Composite Materials Research Progress

258 Yasuhide Shindo and Fumio Narita
understand the electromechanical field concentrations due to electrodes in piezoelectric ceramics
and composites. Recently, Yoshida et al. [5] discussed the electromechanical field
concentrations due to circular electrodes in piezoelectric ceramics through theoretical and
experimental characterizations. Their model quantitatively predicted the nonlinear electromechanical
fields induced by polarization switching near the circular electrode tip. Also,
numerical predictions of strain concentration were in relatively good agreement with measured
values.
The main aim of this work is to evaluate the electromechanical fields in the neighborhood
of surface and internal electrodes in piezoelectric composites. First, we study the
effect of applied voltage on the electromechanical field concentrations near the electrodes
in rectangular piezoelectric composite actuators. A nonlinear finite element analysis is performed
to calculate the strain, stress, electric field and electric displacement by introducing
models for polarization switching in local areas of the field concentrations. Two criteria
based on the work done by electromechanical loads and the internal energy density are
used and compared. Strain measurements are also presented to validate the predictions
using a four layered piezoelectric actuator. A comparison of strain concentration is made
between measurements and calculations, and a nonlinear behavior induced by localized polarization
switching is discussed. The device performance and polarization switching zone
near the electrodes are further predicted for some electrode configurations in the rectangular
piezoelectric composites. Next, we discuss the electromechanical field concentrations due
to circular electrodes in piezoelectric disk composites. The effects of applied voltage and
localized polarization switching on the disk device performance are examined.
2. Basic Equations
Consider a piezoelectric material with no body force and free charge. The governing equations
in the Cartesian coordinates xi(i =1, 2, 3) are given by
σji,j =0 (1)
Di,i =0 (2)
where σij is the stress tensor, Di is the electric displacement vector, a comma denotes
partial differentiation with respect to the coordinate xi, and the Einstein summation convention
over repeated indices is used. The relation between the strain tensor εij and the
displacement vector ui is given by
εij = 1
2 (uj,i
and the electric field intensity vector is
+ ui,j) (3)
Ei = −φ,i (4)
where φ is the electric potential. In a ferroelectric, polarization switching leads to a change
in the remanent strain εr r
ij and remanent polarization Pi . The total strain and electric displacement
are
εij = ε l ij + ε r ij (5)
Di = D l i + P r
i
(6)