Abstract
In multi-layered composite laminates, Lamb wave equations are obtained using the transfer matrix method and global matrix method. These methods have numerical issues (missing roots or spurious roots) while solving the Lamb wave equations especially at high frequencies and for the laminates with a large number of layers. In the present work, an effective stiffness matrix method (ESM) is presented to solve the Lamb wave equations without numerical issues. The proposed ESM method offers a simple and mathematically straightforward formulation as it considers the multi-layered laminate as a single homogenous layer with effective stiffness properties. The Lamb wave equations of a single monoclinic layer are first derived by considering the displacement field in three directions and solved for obtaining dispersion curves. The proposed ESM method is then applied to various laminate configurations to test the effectiveness of the method. The different laminate configurations include quasi-isotropic, cross-ply, generally anisotropic and orthotropic laminates. The efficacy of the proposed method is established in these cases. In addition, the directional dependency of Lamb wave propagation characteristic (wave velocity) with laminate configurations is evaluated and analysed.
Similar content being viewed by others
References
Wang, L., Yuan, F.G.: Lamb wave propagation in composite laminates using a higher-order plate theory. In: Proceedings of SPIE 6531 Nondestructive Characterization for Composite Materials, Aerospace Engineering, Civil Infrastructure, and Homeland Security, vol. 6531, pp. 653101–653112 (2007)
Marzani, A., Viola, E., Bartoli, I., di Lanza, S.F., Rizzo, P.: A semi-analytical finite element formulation for modeling stress wave propagation in axisymmetric damped waveguides. J. Sound Vib. 318, 488–505 (2008)
Thomson, W.T.: Transmission of elastic waves through a stratified solid medium. J. Appl. Phys. 21, 89–93 (1950)
Haskell, N.A.: The Dispersion of surface waves on multilayered media. Bull. Seismol. Soc. Am. 23, 10–23 (1970)
Anderson, D.L.: Elastic wave propagation in layered anisotropic media. J. Geophys. Res. 66, 2953–63 (1961). https://doi.org/10.1029/JZ066i009p02953
Nayfeh, A.H., Chimenti, D.E.: Free wave propagation in plates of general anisotropic media. J. Appl. Mech. 56, 181–188 (1989)
Nayfeh, A.H.: The general problem of elastic wave propagation in multilayered anisotropic media. J. Acoust. Soc. Am. 89, 1521–1531 (1991)
Wang, L., Rokhlin, S.I.: Stable reformulation of transfer matrix method for wave propagation in layered anisotropic media. Ultrasonics. 39, 413–424 (2001)
Rokhlin, S.I., Wang, L.: Stable recursive algorithm for elastic wave propagation in layered anisotropic media: stiffness matrix method. J. Acoust. Soc. Am. 112, 822–834 (2002)
Wang, L., Yuan, F.G.: Group velocity and characteristic wave curves of Lamb waves in composites: modeling and experiments. Compos. Sci. Technol. 67, 1370–1384 (2007)
Knopoff, L.: A matrix method for elastic wave problems. Bull. Seismol. Soc. Am. 54, 431–438 (1964)
Pavlakovic, B., Lowe, M., Alleyne, D., Cawley, P.: Disperse: a general purpose program for creating dispersion curves. Rev. Prog. Quant. Nondestruct. Eval. 185–192 (1997). https://doi.org/10.1007/978-1-4615-5947-4_24
Lowe, M.J.S., Cawley, P.: A General Purpose Computer Model for Calculating Elastic Waveguide Properties, with Application to Non-Destructive Testing. Surf. Waves Anisotropic Laminated Bodies Defects Detect. 241–256 (2005). https://doi.org/10.1007/1-4020-2387-1_14
Pant, S., Laliberte, J., Martinez, M., Rocha, B.: Derivation and experimental validation of Lamb wave equations for an n-layered anisotropic composite laminate. Compos. Struct. 111, 566–579 (2014)
Barazanchy, D., Giurgiutiu, V.: A unified formulation for predictive modeling of guided-ultrasonic wave dispersion curves in metallic and composite materials. J. Intell. Mater. Syst. Struct. 28, 1272–1286 (2017)
Barazanchy, D., Giurgiutiu, V., Ricci, F.: A comparative convergence and accuracy study of composite guided-ultrasonic wave solution methods: comparing the unified analytic method, SAFE method and DISPERSE. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci. 231, 2961–2973 (2017)
Sun, C.T., Li, S.: Three-dimensional effective elastic constants for thick laminates. J. Compos. Mater. 22, 629–639 (1988)
Zhao, J., Ji, H., Qiu, J.: Modeling of Lamb waves in composites using new third-order plate theories. Smart Mater. Struct. 23, 45017 (2014). https://doi.org/10.1088/0964-1726/23/4/045017
Cawley, P.: The rapid non-destructive inspection of large composite structures. Composites 25, 351–357 (1994)
Sharif-Khodaei, Z., Aliabadi, M.H.: Assessment of delay-and-sum algorithms for damage detection in aluminium and composite plates. Smart Mater. Struct. 23, 075007 (2014)
Watkins, R., Jha, R.: A modified time reversal method for Lamb wave based diagnostics of composite structures. Mech. Syst. Signal Process. 31, 345–354 (2012)
Sohn, B.H., Park, H.W., Law, K.H., Farrar, C.R.: Damage detection in composite plates by using an enhanced time reversal method. J. Aerosp. Eng. 20, 141–151 (2004)
Harb, M.S., Yuan, F.G.: Barely visible impact damage imaging using non-contact air-coupled transducer/laser Doppler vibrometer system. Struct. Heal. Monit. 16, 663–673 (2017)
Ewins, D.J.: Modal Testing: Theory. Research Studies Press LTD, Practice and Application (2009)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Nandyala, A.R., Darpe, A.K. & Singh, S.P. Effective stiffness matrix method for predicting the dispersion curves in general anisotropic composites. Arch Appl Mech 89, 1923–1938 (2019). https://doi.org/10.1007/s00419-019-01552-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00419-019-01552-x