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	79
	5.2 Balance Laws for Internal Fields	82
	5.3 Phase-Field Model of Ferroelectrics	85
	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