Wave propagation in an infinite medium

The results of load case 1 are shown in the form of time histories of vertical displacements at nodes 13, 103, and 601, as indicated on the meshes. Figure 6 (node 13), Figure 7 (node 103), and Figure 8 (node 601) show the displacement responses. The wave reflection caused by the free boundaries in the small finite element mesh is evident, while the small finite/infinite element quiet boundary mesh largely succeeds in eliminating this reflection. The results obtained with Abaqus/Standard and Abaqus/Explicit agree well.

The wave pattern produced by a distributed load on an infinite half-space is shown in Figure 1. The majority of the energy introduced into the system by the loading is contained in the straight section of the longitudinal wave. The curved wave fronts and the surface waves are produced by the discontinuity at the edge of the distributed load. The same wave pattern can be identified in the deformed configurations of both load cases. In particular, the deformed configuration for load case 2 just prior to the longitudinal wave leaving the lower boundary of the mesh is shown in Figure 10. The contour plots of the vertical and horizontal displacements for load case 2 (see Figure 11 and Figure 12, respectively) show the lower energy shear waves emanating from the edge of the distributed load more clearly.

The time histories of the whole model energies for load case 2 are shown in Figure 13. It can be seen that the kinetic and internal energies remain constant until the longitudinal wave reaches the infinite elements at the mesh boundary. The viscous dissipation time history represents the energy absorbed by the infinite elements. At 1.07$\mu$s, when the trailing end of the longitudinal pulse reaches the mesh boundary, most of the energy has been absorbed by the infinite elements. The last wave to exit the mesh should be the shear wave generated by the discontinuity at the edge of the distributed load that travels toward the symmetry axis. It will be reflected by the symmetry axes and travel toward the lower-right corner of the mesh. This should occur at 3.92$\mu$s. Any waves remaining in the mesh after this time are due to spurious wave reflection at the infinite boundaries. The kinetic energy associated with these waves is less than 0.2% of the total kinetic energy created by the pulse.

Figure 14 shows the vertical displacement responses of a node positioned 2 mm below the edge of the distributed load. The initial longitudinal pulse reaches the node in 0.324$\mu$s. Because the wave is not completely absorbed by the infinite elements, its reflection can be seen on its way back to the surface and again on its return from the surface after it is has been reflected. The response after 2.4$\mu$s is due to the shear wave reflecting off the symmetry axis. The shear wave traveling directly downward from the edge of the distributed load does not appear in this plot because its motion is completely in the horizontal direction.

Figure 15 shows the horizontal displacement responses of a node positioned 3.2 mm below the edge of the distributed load. The horizontal component of the longitudinal wave reaches the node in 0.519$\mu$s, while the slower traveling shear wave reaches the node at a time of 1.03$\mu$s. The response after approximately 1.7$\mu$s is due to spurious reflections of the longitudinal and shear waves from the lower boundary as well as the shear wave reflected from the symmetry axes. Again, the results obtained with Abaqus/Standard and Abaqus/Explicit agree well.