High-order mixed finite elements for an energy-based model of the polarization process in ferroelectric materials

Pechstein, Astrid S.; Meindlhumer, Martin; Humer, Alexander

doi:10.1177/1045389x20953895

Cited by 7 publications

(4 citation statements)

References 45 publications

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“…In the context of ferroelectric materials, we mention contributions by Kamlah [2], Landis [3], Klinkel [4] and Miehe et al [5]. More recently, mixed-finite elements for ferroelectric continua have been presented in [6,7,8,9]. A geometrically non-linear setting for ferroelectrics has been considered in [10,11].…”

Section: Electro-magneto-mechanical Coupling In Forroicsmentioning

confidence: 99%

Dissipative Response in Ferroic Materials: Continuum Modeling and Simulation of Electro-Magneto-Mechanical Coupling

Humer,

Pechstein,

Krommer

2023

10th ECCOMAS Thematic Conference on Smart Structures and Materials

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Ferroic materials are characterized by the presence of electric/magnetic dipoles that can be irreversibly re-oriented under sufficiently strong loading. Upon poling, an initially random distribution of dipoles on the microscopic domain scale results in a non-vanishing state of remanent polarization/magnetization on the macroscopic level. The remanent switching of dipoles constitutes the "smartness" in materials and their usage both as sensors and as actuators. Once being poled, material properties are typically assumed to be constant and uni-directional in engineering problems. To extend the operational range of ferroic transducers and open perspectives for novel applications, we seek for an accurate understanding of the evolution of the polarization/magnetization, which requires both physical and geometric non-linearities to be included in the modeling. To describe irreversible changes of the remanent state, we transfer phenomenological concepts and algorithms of associative elasto-plasticity to the field of electro-magneto-mechanics. To describe the dissipative response in a thermodynamically consistent manner, the principle of maximum dissipation is adopted. As in elasto-plasticity, constitutive equations for dissipative internal forces that drive the evolution of the remanent polarization/magnetization then follow as associated flow rules. To simplify the algorithmic treatment, we introduce the notion of dissipation functions, by which the constrained optimization problem is converted into an unconstrained problem, which can be efficiently solved by standard means. The vectors of remanent polarization/magnetization enter the model as additional internal variables. The constitutive response is governed by thermodynamic potentials, i.e., the free energy and dissipation functions. We derive a (mixed) variational formulation, identify appropriate function spaces for the dependent fields involved, and introduce a finite-element discretization. Representative numerical examples demonstrate the efficacy of the framework in electro-magneto-mechanically coupled problems.

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Section: Electro-magneto-mechanical Coupling In Forroicsmentioning

confidence: 99%

Dissipative Response in Ferroic Materials: Continuum Modeling and Simulation of Electro-Magneto-Mechanical Coupling

Humer,

Pechstein,

Krommer

2023

10th ECCOMAS Thematic Conference on Smart Structures and Materials

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“…χ i is a susceptibility, P sat the internal saturation polarization and m is a shape parameter. This family of potentials is also used for ferroelectric ceramics, see Pechstein et al [4].…”

Section: Isotropic Relaxor Ferroelectric Ceramicsmentioning

confidence: 99%

Nonlinear Modeling of Ferroelectric Plates as Electro-Elastic Material Surfaces

Krommer,

Pechstein

2023

10th ECCOMAS Thematic Conference on Smart Structures and Materials

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We discuss nonlinear modeling of ferroelectric plates as electro-elastic material surfaces assuming the plate as a two-dimensional continuum with five mechanical degrees of freedom for each material point. Constitutive coupling by means of electrostriction, ferroelectric hysteresis and piezoelectricity are accounted for. For that sake, the augmented free energy is additively decomposed into an elastic part, a dielectric part and an augmentation energy. The elastic part involves an additive decomposition of the plate strain measures into elastic and electrical parts. Here, the electrical parts account for electrostriction and piezoelectricity, where electrostriction is assumed to depend quadratically on the polarization. For the dielectric part, we introduce the internal energy as a function of the polarization and an internal polarization, which allows us to accurately capture nonlinearities, such as saturation. Then, we compute the free energy applying a Legendre transformation, such that the voltage enters the formulation. The augmentation energy accounts for the contribution from vacuum. Finally, we introduce a so-called dissipation function, by means of which irreversible polarization due to domain switching is accounted for. Numerical results computed with the plate model are compared to results using a three-dimensional formulation.

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“…28,41 Applications to other dissipative processes such as modeling of ferroelectrics have been conducted more recently. 42,43 Employing a dissipation function, we propose a mixed formulation that does not require a return-mapping scheme.…”

Section: Introductionmentioning

confidence: 99%

“…The notion of a dissipation function , which was first introduced in the computational modeling of elasto‐plasticity by Caddemi and Martin, 40 is also successfully used in this context 28,41 . Applications to other dissipative processes such as modeling of ferroelectrics have been conducted more recently 42,43 . Employing a dissipation function, we propose a mixed formulation that does not require a return‐mapping scheme.…”

Section: Introductionmentioning

confidence: 99%

A mixed finite element formulation for elastoplasticity

Nagler

Pechstein

Humer

2022

Numerical Meth Engineering

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The present article, provides a novel mixed finite element formulation for elastoplasticity which is suitable for the discretization of thin-walled structures using highly anisotropic volume elements. We present a thermodynamically consistent framework for the modeling of elastoplastic stress response, where dissipative effects are considered through a dissipation function instead of explicit flow rules. Along these lines, a mixed incremental principle, in which stresses are included as independent unknowns, is derived. We propose to employ tangential-displacement normal-normal-stress (TDNNS) elements for a Galerkin discretization of the underlying variational problem. These elements were originally developed for linear elasticity, where it could be shown that they do not suffer from shear locking if the element's aspect ratio deteriorates. Excellent computational performance and accuracy of the proposed method are demonstrated in several benchmark problems.

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High-order mixed finite elements for an energy-based model of the polarization process in ferroelectric materials

Cited by 7 publications

References 45 publications

Dissipative Response in Ferroic Materials: Continuum Modeling and Simulation of Electro-Magneto-Mechanical Coupling

Dissipative Response in Ferroic Materials: Continuum Modeling and Simulation of Electro-Magneto-Mechanical Coupling

Nonlinear Modeling of Ferroelectric Plates as Electro-Elastic Material Surfaces

A mixed finite element formulation for elastoplasticity

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