HALF-SPACE GREEN’S FUNCTION FOR LAMB’S PROBLEM

The wave propagation in an isotropic, homogeneous linear elastic half-space is treated. The half-space is a semiinfinite region that bounded by a horizontal plane. The vibration is generated by an arbitrary direction buried point pulse.It is necessary to consider both of source and receiver point are located in the interior of the three dimensional domain. The solution of this elastodynamic problem, i.e. the so-called Lamb’s problem, is derived by using the method of source image as well as the superposition principle. Accordingly the transient response of the problem in time domain can be considered as the superposition of the responses to the real and imaginary sources in the full-space and some additional vertical loads on the surface of the half-space. The additional vertical loads distributed over a rectangular area on the surface of a half-space are space and time-dependent functions that vary with time as Heaviside step, Dirac delta and derivatives of Dirac delta functions. The motion at depth produced by a point source applied on the surface when the time variation of the pulse is H(t) ,  (t) and  (t) is obtained base on the approach has been used by (Eringen, Suhubi, andBland 1977). To achieve Laplace transform displacement they have employed Helmholtz potentials for displacement field and satisfy Laplace transform wave equation as well as Hankel transform of boundary condition. The time-domain solutions are found via a modified version of Cagniard-Pekeris method by (Emami and Eskandari-Ghadi 2019). The final solutions are amenable to numerical calculations. The solutions obtained in this way satisfy automatically the traction-free boundary conditions over the surface of the half-space. These Green’s functions can be implemented in 3D time-domain BEM and no discretization of the ground surface is needed. The efficiency of 3D site response analysis of topographic features by the BEM is improved and the computational time and cost is reduced.

Keywords:

3D Green’s function , Source image , Half space , Heaviside step function , Dirac delta function