Journal ArticleDOI
Recursive geometric integrators for wave propagation in a functionally graded multilayered elastic medium
L. Wang,S. I. Rokhlin +1 more
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TLDR
In this article, the stiffness matrices of the thin nonhomogeneous layer are combined to obtain the total stiffness matrix for an arbitrary functionally graded multilayered system, and a computationally stable hybrid method is proposed which first starts the recursive computation with the transfer matrices and then, as the thickness increases, transits to the stiffness matrix recursive algorithm.Abstract:
The differential equations governing transfer and stiffness matrices and acoustic impedance for a functionally graded generally anisotropic magneto-electro-elastic medium have been obtained. It is shown that the transfer matrix satisfies a linear 1st order matrix differential equation, while the stiffness matrix satisfies a nonlinear Riccati equation. For a thin nonhomogeneous layer, approximate solutions with different levels of accuracy have been formulated in the form of a transfer matrix using a geometrical integration in the form of a Magnus expansion. This integration method preserves qualitative features of the exact solution of the differential equation, in particular energy conservation. The wave propagation solution for a thick layer or a multilayered structure of inhomogeneous layers is obtained recursively from the thin layer solutions. Since the transfer matrix solution becomes computationally unstable with increase of frequency or layer thickness, we reformulate the solution in the form of a stable stiffness-matrix solution which is obtained from the relation of the stiffness matrices to the transfer matrices. Using an efficient recursive algorithm, the stiffness matrices of the thin nonhomogeneous layer are combined to obtain the total stiffness matrix for an arbitrary functionally graded multilayered system. It is shown that the round-off error for the stiffness-matrix recursive algorithm is higher than that for the transfer matrices. To optimize the recursive procedure, a computationally stable hybrid method is proposed which first starts the recursive computation with the transfer matrices and then, as the thickness increases, transits to the stiffness matrix recursive algorithm. Numerical results show this solution to be stable and efficient. As an application example, we calculate the surface wave velocity dispersion for a functionally graded coating on a semispace.read more
Citations
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An analytical method for free vibration analysis of functionally graded beams with edge cracks
TL;DR: In this article, an analytical method is proposed for solving the free vibration of cracked functionally graded material (FGM) beams with axial loading, rotary inertia and shear deformation.
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The reverberation-ray matrix and transfer matrix analyses of unidirectional wave motion
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Guided waves in functionally graded viscoelastic plates
TL;DR: In this paper, a dynamic solution for the propagating viscoelastic waves in functionally graded material (FGM) plates subjected to stress-free conditions is presented in the context of the Kelvin-Voigt theory.
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Effective Willis constitutive equations for periodically stratified anisotropic elastic media
TL;DR: In this article, a method to derive homogeneous effective constitutive equations for periodically layered elastic media is proposed, where the coefficients of the dynamic effective medium can be associated with the matrix logarithm of the propagator over a unit period.
Journal ArticleDOI
Effective Willis constitutive equations for periodically stratified anisotropic elastic media
TL;DR: In this article, a method to derive homogeneous effective constitutive equations for periodically layered elastic media is proposed, where the coefficients of the dynamic effective medium can be associated with the matrix logarithm of the propagator over a unit period.
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