S1-ZGV Modes of a Linear and Nonlinear Profile for Functionally Graded Material Using Power Series Technique

The present work deals with functionally graded materials (FGM) isotropic plates in the neighborhood of the first-order symmetric zero group velocity (S1-ZGV) point. The mechanical properties of functionally graded material (FGM) are assumed to vary continuously through the thickness of the plate and obey a power law of the volume fraction of the constituents. Governing equations for the problem are derived, and the power series technique (PST) is employed to solve the recursive equations. The impact of the FGM basic materials properties on S1-ZGV frequency of FGM plate is investigated. Numerical results show that S1-ZGV frequency is comparatively more sensitive to the shear modulus. The gradient coefficient does not affect the linear dependence of ZGV frequency as function of cut-off frequency ; only the slope is slightly varied. 1. Introduction A functionally graded material (FGM) is a kind of an inhomogeneous material. The characterization of mechanical properties of materials is important for testing their structural integrity. Lamb waves are frequently employed in the ultrasonic characterization of thin plates [1]. As an important property of Lamb waves, the zero group velocity (ZGV) at the frequency minimum of the first-order symmetric (S1) continues to be of an interest for the scientific community [2, 3]. Tolstoy and Usdin pointed out that for the S1 Lamb mode, group velocity vanishes at a particular point of the dispersion curve and predicted that this zero group velocity point must be associated with a sharp continuous wave resonance and ringing effect [4]. Holland and Chimenti demonstrated the exploitation of this mode for high-sensitivity imaging applications. With air-coupled transducers, they observed the transparency of a plate due to the S1 mode ZGV resonance [5]. The S1-ZGV frequency is obviously sensitive to mechanical properties and to any change in the plate thickness. To exploit this phenomenon recent works evoke the idea that it may be suitable for the measurement of nanometer-scale thickness variations in homogeneous plates [6, 7]. Due to the resulting differential equations of variable coefficients associated with the spatial variation of the material properties, the wave propagation in FGM remains difficult to analyze. Some numerical [8–10] and analytical methods [11–16] have been applied in order to study the wave propagation behavior in an inhomogeneous medium with material properties varying continuously along the depth direction. In an effort to show the interest of ZGV in the study of FGM materials, Bouhdima [15] first