Composite Materials Research Progress

Eletromechanical Field Concentrations and Polarization Switching... 259
where the superscript l denotes the linear contribution to the strain and electric displacement,
and the linear piezoelectric relationships are given by
ε l ij = sijklσkl + dkijEk (7)
D l i = diklσkl + ɛikEk (8)
In Eqs. (7) and (8), sijkl, dkij and ɛik are the elastic compliance tensor, direct piezoelectric
tensor and dielectric permittivity tensor, which satisfy the following symmetry relations:
sijkl = sjikl = sijlk = sklij, dkij = dkji, ɛik = ɛki
σij and D l i are related to εl ij and Ei by
σij = cijklε l kl − ekijEk (10)
D l i = eiklε l kl + ɛikEk (11)
where cijkl and eikl are the elastic and piezoelectric tensors, and
cijkl = cjikl = cijlk = cklij, ekij = ekji
For piezoceramics which exhibit symmetry of a hexagonal crystal of class 6 mm with respect
to principal x1,x2, and x3 axes, the constitutive relations can be written in the following
form:
⎧
⎪⎨
⎪⎩
⎧
⎪⎨
⎪⎩
σ1
σ2
σ3
σ4
σ5
σ6
where
D l 1
D l 2
D l 3
⎫
⎡
⎢
⎪⎬
⎢
= ⎢
⎣
⎪⎭
⎫
⎪⎬
⎪⎭ =
⎡
⎢
⎣
c11 c12 c13 0 0 0
c12 c11 c13 0 0 0
c13 c13 c33 0 0 0
0 0 0 c44 0 0
0 0 0 0 c44 0
0 0 0 0 0 c66
0 0 0 0 e15 0
0 0 0 e15 0 0
e31 e31 e33 0 0 0
⎤ ⎧
⎥ ⎪⎨
⎥
⎦
⎪⎩
σ1 = σ11, σ2 = σ22, σ3 = σ33
ε l 1
ε l 2
ε l 3
ε l 4
ε l 5
ε l 6
(9)
(12)
⎫ ⎡
0
⎢
⎪⎬
⎢ 0
⎢ 0
− ⎢ 0
⎢
⎣
⎪⎭
e15
0
0
0
0
e15
0
0
e31
e31
e33
0
0
0
⎤
⎥ ⎧
⎥ ⎪⎨
⎥ ⎪⎩ ⎥
⎦
E1
E2
E3
⎫
⎪⎬
⎪⎭
(13)
⎧
ε
⎤
⎪⎨
⎥
⎦
⎪⎩
l 1
εl 2
εl 3
εl 4
εl 5
εl ⎫
⎡
⎤ ⎧ ⎫
⎪⎬ ɛ11 0 0 ⎪⎨ E1 ⎪⎬
⎢
⎥
+ ⎣ 0 ɛ11 0 ⎦ E2
⎪⎩ ⎪⎭
0 0 ɛ33 E3
⎪⎭
6
(14)
σ4 = σ23 = σ32, σ5 = σ31 = σ13, σ6 = σ12 = σ21
ε l 1 = ε l 11, ε l 2 = ε l 22, ε l 3 = ε l 33
ε l 4 =2εl 23 =2εl 32 ,εl 5 =2εl 31 =2εl 13 ,εl 6 =2εl 12 =2εl 21
�
�
(15)
(16)