3.3 Application of Harmonic Functions
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An elastic body is given (area D). Three components u, v, w of the movement vector are
required that satisfy equations (3.6) inside D. In each non-specific point of the body surface,
movements must satisfy three boundary conditions.
Boundary conditions can be formulated in three variants of the problem:
movements are set; combinations of stresses are set, which are written through
normal and tangential derivatives from movements; combinations of stresses and
movements are set, which are written through normal and tangential derivatives
from movements and through movements themselves.
3.3 Application of Harmonic Functions
Any function ϕ(x, y, z) having in some area D continuous second derivatives and
satisfying in this area Laplace’s equation [2]
ϕ =
∂ 2 ϕ
∂x 2 +
∂ 2 ϕ
∂y 2 +
∂ 2 ϕ
∂z 2 = 0
(3.7)
is referred to as the function harmonic in D. Speaking of harmonic functions in the
future, we will consider D as a non-enclosed area occupied by a body, excluding
boundary body surfaces.
Let us show that volumetric expansion is the function harmonic in D. For this
purpose, let us differentiate the first Eq. (3.6) upon x, the second equation upon y,
and the third equation upon z and let us sum up differentiated equations. We obtain
∂
∂x
u +
∂
∂y
v +
∂
∂z
w +
1
1 − 2ν
= 0.
By changing the order of differentiation, we have
∂u
∂x
+
∂v
∂y
+
∂w
∂z
+
1
1 − 2ν
= 0
or
2(1 − ν)
1 − 2ν
· = 0.
The multiplier in front of is not converted into zero since for real solid bodies
0 < ν < 0, 5. The last equation follows that = 0, which means that is the
function harmonic in D.
Any function harmonic in D is unrestrictedly differentiated. Therefore, we can
write