A phenomenological constitutive theory for polycrystalline ferroelectric ceramics based on orientation distribution functions
Introduction
The increasing need for viable computational tools to assess the performance of electrode formable devices has motivated the development of various phenomenological theories of ferroelectricity in polycrystalline ceramics with a reduced number of internal variables. A particularly attractive class of multiaxial theories hinges upon an additive decomposition of the deformation and the electric polarization into reversible contributions from elasticity and molecular polarizability, on the one hand, and irreversible contributions from ferroelectric switching, on the other hand (Landis, 2002; McMeeking and Landis, 2002; Mehling et al., 2007; Miehe and Rosato, 2011; Maniprakash et al., 2016). The measures of irreversible deformation and irreversible polarization are thus identified as internal variables, and corresponding evolution laws are derived from postulated thermodynamic potentials in accordance with the framework of generalized standard materials (Bassiouny and Maugin, 1988). The resulting laws are able to reproduce essential features of ferroelectric responses such as electric hysteresis loops, nonlinear stress-strain curves, butterfly loops, and dipole rotation, and are amenable to numerical implementation into efficient finite-element codes (Semenov et al., 2010; Miehe and Rosato, 2011; Schwaab et al., 2012). It has been recently recognized, however, that the thermodynamic potentials often employed in these theories do not conform to a generalized-standard material description in the sense Halphen and Nguyen (1975) and that, consequently, existence and stability of solutions to the resulting evolution equations are not guaranteed (Bottero and Idiart, 2015). Examples where issues of existence and stability of solutions do indeed manifest themselves have been reported by Stark et al. (2016b) and Bottero and Idiart (2015, 2018). Although a few alternative theories are already available in the literature (Kamlah and Wang, 2003; Klinkel, 2006), they are only accurate for restricted loading conditions by design. Motivated by these observations, the purpose of this work is to put forward a new set of thermodynamic potentials based on the same set of internal variables but conforming to a generalized-standard material description.
To that end, we borrow ideas from the various microelectromechanical and hybrid theories for polycrystalline ferroceramics already available in the literature (Huber et al., 1999; Lange and Ricoeur, 2015; Kim, 2015; Stark et al., 2016a; Tan and Kochmann, 2017). These theories consider the switching process as a phase transformation between ferroelectric domain variants, and thus identify the volume fractions of the multiple variants as internal variables and employ dissipation potentials that depend on the rates of transformations between variants. The measures of deformation and polarization are then related indirectly via these volume fractions, whose evolution is dictated by suitably chosen thermodynamic potentials. Realistic predictions can be generated by considering some hundreds of variants, albeit with an equally large number of internal variables. A similar constitutive framework is employed in this work. Thus, the internal state of the material is initially characterized by an infinite-dimensional internal variable representing the orientation distribution function of ferroelectric domains. A class of convex thermodynamic potentials in terms of this internal variable is then posited, and constitutive relations within the framework of generalized standard materials are derived. However, the functional form of the dissipation is selected in such a way that it effects an order reduction of the constitutive description whereby the infinite-dimensional internal variable is reduced to finite-dimensional internal variables representing irreversible polarization and irreversible deformation due to ferroelectric switching, preserving at the same time the generalized standard structure of the theory. Existence and stability of the predicted material response are thus ensured (Halphen and Nguyen, 1975; Germain et al., 1983). The presentation begins with a description of the constitutive framework in Section 2 and of the assumed thermodynamic potentials in Section 3. The potentials are characterized in terms of various functionals of the orientation distribution function. A special choice of functionals is thus proposed in Section 4 and the ensuing constitutive relations are spelled out. Sample predictions for various electromechanical loading programs typically employed in experimental studies are reported and discussed in Sections 5 Sample results and discussion, 6 The influence of local anisotropy, and some concluding remarks are finally given in Section 7.
