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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
xy
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  
xy

 


 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
xy
x
y
2
2
2



L*2  b24 2  (b14  b25 )
 b15 2
xy
x
y
2
2
2
**
L2  d 22 2  2d12
 d11 2
xy
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
xy
y
3
3
3
3
L3  a24 3  (a25  a46 ) 2  (a14  a56 )
 a15 3
2
x
x y
xy
y
3
3
3
3
L  b22 3  (b12  b26 ) 2  (b21  b16 )
 b11 3
2
x
x y
xy
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
xy
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  
xy
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
x1x22
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 iX dX

1
g( X ,Y ) 
2



g~( , Y )e iX 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  ceipx2

 6 (1 p 6   2 p 4   3 p 2   4 )ceipx  0

1 p 6   2 p 4   3 p 2   4  0
2
Characteristic of roots (form I)


p1  i1 , p2  i 2 , p3  i3 , p4  p1 , p5  p2 , p6  p3
Characteristic of roots (form II)



p1    i , p2    i , p3  i3 , 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  ceipx2

 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  ik  F
~
~
~11  a11~11  a12~22  b21D2  (a11 pk2  a12  ib21k ) 2 F

~
2
~
 u1  i(a11 pk  a12  ib21k ) F
~
~
~22  a12~11  a22~22  b22 D2  (a12 pk2  a22  ib22k ) 2 F
~
~
 u2  i(a11 pk  a12 / pk  ib21k / 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 

 p11
p 2 2
p33
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