BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI

14 Ion Crăciun
If we introduce the notations
⎧ ω 2 2
2 4α
2 4α
⎪σi
= , σ3 = σ3 − , σ4 = σ4
− ,
⎪
ci
β+ 2γ γ+
ε
⎪ 1 ρ 1 ρ 1 J 1 J
⎨ = , = , = , = ,
2 2 2 2
⎪c1 λ+ 2μ c2
μ+ α c3
β+ 2γ c4
γ+
ε
⎪
⎪ 2α
2α
p= , s=
,
⎪
⎩ γ+ ε μ+
α
then the operators , , , can be written as
where
1 2 3 4
2 2 2 2
⎧
1= ( λ+ 2 μ)( ∇ + σ1), 2= ( μ+ α)( ∇ + σ ),
⎪
2
⎨
2
2
2 2
⎪ ⎩ 3= ( β+ 2 γ)( ∇ + σ3 ),
4= ( γ+ ε)( ∇ + σ4
).
Another forms of the operators Ω and Θ are given by
2 2 2 2
⎧Ω=
⎪
( μ+ α)( ∇ + k1 )( ∇ + k2),
⎨
2 2 2 2 2 2
⎪Θ= ⎩ ( μ + α)( β+ 2 γ)( ∇ + σ ˆ
2
+ σ3) − ( μ+ α)( γ+ ε)( ∇ + k1 + k2),
2
k 1
and are the quadratic roots of the equation
(17)
(18)
(19)
k − ( σ + σˆ
+ ps)
k + σσˆ
=0. (20)
4 2 2 2 2 2
2 4 2 4
T h e o r e m 1 (Nowacki, 1986). If the functions (13) satisfy the
differential system
⎧ 1Ω Φ + X=
0,
⎨ (21)
⎩ 3 Ω Ψ + Y = 0,
then the set of functions (11) represents a solution of the basic differential
system (9).
Eqs. (11) and (13) are useful in the determination of the fundamental
solutions for the system of differential equations (9) in the unbounded elastic
space, and they have been obtained by using different methods by Nowacki
(1986), Kupradze (1986), Stefaniak (1968), Şandru (1966), and others.
3. The Basic Equations in Cylindrical Coordinates
For the analysis of specific problems of micropolar elasticity,
orthogonal curvilinear coordinates often lead to simplification of the
mathematical treatment.
In the study of some axially-symmetric problems of the micropolar
elasticity it is convenient to use the cylindrical coordinate system ( r, θ , z).
In
3
every point of the vector base of this coordinate system is formed by the
unit vectors er
, e
θ
, ez.
The first unit vector, orthogonal both to e
θ
and e z
, is