View source for Constitutive Models for ferroelectric materials under combined mechanical and electrical loading

== Abstract ==
Ferroelectric materials, often used as actuators and sensors in practice, demonstrate complex nonlinear hysteretic behavior when subjected to electro-mechanical loading. In the attempts to capture quantitatively their response, numerous constitutive models were motivated, ranging from most simple linear phenomenological to complex self-consistent, multiscale ones. In the current paper some of the most notable models for ferroelectrics are critically reviewed, with emphasis on the evolution of assumptions and limitations. A discussion on different constitutive models’ strengths and weaknesses, experimental validation is presented as well as suggestions for future work.



== Introductrion ==
Ferroelectrics are used in a broad range of engineering applications as actuators and sensors. Those materials typically exhibit electro-mechanical coupling, hysteresis, memory effects, anisotropy, saturation of strain and electric displacement, which complicates the constitutive description <ref name="huber2001"> Huber, J.E., Fleck, N.A., ''Multi-axial electric switching of a ferroelectric: theory versus experiment.'' J. Mech. Phys. Solids 49, 785–811,  2001. </ref>. The fundamental physical laws of mechanical equilibrium, geometric compatibility, Gauss’ and Maxwell’s laws fail to provide enough equations to explicitly define each independent variable, which, combined with the scarcity of reliable experimental data, gave birth to wide variety of theories and models. In the current paper some of the most notable attempts to mathematically describe the electro-mechanical response of ferroelectrics are critically reviewed, with emphasis on their strengths and weaknesses, assumptions and limitations.

== Background ==
To describe the nonlinear and hysteretic nature of the ferroelectric response, many phenomenological, self-consistent, micromechanical, multi-scale etc. models were motivated, exploring the boundaries of the available knowledge and experimental data.  The internal state variables (ISV) theory was widely implemented with debate between the use of one variable - volume fraction of domain switching 
<ref name="yu"> Yu L., Yu S., Feng X., ''A Simple Constitutive Model For Ferroelectric Ceramics Under Electrical/Mechanical Loading.'' Acta Mechanica Solida Sinica, Vol.20, No.1, March 2007. </ref>
<ref name="huber99"> Huber, J.E., Fleck, N.A., Landis, C.M., McMeeking, R.M., ''A constitutive model for ferroelectric polycrystals.'' J.Mech.Phys.Solids 47, 1663–1697, 1999.</ref> or two “semi-independent” variables - the remanent polarization and remanent strain
<ref name="kamlah">Kamlah, M., Tsakmakis, C., ''Phenomenological modeling of the non-linear electro-mechanical coupling in ferroelectrics.'' Int.J.Solids Struct.36, 669–695, 1999.</ref>
<ref name="landis2002">Landis C.M., ''Fully coupled, multi-axial, symmetric constitutive laws for polycrystalline ferroelectric ceramics.'' J.Mech Phys Solids 2002:50:127–52.</ref>. 
Even models that did not claim to use ISVs, like the case in 
<ref name="daniel">Daniel, L., Hall, D.A., Withers, P.J., ''A multiscale model for reversible ferroelectric behavior of polycrystalline ceramics.'' Mechanics of Materials 71 (2014) 85–100</ref>, still relied on the same physical entities, but described them with probability functions. The models of <ref name="landis2002"/>, <ref name="huber99"/>, 
<ref name="kim">Kim, S., ''A constitutive model for thermo-electro-mechanical behavior of ferroelectric polycrystals near room temperature.'' Int.J.Solids and Structures 48 (2011) 1318–1329</ref> implemented thermodynamics framework and the Helmholz free energy inequation to add supplementary relations between variables.  <ref name="kim"/> added temperature dependence to the phenomenological model, proposed by <ref name="huber2001"/>. Daniel et al (2014) <ref name="daniel"/> motivated an anhysteretic model, though they claimed the model still captured some basic mechanisms of electromechanical coupling.

==Ferroelectric Domain Switching==
Figure 1 represents the domain switching mechanism in lead lanthanum zirconate titanate (PLZT) as a dislocation-based phenomenon <ref name="huber99"/>. <ref name="landis2002"/> and <ref name="huber99"/> employed the similarities between domain wall motion in ferroelectrics and plastic slip in metals to motivate a switching mechanism, similar to J2 flow theory in metal plasticity.

[[Image:Domain_switching_mechanism.jpg|thumb|center|380px|Figure 1. Domain switching mechanism in lead lanthanum zirconate titanate (PLZT)<ref name="huber99"/>.]]

==Conclusions==
The proposed constitutive models from the phenomenological family <ref name="yu"/><ref name="huber2001"/><ref name="kamlah"/> and <ref name="landis2002"/> are physically sound, computationally simple and easy to implement into finite element analysis. Their simplicity (and precision) varies significantly from model to model and can be chosen according to the engineering application. Yet their authors did not provide statistical evidence on the model accuracy.
Multi-scale self-consistent models, as proposed by <ref name="huber99"/> and <ref name="kim"/>, were considered by <ref name="huber2001"/> superior in precision to macroscale phenomenological ones (though with no clear statistical evidence to support this statement). With proper calibration, those self-consistent models have the capability to predict ferroelectric response with desired accuracy at the cost of numerical solution complexity and computational time. 
The micro-mechanical anhysteretic model, proposed in <ref name="daniel"/> surpasses the limitations of elastic isotropy, yet provides simple solutions - experimental validation is yet to be done. 
From the review of the existing models for ferroelectric response, the following fields for improvement and suggestions for future work are becoming evident:

•	Formulation of physically sound, yet simple and computationally-friendly constitutive laws with predictable errors and uncertainties.

•	Experiments with multi-axial loading, real-time electromechanical coupling and/or high temperatures.

•	Statistical calculations on experimental or numerical solution errors, model uncertainty, parameter sensitivity analysis.

•	Analog ferroelectric solutions to mechanical problems (and vice versa)

==Acknowledgements==
The author gratefully acknowledges the support from Mississippi State University.

==References==
<references/>

[[Category: Ceramics]]
[[Category: Macroscale]]