Mechanical and Civil Engineering Seminar

Since the middle of the 19th century, scientists in mathematical physics have provided exact formulas for the response of a class of idealized elastic bodies to dynamic point sources applied at some specific locations. The most well-known of these canonical formulas are those of Kelvin-Stokes, Lamb, and many others who followed in the 20th century. These fundamental solutions have played a vital role in various fields, which include seismology, soil dynamics, and the Boundary Element Method, just to name a few application areas. They have also served as reference point to assess the quality of numerical models for those same bodies when analyzed with discrete methods such as finite elements or finite differences.

In this talk we shall review a small subset of these formulas and discuss some peculiar even if non-obvious characteristics. We shall also elaborate on apparent pathologies that arise when the response to point loads are used to infer the response to dipoles in 2 or 3 dimensions, which include both seismic moments and cracks. In addition, we shall report on some erroneous, classical formulas for moving loads that are widely quoted in the literature and are taken for granted to be correct, but which regrettably they are not.

Finally, we wish to report on a most surprising and paradoxical finding concerning a certain discrete (i.e. FE) solid for which the fundamental solution is undistinguishable from that of the continuous solid that it replaces. This would seem to be in obvious contradiction to the so-called "representation theorems" and the widespread dogma that a discrete model can never, ever perfectly substitute a continuous medium, and can only do so in the limit of an infinitesimal grid, i.e. as an approximation.