5.3 Strain of Elastic Half-Space
49
f (x, y, z) =
1 − ν 2
πE
(∞)
p(ξ, η)dξ dη
(x − ξ) 2 + (y − η) 2 + z 2
.
(5.15)
To perform similar steps relative to the functions ϕ and ψ from formulas (5.4)
and (5.6), we find
∂ϕ
∂z
= −
1 − 2ν
2 − 2ν
∂f
∂x
??? z = 0.
(5.16)
Using formula (5.15) and the designation
ρ =
x − ξ) 2 + (y − η) 2 ,
(5.17)
we find
∂ϕ
∂z
z=0
=
1 − 2ν
πG
∂
∂x
(∞)
p(ξ, η)dξ dη
ρ
.
(5.18)
Let us find in a totally similar way as follows
∂ψ
∂z
z=0
=
1 − 2ν
πG
∂
∂y
(∞)
p(ξ, η)dξ dη
ρ
.
(5.19)
Noting that w(x, y, 0) = f (x, y, 0), let us find the half-space surface subsidence
from formula (5.15):
w(x, y, 0) =
1 − ν 2
πE
(∞)
p(ξ, η)dξ dη
(x − ξ) 2 + (y − η) 2
.
(5.20)
The last formula has a number of important applications in both strain mechanics
and engineering.
5.3.1 Integral Operator of Formulas (5.18)–(5.20)
Formulas (5.18)–(5.20) include an integral
=
(∞)
p(ξ, η)dξ dη
ρ
,
(5.21)