m_piezo, p

Contents Functions

m_piezo, p_piezo

Purpose

Material function for piezoelectric solids

Syntax

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

Sample calls are given at the end of this page and in section 6.6.

Accepted electrode commands are

ElectrodeMPC

[model,InputDOF(end+1,1)]=p_piezo('ElectrodeMPC Name',model,'z==5e-5'); defines the isopotential constraint as a case entry Name associated with FindNode command z==5e-5. An illustration is given in the MPC_actuator demo.

ElectrodeView

p_piezo('electrodeview',cf) outlines the electrodes in the model and prints a clear text summary of electrode information. To only get the summary, pass a model model rather than a pointer cf to a feplot figure.

Material

Theoretical details on piezoelectric materials are given in section 6.1.5. 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.

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 (6.28) 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 є^T (permitivity at zero stress) should be given (and not є^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 (see equation (6.29)), and epsT_1:9 are the 9 constants of the [є^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 [є^T] matrix in the constitutive law (6.28).

Shell element properties

Piezo shell elements with electrodes are declared by a combination of a mechanical definition as a layered composite, see p_shell 2, and an electrode definition with element property rows of the form

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

Type typically fe_mat('p_piezo','SI',1)
UnderlyingProId : ProId of underlying element p_shell 2 composite entry. The MatIdi for piezo layers must be associated with piezo electric material properties.
ElNodId1 : NodeId for electrode 1. This needs to be a node declared in the model but its position is not used since only the value of the electric potential (DOF 21) is used.
LayerId : layer number as declared in the composite entry
UNU1 : currently unused property (angle for polarization)

The constitutive law for a piezoelectric shell is given in equation (6.39). The following gives a sample declaration.

model=femesh('testquad4'); % Shell MatId 100 ProdId 110

% MatId 1 : steel, MatId 12 : PZT elastic prop
model.pl=m_elastic('dbval 1  Steel');
model=feutil('setmat',model,[12 fe_mat('m_elastic','SI',1) 65e9 .3 7800]);
% Sample ULB piezo material, sdtweb m_piezo('sample_ULB')
model.pl=m_piezo(model.pl,'dbval 3 -elas 12 Sample_ULB');

% ProId 111 : 3 layer composite (mechanical properties)
model.il=p_shell(model.il,['dbval 111 laminate ' ...
    '3 1e-3 0 ' ...  % MatID 3 (PZT),   1 mm piezo, 0
    '1 2e-3 0  ' ... % MatID 1 (Steel), 2 mm 
    '3 1e-3 0']);    % MatID 3 (PZT),   1 mm piezo, 0
% ProId 110 : 3 layer piezo shell with electrodes on nodes 1682 and 1683
model.il=p_piezo(model.il,'dbval 110 shell 111 1682 1 0 1683 3 0');

p_piezo('viewdd',model) % Details about the constitutive law