Section snippets
Constitutive framework
The phenomenology of ferroelectricity and ferroelasticity in polycrystalline ceramics has been described with varying detail in many works (Jaffe et al., 1971; Lines and Glass, 1977). For conciseness, we restrict attention to ceramic systems with tetragonal perovskite structure below their Curie temperature and subject to isothermal electromechanical loadings such that the influence of inertia, magnetism and the potential presence of mobile charges can be neglected. It is further assumed that
Free-energy density and dissipation potential
The constitutive theory proposed in this work is based on a class of free-energy density functionals of the formwhere the functionals , and represent macroscopic tensors of polarizability, elasticity and piezoelectricity moduli, while the functionals and represent the irreversible polarization and strain due to ferroelectric switching, respectively. Conceptually, the form (5) hinges upon a series model
A special choice of constitutive functions and functionals
Further progress requires specific choices of the various constitutive functions and functionals describing the material response at the macroscopic and microscopic levels. In practice, ferroelectric behavior is often characterized by the application of electric fields and mechanical stresses. Thus, it is more convenient to describe the material response in terms of the free-enthalpy density and force potential introduced in Section 3.3 rather than in terms of the free-energy density and
Sample results and discussion
By way of example, the constitutive theory proposed above is employed here to generate predictions for specimens subject to various electromechanical loading programs. The numerical values adopted for the material parameters are given in Table 2. These values do not correspond to any particular material system but are found below to emulate material responses observed in common lead zirconate titanates. Note that by adopting and , the influence of local anisotropy on the overall elastic
The influence of local anisotropy
The predictions reported in the previous section do not account for the expected change of the overall permittivity and elasticity of the material due to reorientation of domains during ferroelectric switching. Consequently, the initial and saturated slopes of the electric and mechanical curves upon loading agree exactly, see Fig. 1, Fig. 2. The constitutive theory proposed in this work does, however, allow to incorporate the influence of local anisotropy on those properties via the material
Concluding remarks
We have proposed a phenomenological theory for the electromechanical response of polycrystalline ferroceramics subject to general loading conditions. To the best of our knowledge, it is the first phenomenological theory to fully comply with the framework of generalized standard materials with convex thermodynamic potentials. The specific forms of the thermodynamic potentials employed in this work contain nineteen material parameters that can be used to calibrate the theory from typical
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This work was funded by the Agencia Nacional de Promoción Científica y Tecnológica through grant PICT-2014-1988. Additional funding was received from UNLP through grant I-2017-225.
References (40)
- et al.
Formulation for coupled poling of ceramics
Int. J. Eng. Sci.
(1988) - et al.
Multi-axial electrical switching of a ferroelectric: theory versus experiment
J. Mech. Phys. Solid.
(2001) - et al.
A constitutive model for ferroelectric polycrystals
J. Mech. Phys. Solid.
(1999) - et al.
A thermodynamically and microscopically motivated constitutive model for piezoceramics
Comput. Mater. Sci.
(2003) A phenomenological constitutive model for ferroelastic and ferroelectric hysteresis effects in ferroelectric ceramics
Int. J. Solid Struct.
(2006)Fully coupled, multi-axial, symmetric constitutive laws for polycrystalline ferroelectric ceramics
J. Mech. Phys. Solid.
(2002)- et al.
A condensed microelectromechanical approach for modeling tetragonal ferroelectrics
Int. J. Solid Struct.
(2015) - et al.
Nonlinear electric-mechanical behavior and micromechanics modelling of ferroelectric domain evolution
Acta Mater.
(1999) The effect of uniaxial stress on the electro-mechanical response of 8/65/35 PLZT
Acta Mater.
(1996)- et al.
A phenomenological multi-axial constitutive law for switching in polycrystalline ferroelectric ceramics
Int. J. Eng. Sci.
(2002)
Phenomenological model for the macroscopical material behavior of ferroelectric ceramics
J. Mech. Phys. Solid.
A rate-dependent incremental variational formulation of ferroelectricity
Int. J. Eng. Sci.
Micromechanical modelling of remanent properties of morphotropic PZT
J. Mech. Phys. Solid.
Macroscopical non-linear material model for ferroelectric materials inside a hybrid finite element formulation
Int. J. Solid Struct.
On the evolution of the linear material properties of PZT during loading history–an experimental study
Int. J. Solid Struct.
An evaluation of switching criteria for ferroelectrics under stress and electric field
Acta Mater.
A hybrid phenomenological model for ferroelectroelastic ceramics. Part I: single phased materials
J. Mech. Phys. Solid.
Some aspects of macroscopic phenomenological material models for ferroelectroelastic ceramics
Int. J. Solid Struct.
An effective constitutive model for polycrystalline ferroelectric ceramics: theoretical framework and numerical examples
Comput. Mater. Sci.
In-situ observation of evolving microstructural damage and associated effective electro-mechanical properties of PZT during bipolar electrical fatigue
Acta Mater.
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