smart technologies for safety engineering

Smart Technologies in Vibroacoustics 283
displacements (u pz
i
), and the Maxwell’s law for electrostatics, E pz
i
=−V pz
|i , relating the electric
field (E pz
i ) with its potential (V pz ), have been explicitly used in Equations (34).
The constitutive equations (34) are given for a general case of anisotropic piezoelectricity.
However, piezoelectric materials are usually treated as orthotropic or even transversally
isotropic. This is certainly valid for thin piezoelectric patches with through-thickness polarization,
which are often used in smart structures as fixed-on-surface piezoelectric actuators and
sensors. In this case, assuming that the thickness (and therefore, the polarization) of the piezopatch
is directed along the x3 axis, the orthotropic constitutive equations of the piezoelectric
material can be expressed as follows, where the formulas for stresses are
σ pz
11 = Ĉpz
11 upz
1|1 + Ĉpz
12 upz
2|2 + Ĉpz
13 upz
pz
3|3 − êpz
31 V|3 σ pz
22 = Ĉpz
12 upz
1|1 + Ĉpz
22 upz
2|2 + Ĉpz
23 upz
pz
3|3 − êpz
32 V|3 σ pz
33 = Ĉpz
13 upz
1|1 + Ĉpz
23 upz
2|2 + Ĉpz
33 upz
pz
3|3 − êpz
33 V|3 σ pz
23 = Ĉpz
44 (upz
pz
2|3 + upz
3|2 ) − êpz
24 V|2 σ pz
13 = Ĉpz
55 (upz
pz
1|3 + upz
3|1 ) − êpz
15 V|1 σ pz
12 = Ĉpz
66 (upz
1|2 + upz
2|1 )
and for the electric displacements are
D pz
1
D pz
2
D pz
3
= êpz
15 (upz
1|3 + upz
3|1
= êpz
24 (upz
2|3 + upz
3|2
) + ɛpz
11
) + ɛpz
22
pz
V|1 pz
V|2 = êpz
31 upz
1|1 + êpz
32 upz
2|2 + êpz
33 upz
3|3
pz
+ ɛpz
33 V|3 Moreover, for transversal isotropy the following relations are satsfied by the elastic material
constants:
Ĉ pz
22
= Ĉpz
11 , Ĉpz
23
by the piezoelectric constants:
and by the dielectric constants:
= Ĉpz
13 , Ĉpz
55
ê pz
24
= êpz
15 , êpz
32 = êpz
33
ɛ pz
22 = ɛpz
11
Ĉpz
11 − Ĉpz
12
= Ĉpz
44 , Ĉpz
66 =
2
In the above relations, some pairs of indices have been replaced by new subscripts as
given by the following rule of change (known from the so-called Kelvin–Voigt notation):
11 ↦→ 1, 22 ↦→ 2, 33 ↦→ 3, 23 ↦→ 4, 13 ↦→ 5 and 12 ↦→ 6. Thus, C pz
ijkl is represented as Ĉpz
IJ
and e pz
kij as êpz
kI , where i, j, k, l = 1, 2, 3 and I, J = 1,...,6. This contracted notation allows 21
independent anisotropic elastic material constants to be grouped in a 6 × 6 symmetrical matrix
and the third-order tensor of piezoelectric effects to be represented as a 3 × 6 matrix. Notice
(35)
(36)
(37)
(38)
(39)