CN104361167B - A kind of finite element prediction method of monocrystalline containing electrode ferroelectric analyzed based on Phase Field - Google Patents
A kind of finite element prediction method of monocrystalline containing electrode ferroelectric analyzed based on Phase Field Download PDFInfo
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Abstract
The invention discloses a kind of finite element prediction method of monocrystalline containing electrode ferroelectric based on Phase Field analysis under the effect of power electric heating multi- scenarios method, comprise the following steps：Based on the theoretical cenotype field models that multi- scenarios method is set up with diffusion interface model of Landau；With ANSYS finite element softwares, the model modeling and mesh generation of the monocrystalline containing electrode ferroelectric are obtained, nodal information and unit information is taken out；The various boundary information of obtained nodal information, unit information and consideration is imported, the solution of the power electro thermal coupling nonlinear problem of monocrystalline containing electrode ferroelectric is realized using FEAP rapid solving devices；The field variable that FEAP finite element solvings are obtained is directed into the post processor that Fortran writes, and obtains visualizing field variable result with TECPLOT softwares.Not only process is simple for the method for the present invention, and with good adaptability and stability, can be developed into commercial finite element program with the change of flexible adaptation problem.
Description
Technical field
The invention belongs to a kind of method of single crystal analysis containing electrode ferroelectric, more particularly to it is a kind of based on Phase Field analyze containing electric
Finite element prediction method of the pole ferro-electricity single crystal under the effect of power-electric-thermal multi- scenarios method, a kind of the pre- of this method is realized while being related to
Examining system.
Technical background
Ferroelectric material refers to that with a ferroelectric class material ferroelectricity is that nineteen twenty Frenchman Valasek has found first
, it is referred to as ferroelectric with ferroelectric material.In 32 crystallographic point groups of solid-state physics, only l, 2, m, mm2,4,4mm,
There is 3m, 6,6m this 10 point groups specific polarity direction can produce spontaneous polarization.Generally it is accompanied by out in ferroelectric crystal
Existing domain structure, is mutually referred to as domain wall between farmland and farmland.Domain structure can change direction under External Electrical Field, in alternation dispatch from foreign news agency
In the presence of the E of field, there is loop line in ferroelectric macroscopic polarization intensity P and E relation.Ferroelectric spontaneous polarization is ferroelectric material
A key character, under External Electrical Field occur polarization upset be ferroelectric material performance it is most basic embodiment.
Ferroelectric material with good dielectric, piezoelectricity and pyroelecthc properties, therefore in engineering due to having a wide range of applications
Prospect.It is well known that ferroelectric material such as BaTiO3, PbTiO3, PZT etc., its intrinsic electro-mechanical coupling effect, rapidly response
Higher rigidity, has become the study hotspot of new function material and component.But ferroelectric material, mostly fragility material
Material, therefore it is to crackle, the defect such as to be mingled with especially sensitive.Laminar electric capacity especially containing electrode has important to the property of ferroelectricity matrix
Influence.The presence of these electrodes destroys the geometry or physical continuity of structure, and the point of electrode is made under DC Electric Field
Nearby area will produce local stress concentration at end, so as to cause the mechanics of structure to fail.Therefore, the ferroelectricity containing electrode is furtherd investigate
Mechanics of materials failure behaviour plays vital effect to the stability of a system of laminar capacitor element and the safe design of device.
When outer load level is relatively low, the evolution of ferroelectricity/piezoelectricity bulk microstructure is reversible, and its machine-electroresponse is carried into outer
Linear relationship, now linear theory can for predict material failure behaviour.Between past more than two decades, ferroelectric material it is disconnected
Split with Problem of Failure received since Parton, Deeg, McMeeking, Pak, the Suo, Sosa and Yang et al. work it is wide
Pay close attention to generally.These above-mentioned achievements in research are largely summarized in Yang, Chen and Lu, Kamlah and Zhang and Gao summary opinions
Wen Zhong.Test result indicates that, the fracture toughness of material is the tens of of fracture toughness under mechanical load function in the presence of the electric field
Times, but existing linear theory is without this phenomenon of method interpretation.In fact, under additional machine-electric load effect, electrode is modeled to
Crackle is conducted, the electric field near it is reinforced, so that cause the domain structure near crackle to overturn, i.e., it is switching.It is switching again
Change the electric field near crackle in turn, produce internal stress, ultimately result in material performance machine-electroresponse non-linear.
In order to describe the switching non-linear effects to ferroelectricity/piezoelectric failure behaviour, there are two kinds of conventional micro- knots at present
Structure model.A kind of is the electricdomain upset model based on thermodynamic principles, and the model hypothesis are when the electric work and mechanical work sum of accumulation
When reaching a critical value, domain structure is to occur 180o or 90o upsets.Using the model, people can be easy to capture ferroelectricity
Some essential characteristics of material, such as dielectric hysteresis effect, therefore the model is widely used.Another micro-structural mould
Type is exactly phase field model.
Phase field theory is built upon on the basis of statistical physics, based on Ginzburg-Landau phase transformation theories, is passed through
The comprehensive function of differential equation Reaction-diffusion terms, ordering gesture and thermodynamic driving force.This method about originates from 20th century 70
Age, phase field model is proposed to get around the difficult of tracking liquid-solid boundary in numerical simulation of solidification microstructure, thereafter all kinds of
It is used widely in microstructure Evolution problem.Phase field model substantially describes domain wall using the concept at diffusion interface, by drawing
Enter in interface drastically to change but continuous phase field variable --- S order parameter is mutually tied to describe different phases with other field variables
The evolution problem of tissue is described altogether.These field variables are consecutive variations on whole zoning, it also avoid mathematics
The difficulty of upper description abrupt interface.The model major advantage is, system gross energy is expressed as into the order parameter with time correlation
Function, the real-time evolutionary process of micro-structural can obtain by solving the kinematical equation related to order parameter, and domain structure polarize instead
The criterion that journey does not rely on any hypothesis is turned over, but state in which is determined when can measure minimum value by system.Therefore,
Phase field model provides a kind of effective way to describe the real-time process of ferroelectricity/piezoelectric Microstructure Evolution.
It is the hot issue of recent years about the theory of Phase Field and application study for the ferroelectric material containing electrode.
It there is problems at present：
(1) ferro-electricity single crystal containing electrode is mainly the bad border Imitating in extra electric field, for power-electric-thermal complex environment
Under calculating simulation, calculate it is more difficult and calculate the time it is longer.
(2) the most ferro-electricity single crystal model based on phase-field model is mainly for single electrode at present or single splits
The finite element modelling of line, does not account for influencing each other between multiple electrodes in ferroelectricity laminar electric capacity.So in the presence of can not be calibrated
Really the problem of reflection physical substance.
(3) at present for ferroelectricity multi- scenarios method problem of the finite element model for solving containing electrode, existing business finite element software
The ferroelectricity Microstructure Evolution using spontaneous polarization as the first S order parameter can not be simulated, it is impossible to clear, the whole physical map of description of system
Picture.
The content of the invention
Technical problem
In view of the above-mentioned deficiencies of the prior art, the technical problem to be solved by the present invention is to provide one kind be based on Phase Field
Finite element prediction method of the monocrystalline containing electrode ferroelectric of analysis under power-electric-thermal multi- scenarios method effect, in phase field model, heat
Mechanical energy is represented as one group of continuous parameter, by the dynamics for solving the spatially skewness S order parameter with time term
Equation obtains the microstructure of material temporal-spatial evolution.One phase field model to microstructure it is possible that phase transformation path and
Switching upset criterion is not made any it is assumed that the direct result of phase transformation is total minimization of free energy process of whole simulated system.This
Invention provides the temporal-spatial evolution that the powerful method of One function predicts the micro- microscopical structure of material.
Technical scheme
In order to solve above-mentioned technical problem, method of the invention comprises the following steps：
Step one：Based on the theoretical expressions of free energy provided with diffusion interface model of Landau and wall energy expression formula,
Using spontaneous polarization as the first S order parameter, with reference to Ginzburg-Landau equations and mechanical balance equation, by flexible electrode mould
The wedge type crackle with three-dimensional dimension is intended to be, the electricdomain interaction of surrounding them is exchanged with each other energy, reaches poised state,
The cenotype field model of multi- scenarios method is set up with this.
Step 2：With ANSYS finite element softwares, the model modeling and mesh generation of the monocrystalline containing electrode ferroelectric are obtained, is taken
Egress information and unit information.
Step 3：Compared to other finite element softwares such as ANSYS, FEAP finite elements of increasing income provide preferable secondary development
Platform.One eight complete node isoparametric element ferroelectric cell program has been write by FORTRAN, import obtain nodal information,
Unit information and the various boundary of consideration, the power of monocrystalline containing electrode ferroelectric-electric-thermal is realized using FEAP rapid solving devices
The solution of coupled problem.
Step 4：The field variable that FEAP finite element solvings are obtained is directed into the post processor that Fortran writes, fortune
Obtained visualizing field variable result with TECPLOT softwares.
It is the sequence with system gross energy to be represented to time correlation using spontaneous polarization as the first S order parameter in step one
The function of parameter, introduces Micro-force systems, base is derived based on second law of thermodynamics Clausius-Duhem inequality
In the Ginzburg-Landau equations of time, the real-time evolutionary process of micro-structural can be by solving the kinematics related to order parameter
Equation is obtained, and the equilbrium position of final system is exactly state in which when system can measure minimum value.Domain structure upset need not
Given upset criterion.
ANSYS finite element softwares are used in step 2, flexible electrode is modeled as the wedge type crackle with three-dimensional dimension,
Mesh refinement processing twice is carried out near eletrode tip.The model modeling and mesh generation of the monocrystalline containing electrode ferroelectric are obtained, is taken
Egress information and unit information.
Compared to other finite element softwares such as ANSYS in step 3, FEAP finite elements of increasing income provide preferably secondary
Development platform.With variation principle, the Nonlinear System of Equations of a sign ferroelectric single crystal has been derived by.Pass through FORTRAN
An eight complete node isoparametric element ferroelectric cell programs have been write, obtained nodal information, unit information and difference is imported
Boundary condition, the nonlinear equation is solved using the method for Newton interpolation, realized using FEAP rapid solving devices containing electrode
The solution of ferro-electricity single crystal power-electric-thermal coupled problem.
The field variable for obtaining FEAP finite element solvings in step 4 is directed into the post processor that Fortran writes
In, obtain visualizing field variable result with TECPLOT softwares.
Beneficial effect
The method of the present invention expresses the coupling of ferroelectric material power, electricity, heat with effective Unified Model, and then deeply
Study the inefficacy mechanism under coupling.Continuous media yardstick is combined with atomic scale, macroscopic view and microcosmic phase binding
Mechanics field, electric field, the coupling in temperature field, the apparent Basic Mechanism for understanding ferroelectric material failure behaviour.The side of the present invention
Method is by the advantage reasonable utilization of a variety of finite element softwares, and system gives solution monocrystalline containing electrode ferroelectric in many of power-electric-thermal
Finite element prediction method under coupling, not only process is simple for this method, and with good adaptability and stability, can
Commercial finite element program is developed into the change of flexible adaptation problem.
Brief description of the drawings
Fig. 1 is the system diagram for realizing the inventive method；
Fig. 2 is one specific embodiment of the inventive method：(a) model schematic；(b) mesh generation figure；
Fig. 3 is domain structure distribution of the monocrystalline containing electrode ferroelectric under additional biaxial tensile loading, extra electric field, different temperatures
Cloud atlas and corresponding eletrode tip stress distribution cloud atlas.
Embodiment
Technical scheme is described further below in conjunction with accompanying drawing.
The present embodiment comprises the following steps：
First, DiFor dielectric displacement, uiFor displacement vector.
(1) the phase field model of ferro-electricity single crystal is set up, its electric enthalpy form is：
In formula, αi,αijkIt is dielectric coefficient,As without the material high-order dielectric constant under strain regime, it can use certainly
Parameter material parameter α under stress stateijTo represent, PiFor spontaneous polarization strength, GijklFor gradient energy coefficient, cijklFor material
Expect elastic stiffness, εijFor elastic strain and εij=1/2 (ui,j+uj,i), electrostriction occurs in ferroelectric material in polarization process
Phenomenon qijklFor electrostriction coefficient, κ0For permittivity of vacuum, EiFor electric field.
(2) mechanical balance equation, electrostatic equilibrium equation, and second law of thermodynamics Clausius-Duhem inequality are pushed away
Exporting time-based Ginzburg-Landau equations is：
σij,j+fi=0 (2)
Di,i- q=0 (3)
Wherein DiFor dielectric displacement, fiFor body force, q is volume charge, and L is kinetic coefficient
(3) in order to realize finite element control, based on variation principle, above governing equation is rewritten into weak solution shape
Formula：
WhereinFor gradient current, ξij=Pi,j TiFor surface reaction forces, ω is surface charge density.
(4) in dimensional finite element method, we carry out spatial spreading using 8 node units, using the side of Newton interpolation
Method solves three groups of nonlinear equations as derived from equation (5).The process diagram such as the iterative processing module in Fig. 1 system diagrams.
(5) ANSYS finite element softwares are used, the model modeling and mesh generation of the monocrystalline containing electrode ferroelectric are obtained, such as Fig. 2 institutes
Show, take out nodal information and unit information.
(6) an eight complete node isoparametric element ferroelectric cell programs are write by FORTRAN, imports obtained node letter
Breath, the various boundary of unit information and consideration, using FEAP rapid solving devices realize the power of monocrystalline containing electrode ferroelectric-
The solution of electric-thermal coupled problem.
(7) field variable for obtaining FEAP finite element solvings is directed into the post processor that FORTRAN writes, and is used
TECPLOT softwares obtain visualizing field variable result.
Implement example：
This implementation example carries out stress field, electric field, temperature using the ferro-electricity single crystal containing electrode shown in Fig. 2 as research object
The coupling of field is calculated.Step is as follows：
The known ferro-electricity single crystal structural parameters containing electrode, set up FEM model as shown in Figure 2 using ANSYS.Known iron
The material properties of electric monocrystalline, the stress loading σ under dimensionless*=2, electric load V*=200, T=200 ° at different temperatures,
400 °, 600 ° obtain the spontaneous polarization of each unit in FEM model, stress field using rapid solving device FEAP, especially electric
Spontaneous polarization and stress field near pole tip, as shown in Figure 3.Obtain drawing a conclusion：
(1) influencing each other to domain structure evolution effect substantially between extra electric field and electrode.
(2) with the rise of temperature, the domain structure near eletrode tip, which is developed, reduces the stress concentration journey of eletrode tip
Degree, generates switching toughness reinforcing.
The method of the conclusion display present invention provides to describe the real-time process of the laminated capacitor element Microstructure Evolution of ferroelectricity
A kind of effective way.
Claims (2)
1. a kind of finite element prediction method of monocrystalline containing electrode ferroelectric analyzed based on Phase Field, it is characterised in that it includes following step
Suddenly：
Step one：Based on the theoretical expressions of free energy provided with diffusion interface model of Landau and wall energy expression formula, use
Spontaneous polarization, with reference to Ginzburg-Landau equations and mechanical balance equation, flexible electrode is modeled as the first S order parameter
Wedge type crackle with three-dimensional dimension, the electricdomain interaction of surrounding them, is exchanged with each other energy, reaches poised state, with this
Set up the cenotype field model of multi- scenarios method；In step one, using spontaneous polarization as the first S order parameter, be with by system total energy
Amount represents the function of the order parameter of time correlation, Micro-force systems is introduced, based on second law of thermodynamics Clausius-
Duhem inequality derives time-based Ginzburg-Landau equations, the real-time evolutionary process of micro-structural can by solution with
The related kinematical equation of order parameter is obtained, and the equilbrium position of final system is exactly that system can measure shape residing during minimum value
State；
Step 2：With ANSYS finite element softwares, the model modeling and mesh generation of the monocrystalline containing electrode ferroelectric are obtained, section is taken out
Point information and unit information；In step 2, with ANSYS finite element softwares, flexible electrode is modeled as with three-dimensional dimension
Wedge type crackle, carried out near eletrode tip twice mesh refinement processing, obtain the monocrystalline containing electrode ferroelectric model modeling and
Mesh generation, takes out nodal information and unit information；
Step 3：One eight complete node isoparametric element ferroelectric cell program has been write by FORTRAN, obtained node is imported
Information, unit information and the various boundary of consideration information, list containing electrode ferroelectric is realized using FEAP rapid solving devices
The solution of brilliant power-electric-thermal coupled problem；
Step 4：The field variable that FEAP finite element solvings are obtained is directed into the post processor that Fortran writes, and is used
TECPLOT softwares obtain visualizing field variable result.
2. monocrystalline containing the electrode ferroelectric finite element prediction method according to claim 1 analyzed based on Phase Field, its feature
It is：In step 3, with variation principle, the Nonlinear System of Equations for characterizing ferroelectric single crystal is derived by, passes through FORTRAN
Eight complete node isoparametric element ferroelectric cell programs are write, obtained nodal information, unit information and different borders is imported
Conditional information, the nonlinear equation is solved using the method for Newton interpolation, and iron containing electrode is realized using FEAP rapid solving devices
The solution of electric monocrystalline power-electric-thermal coupled problem.
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| CN104809273B (en) * | 2015-04-03 | 2017-12-15 | 北京航空航天大学 | A kind of method for describing the deformation of creep |
| CN107423463B (en) * | 2017-02-23 | 2020-10-27 | 湘潭大学 | Method and system for establishing ferroelectric field effect transistor model |
| CN107423516A (en) * | 2017-08-01 | 2017-12-01 | 元计算（天津）科技发展有限公司 | A kind of coding method based on finite element language, compiler and method for solving |
| CN109063275B (en) * | 2018-07-12 | 2020-07-17 | 西安电子科技大学 | Method for constructing three-dimensional polycrystalline microstructure material model based on FEAP |
| CN110866349B (en) * | 2019-12-03 | 2023-04-14 | 湘潭大学 | Phase field analysis method of hafnium oxide based ferroelectric film based on polymorphic coexistence |
| CN111859740A (en) * | 2020-07-06 | 2020-10-30 | 西安建筑科技大学 | Finite element model modeling method for amplitude modulation decomposition of tissue |
| CN117574682B (en) * | 2024-01-11 | 2024-03-26 | 华中科技大学 | Method and system for predicting mechanical failure of core-shell NCM electrode particles |
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