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三維壓電彈性力學
Chien-Ching Ma Ru-Li Lin
92.03.31
內容大綱
三維問題
二維問題_平面問題
二維問題_反平面問題
二維問題_壓電問題
二維問題_解的架構
二維問題_全平面的基本解
二維問題_半平面的解函數
三維(3–D)問題
y
y
f
x
z
z
x
f
二維(2–D)問題
y
f
A
B
C
D
x
f
y
A
f
B
C
D
x
A
B
平 面 問 題
D
C
A
z
B
D
C
反 平 面 問 題
x
Constitutive Equation of Piezoelectric Material
Di : 電位移 (Electric displacement)
ij cijkluk ,l eijk Ek
Di eikl u k ,l ik Ek
Ek : 電場 (Electric field)
C ijkl : 彈性模數 (Elastic modulus)
eijk : 壓電係數 (Piezoelectric coefficient)
ij : 介電常數 (Dielectric constant)
ij sijkl kl g kij Dk
sijkl : 彈性柔度 (Elastic compliance)
g kij : 壓電常數 (Piezoelectric constant)
Ei g ikl kl ik Dk
ik : 反誘電常數 (Dielectric
impermeability constant)
構 造 方 程 式 的 矩 陣 形 式
xx s11
yy s12
zz s13
2
yz s14
2 xz s15
2 xy s16
Ex
g11
E y g 21
E
g 31
z
s12
s13
s14
s15
s22
s23
s24
s25
s23
s33
s34
s35
s24
s34
s44
s45
s25
s35
s45
s55
s26
s36
s46
s56
g12
g13
g14
g15
g 22
g 23
g 24
g 25
g 32
g 33
g 34
g 35
s16 xx g11
s26 yy g12
s36 zz g13
s46 yz g14
s56 xz g15
s66 xy g16
xx
yy
g16 11
zz
g 26 12
yz
g 36
xz 13
xy
g 21
g 22
g 23
g 24
g 25
g 26
g 31
g 32
Dx
g 33
Dy
g 34
Dz
g 35
g 36
12 13 Dx
22 23 D y
23 33 Dz
Generalized Plane Problem (infinite at z)
xx a11 a12
yy a12 a22
yz a14 a24
xz a15 a25
a
a26
xy
16
E x b11 b12
E b
y 21 b22
aij sij
bij g ij
si 3 s j 3
s33
s j3 gi3
a14
a24
a44
a45
a46
b14
b24
a15
a25
a45
a55
a56
b15
b25
a16
a26
a46
a56
a66
b16
b26
b11
b21 xx
b12
b22 yy
b14
b24 yz
b15
b25 xz
b16
b26 xy
d11 d12 Dx
d12 d 22 Dy
( g 3i si 3 g 33 / s33 )( g 3 j s j 3 g 33 / s33 )
33 g 33 g 33 / s33
( i 3 g i 3 g 33 / s33 )( g 3 j s j 3 g 33 / s33 )
s33
33 g 33 g 33 / s33
g i 3 g i 3 ( i 3 g i 3 g 33 / s33 )( 3 j s j 3 g 33 / s33 )
d ij ij
s33
33 g 33 g 33 / s33
i, j 1, 2, , 6
i 1, 2, j 1, 2, , 6
i, j 1, 2
Equilibrium Equations
ij , j f i 0 : Elastic Equilibrium, (fi : body force)
Di ,i q 0 : Gauss’s law of Electrostatics or Maxwell’s equation
(q: electric charge density)
Compatibility Equations
2
2 xy
2 xx yy
2
0
2
2
xy
y
x
E x E y
0
y
x
Stress and Electric Displacement function (F , , )
xx
2F
2
y
2F
yy 2
x
2F
xy
xy
xz
y
yz
x
y
Dy
x
In-Plane
Anti-Plane
In-Plane
Dx
Governing Equation of Generalized Plane Problem
L4 F L3 L*3 0
*
L
L
F
L
3
2 0
2
*
**
*
L
L
F
L
2 0
2
3
L8 F 0
2
2
2
L2 a44 2 2a45
a55 2
xy
x
y
2
2
2
L*2 b24 2 (b14 b25 )
b15 2
xy
x
y
2
2
2
**
L2 d 22 2 2d12
d11 2
xy
x
y
L8 L4 L2 L*2 2 L3 L*3 L*2 L*3 L*3 L2 L4 L*2 L*2 L3 L*2 L*2
