Vol. 5, No. 6, 2010

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ISSN: 1559-3959
Time-harmonic elastodynamic Green's function for the half-plane modeled by a restricted inhomogeneity of quadratic type

Tsviatko V. Rangelov and George D. Manolis

Vol. 5 (2010), No. 6, 909–924
Abstract

We derive closed-form solutions for point-force generated motions in a continuously inhomogeneous half-plane, which represent the complete elastic wave-train in the interior domain obeying traction-free boundary conditions at the horizontal surface. More specifically, a special type of material inhomogeneity is studied, where the shear modulus varies quadratically with respect to the depth coordinate. Furthermore, the material density profile varies proportionally to the aforementioned profile, while Poisson’s ratio remains fixed at one-quarter. Limit forms for the Green’s functions are derived for both zero frequency and for the equivalent homogeneous medium. Next, a series of numerical results serve to validate this mechanical model, and to show the differences in the wave motion patterns developing in media that are inhomogeneous as compared to a reference homogeneous background. These singular solutions are useful within the context of boundary element formulations for the numerical solution of problems involving nonhomogeneous continua, which find applications in fields as diverse as composite materials, geophysical prospecting, petroleum exploration and earthquake engineering.

Keywords
inhomogeneous media, elastic waves, Fourier transforms, singular solutions
Milestones
Received: 4 February 2010
Revised: 15 September 2010
Accepted: 29 September 2010
Published: 1 January 2011
Authors
Tsviatko V. Rangelov
Department of Mathematical Physics
Institute of Mathematics and Informatics
Bulgarian Academy of Sciences
acad. G. Bonchev str. bl. 8
1113 Sofia
Bulgaria
George D. Manolis
Department of Civil Engineering
Aristotle University
54124 Thessaloniki
Greece