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An efficient analytical global–local (AGL) analysis of the Lamb wave scattering problem for detecting a horizontal crack in a stiffened plate

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Abstract

This paper addresses a forward problem using an analytical global–local (AGL) method based on the physics of the Lamb wave propagation in the presence of a crack in the complex structure which would be beneficial for the inverse problem of damage detection. Discontinuity in a structure acts as a damage source and interacts with a Lamb wave. The global analytical solution determines the wave generation by a transmitter, wave propagation through the structure, and detection by a receiver. The local analytical solution determines the scattering coefficients for the Lamb waves at the discontinuity location. These frequency-dependent scatter coefficients are calculated considering transmission, reflection, and mode conversion. An analytical method called “complex mode expansion with vector projection (CMEP)” is used to calculate the scattering coefficients of Lamb wave modes from geometric discontinuities. The scattered wave field from a discontinuity is expanded in terms of complex Lamb wave modes with unknown scatter coefficients. These unknown coefficients are obtained from the boundary conditions using a vector projection utilizing the power expression. Two test cases are considered in this paper: (a) a plate with a pristine stiffener and (b) a plate with a cracked stiffener. Complex-valued scattering coefficients are calculated from 50 to 350 kHz for S0 incident waves. Scatter coefficients are compared for both cases to identify the suitable frequency range to excite a Lamb wave to detect the crack. The frequency-dependent complex-valued scattering coefficients are then inserted into the global analytical model. Therefore, in combination AGL method provides the exact analytical Lamb wave solution for the simulation of Lamb wave propagation and interaction with a discontinuity. By comparing the waveforms for both pristine stiffener and cracked stiffener, the crack can be detected. An FEM transient analysis was also performed to calculate the scattered wave signals. FEM results agree well with the AGL predicted results. An experiment was also performed to validate the AGL result. The obtained experimental results match well with the CMEP analytical predictions. The present AGL method is a highly computationally efficient simulation approach which allows performing virtual experiments for structural health monitoring applications.

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References

  1. Farrar, C.R., Worden, K.: An introduction to structural health monitoring. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 365, 303–315 (2007)

    Article  Google Scholar 

  2. Shen, Y., Giurgiutiu, V.: Combined analytical FEM approach for efficient simulation of Lamb wave damage detection. Ultrasonics 69, 116–128 (2016)

    Article  Google Scholar 

  3. Liu, M., Schmicker, D., Su, Z., Cui, F.: A benchmark study of modeling Lamb wave scattering by a through hole using a time-domain spectral element method. J. Nondestruct. Eval. Diagn. Progn. Eng. Syst. 1(2), 21006 (2018)

    Google Scholar 

  4. Lee, B.C., Staszewski, W.J.: Modelling of Lamb waves for damage detection in metallic structures: part I. Wave propagation. Smart Mater. Struct. 12, 804–814 (2003)

    Article  Google Scholar 

  5. Tua, P.S., Quek, S.T., Wang, Q.: Detection of cracks in plates using piezo-actuated Lamb waves. Smart Mater. Struct. 13(4), 643–660 (2004)

    Article  Google Scholar 

  6. Gaul, L., Sprenger, H., Schaal, C., Bischoff, S.: Structural health monitoring of cylindrical structures using guided ultrasonic waves. Acta Mech. 223(8), 1669–1680 (2012)

    Article  Google Scholar 

  7. Ru, Y., Wang, G.F., Su, L.C., Wang, T.J.: Scattering of vertical shear waves by a cluster of nanosized cylindrical holes with surface effect. Acta Mech. 224(5), 935–944 (2013)

    Article  MathSciNet  Google Scholar 

  8. Haider, M.F., Giurgiutiu, V.: Analysis of axis symmetric circular crested elastic wave generated during crack propagation in a plate: a Helmholtz potential technique. Int. J. Solids Struct. 134, 130–150 (2018)

    Article  Google Scholar 

  9. Grahn, T.: Lamb wave scattering from a circular partly through-thickness hole in a plate. Wave Motion 37(1), 63–80 (2003)

