One-Dimensional Transient Dynamic Response in Functionally Graded Layered Media

Abstract

In this study, one-dimensional transient wave propagation in multilayered functionally graded media is investigated. The multilayered medium consists of N different layers of functionally graded materials (FGMs), i.e., it is assumed that the stiffness and the density of each layer are varying continuously in the direction perpendicular to the layering, but isotropic and homogeneous in the other two directions. The top surface of the layered medium is subjected to a uniform dynamic in-plane time-dependent normal stress; whereas, the lower surface of the layered medium is assumed free of surface tractions or fixed. Moreover, the multilayered medium is assumed to be initially at rest and its layers are assumed to be perfectly bonded to each other. The method of characteristics is employed to obtain the solutions of this initial-boundary-value problem. The numerical results are obtained and displayed in curves showing the variation of the normal stress component with time. These curves reveal clearly the scattering effects caused by the reflections and refractions of waves at the boundaries and at the interfaces of the layers. The curves also display the effects of non-homogeneity in the wave profiles. The curves further show that the numerical technique applied in this study is capable of predicting the sharp variations in the field variables in the neighborhood of the wave fronts. By suitably adjusting the material constants, solutions for the case of isotropic, homogeneous and linearly elastic multilayered media and for some special cases including two different functionally graded layers are also obtained. Furthermore, solutions for some special cases are compared with the existing solutions in the literature; very good agreement is found.

Analysis of complicated one‐dimensional stress oscillation behaviour in a functionally graded material plate

“…3 and 4 should be verified, but comparable experimental data or numerical results could not be found in the literature. Therefore, the one-dimensional elastodynamic problem considered in this paper has been analyzed numerically for the cases ofp = −δ andq =δ as well as ofp =δ andq = 3δ by employing the method of characteristics [3,15]. The time histories of stresses obtained from the numerical solutions are illustrated in Figs.…”

“…A similar problem was later solved with the Laplace transform and numerical Laplace inversion techniques [2]. Abu-Alshaikh and Köklüce studied a one-dimensional elastodynamic behaviour in a multilayer FGM plate by means of the method of characteristics [3]. Numerical solutions to plane problems in FGM rectangular plates were obtained using FEM [4] or the improved composite wave propagation scheme [5].…”

An operator-difference scheme for abstract Cauchy problems

“…Hyperbolic partial differential equations with both constant/variable coefficients are of common occurrence in many branches of physics and several other areas of science and engineering, e.g., electromagnetic, electrodynamics, thermodynamics, hydrodynamics, elasticity, fluid dynamics, wave propagation, materials science and biological systems [1][2][3][4][5][6][7][8][9][10][11][12]. In numerical techniques for solving these equations, the problem of stability in various functional spaces has received a great deal of importance and attention (see [13][14][15][16][17][18][19][20][21]).…”