3.2 Piezoelectric shell finite elements
Shell strain is defined by the membrane, curvature and transverse shear as well as the electric field components. In the piezoelectric multi-layer shell elements implemented in SDT, 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=−Δ φi/hi where
Δ φi 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.
| Figure 3.1: Multi-layer shell piezoelectric element |
The discretized strain and electric fields of a piezoelectric shell take the form
There are thus n additional degrees of freedom Δ φi,
n being the number of piezoelectric layers in the laminate shell.
The constitutive laws are obtained by using the "piezoelectric plates" hypothesis (16) and the definitions of the generalized forces N,M,Q and strains є,κ,γ for shells:
Dzi is the electric displacement in piezoelectric layer , zmi is the
distance between the midplane of the shell and the midplane of
piezoelectric layer i (Figure 3.1), Gi is given by
where * refers to the piezoelectric properties under the piezoelectric plate assumption as detailed in section 2.2
and [Rs]i are rotation matrices associated to the
angle θ of the principal axes 1,2 of the piezoelectric layer given by:
Plugging (5) into (1) leads again to:
where {qmech } contains the mechanical degrees of freedom (5 per node corresponding to the displacements u,v,w and rotations rx,ry), and {V } contains the electrical degrees of freedom. The electrical dofs are defined at the element level, and there are as many as there are active layers in the laminate. Note that the electrical degree of freedom is the difference of the electric potential between the top and bottom electrodes Δ φ.
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