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.
Department of Mathematical
Physics
Institute of Mathematics and Informatics
Bulgarian Academy of Sciences
acad. G. Bonchev str. bl. 8
1113 Sofia
Bulgaria