Green’s Functions for the Biharmonic Equation: Bonded Elastic Media

1987 ◽  
Vol 47 (5) ◽  
pp. 982-997 ◽  
Author(s):  
Sinnathurai Vijayakumar ◽  
Donald E. Cormack
Keyword(s):  
Biharmonic Equation ◽  
Green's Functions ◽  
Green’S Functions ◽  
Elastic Media

Exact Green’s functions using leaking modes for axisymmetric boreholes in solid elastic media

Geophysics ◽  
1989 ◽  
Vol 54 (5) ◽  
pp. 609-620 ◽  
Author(s):  
R. A. W. Haddon
Keyword(s):  
Wave Propagation ◽  
Fourier Transform ◽  
Elastic Wave ◽  
Fast Fourier Transform ◽  
Numerical Method ◽  
Green's Functions ◽  
Green’S Functions ◽  
Elastic Media ◽  
Complex Frequency ◽  
Residue Theory

By choosing appropriate paths of integration in both the complex frequency ω and complex wavenumber k planes, exact Green’s functions for elastic wave propagation in axisymmetric fluid‐filled boreholes in solid elastic media are expressed completely as sums of modes. There are no contributions from branch line integrals. The integrations with respect to k are performed exactly using Cauchy residue theory. The remaining integrations with respect to ω are then carried out partly by using the fast Fourier transform (FFT) and partly by using another numerical method. Provided that the number of points in the FFT can be taken sufficiently large, there are no restrictions on distance. The method is fast, accurate, and easy to apply.

scholarly journals Paradox of modelling curved faults revisited with general non-hypersingular stress Green’s functions

2020 ◽  
Vol 223 (1) ◽  
pp. 197-210
Author(s):  
Dye S K Sato ◽  
Pierre Romanet ◽  
Ryosuke Ando
Keyword(s):  
Numerical Methods ◽  
Green’S Function ◽  
Green's Function ◽  
Green's Functions ◽  
Green’S Functions ◽  
Integral Kernel ◽  
Smooth Curve ◽  
Displacement Discontinuity ◽  
Elastic Media ◽  
Alternative Expression

SUMMARY In a dislocation problem, a paradoxical discordance is known to occur between an original smooth curve and an infinitesimally discretized curve. To solve this paradox, we have investigated a non-hypersingular expression for the integral kernel (called the stress Green’s function) which describes the stress field caused by the displacement discontinuity. We first develop a compact alternative expression of the non-hypersingular stress Green’s function for general 2-D and 3-D infinite homogeneous elastic media. We next compute the stress Green’s functions on a curved fault and revisit the paradox. We find that previously obtained non-hypersingular stress Green’s functions are incorrect for curved faults, and that smooth and infinitesimally segmented faults are equivalent. Their compatibility bridges the gap between analytical methods featuring curved faults and numerical methods using subdivided flat patches.

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