Scattering of Lamb Waves in a Composite Plate

Abstract

Recent investigations of space construction techniques have explored the use of composite materials in the construction of space stations and platforms. These composites offer superior strength to weight ratio and are thermally stable. Examples of these materials are laminates of graphite fibers in an epoxy or a metal (Al, Mg) matrix and boron fibers in an aluminum matrix. The overall effective elastic constants of such a medium can be calculated from fiber and matrix properties by using an effective modulus theory as shown in [1] and [2]. The investigation of propagation and scattering of elastic waves in composite materials is necessary in order to develop an ability to characterize cracks and predict the reliability of composite structures. The objective of this investigation is the characterization of a surface breaking crack by ultrasonic techniques. In particular, the use of Lamb waves for this purpose is studied here. The Lamb waves travel through the plate, encountering a crack, and scatter. Of interest is the modeling of the scattered wave in terms of the Lamb wave modes. The direct problem of propagation and scattering of Lamb waves by a surface breaking crack has been analyzed. This would permit an experimentalist to characterize the crack by comparing the measured response to the analytical model. The plate is assumed to be infinite in the x and y directions with a constant thickness in the z direction. The top and bottom surfaces are traction free. Solving the governing wave equations and using the stress-free boundary conditions results in the dispersion equation. This equation yields the guided modes in the homogeneous plate. The theoretical model is a hybrid method that combines analytical and finite elements techniques to describe the scattered displacements. A finite region containing the defects is discretized by finite elements. Outside the local region, the far field solution is expressed as a Fourier summation of the guided modes obtained from the dispersion equation. Continuity of tractions and displacements at the boundaries of the two regions provides the necessary equations to determine the expansion coefficients and the nodal displacements. This method was used for out-of-plane (SH) wave scattering in an isotropic plate[3]. A combined analytical and finite element formulation for a single layered isotropic plate in the state of plane strain was investigated in [4]. In this study the authors considered only the lowest symmetric mode and geometrically symmetric cracks. In [5] a variational approach was used to investigate scattering by a symmetric pair of surface breaking thin slots. Employing standard elastostatic crack solutions as trial functions the authors examined the scattering by the first symmetric mode. A finite difference method was used in [6] to calculate the scattering of Lamb and shear waves from surface breaking cracks. In [7] a modified Wiener-Hopf technique was used to analyze scattering of Lamb waves by a crack. Applying this technique, the authors in [8] studied quantitative sizing of spot welds in joined sheets. Besides the finite difference and finite element techniques, the analytical approaches are not suitable for analyzing arbitrarily shaped defects and anistropic media. In the hybrid method used here these defects can be of arbitrary shapes as well as inclusions of different material. Recently, using the hybrid method, the scattering by surface-breaking cracks in isotropic homogeneous and welded plates has been examined in [9].