A constitutive model for rate dependent behavior of ferroelectric materials is developed from a one-dimensional
switching model [Ikeda et al., Proc. SPIE, 7289 (2009), 728905]. The one-dimensional switching model has the
following three features. (i) Several ferroelectric variants can be considered, such as 0-degree, 90-degree, 180-degree,
and initial mixed variants. (ii) Required switching energy is approximated as a sum of two exponential functions of
volume fraction of the variants. (iii) A specimen is assumed to be comprised of grains with infinitesimal size, and
relationship between two grains regarding the required switching energy is unchanged independently of switching
directions. Accordingly, the switching proceeds one-dimensionally. To take into account loading rate effects, a function
of volume fraction rate is added to the required switching energy. That makes energy barrier higher at higher rates. To
verify validity of the present model, electro-mechanical behavior of a thin PZT plate is measured at various loading rates
and simulated using the present model. Result shows the present model can capture the influence of electric loading rate
on responses of electric displacement and strain, such that remnant polarization decreases and coercive field increases
with increasing the loading rate.
A simple constitutive model for temperature dependent behavior of ferroelectric materials is developed. This model is
based on the one-dimensional phase transformation model of shape memory alloys. To model the temperature dependent
behavior of the ferroelectric materials, a paraelectric phase is considered in addition to four ferroelectric variants in a
ferroelectric phase. These ferroelectric variants are connected in series to each other, whereas the paraelectric phase is
connected in parallel to the ferroelectric phase. The internal stress is induced in the material due to this parallel
connection, which increases or decreases the driving energy for the switching depending on the switching direction. As
the temperature increases up to the Curie temperature, the volume fraction of the paraelectric phase is assumed to
increase and the required switching energy is assumed to decrease as observed in experiments. The temperature
dependence of the relationships among the electric field, electric displacement, stress, and strain are simulated and
compared with published experimental data for a soft PZT. The comparison indicates that the present constitutive model
can predict the temperature dependent behavior well. This implies that the proposed model can provide a convenient tool
to understand the physical mechanism of the ferroelectric materials and to design smart structures containing the
ferroelectric materials.
The one-dimensional phase transformation model of shape memory alloys [Ikeda et al., Smart Materials and Structures,
13, 916-925 (2004)] is applied to expressing the major and minor hysteresis loops in ferroelectric materials. An analogy
between the phase transformation in the shape memory alloys and the switching in the ferroelectric materials is involved.
The one-dimensional phase transformation model has the following two features. (i) A specimen is assumed to be
comprised of grains with infinitesimal sizes, and the order of the energy required for the transformation of the grains is
unchanged independently of the transformation directions. Accordingly, the phase transformation occurs onedimensionally.
(ii) The required transformation energy is approximated as a sum of two exponential functions of phase
volume fraction. To express the ferroelectric behavior, four phases (variants) are considered, namely, the 0° variant, 90°
variant, 180° variant, and initial mixed variant. Electro-mechanical behavior of a ferroelectric material is simulated
numerically. The result shows the model can approximately duplicate the electro-mechanical behavior observed in the
ferroelectric material.
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