4
4
4
4
4
L4 a22 4 2a26 3 (2a12 a66 ) 2 2 2a16
a11 4
3
x
x y
x y
xy
y
3
3
3
3
L3 a24 3 (a25 a46 ) 2 (a14 a56 )
a15 3
2
x
x y
xy
y
3
3
3
3
L b22 3 (b12 b26 ) 2 (b21 b16 )
b11 3
2
x
x y
xy
y
*
3
Special Cases
Case I : Monoclinic (symmetric w.r.t. x-y plane, class m)
s11
s
12
s13
sij
0
0
s16
s12
s13
0
0
s22
s23
0
0
s 23
s33
0
0
0
0
s 44
s45
0
0
s45
s55
s26
s36
0
0
L3 L*2 0
s16
s26
s36
0
0
s66
g11
g ij g 21
0
11
ij 12
0
L4 F L*3 0
L2 0
*
**
L
F
L
2 0
3
g12
g13
0
0
g 22
0
g 23
0
0
g 34
0
g 35
12
22
0
0
33
0
( L4 L*2 L*3 L*3 ) F 0
L2 0
In-plane problem of piezoelectric material
g16
g 26
0
Case II : Transversely Isotropic (Hexagonal, class 6mm)
s11
s
12
s13
sij
0
0
0
s12
s13
0
0
s12
s13
0
0
s13
s33
0
0
0
0
s44
0
0
0
0
s44
0
0
0
0
L3 L*3 0
0
0
0
0
g ij 0
0
0 g15
0
g 31 g 31 g 33 0
0
0
0
11 0
0
0
0
11
ij
2( s11 s12 )
0
0 33
0
L4 F 0
L2 L*2 0
*
**
L
L
2
2 0
L4 F 0
**
* *
(
L
L
L
2 L2 ) 0
2 2
Anti-plane problem of piezoelectric material
g15
0
0
0
0
0
Case III : Orthotropic (Orthorhombic, class mmm)
L L 0
*
2
*
3
L4 F L3 0
L3 F L2 0
**
L2 0
( L4 L2 L3 L3 ) F 0
**
L2 0
No piezoelectric effect
Case IV : Orthotropic (Principal axis = axis of coordinate )
L*2 L*3 L3 0
L4 F 0
L2 0
**
L2 0
No piezoelectric effect
Plane problem of transversely isotropic material
Symmetric plane = x-y plane, poling direction = z axis
xx s11
yy s12
zz s13
2
yz 0
2 xz 0
2 xy 0
s12
s13
0
0
s11
s13
0
0
s13
s33
0
0
0
0
s44
0
0
0
0
s44
0
0
0
0
0 xx 0
0 yy 0
0 zz 0
0 yz 0
0 xz g15
s66 xy 0
0
0
0
g 24
0
0
g 31
g 32
Dx
g 33
D y
0
Dz
0
0
s66 2(s11 s12 )
Ex
0
E y 0
E
g 31
z
0
0
0
g15
0
0
g15
0
g 32
g 33
0
0
xx
yy
0 11
zz
0 0
yz
0
0
xz
xy
0
11
0
0 Dx
0 D y
33 Dz
Complete State of Electromechanical Interaction
for Two-Dimensional Model
fy
f2
fx
Dn
f1
Dn
fy
fx
z
Dn
f2
f1
Dn
fy
f2
fx
Dn
f1
x
Dn
y
x x1
y
yy yz xz E y 0
z x2
Constitutive Equations
of Two-Dimension Transversely isotropic material
a12
11 a11
a
a22
22 12
0
2 12 0
E 0
0
1
E2 b21 b22
0
0
a33
b13
0
0 b21 11
0 b22 22
b13 0 12
d11 0 D1
0 d 22 D2
b21 (1 s12 / s11 ) g 31
d11 11
b22 g 33 g 31s13 / s11
2
d 22 33 g 31
/ s11
b13 g15
s122
a11 s11
s11
a12 s13
s12 s13
s11
s132
a22 s33
s11
a33 s44
Equilibrium equation (absence of body force and free electric volume charge)
11 12
0
x1
x2
12 22
0
x1
x2
Compatibility equation
2
2 xy
2 xx yy
2
0
2
2
xy
y
x
E x E y
0
y
x
Stress function and electric displacement function
