5.2 Stress on Surface
47
or
P = −
f
2 − 2ν
.
(5.6)
The third condition (5.3) when using the second formula from (3.4) looks as
follows:
2G
∂f
∂z
+
∂P
∂z
z=0
+ λλ| z=0 = −p(x, y).
By substituting here the expression f from formula (4.3), we obtain
2G(1 − 2ν)
∂P
∂z
z=0
+ λλ(x, y, 0) = −p(x, y).
(5.7)
Let us note that the comparison of formulas (5.1) and (3.17) shows
= −2(1 − 2ν)P .
Having this in mind, we now use formula (3.12) to determine volumetric expansion
on the surface y, 0). We have
= −2(1 − 2ν)
∂P
∂z
.
(5.8)
In this manner, the condition (5.7) taking into account formula (5.8) results in the
following formula for volumetric expansion on the surface of the elastic half-space
y, 0) = −
1 − 2ν
G
p(x, y).
(5.9)
5.2 Stress on Surface
The representation of volumetric expansion as (3.17) allows formula (5.9) to look
as follows
y, 0) =
1 − 2ν
E
σ x (x, y, 0) + σ y (x, y, 0) − p(x, y)
= −
1 − 2ν
G
p(x, y),
e.g.
σ x (x, y, 0) + σ y (x, y, 0) = −(1 + 2ν)p(x, y).
(5.10)