    Article  MathSciNet  Google Scholar 

  10. Koshiba, M., Karakida, S., Suzuki, M.: Finite-element analysis of Lamb wave scattering in an elastic plate waveguide. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 31, 18–24 (1984)

    Google Scholar 

  11. Friswell, M.I.: Damage identification using inverse methods. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 365, 393–410 (2007)

    Article  Google Scholar 

  12. Cho, Y., Rose, J.L.: A boundary element solution for a mode conversion study on the edge reflection of Lamb waves. J. Acoust. Soc. Am. 99(4), 2097–2109 (1996)

    Article  Google Scholar 

  13. Cho, Y., Rose, J.L.: An elastodynamic hybrid boundary element study for elastic guided wave interactions with a surface breaking defect. Int. J. Solids Struct. 37(30), 4103–4124 (2000)

    Article  Google Scholar 

  14. Galán, J.M., Abascal, R.: Lamb mode conversion at edges. A hybrid boundary element-finite-element solution. J. Acoust. Soc. Am. 117(4), 1777–1784 (2005)

    Article  Google Scholar 

  15. Chakraborty, A., Gopalakrishnan, S.: Wave propagation in inhomogeneous layered media: solution of forward and inverse problems. Acta Mech. 169(1–4), 153–185 (2004)

    Article  Google Scholar 

  16. Mackerle, J.: Finite-element modelling of non-destructive material evaluation, an addendum: a bibliography (1997–2003). Model. Simul. Mater. Sci. Eng. 12(5), 799–834 (2004)

    Article  Google Scholar 

  17. Moser, F., Jacobs, L.J., Qu, J.: Modeling elastic wave propagation in waveguides with the finite element method. NDT E Int. 32(4), 225–234 (1999)

    Article  Google Scholar 

  18. Glushkov, E., Glushkova, N., Eremin, A., Giurgiutiu, V.: Low-cost simulation of guided wave propagation in notched plate-like structures. J. Sound Vib. 352, 80–91 (2015)

    Article  Google Scholar 

  19. Bai, H., Zhu, J., Shah, A.H., Popplewell, N.: Three-dimensional steady state Green function for a layered isotropic plate. J. Sound Vib. 269(1–2), 251–271 (2004)

    Article  Google Scholar 

  20. Ahmad, Z.A.B., Gabbert, U.: Simulation of Lamb wave reflections at plate edges using the semi-analytical finite element method. Ultrasonics 52(7), 815–820 (2012)

    Article  Google Scholar 

  21. Shen, Y., Cesnik, C.E.S.: Local interaction simulation approach for efficient modeling of linear and nonlinear ultrasonic guided wave active sensing of complex structures. J. Nondestruct. Eval. Diagn. Progn. Eng. Syst. 1(1), 11008 (2017)

    Google Scholar 

  22. Flores-López, M.A., Douglas Gregory, R.: Scattering of Rayleigh–Lamb waves by a surface breaking crack in an elastic plate. J. Acoust. Soc. Am. 119(4), 2041–2049 (2006)

    Article  Google Scholar 

  23. Doyle, J.F.: Wave Propagation in Structures: Spectral Analysis Using Fast Discrete Fourier Transforms, vol. 2. Springer, New York (1997)

    Book  Google Scholar 

  24. Chakraborty, A., Gopalakrishnan, S.: A spectral finite element model for wave propagation analysis in laminated composite plate. J. Vib. Acoust. 128(4), 477 (2006)

    Article  Google Scholar 

  25. Ajith, V., Gopalakrishnan, S.: Wave propagation in stiffened structures using spectrally formulated finite element. Eur. J. Mech. A Solids 41, 1–15 (2013)

    Article  MathSciNet  Google Scholar 

  26. Ostachowicz, W.M.: Damage detection of structures using spectral finite element method. Comput. Struct. 86(3–5), 454–462 (2008)

