BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI

16 Ion Crăciun
where the scalar Laplace operator of a scalar f , denoted by
and
2 2
2 1 f 1 f f
f
∂ ⎛
r
∂ ⎞
∇ = + ∂ +
∂
2 2 2
⎜ ⎟
r ∂r⎝
∂r ⎠ r ∂θ ∂z
,
2
∇ f , is
(29)
2 2 1
∇
0
=∇ −
2 .
(30)
r
By starting with (9), and by taking into account (26)-(28) we arrive to the
differential system of steady vibrations of micropolar elasticity in cylindrical
coordinates
⎧ ⎛ 0 2 ∂uθ
⎞ ∂( ∇· u) ⎛1∂φz
∂φθ
⎞
⎪( μ+ α) ⎜ 2
ur
− ( λ μ α) 2 X 0,
2
r
r θ
⎟+ + − + α
r
⎜ −
r θ z
⎟+ =
⎪ ⎝ ∂ ⎠ ∂ ⎝ ∂ ∂ ⎠
⎪
⎪ ⎛ 0 2 ∂ur
⎞ 1 ∂( ∇· u)
⎛∂φr
∂φz
⎞
( μ+ α) 2
uθ
+ + ( λ+ μ− α) + 2α
− + X 0,
2
θ
=
⎪
⎜
r θ
⎟
r θ
⎜
z r
⎟
⎝ ∂ ⎠ ∂ ⎝ ∂ ∂ ⎠
⎪
⎪
∂( ∇· u) 2α
⎛ ∂ ∂φr
⎞
⎪( μ+ α) 2uz
+ ( λ+ μ− α) + ⎜ ( rφθ
) − ⎟+ X
z
= 0,
⎪
∂z r ⎝∂r ∂θ
⎠
⎨
⎪ ⎛ 0 2 ∂φθ
⎞ ∂( ∇ϕ · ) ⎛1∂uz
∂uθ
⎞
⎪
( γ+
ε)
⎜ 4
φr
− + ( β+ γ− ε) + 2α − + Y 0,
2
r
=
r ∂θ ⎟
∂r ⎜
r ∂θ
∂z
⎟
⎪
⎝ ⎠ ⎝ ⎠
⎪
⎛ 0 2 ∂φur
⎞ 1 ∂( ∇· φ)
⎛∂ur
∂uz
⎪
⎞
( γ+ ε) ⎜ 4
φθ
+ ( β γ ε) 2α Y 0,
2 ⎟+ + − + ⎜ − ⎟+ θ
=
⎪ ⎝ r ∂θ ⎠ r ∂θ ⎝ ∂z ∂r
⎠
⎪
⎪
∂( ∇· φ) 2α
⎛ ∂ ∂ur
⎞
⎪( γ+ ε) 4φz
+ ( β+ γ− ε) + ⎜ ( ruθ
) − ⎟+ Yz
= 0,
⎩
∂z r ⎝∂r ∂θ
⎠
(31)
where u= urer + uθeθ + uzez,
φ = φrer + φθeθ + φze
z.
are the amplitudes of the
displacement vector and rotation vector, and X = X
rer + Xθeθ + Xze z
and
Y= Ye
+ Y e + Ye
are the amplitudes of body and couple-body loadings,
r r θ θ z z .
respectively, while the operators
0 0
2,
4
are determined by
4. Axially-symmetric Problems
2 2 1
∇
0
=∇ −
2 .
r
Consider now the case of axially-symmetric deformations of the body B
assuming that all the causes and effects are independent of the variable θ .
Consequently, we find that system (28) splits into two independent systems.
The first of them contains the displacement u= urer + uzezand the rotation
φ = φ θ
e θ
, while in the second system there appears the displacement u= uθeθ
and the rotation φ = φre+ φze
z.
The first system is