52
5 Elastic Half-Space
=
π 2
4
(2a
2
− x
2
− y
2 )
for x 2 + y 2 a 2 ,
π
2
(2a
2
− x
2
− y
2 ) arcsin
a
x 2 + y 2
+
for x 2 + y 2 > a 2
,
(5.29)
where
= a
x 2 + y 2 + a 2 .
Case 2. By generalizing case 1, assume that the pressure curve p(x, y) is limited
by the plane xOy and half of the ellipsoid located under this plane, e. g.
p(x, y) =
1 −
x 2
a 2 −
y 2
b 2 for
x 2
a 2 +
y 2
b 2 1,
0
f o r
x 2
a 2 +
y 2
b 2 > 1.
(5.30)
Let us determine the value of the operator at
x 2
a 2 +
y 2
b 2 1.
(5.31)
Let us introduce the polar coordinates (ρ, α) upon formulas
ξ = x + ρ cos α, η = y + ρ sin α.
(5.32)
When replacing (5.32), the function p inside the ellipse (5.31) is written as
p(ξ, η) = p[ρ, α] =
1 −
(x + ρ cos α) 2
a 2
−
(y + ρ sin α) 2
b 2
.
This equation can be converted as follows:
p[ρ, α] = A
1 −
ρ + ε
B
2
,
(5.33)
where
A =
1 −
(x sin α − y cos α) 2
a 2 sin
2 α + b 2 cos 2 α
;
B = A
ab
a 2 sin
2 α + b 2 cos 2 α
;