Derivation and experimental validation of Lamb wave equations for an n-layered anisotropic composite laminate
Lamb waves are ultrasonic guided waves that propagate between two parallel free surfaces and their use for damage detection has been widely explored and demonstrated. Damage in materials/structures can be detected by analyzing the difference between the phase/group velocity and the loss of amplitude of Lamb waves on damaged and un-damaged specimens. The propagation characteristics of Lamb waves are described in the form of dispersion curves, which are plots of phase/group velocities versus the product of frequency-thickness generated by solving the Lamb wave equations. Lamb waves' dispersion behaviors for isotropic materials are well established in the literature; however, such is not the case for the laminated composites. The most common methods for solving the Lamb wave equations in composites consist of using laminated plate theory or 3D linear elasticity by assuming an orthotropic and/or higher symmetry. This assumption may not be true, if the actuators and sensors in an orthotropic or transversely isotropic laminates are installed in a non-principle direction or the layup is symmetric but not balanced. This paper presents a full derivation of Lamb wave equations for n-layered monoclinic composite laminates based on linear 3D elasticity by considering the displacement fields in all three directions using the partial wave technique in combination with the Global Matrix (GM) approach. In the partial wave technique, the principle of superposition of three upward and three downward travelling plane waves are assumed in order to satisfy the associated boundary conditions. The bounded upper and lower surfaces reflect the waves and the combination of these reflections going towards the upper or lower interfaces results in the propagating guided waves. The GM approach is used to assemble all the equations from each layer to form a global, unified matrix that describes the displacement and stress fields along the entire laminate associated with the wave propagation. A robust method for numerically solving the Lamb wave equations is also presented. The presented method was verified experimentally by analyzing the propagation of Lamb waves in two different composite panels constructed out of unidirectional carbon-fiber epoxy prepreg and fiber-metal laminate (GLARE 3-3/4). The panels were instrumented with lead zirconate titanate (PZT) piezoelectric sensors, which were excited at different frequencies ranging from 20. kHz to 500. kHz to generate and acquire the waves. The waves were excited and gathered at three different propagation angles of 0°, 45°, and 90° for the carbon-fiber epoxy laminate panel and at six different angles of 0°, 20°, 45°, 70° and 90° for the fiber-metal laminates (GLARE). The phase and group velocities of the fundamental symmetric ( So) and anti-symmetric ( Ao) Lamb waves were extracted by tracking the peaks of each individual wave phase and the wave envelope respectively using an in-house code developed in MATLAB. It was found that the presented 3D linear elasticity model followed the experimental data closely for both symmetric and anti-symmetric Lamb modes. The analytical method presented in this paper was able to predict the Lamb wave dispersion for both the carbon-fiber epoxy laminate and the hybrid fiber-metal laminate proving the robustness and versatility of the solution method.
|Keywords||Acoustic-ultrasonic, Composite, Dispersion curve, Fiber-metal laminate, Guided wave, Lamb wave|
Pant, S., Laliberte, J, Martinez, M, & Rocha, B. (2014). Derivation and experimental validation of Lamb wave equations for an n-layered anisotropic composite laminate. Composite Structures, 111(1), 566–579. doi:10.1016/j.compstruct.2014.01.034