BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI
buletinul institutului politehnic din iaşi - Universitatea Tehnică ...
buletinul institutului politehnic din iaşi - Universitatea Tehnică ...
- No tags were found...
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
16 Ion Crăciun<br />
where the scalar Laplace operator of a scalar f , denoted by<br />
and<br />
2 2<br />
2 1 f 1 f f<br />
f<br />
∂ ⎛<br />
r<br />
∂ ⎞<br />
∇ = + ∂ +<br />
∂<br />
2 2 2<br />
⎜ ⎟<br />
r ∂r⎝<br />
∂r ⎠ r ∂θ ∂z<br />
,<br />
2<br />
∇ f , is<br />
(29)<br />
2 2 1<br />
∇<br />
0<br />
=∇ −<br />
2 .<br />
(30)<br />
r<br />
By starting with (9), and by taking into account (26)-(28) we arrive to the<br />
differential system of steady vibrations of micropolar elasticity in cylindrical<br />
coordinates<br />
⎧ ⎛ 0 2 ∂uθ<br />
⎞ ∂( ∇· u) ⎛1∂φz<br />
∂φθ<br />
⎞<br />
⎪( μ+ α) ⎜ 2<br />
ur<br />
− ( λ μ α) 2 X 0,<br />
2<br />
r<br />
r θ<br />
⎟+ + − + α<br />
r<br />
⎜ −<br />
r θ z<br />
⎟+ =<br />
⎪ ⎝ ∂ ⎠ ∂ ⎝ ∂ ∂ ⎠<br />
⎪<br />
⎪ ⎛ 0 2 ∂ur<br />
⎞ 1 ∂( ∇· u)<br />
⎛∂φr<br />
∂φz<br />
⎞<br />
( μ+ α) 2<br />
uθ<br />
+ + ( λ+ μ− α) + 2α<br />
− + X 0,<br />
2<br />
θ<br />
=<br />
⎪<br />
⎜<br />
r θ<br />
⎟<br />
r θ<br />
⎜<br />
z r<br />
⎟<br />
⎝ ∂ ⎠ ∂ ⎝ ∂ ∂ ⎠<br />
⎪<br />
⎪<br />
∂( ∇· u) 2α<br />
⎛ ∂ ∂φr<br />
⎞<br />
⎪( μ+ α) 2uz<br />
+ ( λ+ μ− α) + ⎜ ( rφθ<br />
) − ⎟+ X<br />
z<br />
= 0,<br />
⎪<br />
∂z r ⎝∂r ∂θ<br />
⎠<br />
⎨<br />
⎪ ⎛ 0 2 ∂φθ<br />
⎞ ∂( ∇ϕ · ) ⎛1∂uz<br />
∂uθ<br />
⎞<br />
⎪<br />
( γ+<br />
ε)<br />
⎜ 4<br />
φr<br />
− + ( β+ γ− ε) + 2α − + Y 0,<br />
2<br />
r<br />
=<br />
r ∂θ ⎟<br />
∂r ⎜<br />
r ∂θ<br />
∂z<br />
⎟<br />
⎪<br />
⎝ ⎠ ⎝ ⎠<br />
⎪<br />
⎛ 0 2 ∂φur<br />
⎞ 1 ∂( ∇· φ)<br />
⎛∂ur<br />
∂uz<br />
⎪<br />
⎞<br />
( γ+ ε) ⎜ 4<br />
φθ<br />
+ ( β γ ε) 2α Y 0,<br />
2 ⎟+ + − + ⎜ − ⎟+ θ<br />
=<br />
⎪ ⎝ r ∂θ ⎠ r ∂θ ⎝ ∂z ∂r<br />
⎠<br />
⎪<br />
⎪<br />
∂( ∇· φ) 2α<br />
⎛ ∂ ∂ur<br />
⎞<br />
⎪( γ+ ε) 4φz<br />
+ ( β+ γ− ε) + ⎜ ( ruθ<br />
) − ⎟+ Yz<br />
= 0,<br />
⎩<br />
∂z r ⎝∂r ∂θ<br />
⎠<br />
(31)<br />
where u= urer + uθeθ + uzez,<br />
φ = φrer + φθeθ + φze<br />
z.<br />
are the amplitudes of the<br />
displacement vector and rotation vector, and X = X<br />
rer + Xθeθ + Xze z<br />
and<br />
Y= Ye<br />
+ Y e + Ye<br />
are the amplitudes of body and couple-body loadings,<br />
r r θ θ z z .<br />
respectively, while the operators<br />
0 0<br />
2,<br />
4<br />
are determined by<br />
4. Axially-symmetric Problems<br />
2 2 1<br />
∇<br />
0<br />
=∇ −<br />
2 .<br />
r<br />
Consider now the case of axially-symmetric deformations of the body B<br />
assuming that all the causes and effects are independent of the variable θ .<br />
Consequently, we find that system (28) splits into two independent systems.<br />
The first of them contains the displacement u= urer + uzezand the rotation<br />
φ = φ θ<br />
e θ<br />
, while in the second system there appears the displacement u= uθeθ<br />
and the rotation φ = φre+ φze<br />
z.<br />
The first system is