Typical Boundary Value Problems Research Articles

On the rigorous foundation of short-wave asymptotics

AbstractCertain physical theories are short-wave limits of more general theories. Thus ray optics is the short-wave limit of wave optics, and classical mechanics is the short-wave limit of wave mechanics. In principle it must be possible to deduce the former from the latter theories by a rigorous mathematical limiting process; in fact the arguments found in the literature are formal, plausible and non-rigorous. (We are here concerned with linear wave equations and time-periodic phenomena.) For some wave equations there are, however, a few explicit rigorous canonical solutions relating to simple geometrical configurations, e.g. to conics in two dimensions for the equations of acoustics, and for these the asymptotics can be found rigorously. For more general configurations the solution of a typical boundary-value problem can be reduced to the solution of a Fredholm integral equation of the second kind.

Stability of rigid-plastic solids

The general problem of stability under dead loading is treated for a body of any shape, the material being workhardening and rigid-plastic. It is shown that when the plastic potential and yield function coincide it is only necessary to examine in detail those neighbouring positions which can be reached by strain paths beginning as virtual modes of the yield-point state under the given loading. The least energy dissipated in passing to any such position has to be determined in constructing a sufficient condition for stability. The functional concerned turns out to be identical with that in the uniqueness condition for the typical boundary-value problem. On the other hand, the classes of admissible functions are different, so that failures of uniqueness and stability need not coincide; this is illustrated by the compression of a strut.