Experimental Mechanics

, Volume 53, Issue 4, pp 635–648 | Cite as

Bayesian Identification of Elastic Constants in Multi-Directional Laminate from Moiré Interferometry Displacement Fields

  • C. GoguEmail author
  • W. Yin
  • R. Haftka
  • P. Ifju
  • J. Molimard
  • R. Le Riche
  • A. Vautrin


The ply elastic constants needed for classical lamination theory analysis of multi-directional laminates may differ from those obtained from unidirectional laminates because of three dimensional effects. In addition, the unidirectional laminates may not be available for testing. In such cases, full-field displacement measurements offer the potential of identifying several material properties simultaneously. For that, it is desirable to create complex displacement fields that are strongly influenced by all the elastic constants. In this work, we explore the potential of using a laminated plate with an open-hole under traction loading to achieve that and identify all four ply elastic constants (E 1 , E 2 , ν 12 , G 12 ) at once. However, the accuracy of the identified properties may not be as good as properties measured from individual tests due to the complexity of the experiment, the relative insensitivity of the measured quantities to some of the properties and the various possible sources of uncertainty. It is thus important to quantify the uncertainty (or confidence) with which these properties are identified. Here, Bayesian identification is used for this purpose, because it can readily model all the uncertainties in the analysis and measurements, and because it provides the full coupled probability distribution of the identified material properties. In addition, it offers the potential to combine properties identified based on substantially different experiments. The full-field measurement is obtained by moiré interferometry. For computational efficiency the Bayesian approach was applied to a proper orthogonal decomposition (POD) of the displacement fields. The analysis showed that the four orthotropic elastic constants are determined with quite different confidence levels as well as with significant correlation. Comparison with manufacturing specifications showed substantial difference in one constant, and this conclusion agreed with earlier measurement of that constant by a traditional four-point bending test. It is possible that the POD approach did not take full advantage of the copious data provided by the full field measurements, and for that reason that data is provided for others to use (as on line material attached to the article).


Bayesian identification Composites Full-fields Elastic constants Uncertainty quantification 



This work was supported in part by the NASA grant NNX08AB40A. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Aeronautics and Space Administration.


