Three-dimensional Green’s function for time-harmonic dynamics in a viscoelastic layer

Abstract

This paper presents Green's function for time-harmonic elastodynamic problems for a single layer domain (three-dimensional region bounded by two parallel planes with traction-free boundary conditions). The semi-analytic solution is built in three steps: (a) potential displacement representation; (b) angular Fourier series; (c) radial Hankel transform. Reflection matrices are presented for the plate domain. Kernels are integrally split into a singular closed-form term (the static half-space solution) plus an incremental solution. In order to compute the inverse Hankel transform for displacements and stress components, a modified complex integration path is required. Theoretical considerations allow an adequate delimitation of such a complex path. A specific treatment is proposed for low excitation frequencies where asymmetric Lamb waves play a major role. A series of numerical benchmarks are presented to validate the implementation of the functions (displacements and tractions).

Numerical model for the dynamic and seismic analysis of pile-supported structures with a meshless integral representation of the layered soil

“…These matrices do not contain any unbounded exponential terms in order to avoid numerical instabilities. A particular integration procedure, as described by Pak and Guzina (2002) and Pak (1999, 2001) is employed to evaluate the inverse Hankel transform, which has been modified for low-frequencies based on the single layer problem (Martínez-Castro and Gallego (2007)). Note that, as these Green's functions are obtained assuming a layered half space domain, the boundary conditions of the free-surface and layer interfaces are intrinsically satisfied by the fundamental solution.…”

Three-dimensional modeling of elastic guided waves excited by arbitrary sources in viscoelastic multilayered plates

“…Given the expansion form (32),F p (k) is also equal to the Hankel transform of order p of F p (r), which is defined by the expansion: F(r, θ) = +∞ p=−∞ F p (r)e ipθ where F p (r) = 1 2π 2π 0 F(r, θ)e −ipθ dθ (see Appendix B). The expression (30)…”

“…Furthermore, the proposed modal solution is shown to be applicable to viscoelastic solids as well as in the near field region. It should be mentioned that, while the integral transform approach still applies with complex poles, and thereby to lossy waveguides [29,24,30], the validity of modal techniques with complex modes might be unclear. Complex modes typically occur with viscoelastic materials or in near-field calculations, involving evanescent or inhomogeneous modes.…”