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m_piezo, p_piezo



Purpose

Material function for piezoelectric solids

Syntax
 mat= m_piezo('database name')
 pl = m_piezo('dbval MatId name');
Material

The m_piezo Const and BuildConstit commands support integration constant building for piezo electric volumes integrated in the standard volume elements. Element properties are given by p_solid entries, while materials formats are detailed here. Theoretical details are given in  section 5.2.5.

The elastic constitutive law is declared using ElasMatId referencing an elastic material declared as a row of model.pl with this identifier, see m_elastic for input formats. The piezoelectric constants can be declared using the following sub-types

1 : Simplified 3D piezoelectric properties

[ProId Type ElasMatId d31 d32 d33 eps1T eps2T eps3T EDType]

These simplified piezoelectric properties can be used for PVDF, but also for PZT if shear mode actuation/sensing is not considered (d24=d15=0). For EDType==0 on assumes d is given. For EDType==1, e is given. Note that the values of T (permitivity at zero stress) should be given (and not S).

2 : General 3D piezo

[ProId Type ElasMatId d_1:18 epsT_1:9]

d_1:18 are the 18 constants of the [d] matrix, and epsT_1:9 are the 9 constants of the [T] matrix. One reminds that strains are stored in order xx,yy,zz,yz,zx,yx.

3 : General 3D piezo, e matrix

[ProId Type ElasMatId e_1:18 epsT_1:9]

e_1:18 are the 18 constants of the [d] matrix, and epsT_1:9 are the 9 constants of the [T] matrix.

Examples are given in section 6.5.

Shell element properties

Piezo shell elements with electrodes are associated with element property rows of the form

[ProId Type UnderlyingProId ElNodeId1 LayerId1 UNU1 ElNodeId2...]

Type=fe_mat('p_piezo','SI',1)

UnderlyingProId : ProId of underlying element p_shell composite entry piezo layers must be associated with piezo electric material properties as detailed above. ElNodId1 : NodeId for electrode 1 LayerId : layer number as declared in the composite entry UNU1 : unused property for now (angle for polarization)

Examples are given in section 6.5.

Shell strain is defined by the membrane, curvature and transverse shear as well as the electric field components. It is assumed that in each piezoelectric layer i=1...n, the electric field takes the form E®= (0    0    Ezi). Ezi is assumed to be constant over the thickness hi of the layer and is therefore given by Ezi=-D i/hi where D i is the difference of potential between the electrodes at the top and bottom of the piezoelectric layer i. It is also assumed that the piezoelectric principal axes are parallel to the structural orthotropy axes.
The strain state of a piezoelectric shell takes the form
{
xx
yy
2 xy
xx
yy
2 xy
xz
yz
-Ez1
...
-Ezn
}=[
N,x 0 0 0 0 0 ... 0
0 N,y 0 0 0 0 ... 0
N,y N,x 0 0 0 0 ... 0
0 0 0 0 -N,x 0 ... 0
0 0 0 N,y 0 0 ... 0
0 0 0 N,x -N,y 0 ... 0
0 0 N,x 0 N 0 ... 0
0 0 N,y -N 0 0 ... 0
0 0 0 0 0
-
1
h1
... 0
... ... ... ... ... 0 ...
-
1
hn
] {
u
v
w
ru
rw
D 1
...
D n
}     (8.7)

There are thus n additional degrees of freedom D i, n being the number of piezoelectric layers in the laminate shell

The constitutive law associated to this strain state is given by :
{
N
M
Q
Dz1
...
Dzn
} = [
A B 0 G1T ... GnT
B D 0 zm1 G1T ... zmn GnT
0 0 F H1T ... HnT
G1 zm1 G1 H1 -1 ... 0
... ... ... 0 ... 0
Gn zmn Gn Hn 0 ... -n
] {
-Ez1
...
-Ezn
}     (8.8)

where Dzi is the electric displacement in piezoelectric layer (assumed constant and in the z-direction), zmi is the distance between the midplane of the shell and the midplane of piezoelectric layer i, and Gi, Hi are given by :
Gi = {
e.1 e.2 0
}i [Rs]i     (8.9)
Hi = {
e.4 e.5
}i [R]i     (8.10)
. denotes the direction of polarization. If the piezoelectric is used in extension mode, the polarization is in the z-direction, therefore Hi =0 and Gi ={
e31 e32 0
}i . If the piezoelectric is used in shear mode, the polarization is in the x or y-direction, therefore Gi=0, and Hi = {
0 e15
}i or Hi = {
e24 0
}i . It turns out however that the hypothesis of a uniform transverse shear strain distribution through the thickness is not satisfactory, a more elaborate shell element would be necessary. Shear actuation should therefore be used with caution.

[Rs]i and [R]i are rotation matrices associated to the angle of the piezoelectric layer.
[Rs] = [
cos2 sin2 sin cos
sin2 cos2 - sin cos
-2 sin cos 2 sin cos cos2 - sin2
]     (8.11)

[R] = [
cos -sin
sin cos
]     (8.12) 
 model=femesh('testquad4'); 
 model.pl=m_elastic('dbval 1 steel','dbval 2 steel', ...
 'dbval 12 ',[12 fe_mat('m_elastic','SI',1) 65e9 .3 7800]) % PZT elastic prop
 model.pl=m_piezo(model.pl,'dbval 3 -elas 12 Sample_ULB');
 model.il=p_shell('dbval 100 kirchoff .05 -f5','dbval 1 kirchoff .05 -f5');
 % 3 layer composite, .01 mm piezo, .2 mm steel, .01mm piezo
 model.il=p_shell(model.il,'dbval 111 laminate 3 1e-3 0 1 .05 0 3 1e-3 0');
 % piezo shell : 3 layer
 model.il=p_piezo(model.il,'dbval 110 shell 111 1682 1 0 1683 3 0');

 sdtdef('diag',2) % show constitutive properties
 [Case,model.DOF]=fe_mknl('init',model);fe_c(model.DOF,.21,'dofs')
 sdtdef('diag',0);
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