50
5 Elastic Half-Space
where ρ is defined by formula (5.17). The calculation of is reasonable to be
performed in polar coordinates. To do it, apart from the polar radius ρ, let us
introduce the angle α between some fixed beam and the beam ρ connecting the
points (x, y) and (ξ, η). Then formula (5.21) can be written as follows
=
(∞)
p[ρ, α]dS
ρ
,
where
p[ρ, α] = p[ξ(ρ, α), η(ρ, α)].
Since the area element dS in polar coordinates is expressed through the polar
coordinates ρ, α under the formula dS = ρdρdα, then
=
(∞)
p[ρ, α]dρdα.
(5.22)
Let us note that the integral
I =
(∞)
pdρ,
taken upon an unlimited straight line, represents an area (ω) of cross-section of
some body limited by the coordinate plane z = 0 and surface z = p[ρ, α]. With
the parameters x, y being constant, the area (ω) is a function of only the angle α.
Consequently, formula (5.22) can be written as follows
=
π
0
ω(α)dα,
(5.23)
where
ω(α) =
∞
0
|p[ρ, α] + p[ρ, α + π ]| dρ.
(5.24)
5.4 Examples
Case 1. Let the pressure p on the half-space boundary be set by the formula