XVI
ON THE EQUILIBRIUM OF FLOATING ELASTIC
PLATES
(Wiedemann's Annalen, 22, pp. 449-455, 1884.)
SUPPOSE an infinitely extended elastic plate, e.g. of ice, to float
on an infinitely extended heavy liquid, e.g. water; on the plate
rest a number of weights without production of lateral tension;
the position of equilibrium of the plate is required. The
solution of this problem leads to certain paradoxical results, on
account of which it is given here.
If we confine ourselves to small displacements, we may
regard the effects of the separate weights as superposed, and
need only consider the case of a single weight P.
We suppose
it placed at the origin of coordinates of x, y, of which the
plane coincides with the plate, supposed infinitely thin.
Further we write ▼² = Ə²/ǝx² +ə²/ǝy², p² = x² + y², and denote
by E and μ in the usual notation the elastic constants of the
material of the plate, by h its thickness, and by s the density
of the liquid. Let z denote the vertical displacement of the
deformed plate from the plane of x, y, reckoned positive when
downwards; then on the one hand {Eh³/12(1 - μ²)} vʻz is
the upward pressure per unit area due to the elastic stresses,¹
on the other hand sz is the upward hydrostatic pressure per
unit area.
The sum of both pressures must vanish every-
where except at the origin. Here the integral of that sum
taken over a very small area must be equal to P. But since
the integral of the hydrostatic pressure over such an area is
infinitesimal, that condition must be satisfied by the integral
of the elastic reaction alone. If we write for shortness
¹ Clebsch, Theorie der Elasticität, § 73, 1862.