Ferroic Functional Materials: Experiment, Modeling and Simulation

293 Pages·2018·8.517 MB·English

by  Jörg Schröder,Doru C. Lupascu (eds.)

CISM International Centre for Mechanical Sciences 581
Courses and Lectures
Jörg Schröder
Doru C. Lupascu Editors
Ferroic
Functional
Materials
Experiment, Modeling and Simulation
International Centre
for Mechanical Sciences
CISM International Centre for Mechanical
Sciences
Courses and Lectures
Volume 581
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ö ö
J rg Schr der Doru C. Lupascu
(cid:129)
Editors
Ferroic Functional Materials
Experiment, Modeling and Simulation
123
Editors
Jörg Schröder Doru C.Lupascu
Institute of Mechanics Instiute forMaterials Science
University of Duisburg-Essen University of Duisburg-Essen
Essen Essen
Germany Germany
ISSN 0254-1971 ISSN 2309-3706 (electronic)
CISMInternational Centre for MechanicalSciences
ISBN978-3-319-68881-7 ISBN978-3-319-68883-1 (eBook)
https://doi.org/10.1007/978-3-319-68883-1
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Preface
Functional materials play a key role in many modern technical devices ranging
fromconsumermarketitemstoapplicationsinhigh-endequipmentforautomotive,
aircraft and spacecraft, medical, and information technologies. Among functional
materials, smart materials represent a class that transforms one basic physical
propertyintoanother.Thedevelopmentofdevicesutilizingsmartmaterials,aswell
astheirtesting,isgenerallyveryexpensive.Therefore,considerableefforthasbeen
made to develop modeling tools that allow bypassing many of the experimental
steps previously required in design. Important smart materials are ferroelectrics
(couplingbetweenelectricpolarizationandstrain),ferromagnets(couplingbetween
magnetizationandstrain),shape-memoryalloys(couplingbetweentemperatureand
strain), and magnetoelectric multiferroics (coupling between electric polarization
andmagnetization).Thelatteronescombinethemutualcontrollabilityofmagnetic
and electric state variables. They are of great interest in the development of mul-
tifunctionaldevices.Insingle-phasemultiferroics,themagnetoelectricinteractionis
generally very weak and mostly occurs at cryogenic temperatures. Therefore, the
experimental preparation and characterization of composite materials, as well as
their constitutive description based on homogenization strategies, are key chal-
lenges for the optimization of such magnetoelectric composites. The development
of such composites made from two different ferroics is based on a comprehensive
understanding of both the experimental and theoretical details of these materials.
Thus, this CISM course covers the modeling of ferroelectric materials, ferromag-
netic materials and shape-memory alloys, the formation offerroic microstructures
and their continuum-mechanical modeling, the experimental preparation and
characterization of magnetoelectric multiferroics, computational homogenization,
and the algorithmic treatment in the framework of numerical solution strategies.
TheCISMcourseon“FerroicFunctionalMaterials:Experiment,Modeling,and
Simulation,”heldinUdinefromSeptember8to12,2014,wasaddressingdoctoral
studentsandpostdoctoralresearchersincivilandmechanicalengineering,materials
science,physicsandappliedmathematics,andindustrialresearcherswhowishedto
broaden their knowledge in experiments and theory offerroic materials. The main
focus was on the state-of-the-art experimental methods and advanced modeling
v
vi Preface
techniques,whichareessentialtoqualifyyoungscientistsforhigh-qualityresearch,
and the development of innovative products and applications.
It is our pleasure to thank the lecturers of the CISM course Kaushik
Bhattacharya (Pasadena, USA), Manfred Fiebig (Zurich,Switzerland), JohnHuber
(Oxford, UK), Christopher Lynch (Los Angeles, USA), Marc-André Keip
(Stuttgart, Germany) as well as the additional contributors to these CISM lecture
notes Dorinamaria Carka (Old Westbury, USA), Irina Anusca (Essen, Germany),
Morad Etier (Essen, Germany), Yanling Gao (Essen, Germany), Gerhard Lackner
(Essen, Germany), Ahmadshah Nazrabi (Essen, Germany), Mehmet Sanlialp
(Essen, Germany), Harshkumar Trivedi (Essen, Germany), Matthias Labusch
(Essen, Germany), and Naveed Ul-Haq (Essen, Germany). We furthermore thank
theparticipantswhomadethecourseasuccess.Finally,weextendourthankstothe
Rectors, the Board, and the staff of CISM for the excellent support and kind help.
Essen, Germany Jörg Schröder
Doru C. Lupascu
Contents
FundamentalsofMagneto-Electro-MechanicalCouplings:Continuum
Formulations and Invariant Requirements. . . . . . . . . . . . . . . . . . . . . . . 1
Jörg Schröder
Ferroelectric and Ferromagnetic Phase Field Modeling . . . . . . . . . . . . . 55
Dorinamaria Carka and Christopher S. Lynch
Semiconductor Effects in Ferroelectrics . . . . . . . . . . . . . . . . . . . . . . . . . 97
Doru C. Lupascu, Irina Anusca, Morad Etier, Yanling Gao,
Gerhard Lackner, Ahmadshah Nazrabi, Mehmet Sanlialp,
Harshkumar Trivedi, Naveed Ul-Haq and Jörg Schröder
