On the analysis of wave motions in a multi-layered solid (original) (raw)
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Journal of Computational Acoustics, 2008
A semi-analytic method based on the propagation matrix formulation of indirect boundary element method to compute response of elastic (and acoustic) waves in multi-layered media with irregular interfaces is presented. The method works recursively starting from the top-most free surface at which a stress-free boundary condition is applied, and the displacement-stress boundary conditions are then subsequently applied at each interface. The basic idea behind this method is the matrix formulation of the propagation matrix (PM) or more recently the reflectivity method as wide used in the geophysics community for the computation of synthetic seismograms in stratified media. The reflected and transmitted wave fields between arbitrary shapes of layers can be computed using the indirect boundary element (BEM) method. Like any standard BEM methods, the primary task of the BEM-based propagation matrix method (thereafter called PM-BEM) is the evaluation of element boundary integral of the Green's function, for which there are standard method that can be adapted. In addition, effective absorbing boundary conditions as used in the finite difference numerical method is adapted in our implementation to suppress the spurious arrivals from the artificial * 381 382 E. Liu et al.
Layerwise fundamental solutions and three-dimensional model for layered media
Applied Composite Materials, 1996
A hybrid method is presented for the analysis of layers, plates, and multilayered systems consisting of isotropic and linear elastic materials. The problem is formulated for the general case of a multilayered system using a total potential energy formulation and employing the layerwise laminate theory of Reddy. The developed boundary integral equation model is two-dimensional, displacement based and assumes piecewise continuous distribution of the displacement components through the system's thickness. A one-dimensional finite element model is used for the analysis of the multilayered system through its thickness, and integral Fourier transforms are used to obtain the exact solution for the in-plane problem. Explicit expressions are obtained for the fundamental solution of a typical infinite layer (element), which can be applied in a two-dimensional boundary integral equation model to analyze layered structures. This model describes the three-dimensional displacement field at arbitrary points either in the domain of the layered medium or on its boundary. The proposed method provides a simple, efficient, and versatile model for a three-dimensional analysis of thick plates or multilayered systems.
Three-Dimensional Green’s Functions for a Multilayered Half-Space in Displacement Potentials
Journal of Engineering Mechanics, 2002
To advance the mathematical and computational treatments of mixed boundary value problems involving multilayered media, a new derivation of the fundamental Green's functions for the elastodynamic problem is presented. By virtue of a method of displacement potentials, it is shown that there is an elegant mathematical structure underlying this class of three-dimensional elastodynamic problems which warrant further attention. Constituted by proper algebraic factorizations, a set of generalized transmission-reflection matrices and internal source fields that are free of any numerically unstable exponential terms common in past solution formats are proposed for effective computations of the potential solution. To encompass both elastic and viscoelastic cases, point-load Green's functions for stresses and displacements are generalized into complex-plane line-integral representations. An accompanying rigorous treatment of the singularity of the fundamental solution for arbitrary source-receiver locations via an asymptotic decomposition of the transmission-reflection matrices is also highlighted.
A layerwise boundary integral equation model for layers and layered media
Journal of Elasticity, 1995
A hybrid method is presented for the analysis of layers, plates, and multilayered systems consisting of isotropic and linear elastic materials, The problem is formulated for the general case of a multilayered system using a total potential energy formulation. The layerwise laminate theory of Reddy is employed to develop a layerwise, two-dimensional, displacement-based, hybrid boundary element model that assumes piecewise continuous distribution of the displacement components through the system's thickness. A one-dimensional finite element model is used for the analysis of the multilayered system through its thickness, and integral Fourier transforms are used to obtain the exact solution for the in-plane problem. Explicit expressions are obtained for the fundamental solution of a typical infinite layer (element) assuming linear displacement distribution through its thickness. This fundamental solution is given in a closed form in the cartesian space, and it can be applied in the two-dimensional boundary integral equation model to analyze layered structures with finite dimensions. The proposed method provides a simple, efficient, and versatile model for a three-dimensional analysis of thick plates or multilayered systems.
Wave Motion, 2004
An integral equation method is applied for the calculation of elastic wave fields in unbounded solids containing general anisotropic inclusions and voids. The domain of the integral equation involves the volume of the inclusions as well as the surface of the voids. In contrast to the conventional boundary integral equation method (BIEM), where the infinite medium Green's functions for both the matrix material and the inclusion material are needed, the present method does not require the latter. Since the elastodynamic Green's functions for anisotropic media are extremely difficult to calculate, the present method offers a definite advantage over methods based on boundary integral equation (BIE) alone. The newly developed mixed method takes full advantage of the volume integral equation method that is effective for problems with anisotropic inclusions and the BIEM that is effective for problems involving voids and cracks. In this paper, the mixed method is used to calculate the interaction of plane, time-harmonic elastic waves with an isotropic and an orthotropic cylindrical inclusion in absence or presence of a parallel cylindrical void in its vicinity, for waves incident normal to the cylinder axis. Numerical results are presented for the displacement and stress fields at the interfaces of the inclusions in a broad frequency range of practical interest. The new method is shown to be very accurate and efficient for solving this class of problems.
Pure and Applied Geophysics, 2014
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Recursive Green functions technique applied to the propagation of elastic waves in layered media
2002
Guided by similarities between electronic and classical waves, a numerical code based on a formalism proven to be very effective in condensed matter physics has been developed, aiming to describe the propagation of elastic waves in stratified media (e.g. seismic signals). This so-called recursive Green function technique is frequently used to describe electronic conductance in mesoscopic systems. It follows a space-discretization of the elastic wave equation in frequency domain, leading to a direct correspondence with electronic waves travelling across atomic lattice sites. An inverse Fourier transform simulates the measured acoustic response in time domain. The method is numerically stable and computationally efficient. Moreover, the main advantage of this technique is the possibility of accounting for lateral inhomogeneities in the acoustic potentials, thereby allowing the treatment of interface roughness between layers. Ó
Composites Science and Technology, 2000
In this paper we illustrate a procedure for obtaining an approximate solution of the integral equation form of the two-dimensional Lame equation for a two-layer problem, based on collocation via use of Sinc approximation. The Sinc approach automatically concentrates points near corners of the boundary where the solution has singularities and yields exponential convergence. A model two-layer bi-material elasticity problem is numerically investigated here as an illustration of this novel approach.
Green's functions for 2.5D elastodynamic problems in a free solid layer formation
Engineering Structures, 2002
This work presents analytical Green's functions for the steady state response of a homogeneous three-dimensional free solid layer formation (slab) subjected to a spatially sinusoidal harmonic line load, polarized along the horizontal, vertical and z directions. The equations presented here are not only themselves very interesting but are also useful for formulating three-dimensional elastodynamic problems in a slab-type formation, using integral transform methods and/or boundary elements. The final expressions are validated by comparing them with the results obtained by using the Boundary Element Method solution, for which both free surfaces of the slab are discretized with boundary elements.
Fundamental solutions for singularities within a layered solid
European Journal of Mechanics - A/Solids, 2012
In this paper, the generic fundamental solution for the elastic layered solid with a singularity contained within the solid is derived with the help of the superposition principle. The singularity can be a point force, an edge dislocation, a point moment, a point residual strain nucleus, and so on. The degenerate cases are also derived. The solutions can be used as kernel functions (ie, Green's functions) to formulate integral equations for problems of the layered solids using the Green's function method. The application of the solutions to ...