimation with kerdiest: an R package for bandwidth choice and applications. J. Stat. Softw. 50, 1–21 (2012)CrossRefGoogle Scholar
  25. 25.
    Ramsey, J.: pspline: penalized smoothing splines (2013). R package version 1.0-16. http://CRAN.R-project.org/package=pspline
  26. 26.
    Shariff, A.: Simulation of Paris-Erdogan crack propagation model with power value, m = 3: The impact of applying discrete values of stress range on the behaviour of damage and lifetime of structure. Asian J. Appl. Sci. 2, 91–95 (2009)CrossRefGoogle Scholar
  27. 27.
    Verbeke, G., Molenberghs, G.: Linear Mixed Models for Longitudinal Data. Springer (2009)Google Scholar
  28. 28.
    Wand, M.P., Jones, M.C.: Kernel Smoothing. Chapman & Hall/CRC (1995)Google Scholar
  29. 29.
    Wang, K., Wang, F., Cui, W., Hayat, T., Ahmad, B.: Prediction of short fatigue crack growth of Ti-6Al-4V. Fatigue Fract. Eng. Mater. Struct. 37, 1075–1086 (2014)CrossRefGoogle Scholar
  30. 30.
    Wood, S.N.: Generalized Additive Models: An Introduction with R. Chapman & Hall/CRC (2006)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Antonio Meneses
    • 1
  • Salvador Naya
    • 3
    Email author
  • Ignacio López-de-Ullibarri
    • 2
  • Javier Tarrío-Saavedra
    • 3
  1. 1.Universidad Nacional de ChimborazoRiobambaEcuador
  2. 2.Universidade da Coruña. Escola Universitaria PolitécnicaFerrolSpain
  3. 3.Universidade da Coruña. Escola Politécnica SuperiorFerrolSpain

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