Fracture Mechanics of Piezoelectric and Ferroelectric Solids 压电与铁电体的断裂力学

Chapter 1 Introduction 1

1.1 Background of the research on fracture mechanics of piezoelectric/ferroelectric materials 1

1.2 Development course and trend 3

1.3 Framework of the book and content arrangements 4

References 6

Chapter 2 Physical and Material Properties of Dielectrics 9

2.1 Basic concepts of piezoelectric/ferroelectric materials 9

2.2 Crystal structure of dielectrics 12

2.3 Properties of electric polarization and piezoelectricity 16

2.3.1 Microscopic mechanism of polarization 17

2.3.2 Physical description of electric polarization 17

2.3.3 Dielectric constant tensor of crystal and its symmetry 20

2.4 Domain switch of ferroelectrics 21

2.4.1 Electric domain and domain structure 21

2.4.2 Switching of electric domain and principles for domain switch 26

References 31

Chapter 3 Fracture of Piezoelectric/Ferroelectric Materials—Experiments and Results 33

3.1 Experimental approaches and techniques under an electromechanical coupling field 35

3.1.1 High-voltage power supply 35

3.1.2 High voltage insulation 35

3.1.3 Moire interferometry 40

3.1.4 Digital speckle correlation method 42

3.1.5 Method of polarized microscope 43

3.1.6 Experimental facilities 44

3.2 Anisotropy of fracture toughness 45

3.3 Electric field effect on fracture toughness 46

3.4 Fracture behavior of ferroelectric nano-composites 52

3.5 Measurement of strain field near electrode in double-layer structure ofpiezoelectric ceramics 55

3.6 Observation of crack types near electrode tip 58

3.7 Experimental results and analysis related to ferroelectric single crystal out-of-plane polarized 60

3.7.1 Restorable domain switch at crack tip driyen by low electric field 61

3.7.2 Cyclic domain switch driven by cyclic electric field 64

3.7.3 Electric crack propagation and evolution of crack tip electric domain 65

3.8 Experimental results and analysis concerning in-plane polarized ferroelectric single crytal 67

3.8.1 Response of specimen under a positive electric field 67

3.8.2 Crack tip domain switch under low negative electric field 68

3.8.3 Domain switching zone near crack tip under negative field 69

3.8.4 Evolution ofelectric domain near crack tip under alternating electric field 72

References 75

Chapter 4 Basic Equations of Piezoeleetrie Materials 77

4.1 Basic equations 77

4.1.1 Piezoelectric equations 77

4.1.2 Gradient equations and balance equations 83

4.2 Constraint relations between various electroelastic constants 84

4.3 Electroelastic constants of piezoelectric materials 85

4.3.1 Coordinate transformation between vector and tensor of the second order 85

4.3.2 Coordinate transformation of electroelastic constants 86

4.3.3 Electroelastic constant matrixes of piezoelectric crystals vested in 20 kinds ofpoint groups 88

4.4 Governing differential equations and boundary conditions of electromechanical coupling problems 92

4.4.1 Governing difierential equations of electromechanical coupling problems 92

4.4.2 Boundary conditions of electromechanical coupling 95

References 95

Chapter 5 General Solutions to Electromeehanical Coupling Problems of Piezoelectric Materials 97

5.1 Extended Stroh formalism for piezoelectricity 97

5.1.1 Extended Stroh formalism 98

5.1.2 Mathematical properties and important relations of Stroh formalism 102

5.2 Lekhniskii formalism for piezoelectricity 107

5.3 General solutions to two-dimensional problems of transversely isotropic piezoelectric materials 112

5.3.1 The general solutions to the anti-plane problems of transversely isotropic piezoelectric materials 112

5.3.2 The general solutions to the in-plane problems of transversely isotropic piezoelectric materials—Stroh method 113

5.3.3 The general solutions to the in-plane problems of transversely isotropic piezoelectric materials—Lekhniskii method 116

