58
6 Herz’s Task
Hence the following can be represented within a sufficiently small vicinity of the
reference point
F 1 [r] ≈
r 2
2ρ 1
, F 2 [r] ≈ −
r 2
2ρ 2
,
(6.11)
where
1
ρ 1
=
d 2 F 1
dr 2
r=0
,
1
ρ 2
=
d 2 F 2
dr 2
r=0
,
whereas ρ 1 and ρ 2 are the curvature radii of the forming surfaces F 1 and F 2 in the
point r = 0. These curvature radii are positive if the body in the contact point is
convex and are negative if the body is concave. Taking into account the comments
from formulas (6.11), it follows that
F 1 [r] − F 2 [r] =
1
2
1
ρ 1
+
1
ρ 2
(x
2
+ y
2 ).
(6.12)
Then formula (6.8) will look like
(k 1 + k 2 )(x, y) = −
1
2
1
ρ 1
+
1
ρ 2
(x
2
+ y
2 ) + δ,
(6.13)
where (x, y) is the integral operator defined by formula (5.21). A single continuous solution of Eq. (6.13) can be represented as
p(x, y) = c
a 2 − x 2 − y 2 ,
(6.14)
where c and a are the constant values to be found. To do it, let us use formula (5.29);
we obtain
(x, y) = c
π 2
4
(2a
2
− x
2
− y
2 ), (x
2
+ y
2
a
2 ).
By substituting the last equation to Eq. (6.13) and equaling the coefficients with the
same degrees of x and y, we obtain
c =
2
π 2
1
ρ 1
+
1
ρ 2
k 1 + k 2
, δ =
π 2
2
(k 1 + k 2 )ca
2 .
(6.15)
The constant value a (the radius of the circular contact area) is determined [1]
from the equilibrium condition