Electromechanical Models of Ferroelectric Materials. . . . . . . . . . . . . . . 179
J. E. Huber
An FE2-Scheme for Magneto-Electro-Mechanically Coupled
Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
Matthias Labusch, Jörg Schröder and Marc-André Keip
Multiscale Modeling of Electroactive Polymer Composites. . . . . . . . . . . 263
Marc-André Keip and Jörg Schröder
vii
Fundamentals of
Magneto-Electro-Mechanical Couplings:
Continuum Formulations and Invariant
Requirements
JörgSchröder
Abstract Couplings of magnetic and electric fields in materials could allow for
promising applications in medical and information technology. In this contribu-
tion,werecapitulatewell-knownaspectsofmagneto-electro-mechanicalproperties
and their couplings. At first, we echo basic aspects of electricity and magnetism
and Maxwell’s equations. Secondly, we summarize the governing equations for
electrostatics and magnetostatics, point out the properties of physical fields across
internalsurfaces,anddiscussthework-energytheoremofelectrodynamics,theso-
calledPoynting‘stheorem.Thirdly,wewilldiscusssomefundamentalconceptsof
magneto-electro-mechanical couplings in matter. Here, we will formulate thermo-
dynamic potentials depending on different basic variables in order to be flexible
withaviewtodifferentmodelingaspects.Afterwards,wediscussaspectsofform-
invariance of physical laws under coordinate transformations: Lorentz invariance,
Galileantransformationandtimereversal.Here,wefocusonpiezoelectricaswellas
onmagneticsymmetrygroupsandgiveremarksonclassicalinvarianttheorysuitable
forcoordinate-invariantmodelingofthermodynamicalpotentials.
Partsofworkpresentedinthiscontributionaretakenfromcommonworkstogetherwithmy
former co-workers Holger Romanowski, Ingo Kurzhöfer, Marc-André Keip, and Matthias
Labusch.Theauthorgreatlyappreciatesthe“DeutscheForschungsgemeinschaft”(DFG)for
thefinancialsupportundertheresearchgrantSCHR570/12-1withintheresearchgroupFOR
1509 on “Ferroische Funktionsmaterialien—Mehrskalige Modellierung und experimentelle
Charakterisierung”.
B
J.Schröder ( )
FacultyofEngineering,DepartmentCivilEngineering,InstituteofMechanics,
UniversityofDuisburg-Essen,Essen,Germany
e-mail:[e-mail]
©CISMInternationalCentreforMechanicalSciences2018 1
J.SchröderandDoruC.Lupascu(eds.),FerroicFunctionalMaterials,
CISMInternationalCentreforMechanicalSciences581,
https://doi.org/10.1007/978-3-319-68883-1_1
2 J.Schröder
1 Introduction
In this course, we are interested in the continuum modeling of ferroic functional
materials. Of particular interest are materials which allow for couplings between
differentphysicalquantitiesasforinstancecouplingbetweenelectricandmechanical
(ferroelectrics),mechanicalandmagnetic(ferromagnetics),ormagneticandelectric
fields(multiferroics),whicharedefinedas
• ferroelectricmaterialshaveaspontaneouselectricpolarization,wheretheinternal
electric dipoles are coupled to the material lattice, that can be switched by an
appliedelectricfield,
• ferromagnetic materials, governed by their crystalline structure and microstruc-
ture,haveaspontaneousmagneticpolarizationthatcanbereversedbyamagnetic
field,and
• multiferroicmaterialsexhibittwoormoreferroicproperties,likeferroelectricity,
ferromagnetism,andferroelasticity,inthesamephase.
However, the coexistence of magnetic and electric orderings in the same phase
isratherdifficult.Basedontheoreticalstudies,ithasbeenshownthatusualatomic-
levelmechanismswhicharedrivingferromagnetismandferroelectricityaremutually
exclusive.Forexample,ferromagnetismrequirespartiallyfilledorbitals(“d”-shells)
andferroelectricityemptyorbitalsintheatomicstructure.Ingeneral,theproperties
offunctionalmaterialsemergeondifferentscales.Someexistontheatomicscale,as
forexamplethemagnetization.Others,ase.g.theelectricpolarization,arepresent
ontheunitcelllevelofacrystal.Furthermore,somematerialsobtaintheirfunctional
properties only when the above quantities couple over a larger length scale, as for
instanceincaseofmultiferroiccomposites.
In this course, we will recapitulate well known basic properties, balance equa-
tions and some formulations of thermodynamics. Furthermore, we will focus on
sometransformationproperties:Physicallaws,validinaframeofreference,satisfy
specificsymmetryconditions.Forexample,thesymmetrywhichreflectstheform-
invarianceofphysicallawsundercoordinatetransformations.Invariancemeans,that
somephysicalquantitiesremainunchangedunderspecifictransformations.Inorder
togetadeeperinsightintothemathematicalmodelingofelectro-mechanicallycou-
pledmaterials,wediscussthepropertiesoftheassociatedphysicalquantitiesunder
thecoordinate-transformations:rotations,spatialreflections,andtime-reversal.
AnoverviewofelectromagnetictheoriesisgiveninLandauandLifschitz(1985),
FabrizioandMorro(2003),Tipler(1999),Jackson(2002),Griffiths(2008),Bobbio
(2000), Fließbach (1999), Fließbach (2000), Grehn and Krause (2007), and Zohdi
(2012). For a detailed discussion of Maxwell Equations see Maugin et al. (1991),
EringenandMaugin(1989),EringenandMaugin(1990)orWeileetal.(2014).