2F
11 2 ,
y
2F
22 2 ,
x
D2
,
x2
D1
x1
2F
12
xy
D1 D2
0
x1 x2
Governing Equations
of Two-Dimension Transversely isotropic material
L4 F L3 0
L3 F L2 0
( L4 L2 L3 L3 ) F 0
4
4
4
L4 a22 4 (2a12 a33 ) 2 2 a11 4
x1
x1 x2
x2
3
3
L3 b22 3 (b21 b13 )
x1
x1x22
2
2
L2 d 22 2 d11 2
x1
x2
Explicit form of governing equation
6 F
6 F
6F
6 F
4 6 3 4 2 2 2 4 1 6 0
x1
x1 x2
x1 x2
x2
1 a11d11
2 a11d 22 2a12d11 a33d11 b212 b132 2b21b13
3 a22d11 2a12d 22 a33d 22 2b21b22 2b13b22
4 a22d 22 b222
Fourier transform pair
~
g ( , Y ) g ( X , Y )e iX dX
1
g( X ,Y )
2
g~( , Y )e iX d
Governing equation in transform domain
~
4~
2~
d 6F
F
d
F
~
1 6 2 2 4 3 4 2 4 6 F 0
dx2
dx2
dx1
Let
~
F ceipx2
6 (1 p 6 2 p 4 3 p 2 4 )ceipx 0
1 p 6 2 p 4 3 p 2 4 0
2
Characteristic of roots (form I)
p1 i1 , p2 i 2 , p3 i3 , p4 p1 , p5 p2 , p6 p3
Characteristic of roots (form II)
p1 i , p2 i , p3 i3 , p4 p1 , p5 p2 , p6 p3
Operator in transform domain
2
4
d
d
~
L4 4 a 22 (2a12 a 33 ) 2 2 a11 4
dx 2
dx 2
d2
~
3
L3 i b22 i (2b21 b13 ) 2
dx2
d2
~
2
L2 d 22 d11 2
dx2
~
~
Relations of F and in the transform domain
By
~~ ~~
L3 F L2 0
~
~
F
k
Let
~
F ceipx2
i (b21 b13 ) p 2 b22
where k
d 22 d11 p 2
Or
~ ~ ~~
~ ~
a11 p 4 (a33 2a12 ) p 2 a22
*
L4 F L3 0 k F where k
d 22 d11 p 2
k *k 1 p 6 2 p 4 3 p 2 4 0
Field in the transform domain
~
d F
2 2~
11 2 pk F
dx2
2
~
~ d
~
D1
ip k k 2 F
dx2
~
~22 2 F
~
dF
2~
12 i
pk F
dx2
~
~
2~
D2 i ik F
~
~
~11 a11~11 a12~22 b21D2 (a11 pk2 a12 ib21k ) 2 F
~
2
~
u1 i(a11 pk a12 ib21k ) F
~
~
~22 a12~11 a22~22 b22 D2 (a12 pk2 a22 ib22k ) 2 F
~
~
u2 i(a11 pk a12 / pk ib21k / pk ) F
General solution in the transform domain
~
σ
u
~ P e i p* y c
~
D
~
1
22
~
σ
2 ~12
~
u
1
1
~
u
i u~2
~
1 D1
~
D 2 ~
i D2
1
1
1
1
1
1
p
p
p
p
p
p
2
3
4
5
6
1
( p1 ) ( p2 ) ( p3 ) ( p4 ) ( p5 ) ( p6 )
P
(
p
)
(
p
)
(
p
)
(
p
)
(
p
)
(
p
)
1
2
3
4
5
6
p11
p 2 2
p33
p 4 4
p5 5
p6 6
1
2
3
4
5
6
c1
c
2
c3
c
c 4
c5
c6
Green’s Function of Infinite Plane
~
~
f
σ
σ
1
u
~
~
u
b
~
i
~
D y d D y d
iQ
f2
f
f1
c c
b1
b
b2
1
2
e i p*d q
Q1
Q
Q2
q1
q
2
f
q3
q P 1 b
q4
D
q5
q6
iq j
3
x
p
(
x
d
)
22
1 j 1 1
j
2
Re 3
ip
q
j j
12
j 1 x1 p j ( x2 d )
3
( p j 3 )q j ln x1 p j ( x2 d )
u1
1 j 1
Re
3
u
( p )q ln x p ( x d )
2
j 3
j
1
1
2
j 1
3
D1
1 j 1 x1
D Im 3
2
j 1 x1
iq j
p j ( x2 d )
ip j q j
p j ( x2 d )
Surface Green’s Function of Half-Plane
22 ( x,0) f 2 ( x)
f2
x1
12 ( x,0) 0
D2 ( x,0) 0
x2