CONTACT OF ELASTIC SOLIDS
155
the cube root of the total pressure and as the cube root of
the quantity 9, +9. By the preceding the distance through
which the bodies approach each other under the action of the
given pressure is
3p D1 + D2
α=
8π
a
dz
´(1+h² z²) (1+z²) ·
1
If we perform the multiplication by 9, +92 a splits up into
two portions which have a special meaning. They denote the
distances through which the origin approaches the infinitely
distant portions of the respective bodies; we may call them
the indentations which the respective bodies have undergone.
With a given form of the touching surfaces the distance of
approach varies as the pressure raised to the power and
also as the same power of the quantity 9, +92 When A and
B alter in magnitude while their ratio remains unchanged, the
dimensions of the surface of pressure vary inversely as the
cube roots of the absolute values of A and B, and the distance
of approach varies directly as these roots. When A and B
become infinite, the distance of approach becomes infinite;
bodies which touch each other at sharp points penetrate into
each other.
In connection with this we shall determine what happens
to the element at the origin of our system of coordinates by
de an ar
In the first
dx' dy' dz
finding the three displacements
place we have at the origin
2 Әр
3p
1
σ=
=
K(1+20) Əz
2K(1+20)π аb
"
ay
Əz
=
1
ap
3p
1
4K(1+20)π ab
K(1+20) Əz
Further, at the plane z = 0
II =
1
દુઃ
=
K(1+20)
әп
მე:
∞
"
n=
Pdz=
әп
dy
"
1
2K(1+20))
-8
∞
Pdz.