158
CONTACT OF ELASTIC SOLIDS
each other, and also for the displacement over the part of
the plane z = 0 inside the circle of pressure:-
3
>
a = 3p(I1 + I₂)
16a
3p(+92)
a =
16(p₁+P₂)
१ =
=
3p 2a² - 2
32
6.
as
Outside the circle of pressure is represented by a some-
what more complicated expression, involving an inverse tangent.
Very simple expressions may be got for and n at the plane
= 0. For the compression at the plane z = 0 we find
2=
σ =
-
3p
2K(1+20)π
√ a² - p²
a3
inside the circle of pressure; outside it σ = 0.
pressure Z, inside the circle of pressure we obtain
น
=
3p √ a² - 2
a³
2π
For the
at the centre we have
Zz
=
3p
2πα
"
X₁ = Y₂
=
3p
1+40
4(1 + 20) πα
The formulæ obtained may be directly applied to particular
cases. In most bodies may with a sufficient approximation
be made equal to 1. Then K becomes of the modulus of
elasticity; becomes equal to 32 times the reciprocal of that
modulus; in all bodies 9 is between three and four times this
reciprocal value. If, for instance, we press a glass lens of 100
metres radius with the weight of 1 kilogramme against a
plane glass plate (in which case the first Newton's ring would
have a radius of about 5.2 millimetres), we get a surface of
pressure which is part of a sphere of radius equal to 200
metres. The radius of the circle of pressure is 2.67 millimetres;
the distance of approach of the glass bodies amounts to only
71 millionths of a millimetre. The pressure Z, at the centre
of the surface of pressure is 0.0669 kilogrammes per square
millimetre, and the perpendicular pressures X, and Y, have