a análise de placas laminadas compostas inteligentes
a análise de placas laminadas compostas inteligentes
a análise de placas laminadas compostas inteligentes
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
7.3 Equações constitutivas do laminado piezelétrico 129<br />
⎧<br />
⎪⎨<br />
⎪⎩<br />
=<br />
+<br />
⎧<br />
⎪⎨<br />
⎪⎩<br />
=<br />
+<br />
Nx<br />
Ny<br />
Nxy<br />
⎡<br />
⎢<br />
⎣<br />
⎡<br />
⎢<br />
⎣<br />
Mx<br />
My<br />
Mxy<br />
⎡<br />
⎢<br />
⎣<br />
⎡<br />
⎢<br />
⎣<br />
⎫<br />
⎪⎬<br />
=<br />
⎪⎭<br />
N<br />
k=1<br />
zk<br />
A11 A12 A16<br />
A12 A22 A26<br />
A16 A26 A66<br />
L11 L12 L16<br />
L12 L22 L26<br />
L16 L26 L66<br />
⎫<br />
⎪⎬<br />
=<br />
⎪⎭<br />
N<br />
k=1<br />
zk−1<br />
zk<br />
B11 B12 B16<br />
B12 B22 B26<br />
B16 B26 B66<br />
F11 F12 F16<br />
F12 F22 F26<br />
F16 F26 F66<br />
⎛⎡<br />
⎜<br />
⎜⎢<br />
⎜⎢<br />
⎝⎣<br />
<br />
⎧<br />
<br />
⎪⎨<br />
<br />
+<br />
<br />
⎪⎩<br />
<br />
⎧<br />
⎤<br />
⎪⎨ ⎥<br />
⎦<br />
⎪⎩<br />
⎧<br />
⎤<br />
⎪⎨ ⎥<br />
⎦<br />
zk−1<br />
⎪⎩<br />
⎛⎡<br />
⎜<br />
⎜⎢<br />
⎜⎢<br />
⎝⎣<br />
C11 C12 C16<br />
C12 C22 C26<br />
C16 C26 C66<br />
∂u 0<br />
∂y<br />
e31<br />
e32<br />
e36<br />
∂ψ3x<br />
∂y<br />
<br />
⎧<br />
<br />
⎪⎨<br />
<br />
+<br />
<br />
⎪⎩<br />
<br />
⎧<br />
⎤<br />
⎪⎨ ⎥<br />
⎦<br />
⎪⎩<br />
⎧<br />
⎤<br />
⎪⎨ ⎥<br />
⎦<br />
⎪⎩<br />
∂u 0<br />
∂x<br />
∂v 0<br />
∂y<br />
⎫<br />
⎪⎬<br />
⎪⎭<br />
(k)<br />
+ ∂v0<br />
∂x<br />
∂ψ3x<br />
∂x<br />
∂ψ3y<br />
∂y<br />
<br />
+ ∂ψ3y<br />
∂x<br />
⎫<br />
⎤(k)<br />
⎧⎪<br />
⎥ ⎨<br />
⎥<br />
⎦<br />
⎪⎩<br />
− ϕk−1<br />
hk<br />
⎡<br />
⎪⎬ ⎢<br />
+ ⎢<br />
⎣<br />
⎪⎭<br />
⎫<br />
⎪⎬<br />
−<br />
⎪⎭<br />
C11 C12 C16<br />
C12 C22 C26<br />
C16 C26 C66<br />
∂u 0<br />
∂y<br />
∂ψ3x<br />
∂y<br />
e31<br />
e32<br />
e36<br />
∂u 0<br />
∂x<br />
∂v 0<br />
∂y<br />
⎫<br />
⎪⎬<br />
⎪⎭<br />
(k)<br />
+ ∂v0<br />
∂x<br />
∂ψ3x<br />
∂x<br />
∂ψ3y<br />
∂y<br />
+ ∂ψ3y<br />
∂x<br />
<br />
⎫<br />
⎞<br />
ε 0 x + zκx + z 3 κ3x<br />
ε 0 y + zκy + z 3 κ3y<br />
γ 0 xy + zκxy + z 3 κ3xy<br />
⎟<br />
⎠ dz<br />
B11 B12 B16<br />
B12 B22 B26<br />
B16 B26 B66<br />
npiez <br />
k=1<br />
⎤(k)<br />
⎧⎪<br />
⎥ ⎨<br />
⎥<br />
⎦<br />
⎪⎩<br />
− ϕk−1<br />
hk<br />
⎡<br />
⎪⎬ ⎢<br />
+ ⎢<br />
⎣<br />
⎪⎭<br />
⎫<br />
⎪⎬<br />
−<br />
⎪⎭<br />
⎧<br />
⎪⎨<br />
⎪⎩<br />
⎞<br />
e31<br />
e32<br />
e36<br />
⎫<br />
⎪⎬<br />
⎪⎭<br />
(k)<br />
⎧<br />
⎤<br />
⎪⎨ ⎥<br />
⎦<br />
⎪⎩<br />
ϕk−1<br />
ε 0 x + zκx + z 3 κ3x<br />
ε 0 y + zκy + z 3 κ3y<br />
∂ψx<br />
∂y<br />
γ 0 xy + zκxy + z 3 κ3xy<br />
⎟ z dz<br />
⎠<br />
D11 D12 D16<br />
D12 D22 D26<br />
D16 D26 D66<br />
npiez <br />
k=1<br />
⎧<br />
⎪⎨<br />
⎪⎩<br />
e31<br />
e32<br />
e36<br />
⎫<br />
⎪⎬<br />
⎪⎭<br />
(k)<br />
⎧<br />
⎤<br />
⎪⎨ ⎥<br />
⎦<br />
⎪⎩<br />
ϕk−1<br />
∂ψx<br />
∂y<br />
1<br />
2hk<br />
⎫<br />
<br />
⎪⎬<br />
<br />
<br />
<br />
<br />
⎪⎭<br />
<br />
<br />
<br />
∂ψx<br />
∂x<br />
∂ψy<br />
∂y<br />
+ ∂ψy<br />
∂x<br />
⎫<br />
<br />
⎪⎬<br />
<br />
<br />
<br />
<br />
⎪⎭<br />
<br />
<br />
<br />
∂ψx<br />
∂x<br />
∂ψy<br />
∂y<br />
+ ∂ψy<br />
∂x<br />
⎫<br />
⎪⎬<br />
⎪⎭<br />
(z 2 k − z 2 k−1)<br />
⎫<br />
(7.59)<br />
⎪⎬<br />
(7.60)<br />
⎪⎭