A Physically Based Constitutive Model for Ferroelectrics
Authors: Manzari Majid T. , Zeng Wenjun , Shen Yin-Lin , Lee James D. 
Published: May 2008

Ferroelectric ceramics are widely used to exploit the piezoelectric effect in various electrical and mechanical equipment and devices such as ultrasonic generators, filters, hydrophones, pressure sensors, accelerometers, actuators, and adjustable aerodynamic surfaces. Despite their high potential for the use in many new engineering applications, there are several complications that limit their application to relatively low-level stress, electric field, and temperature conditions. One of the most important limitations is the fact that ferroceramics experience phase transformation or phase twinning that lead to their nonlinear response to electrical and/or mechanical loading. This is the main reason for the common design practice that limits the use of ferroelectric components to relatively small stress/electric field levels where the material behavior is linear and the measured piezoelectric properties of the material can be used with confidence. Hence there is a need for developing accurate nonlinear constitutive models for ferroelectric ceramics. This paper presents a physically based macroscopic constitutive model that can be used in the analysis of devices involving ferroelectric ceramics. It is assumed that the material point at each Gauss point of a typical finite element represents an aggregate of n unit cells distributed in a manner that the c-axis of each cell is oriented in a specific direction. The intial distribution of the orientations of the unit cells depends on whether the material is already poled or not. Once the strain increment for each unit cell is calculated, the corresponding stress tensor is computed by using the linear relationship between the stress and strain tensors in the cell’s local coordinate system. An appropriate domain-switching criterion is then checked for each unit cell. If the criterion is met, then the polarization orientation is switched for that cell and the orientation of the c-axis for that unit cell is updated. The constitutive moduli for the aggregate of unit cells are then calculated by using a homogenization technique. Using this approach it is clear that a nonlinear constitutive relationship is generated as soon as one of the unit cells in the aggregate switches its polarization direction. The nonlinearity of the response becomes more obvious as the polarization directions in more elements switch. This will lead to a hardening regime. As more elements switch and the number of elements that can potentially switch become smaller, the response of the system starts to become mainly affected by the constitutive response of the switched element, albeit in their new directions. This will lead to a stiffening part in the constitutive relationship, which eventually asymptotes to a linear relationship as the tendency to switching of the unit cells saturates and most of the unit cells in the aggregate become fully oriented toward the new polarization direction. This closely mimics the observed response of ferroelectric ceramics. Numerical simulations of compression of a poled PZT specimen by using the aforementioned constitutive model show close correlation of the simulations with the experimental data available for PZT materials.