Piezoelastic Beam
Piezoelastic materials exhibit a coupling between strain and electric polarization or voltage. Thus, these
materials can be used for sensors or actuators. They have been used for active vibration suppression in
elastic structures. They have also been explored for active optics space applications. Many natural materials
exhibit piezoelasticity such as quartz as well as manufactured materials such as barium titanate, lead
zirconate titanate (PZT), and polyvinylidene fluoride (PVDF). Unlike forces on charges and currents (see
below), the electric effect takes place through a change in shape of the material. The modeling of these
devices can be done by modifying the equations for elastic structures.
The following work on piezo-benders is based on the work of Lee and Moon (1989) as summarized
in Miu (1993). One of the popular configurations of a piezo actuator-sensor is the piezo-bender shown
in Figure 6.9. The elastic beam is of rectangular cross section as is the piezo element. The piezo element
can be cemented on one or both sides of the beam either partially or totally covering the surface of the
non-piezo substructure.
In general the local electric dipole polarization depends on the six independent strain components
produced by normal and shear stresses. However, we will assume that the transverse voltage or polarization
is coupled to the axial strain in the plate-shaped piezo layers. The constitutive relations between

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