174
VI
ON HARDNESS
z and an axis of the ellipse of pressure; arrow-heads pointing
towards each other denote a tension, those pointing away from
each other a pressure. The figure relates to the case in which
0=1. The portion ABDC of the body, which originally
formed an elevation above the surface of pressure, is now
pressed into the body like a wedge; hence the pressure is
transmitted not only in the direct line AE, but also, though
with less intensity, in the inclined directions AF and AG.
The consequence is that the element is also powerfully com-
pressed laterally; while the parts at F and G are pressed apart
and the intervening portions stretched. Hence at A on the
element of area perpendicular to the x-axis there is pressure,
which diminishes inwards, and changes to a tension which
rapidly attains a maximum, and then, with increasing distance,
diminishes to zero. Since the part near A is also laterally
compressed, all points of the surface must approach the origin,
and must therefore give rise to stretching in a line with the
origin. In fact the pressure which acts at A parallel to the
axis of x already changes to a tension inside the surface of
pressure as we proceed along the x-axis; it attains a maximum
near its boundary and then diminishes to zero. Calculation
shows that for 1 this tension is much greater than that
in the interior. As regards the third principal pressure which
acts perpendicular to the plane of the diagram, it of course
behaves like the one parallel to the x-axis; at the surface it
is a pressure, since here all points approach the origin.
the material is incompressible the diagram is simplified, for
since the parts near A do not approach each other, the tensions
at the surface disappear.
=
If
We shall briefly mention the simplifications occurring in
the formula, when the bodies in contact are spheres, or are
cylinders which touch along a generating line. In the first
case we have simply kμv = 1, P11 = P12 = P1, P21 = P22P23
hence
=
a =
= b =
1
3p(9₁+9₂)
16(P₂+P₂)
α=
>
3p(9, +9₂)
16a
The formula for the case of cylinders in contact are not
got so directly. Here the major semi-axis a of the ellipse
becomes infinitely great; we must also make the total pressure