BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 17
0
⎧
2
ur + ( λ+ μ−α) ∂re −2α∂ zφθ + X
r
= 0,
⎪
−1
⎨ 2uz + ( λ+ μ−α) ∂
ze + 2 αr ∂
r( rφθ ) + X
z
= 0, (32)
⎪
0
⎪⎩ 4
φθ + 2 α( ∂zur −∂
ruz) + Yθ
= 0.
Here, we used the symbols ∂r,
∂
z
to denote partial derivatives with respect to
the position variable r, z , respectively. The dilatation e = ∇ · u in cylindrical
coordinates and in the absence of variable θ , is given by
−1
e= r ∂
r( rur)
+∂
zuz.
0
The symbols X
r, Yθ , Xz
denote body loadings and the operators
i
are defined
through the operator by
2
∇ 0
0 2 2 0 2
2 0 2 4 0
= ( μ + α)( ∇ + σ ); = ( γ + ε)( ∇ + σ 4
).
The role of the position variable x is played by the coordinates ( rz , ).
The form of the second system in B is
0
⎧
4
ϕr + ( β + γ −ε) ∂rκ −2α∂ zuθ
+ Yr
= 0,
⎪
−1
⎨ 4ϕz + ( β + γ −ε) ∂
zκ + 2 αr ∂ r( ruθ
) + Yz
= 0,
(33)
⎪ 0
⎩ 2
uθ
+ 2 α( ∂zϕr −∂
rϕz) + Xθ
= 0,
−1
where κ = ∇ · ϕ = r ∂
r( rϕr) +∂
zϕz, ϕ = ( ϕr, 0,
ϕz),
and Yr, Xθ
, Yz
0 are body
loadings.
5. The Second Axially-symmetric Problem of Micropolar Elasticity
in the Case of Steady Vibrations
This important problem of micropolar elasticity in the case of steady
vibrations has as object of study the system (30). There are many references
about the second axially-symmetric problem of micropolar elasticity in the case
of steady vibrations. We quoute here Nowacki (1986), Kupradze (1986), and
Dyszlewich (2004). The uncoupled form of system (30) may be found in
Dyszlewicz (2004)
2
0 0
⎧Ω uθ
= 2 α( ∂zYr −∂rYz) −
4
Xθ
,
⎪
0 0 0 0 0 0
⎨ 3Ω φr =∂r( Θ Y ) −2 α∂z 3
Xθ −
2 3Yr
,
⎪
⎩ Ω φ =∂ ( Θ Y ) + 2 α [ r ∂ ( rX )] − Y
0
3 z z 3 −1
r θ 2 3 z
,
(34)
where
−1
Y = ∇ · Y= r ∂ ( rY ) +∂ Y , Y=
( Y , 0,
Y ),
r r z z r z