172
VI
ON HARDNESS
parts of the bodies may be suitably denoted as indentations.
Their values are easily found by multiplying by 9₁+9, and
thus separating a into two portions. Substituting for a its
value, we see that a involves a numerical factor which depends
on the form of the ellipse of pressure; and that for a given value
of this factor a varies as the power of the pressure, as the
power of the mean value of the coefficients 9, and as the cube
root of the mean value of the curvatures. If one or more of these
curvatures become infinitely great, then distance of approach
and indentations become infinitely great-a result sufficiently
illustrated by the penetrating action of points and edges.
=
We assumed the surface of pressure to be so small that
the deformed surfaces could be represented by quadric sur-
faces throughout a region large compared with the surface of
pressure. Such an assumption can no longer be made after
application of the pressure; in fact outside the surface of
pressure the surface can only be represented by a complicated
function. But we find that inside the surface of pressure the
surface remains a quadric surface to the same approximation
as before. Here we have ₁₂a-Ax²- By2 = a -21+%2
Ꭺx
again ₁₁P, S₂ = 9₂P, or 51: S₂ = 1:2, and lastly, the
equation of the deformed surface is z = z₁+5₁ = 2 + 2; whence
neglecting a constant, we easily deduce (9₁+92)≈ = 9231 +91≈2.
This equation expresses what we wished to demonstrate; it
also shows that the common surface after deformation lies
between the two original surfaces, and most nearly resembles
the body which has the greater modulus of elasticity. When
spheres are in contact the surface of pressure also forms part
of a sphere: when cylinders touch with axes crossed it forms
part of a hyperbolic paraboloid.
=
So far we have spoken of the changes of form, now we
will consider the stresses. We have already found for the
normal pressure in the compressed surface
=
3p
2ab
1
-
x² y²
a2 b2
-
•
This increases from the periphery to the centre, as do the
ordinates of an ellipsoid constructed on the ellipse of pressure;
it vanishes at the edge, and at the centre is one and a half
times as great as it would be if the total pressure were