  1. 1.
    Molimard J, Le Riche R, Vautrin A, Lee JR (2005) Identification of the four orthotropic plate stiffnesses using a single open-hole tensile test. Exp Mech 45:404–411CrossRefGoogle Scholar
  2. 2.
    Lecompte D, Sol H, Vantomme J, Habraken AM (2007) Mixed numerical-experimental technique for orthotropic parameter identification using biaxial tensile tests on cruciform specimens. Int J Solids Struct 44:1643–1656CrossRefGoogle Scholar
  3. 3.
    Avril S, Bonnet M, Bretelle AS, Grédiac M, Hild F, Ienny P, Latourte F, Lemosse D, Pagano S, Pagnacco E (2008) Overview of identification methods of mechanical parameters based on full-field measurements. Exp Mech 48(4):381–402CrossRefGoogle Scholar
  4. 4.
    Kaipio JP, Somersalo E (2005) Statistical and computational inverse problems. Springer, New YorkzbMATHGoogle Scholar
  5. 5.
    Gogu C, Haftka RT, Le Riche R, Molimard J, Vautrin A (2010) An introduction to the Bayesian approach applied to elastic constants identification. AIAA J 48(5):893–903CrossRefGoogle Scholar
  6. 6.
    Isenberg J (1979) Progressing from least squares to Bayesian estimation. Proc., ASME Winter Annual Meeting, New York, NY, Paper 79-A/DSC-16Google Scholar
  7. 7.
    Sol H (1986) Identification of anisotropic plate rigidities, PhD dissertation (Vrije Universiteit Brussel)Google Scholar
  8. 8.
    Lai TC, Ip KH (1996) Parameter estimation of orthotropic plates by Bayesian sensitivity analysis. Compos Struct 34(1):29–42CrossRefGoogle Scholar
  9. 9.
    Lecompte D, Sol H, Vantomme J, Habraken AM (2005) Identification of elastic orthotropic material parameters based on ESPI measurements.Proc, SEM Annual Conf Expo Exp Appl MechGoogle Scholar
  10. 10.
    Silva G, Le Riche R, Molimard J, Vautrin A, Galerne C (2007) Identification of material properties using FEMU: application to the open hole tensile test. Appl Mech Mater 7–8:73–78CrossRefGoogle Scholar
  11. 11.
    Noh WJ (2004) Mixed mode interfacial fracture toughness of sandwich composites at cryogenic temperatures, Master’s Thesis, University of Florida. Google Scholar
  12. 12.
    Post D, Han B, Ifju P (1997) High sensitivity moiré: Experimental analysis for mechanics and materials. Springer, New YorkGoogle Scholar
  13. 13.
    Walker CA (1994) A historical review of moiré interferometry. Exp Mech 34(4):281–299CrossRefGoogle Scholar
  14. 14.
    Wood JD, Wang R, Weiner S, Pashley DH (2003) Mapping of tooth deformation caused by moisture change using moire interferometry. Dent Mater 19(3):159–166CrossRefGoogle Scholar
  15. 15.
    Lee JR, Molimard J, Vautrin A, Surrel Y (2006) Diffraction grating interferometers for mechanical characterisations of advanced fabric laminates. Opt Laser Technol 38(1):51–66CrossRefGoogle Scholar
  16. 16.
    Yin W (2009) Automated strain analysis system: Development and applications. PhD dissertation, University of Florida.Google Scholar
  17. 17.
    Gogu C (2009) Facilitating Bayesian identification of elastic constants through dimensionality reduction and response surface methodology, PhD dissertation, University of Florida and Ecole des Mines de Saint Etienne.Google Scholar
  18. 18.
    Myers RH, Montgomery DC (2002) Response surface methodology: Process and product in optimization using designed experiments. Wiley, New YorkzbMATHGoogle Scholar
  19. 19.
    Gogu C, Haftka RT, Le Riche R, Molimard J, Vautrin A, Sankar BV (2009) Bayesian statistical identification of orthotropic elastic constants accounting for measurement and modeling errors, AIAA paper 2009–2258, 11th AIAA Non-Deterministic Approaches Conference, Palm Springs, CA, May 2009Google Scholar
  20. 20.
    Smarslok BP, Haftka RT, Ifju P (2008) A correlation model for graphite/epoxy properties for propagating uncertainty to strain response, 23rd Annual Technical Conference of the American Society for Composites, Memphis, Tenn.Google Scholar
  21. 21.
    Jolliffe IT (2002) Principal component analysis, 2nd edn. Springer, New YorkzbMATHGoogle Scholar
  22. 22.
    Allen DM (1971) Mean square error of prediction as a criterion for selecting variables. Technometrics 13:469–475zbMATHCrossRefGoogle Scholar
  23. 23.
    Gogu C, Haftka RT, Le Riche R, Molimard J (2010) Effect of approximation fidelity on vibration based elastic constants identification. Struct Multidiscip Optim 42(2):293–304CrossRefGoogle Scholar

Copyright information

© Society for Experimental Mechanics 2012

Authors and Affiliations

  • C. Gogu
    • 1
    Email author
  • W. Yin
    • 2
  • R. Haftka
    • 2
  • P. Ifju
    • 2
  • J. Molimard
    • 3
  • R. Le Riche
    • 4
  • A. Vautrin
    • 5
  1. 1.Université de Toulouse; INSA, UPS, Mines Albi, ISAE; ICA (Institut Clément Ader)ToulouseFrance
  2. 2.Mechanical and Aerospace Engineering DepartmentUniversity of FloridaGainesvilleUSA
  3. 3.CIS-EMSE, CNRS:UMR5146, LCGEcole Nationale Supérieure des MinesSaint EtienneFrance
  4. 4.FAYOL-EMSE, CNRS:UMR5146, LCGEcole Nationale Supérieure des MinesSaint EtienneFrance
  5. 5.SMS-EMSE, CNRS:UMR5146, LCGEcole Nationale Supérieure des MinesSaint EtienneFrance

Personalised recommendations