32
3 The Second Primary Problem of Elasticity Theory
∂∂
∂x
=
∂
∂x
e. g. the derivative in the function harmonic in D is also the function harmonic in D.
By directly checking, we can make sure that for the function harmonic in D,
there is equation
[(z + c))] = 2
∂∂
∂z
, (c = const).
(3.8)
Suggesting in (3.8) ≡ , we have
∂∂
∂z
=
1
2
[(z + c))].
By substituting this expression into the last Eq. (3.6), we will find that
w +
z + c
2(1 − 2ν)
·
= 0.
The last equation means that the expression in brackets is the function harmonic
in D, e. g.
w = f z −
zz
2(1 − 2ν)
,
(3.9)
where f z is the function harmonic in D.
By substituting the coordinate z with x, and then with y, we obtain two other
similar ratios:
u = f x −
xx
2(1 − 2ν)
,
v = f y −
yy
2(1 − 2ν)
.
(3.10)
Here, the functions harmonic in D f x , f y , f z ? are connected, according to
formulas (1.13) and (2.2), by the dependency
5 − 4ν
2(1 − 2ν)
+
1
2(1 − 2ν)
x
∂∂
∂x
+ y
∂∂
∂y
+ z
∂∂
∂z
=
∂f x
∂x
+
∂f y
∂y
+
∂f z
∂z
.
(3.11)