    Article  Google Scholar 

  27. Castaings, M., Le Clezio, E., Hosten, B.: Modal decomposition method for modeling the interaction of Lamb waves with cracks. J. Acoust. Soc. Am. 112(6), 2567–2582 (2002)

    Article  Google Scholar 

  28. Terrien, N., Osmont, D., Royer, D., Lepoutre, F., Déom, A.: A combined finite element and modal decomposition method to study the interaction of Lamb modes with micro-defects. Ultrasonics 46(1), 74–88 (2007)

    Article  Google Scholar 

  29. Gregory, R.D., Gladwell, I.: The cantilever beam under tension, bending or flexure at infinity. J. Elast. 12(4), 317–343 (1982)

    Article  MathSciNet  Google Scholar 

  30. Moreau, L., Caleap, M., Velichko, A., Wilcox, P.D.: Scattering of guided waves by flat-bottomed cavities with irregular shapes. Wave Motion 49(2), 375–387 (2012)

    Article  MathSciNet  Google Scholar 

  31. Yang, J., Li, X.: The scattering of the SH wave on a limited permeable crack in a functionally graded piezoelectric substrate bonded to a homogeneous piezoelectric strip. Acta Mech. 226(10), 3205–3219 (2015)

    Article  MathSciNet  Google Scholar 

  32. Du, J.K., Shen, Y.P., Wang, X.: Scattering of anti-plane shear waves by a partially debonded piezoelectric circular cylindrical inclusion. Acta Mech. 158(3–4), 169–183 (2002)

    Article  Google Scholar 

  33. Pau, A., Capecchi, D., Vestroni, F.: Reciprocity principle for scattered fields from discontinuities in waveguides. Ultrasonics 55, 85–91 (2015)

    Article  Google Scholar 

  34. Poddar, B., Giurgiutiu, V.: Scattering of Lamb waves from a discontinuity: an improved analytical approach. Wave Motion 65, 79–91 (2016)

    Article  MathSciNet  Google Scholar 

  35. Poddar, B., Giurgiutiu, V.: Complex modes expansion with vector projection using power flow to simulate Lamb waves scattering from horizontal cracks and disbonds. J. Acoust. Soc. Am. 140(3), 2123–2133 (2016)

    Article  Google Scholar 

  36. Poddar, B.: Physics Based Modeling of Guided Waves for Detection and Characterization of Structural Damage in NDE and SHM. University of South Carolina, Columbia (2016)

    Google Scholar 

  37. Giurgiutiu, V., Faisal Haider, M., Poddar, B.: A novel analytical global local (AGL) approach to determine scattering of Lamb waves from discontinuity, USC Technology ID # 1324, Patent application for U.S. Letters Patent bearing U.S. Serial No. 62/672,179 (2018)

  38. Shen, Y., Giurgiutiu, V.: Effective non-reflective boundary for Lamb waves: theory, finite element implementation, and applications. Wave Motion 58, 22–41 (2015)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors are grateful for the financial support from National Aeronautics and Space Administration (NASA) under Grant No. NNX17CL69P.

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Authors and Affiliations

  1. Department of Mechanical Engineering, University of South Carolina, 300 Main Street, Columbia, SC, 29208, USA

    Mohammad Faisal Haider, Roshan Joseph & Victor Giurgiutiu

  2. Intelligent Automation, Inc., Rockville, MD, USA

    Banibrata Poddar

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  1. Mohammad Faisal Haider
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  2. Roshan Joseph
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  3. Victor Giurgiutiu
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  4. Banibrata Poddar
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Correspondence to Mohammad Faisal Haider.

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Haider, M.F., Joseph, R., Giurgiutiu, V. et al. An efficient analytical global–local (AGL) analysis of the Lamb wave scattering problem for detecting a horizontal crack in a stiffened plate. Acta Mech 231, 577–596 (2020). https://doi.org/10.1007/s00707-019-02555-z

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  • DOI: https://doi.org/10.1007/s00707-019-02555-z

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