Integral Solutions to 3D Lamb’s Problem (I): Theoretical Formulation

The 2D problems considered in the previous chapters can be viewed, from a 3D perspective, as displacement field problems generated by infinitely long line sources distributed along the y axis, where all physical quantities such as force sources and displacements are independent of the y coordinate. This implies two things: first, the y components of all physical quantities are zero, and second, all non-zero physical quantities are independent of y; in other words, \({\partial }/{\partial y}=0\) . Therefore, all quantities related to y can be omitted, simplifying the problem. However, for a truly point source in 3D space, the resulting displacement field obviously depends on all three coordinates. In this case, the formulation of the boundary value problem must include all information related to the y coordinate from the very beginning.