A phenomenological constitutive theory for polycrystalline ferroelectric ceramics based on orientation distribution functions

https://doi.org/10.1016/j.euromechsol.2020.103982Get rights and content

Highlights

  • A phenomenological constitutive theory for the ferro-electro-elastic response of polycrystalline ceramics with tetragonal perovskite structure is proposed.

  • It is the first phenomenological theory to fully comply with the framework of generalized standard materials with convex thermodynamic potentials.

  • Predictions emulate most essential features of ferroelectric and ferroelastic behavior with minimal computational cost and, furthermore, generate stable predictions in contrast to earlier phenomenological theories.

  • We expect the theory to be particularly suitable for numerical implementation into efficient finite-element codes for large-scale structural simulations.

Abstract

A phenomenological constitutive theory for the ferro-electro-elastic response of polycrystalline ceramics with tetragonal perovskite structure is proposed. The state of a material point is characterized by an intrinsic polarization vector, an infinitesimal deformation tensor, and an internal variable representing the orientation distribution of the ferroelectric domains at that point. A class of convex thermodynamic potentials in terms of these state variables is posited, and constitutive relations within the framework of generalized standard materials are then derived. 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 polarization and deformation due to ferroelectric switching, preserving at the same time the generalized standard structure of the theory. By way of example, a specific set of constitutive functions is considered. The resulting constitutive relations are able to emulate most essential features of ferroelectric and ferroelastic behavior with minimal computational cost and, furthermore, generate stable predictions in contrast to earlier phenomenological theories.

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 formw(P,ε,ν)=12(Pp[ν])κ[ν](Pp[ν])+12(εξ[ν])C[ν](εξ[ν])+(εξ[ν])H[ν](Pp[ν])+w[ν],where the functionals κ, C and H represent macroscopic tensors of polarizability, elasticity and piezoelectricity moduli, while the functionals p and ξ 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 λ=+ and ζ=0, 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)

Cited by (9)

  • Variational free energy based macroscopical modeling of ferroelectroelasticity

    2023, Journal of the Mechanics and Physics of Solids
  • Rate-dependent ferroelectric switching in barium titanate ceramics from modified PUND experiments

    2022, Extreme Mechanics Letters
    Citation Excerpt :

    Knowledge of the full-field switching strain tensor could shed light on the orientation distribution of polarization during 90° ferroelastic switching (which is a complex process ultimately involving all six tetragonal variants). A combination of the experimental technique used in this study with detailed texture analysis prior to applying electro-mechanical loading can offer new insight for the calibration and validation of phenomenological models [71–73]. Our experimental data indicate that, as the electric field loading rate increases, (i) domain nucleation dominates over domain wall motion as the primary switching mechanisms due to the finite relaxation times of domain wall motion, (ii) the density of remanent domain walls increases, (iii) the remanent domain configurations are less stable, leading to more back-switching upon electric field removal, (iv) when switching anti-parallel to the pre-poled direction, 90° domains nucleate increasingly with rate (which are removable during the quasistatic probe pulses), (v) when switching parallel to the pre-poled direction, the microstructural evolution changes from primarily 90° domain wall motion at low rates to 180° domain nucleation at high rates.

View all citing articles on Scopus
View full text