Ferroic Functional Materials (eBook)

Preface 6
Contents 8
Fundamentals of Magneto-Electro-Mechanical Couplings: Continuum Formulations and Invariant Requirements 9
1 Introduction 10
2 Foundations of Magneto-Electro Couplings 11
2.1 Preliminaries, Definitions and Units 12
2.2 A Primer in Electrostatics 13
2.3 A Primer in Magnetostatics 19
2.4 Maxwell's Equations 23
2.5 Special Cases 25
2.6 Electromagnetic Waves in Vacuum 26
2.7 Jump Conditions Across Interfaces 27
2.8 Poynting's Theorem 30
2.9 Maxwell Stress Tensor 35
3 Thermodynamics 38
3.1 First Law of Thermodynamics, Balance of Energy 38
3.2 Second Law of Thermodynamics, Entropy Inequality 39
3.3 Thermodynamic Potentials 40
4 Rotation, Spatial Reflection, Time-Reversal 43
4.1 Lorentz Invariance 44
4.2 Galilean Transformation 46
4.3 calT-Symmetry 47
4.4 Crystal Classes and Magnetic Crystal Classes 50
5 Piezoelectricity, Piezomagnetism, Some Foundations 51
5.1 Piezoelectricity 51
5.2 Piezomagnetism 56
5.3 Magnetoelectricity 56
5.4 Anisotropic and Isotropic Tensor Functions 57
6 Summary 60
References 60
Ferroelectric and Ferromagnetic Phase Field Modeling 63
1 Introduction 63
2 Maxwell's Equations and Polarization 64
2.1 Electro-Statics 65
3 Magnetism 71
3.1 Magneto-Statics Review 72
4 Mechano-Statics Review 76
5 Thermodynamics of Ferroelectric and Ferromagnetic Materials 79
5.1 Ferroelectric Materials: External Mechanical, Electrical Work and Heat Addition
5.2 Balance Laws for Internal Fields 82
5.3 Phase-Field Model of Ferroelectrics Dissipative Evolution of Domains
5.4 Internal Energy 88
5.5 Series Expansions for the Energy Functions 90
5.6 Phase-Field Modeling of Ferromagnetics 95
5.7 Finite Element Implementation 99
5.8 Example of Strain-Mediated Multiferroic Phase-Field Modeling 102
References 104
Semiconductor Effects in Ferroelectrics 105
1 Introduction 106
2 Thermodynamics of a Ferroelectric 107
2.1 Material Properties, Tensors, and Summation Rules 107
2.2 The Thermodynamic Energy Approach 109
2.3 The Landau-Devonshire Polynomial Approximation 116
2.4 The Depolarizing Field 122
3 Energetics of a Semiconductor 126
3.1 The Band Structure 126
3.2 Electron Statistics 130
3.3 Semiconductors with Impurities 132
3.4 Semiconductor with both Donors and Acceptors 134
3.5 Transport of Charge Carriers 135
3.6 Metal-Semiconductor Junctions 140
3.7 Heterojunctions 143
3.8 Structural Defects 145
4 The Ferroelectric Semiconductor 154
4.1 Energy Value Considerations 155
4.2 A Joint Energy Function 155
4.3 Screening 158
4.4 Maxwell-Wagner-Relaxation 162
4.5 The Electronic Impact of Defects 164
5 Case Studies 168
5.1 The Domain Wall 168
5.2 The PTCR-Effect at the Grain Boundary 170
5.3 Magnetoelectric Composites 174
5.4 Polarization Stability in Heterostructures 177
6 Conclusion and Outlook 178
References 181
Electromechanical Models of Ferroelectric Materials 187
1 Introduction 187
2 Origins of Ferroelectricity and Piezoelectricity 188
3 Piezoelectric Composites 192
3.1 Example of a Piezoelectric Composite 196
4 Models of Ferroelectric Switching 198
4.1 Classical Plasticity Model 199
4.2 Crystal Plasticity Model 202
4.3 Example of Crystal Plasticity Model 205
5 Models of Ferroelectric Domain Patterns 209
5.1 Theory of Compatibility 211
5.2 Average Compatibility 212
5.3 Exact Compatibility 216
5.4 Examples of Compatible Laminates 219
5.5 Evolution of Laminate Domain Patterns 224
5.6 Example of Domain Pattern Evolution 227
6 Summary and Outlook 231
References 232
An FE2-Scheme for Magneto-Electro-Mechanically Coupled Boundary Value Problems 235
1 Introduction 236
2 Theory of the Two-Scale Homogenization Scheme 240
2.1 Boundary Value Problems and Scale Transition 242
2.2 Discretizations of the Boundary Value Problems 245
2.3 Consistent Linearization of Macroscopic Field Equations 246
3 Magneto-Electro-Mechanical Material Models 248
3.1 Linear Piezoelectric and Piezomagnetic Model 249
3.2 Nonlinear Electrostrictive Model 250
3.3 Piezoelectric Model with Tetragonal Symmetry 251
4 Numerical Examples 257
4.1 Electrostrictive/Piezomagnetic Cantilever Beam 257
4.2 Piezoelectric/Piezomagnetic Composites 258
4.3 Ferroelectric Matrix with Cylindrical Magnetic Inclusions 261
4.4 Ferroelectric Matrix with Ellipsoidal Magnetic Inclusions 263
5 Summary 264
References 266
Multiscale Modeling of Electroactive Polymer Composites 271
1 Introduction 272
2 Governing Equations of Electro-Elasto-Statics at Finite Strains 274
3 Electro-Elasto-Static Boundary Value Problems on the Macro- and the Micro-scale 276
3.1 Boundary Value Problem on the Macroscopic Scale 276
3.2 Definition of Macroscopic Quantities via Homogenization 277
3.3 Boundary Value Problem on the Microscopic Scale 278
3.4 Consistent Linearization of Macroscopic Field Equations 282
4 Numerical Examples 286
4.1 Determination of the Effective Response of Electroactive Polymers with Spherical and Ellipsoidal Inclusions 286
4.2 Multiscale Simulation of Electromechanical Actuator with Composite Microstructure 288
5 Summary and Outlook 289
References 289