V
157
CONTACT OF ELASTIC SOLIDS
softer metals, this transgression will at first consist in a lateral
deformation accompanied by a permanent compression; so that
it will not result in an infinitely increasing disturbance of
equilibrium, but the surface of pressure will increase beyond
the calculated dimensions until the pressure per unit area is
sufficiently small to be sustained. It is more difficult to de-
termine what happens in the case of brittle bodies, as hard
steel, glass, crystals, in which a transgression of the elastic
limit occurs only through the formation of a rent or crack, i.e.
only under the influence of tensional forces. Such a crack
cannot start in the element considered above, which is com-
pressed in every direction; and with our present-day knowledge
of the tenacity of brittle bodies it is indeed impossible exactly
to determine in which element the conditions for the production
of a crack first occur when the pressure is increased. However,
a more detailed discussion shows this much, that in bodies
which in their elastic behaviour resemble glass or hard steel,
much the most intense tensions occur at the surface, and in
fact at the boundary of the surface of pressure. Such a dis-
cussion shows it to be probable that the first crack starts at
the ends of the smaller axis of the ellipse of pressure, and
proceeds perpendicularly to this axis along that ellipse.
The formulæ found become especially simple when both
the bodies which touch each other are spheres. In this case
the surface of pressure is part of a sphere. If p is the recip-
rocal of its radius, and if p, and p, are the reciprocals of the
radii of the touching spheres, then we have the relation
(F₁ + F₂)p = √ ¿P₂+1P2; which for spheres of the same material
takes the simpler form 2p P₁+ P₂
The curve of pressure is
If we put
Ꮽ
=
a circle whose radius we shall call a.
x² + y² = r²,
then will
3p
P =
16π
-
2.2
z2
-=
1,
+
a² + u И
2
du
-
a² + u
u / (a²+u) √u'
16
which may also be expressed in a form free of integrals.
We easily find for a, the radius of the circle of pressure,
and for a, the distance through which the spheres approach