154
CONTACT OF ELASTIC SOLIDS
The equations for the axes of the ellipse of pressure
written explicitly are
du
A 16π
=
/ (a²+u)³(b²+u)u¯¯d₁+d½ 3p
du
B 16π
/ (a² + u) (b² + u)³u ¯¯Ð½+d½ 3p
2
or introducing the ratio k = a/b, and transforming,
∞
1
dz
8 п A
=
(1 + h² z ²)³ (1+z²) 3p d₁ + d₂
dz
}{µ√@+8=X1+z) =
8,π B
9, +9
2
k²√(1+h²z²)(1+z2)³ 3p d₁ + d2
By division we obtain a transcendental equation for the
ratio k.¹ This depends only on the ratio A: B, and it follows
at once, from the meaning we have attached to the forces and
displacements, that the ellipse of pressure is always more
elongated than the ellipses at which the distance between the
bodies is constant. As regards the absolute magnitude of the
surface of pressure for a given form of the surfaces it varies as
¹ The solution of this equation and the evaluation of the integrals required
for the determination of a and b may be performed by the aid of Legendre's
tables without necessitating any new quadratures. The calculation, usually
somewhat laborious, may in most cases be avoided by the use of the following
small table, of which the arrangement is as follows. If we express A and B in
the equations for a and b in terms of the principal curvatures and the auxiliary
angle introduced in a previous note, the solutions of these equations are
expressible in the form
where,
α=μ
3p(9₁+9)
8(P11+ P12+ P21 + P22)'
3p(9+92)
b= 8(p11 + P12+ P21+ P22)
are transcendental functions of the angle r. The table gives the
values of these functions for ten values of the argument r expressed in degrees.
T
90 80 70 60 | 50 | 1
40
30
20
10
0
μ 1.0000 1.1278 1.2835 1.4858 1-7542 2-1357 2-7307 3.7779 6.6120
∞
V 1.0000 0.8927 0-8017 0-7171 0.6407 0.5673 0.4930 0-4079 0·3186 0.0000