34
3 The Second Primary Problem of Elasticity Theory
u = ϕ x −
z
2(1 − 2ν)
∂∂
∂x
,
v = ϕ y −
z
2(1 − 2ν)
∂∂
∂y
,
w = ϕ z −
z
2(1 − 2ν)
∂∂
∂z
,
(3.17)
based on formula (3.12), the functions harmonic in D ϕ x , ϕ y , ϕ z , and are
connected by the dependency
∂∂
∂z
= =
∂u
∂x
+
∂v
∂y
+
∂w
∂z
=
∂ϕ x
∂x
+
∂ϕ y
∂y
+
∂ϕ z
∂z
−
1
2(1 − 2ν)
∂∂
∂z
,
or
∂∂
∂z
=
2(1 − 2ν)
3 − 4ν
∂ϕ x
∂x
+
∂ϕ y
∂y
+
∂ϕ z
∂z
.
(3.18)
The total integral of the equation system (3.6) in the form of formulas (3.17)
provided that (3.18) is referred to as the Trefftz integral
We note that formulas (3.17) can be obtained directly from Eqs. (3.6). Indeed, by
using expression (3.14), we have
∂∂
∂x
=
∂ 2
∂x∂z
=
1
2
z
∂∂
∂x
.
In this case, the first Eq. (3.6) will look as:
u +
z
2(1 − 2ν)
∂∂
∂x
= 0,
hence we obtain the first of formulas (3.17). The second formula is obtained in the
case of respective substitution of the axes Ox and Oy, and the third formula from
(3.17) is identical to the previously obtained formula (3.9).
3.5 Grodsky–Neyber–Papkovich Integral
Let F be the function, which is an arbitrary solution of the equation
F = .
(3.19)
Strain equations (3.6) are represented as: