Point force and dipole solutions in the inhomogeneous half-plane under time-harmonic conditions

• Considered is inhomogeneous half plane under anti-plane strain state.
• Time-harmonic conditions are assumed to hold.
• The boundary conditions comprise a traction free horizontal surface.
• Point force and dipole fundamental solutions are derived.
• Numerical examples serves to illustrate the evaluation of the fundamental solutions.

In this paper, we present fundamental solutions for the inhomogeneous half-plane under anti-plane strain conditions subjected to a point force and two dipoles. Time-harmonic conditions are assumed to hold, while the boundary conditions comprise a traction-free horizontal surface plus the Sommerfeld radiation condition. The aforementioned fundamental solutions are derived for two special types of continuous material inhomogeneity, whereby the shear modulus and the density vary either as an exponential function or as a quadratic polynomial with respect to depth. These solutions converge to their static equivalents as the frequency of vibration approaches zero, and collapse to the ones corresponding to the homogeneous half-plane when the inhomogeneity parameter is set to zero. Finally, a numerical example serves to illustrate the fundamental solutions obtained herein.