A Method to Evaluate Three-Dimensional Time-Harmonic Elastodynamic Green’s Functions in Transversely Isotropic Media

1992 ◽  
Vol 59 (2S) ◽  
pp. S96-S101 ◽  
Author(s):  
H. Zhu
Keyword(s):  
Green's Functions ◽  
Green’S Functions ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Isotropic Case ◽  
Phase Method ◽  
Stationary Phase Method ◽  
Transversely Isotropic Media ◽  
Time Harmonic ◽  
Isotropic Media

The three-dimensional time-harmonic elastodynamic Green’s functions in infinite transversely isotropic media have been derived explicitly. The Green’s functions consist of the corresponding static Green’s functions and double integral representations over a finite domain with the integrands being continuous. The Green’s functions will reduce to those for the isotropic case when the isotropic elastic constants are substituted. The singular parts of the Green’s functions have been shown to be the same as those of the static ones. The far-field approximations have been obtained by using the stationary phase method. In addition, a simpler method to construct wave front curves has been presented.

Three-dimensional analysis of cracks in layered transversely isotropic media

1989 ◽  
Vol 424 (1867) ◽  
pp. 307-322 ◽  
Keyword(s):  
Integral Equation ◽  
Green's Functions ◽  
Green’S Functions ◽  
Crack Opening ◽  
Mathematical Formulation ◽  
Transversely Isotropic ◽  
Boundary Integral ◽  
Opening Displacement ◽  
Transversely Isotropic Media ◽  
Isotropic Media

A general mathematical formulation to analyse cracks in layered transversely isotropic media is developed in this paper. By constructing the Green’s functions, an integral equation is obtained to determine crack opening displacements when an applied crack face traction is specified. For the infinite body, the Green’s functions have solutions in a closed form. For layered media, a flexibility matrix in the integral transformed domain is formed that establishes the relation between the traction and the displacement for a single layer; the global matrix is formed by assembling all of the flexibility matrices constructed for each layer. The Green’s functions in the spatial domain are obtained by inversion of the Hankel transform. Finally, the crack opening displacement and the crack-tip opening displacement for a vertical planar crack in a layered transversely isotropic medium are obtained numerically by the boundary integral equation method.

A point dislocation in a layered, transversely isotropic and self-gravitating Earth — Part II: accurate Green's functions

2019 ◽  
Vol 219 (3) ◽  
pp. 1717-1728 ◽  
Author(s):  
J Zhou ◽  
E Pan ◽  
M Bevis
Keyword(s):  
Green's Functions ◽  
Green’S Functions ◽  
Transversely Isotropic ◽  
Earth Model ◽  
Transversely Isotropic Media ◽  
Love Numbers ◽  
Isotropic Media ◽  
Point Dislocation ◽  
High Degree ◽  
Harmonic Degree

SUMMARY We present an accurate approach for calculating the point-dislocation Green's functions (GFs) for a layered, spherical, transversely-isotropic and self-gravitating Earth. The formalism is based on the approach recently used to find analytical solutions for the dislocation Love numbers (DLNs). However, in order to make use of the DLNs, we first analyse their asymptotic behaviour, and then the behaviour of the GFs computed from the DLNs. We note that the summations used for different GF components evolve at different rates towards asymptotic convergence, requiring us to use two new and different truncation values for the harmonic degree (i.e. the index of summation). We exploit this knowledge to design a Kummer transformation that allows us to reduce the computation required to evaluate the GFs at the desired level of accuracy. Numerical examples are presented to clarify these issues and demonstrate the advantages of our approach. Even with the Kummer transformation, DLNs of high degree are still needed when the earth model contains very fine layers, so computational efficiency is important. The effect of anisotropy is assessed by comparing GFs for isotropic and transversely isotropic media. It is shown that this effect, though normally modest, can be significant in certain contexts, even in the far field.

Complete solutions of three-dimensional problems in transversely isotropic media

2018 ◽  
Vol 32 (3) ◽  
pp. 775-802 ◽  
Author(s):  
Francesco Marmo ◽  
Salvatore Sessa ◽  
Nicoló Vaiana ◽  
Daniela De Gregorio ◽  
Luciano Rosati
Keyword(s):  
Three Dimensional ◽  
Transversely Isotropic ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Complete Solutions
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