4.5 Electroelastic Nonlinear Constitutive Relations
73
σ =
⎧
⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎩
σ 11
σ 22
τ 12
τ 23
τ 13
⎫
⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎭
, ε =
⎧
⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎩
ε 11
ε 22
γ 12
γ 23
γ 13
⎫
⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎭
, D =
⎧
⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎩
D
(1)
3
D
(2)
3
. . .
D
(N )
3
⎫
⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎭
, E =
⎧
⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎩
E
(1)
3
E
(2)
3
. . .
E
(N )
3
⎫
⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎭
,
(4.90)
c =
⎡
⎢
⎢
⎢
⎢
⎣
c 11 c 12 0 0 0
c 12 c 22 0 0 0
0 0 c 66 0 0
0 0 0 c 44 0
0 0 0 0 c 55
⎤
⎥
⎥
⎥
⎥
⎦
, ¯
E =
⎡
⎢
⎢
⎢
⎣
E
(1)
3
0 · · · 0
0 E
(2)
3 · · · 0
. . .
. . .
. . .
. . .
0 0 · · · E
(N )
3
⎤
⎥
⎥
⎥
⎦
,
(4.91)
e =
⎡
⎢
⎢
⎢
⎣
e
(1)
31 e
(1)
32 0 0 0
e
(2)
31 e
(2)
32 0 0 0
. . .
. . .
. . .
. . .
. . .
e
(N )
31 e
(N )
32 0 0 0
⎤
⎥
⎥
⎥
⎦
, b =
⎡
⎢
⎢
⎢
⎣
b
(1)
331 b
(1)
332 0 0 0
b
(2)
331 b
(2)
332 0 0 0
. . .
. . .
. . .
. . .
. . .
b
(N )
331 b
(N )
332 0 0 0
⎤
⎥
⎥
⎥
⎦
,
(4.92)
g =
⎡
⎢
⎢
⎢
⎣
g
(1)
33
0 · · · 0
0 g
(2)
33 · · · 0
. . .
. . .
. . .
. . .
0 0 · · · g
(N )
33
⎤
⎥
⎥
⎥
⎦
, h =
⎡
⎢
⎢
⎢
⎣
h
(1)
333 0 · · · 0
0 h
(2)
333 · · · 0
. . .
. . .
. . .
. . .
0 0 · · · h
(N )
333
⎤
⎥
⎥
⎥
⎦
,
(4.93)
In the above equations, N represents the total number of piezoelectric layers, σ ,
ε, D, E are the stress vector, the strain vector, the electric displacement vector, the
electric field vector; c is the elasticity constant matrix; ¯
E is the nonlinear electric
field coefficient matrix; e and g denote, respectively, the piezoelectric constant and
dielectric constant matrices; b and h are the nonlinear electroelastic strain constant
and susceptibility constant matrices.
Considering structures undergoing large displacements but in elastic range and
under strong electric field, both geometrically nonlinear and electro-elastic nonlinear
effects should be taken into account. The resulting nonlinear models are abbreviated
as RVK5SE, MRT5SE, LER5SE, LRT56SE, where SE represents strong electric
field. The model including geometrically linear and electro-elastic nonlinear phenomena is denoted by LIN5SE. If linear constitutive equations are considered, the resulting models are denoted by LIN5WE, RVK5WE, MRT5WE, LER5WE, LRT56WE,
in which WE is shortened by weak electric filed. In most of this report, the WE is
not always appearing in the model abbreviations. If the model abbreviations exclude
WE, then the model considers only linear constitutive equations.