VI
ON THE CONTACT OF RIGID ELASTIC SOLIDS
AND ON HARDNESS
(Verhandlungen des Vereins zur Beförderung des Gewerbefleisses, November 1882.)
WHEN two elastic bodies are pressed together, they touch each
other not merely in a mathematical point, but over a small
but finite part of their surfaces, which part we shall call the
surface of pressure.
The form and size of this surface and
the distribution of the stresses near it have been frequently
considered (Winkler, Lehre von der Elasticität und Festigkeit,
Prag. 1867, I. p. 43; Grashof, Theorie der Elasticität und
Festigkeit, Berlin, 1878, pp. 49-54); but hitherto the results
have either been approximate or have even involved unknown
empirical constants. Yet the problem is capable of exact
solution, and I have given the investigation of the problem in
vol. xcii. of the Journal für reine und angewandte Mathematik,
p. 156.¹
As some aspects of the subject are of considerable
technical interest, I may here treat it more fully, with an
addition concerning hardness. I shall first restate briefly the
proof of the fundamental formula.
We first imagine the two bodies brought into mathematical
contact; the common normal coincides with the line of action
of the pressure which the one body exerts upon the other.
In the common tangent plane we take rectangular rectilinear
axes of xy, the origin of which coincides with the point of
contact; the third perpendicular axis is that of z. We can
confine our attention to that part of each body which is very
close to the point of contact, since here the stresses are
extremely great compared with those occurring elsewhere, and
1 See V. p. 146.