152
V
CONTACT OF ELASTIC SOLIDS
whence we have for z= 0
S₁ = J₁P, S₂ = − J₂P, Z₂1 -2-
-
=
ӘР
ӘР
Z=2
"
дъ
дъ
This assumption satisfies the conditions (1), (2 a), and (2 b)
according to the explanations given. Since OP/ǝz has on the
two sides of the plane z = 0 values equal but of opposite sign,
and since it vanishes outside the electrically charged surface
whose potential is P, the conditions (2 c) also are fulfilled, pro-
vided the surface of pressure coincides with the electrically
charged surface. From the fact that P is continuous across
the plane z = 0, it follows that for 2 = 0,9251 +9152 = 0. But
according to the condition (2 d) we have for the surface of
pressure, ₁₂-a-1+22; here therefore
92
---
91
S₁
=
·(a− z₂+22), S₂ =
D1 + D2
-(a − z₁ + z₂) .
D1 + D2
Apart from a constant which depends on the choice of the
system of coordinates, and need therefore not be considered, the
equation of the surface of pressure is z=21 + $1=22 + 52, or
(D1 + D2)² = 921+912 Thus the surface of pressure is part
of a quadric surface lying between the undeformed positions of
the surfaces which touch each other; and is most like the
boundary of the body having the greater coefficient of elasticity.
If the bodies are composed of the same material it is the
mean surface of the surfaces of the two bodies, since then
2x=2₂+22
We now make a definite assumption as to the distribution
of the electricity whose potential is P. Let it be distributed
over an ellipse whose semi-axes a and b coincide with the axes of
2 and y, with a density
3p
8m2ab
شمرد
1
-
--
a² b²
>
so that it can be
regarded as a charge which fills an infinitely flattened ellipsoid
with uniform volume density. Then
P
=
3p
16π
9
-
x2
a² + λ
-
y2
b² +λ
αλ
λ √(a² +λ)(b² +λ)λ