5.4 General solutions to three-dimensional problems of transversely isotropic piezoelectric materials 119

References 123

Chapter 6 Fracture Mechanics of Homogeneous Piezoelectric Materials 125

6.1 Anti-plane fracture problem 127

6.2 In-plane fracture problem 130

6.3 Three dimensional fracture problem 135

6.3.1 Description of problem 136

6.3.2 Derivation of electroelastic fields 138

6.4 Electromechanical coupling problem for a dielectric elliptic hole 142

6.4.1 Anti-plane problem of transversely isotropic piezoelctric material containing dielectric ellipic holes 142

6.4.2 Generalized plane problems of piezoelectric materials containing a dielectric elliptic hole 149

6.5 Influence on crack tip field imposed by electric boundary conditions along the crack surface 158

References 158

Chapter 7 Interface Fracture Mechanics of Piezoelectric Materials 161

7.1 Interfacial cracks in piezoelectric materials under uniform electromechanical loads 163

7.1.1 Tip field of interfacial crack 163

7.1.2 Full field solutions for an impermeable interfacial crack 167

7.2 Effect of material properties on interfacial crack tip field 170

7.3 Green's functions for piezoelectric materials with an interfacial crack 172

7.3.1 Brief review of Green's functions for piezoelectric materials 172

7.3.2 Green's functions for anti-plane interfacial cracks 174

References 179

Chapter 8 Dynamic Fracture Mechanics of Piezoelectric Materials 183

8.1 Scattering of elastic waves in a cracked piezoelectrics 185

8.1.1 Basic concepts concerning propagation of elastic wave in a piezoelectrics 185

8.1.2 Dominant research work on elastic wave scattering caused by cracks in piezoelectrics 188

8.1.3 Scattering of Love wave caused by interficial cracks in layered elastic half-space ofpiezoelectrics 190

8.2 Moving cracks in piezoelectric medium 197

8.2.1 Anti-plane problems of moving interficial cracks 198

8.2.2 The plane problem of moving cracks 203

8.3 Transient response of a cracked piezoelectrics to electromechanical impact load 210

8.3.1 Anti-plane problems of cracked piezoelectrics under impact electromechanical loads 211

8.3.2 Transient response of crack mode-Ⅲ in strip-shaped piezoelectric medium 216

8.3.3 In-plane problems of cracked piezoelectrics under the action of impact electromechanical loads 217

8.4 Dynamic crack propagation in piezoelectric materials 222

8.4.1 Dynamic propagation of conducting crack mode-Ⅲ 223

8.4.2 Dynamic propagation of dielectric crack mode-Ⅲ 229

References 233

Chapter 9 Nonlinear Fraeture Mechanics of Ferroelectric Materials 235

9.1 Nonlinear fracture mechanical model 236

9.1.1 Electrostriction model 236

9.1.2 Dugdale model(striP saturation mode) 244

9.2 Domain switching toughening model 248

9.2.1 Decoupled isotropy model 249

9.2.2 Anisotropy model for electromechanical coupling 252

9.3 Nonlinear crack opening displacement model 262

9.3.1 Definition of crack opening displacement 263

9.3.2 Crack opening displacement δ0 caused by piezoelectric effect 265

9.3.3 Effect △δof domain switching on crack opening displacement 266

9.4 Interaction between crack tip domain switching of BaTi03 single crystal and crack growth under electromechanical load 272

9.4.1 Experiment principle and technology 273

9.4.2 Experimental phenomena 273

9.4.3 Analysis ofdomain switching zone 276

9.4.4 Ferroelastic domain switching toughening 285

References 289

Chapter 10 Fracture Criteria 293

10.1 Stress intensity factor criterion 294

10.2 Energy release rate critefrion 294

10.2.1 Total energy release rate criterion 294

10.2.2 Mechanical strain energy release rate criterion 297

10.3 Energy density factor criterion 305

10.4 Further discussion on stress intensity factor criterion 305

10.5 COD criterion 308

References 310

Chapter 11 Electro-elastic Concentrations Induced by Electrodes in Piezoelectric Materials 313

111 Electroelastic field near surface electrodes 314

11.1.1 Electroelastic field near stripe-shaped surface electrodes 314

11.1.2 Electroelastic field near circular surface electrodes 322

11.2 Electroelastic field near interface electrode 328

11.2.1 General solution to the interface electrode of anisotropic piezoelectric bi-materials 329

11.2.2 Electroelastic field near the interface electrode in transversely isotropic piezoelectric bi-materials 332

11.3 Electroelastic field in piezoelectric ceramic-electrode layered structures 334

11.3.1 Laminated structure model,experimental set-up and finite element calculation model 334

11.3.2 Numerical calculation and experimentally measured results 337

References 340

Chapter 12 Electric-Induced Fatigue Fracture 343

12.1 Experimental observation and results 344

12.1.1 Electrically induced fatigue experiment by Cao and Evans(1994) 344

12.1.2 Electrically induced fatigue experiment of samples containing penetrating cracks 346

12.2 Phenomenological model 356

12.2.1 Model Ⅰ 356

12.2.2 Model Ⅱ 360

12.3 Domain switching model 361

12.3.1 Electrically induced fatigue investigated by means of crack tip intensity factor 361

12.3.2 Investigation of electrically induced fatigue by means of crack opening displacement(COD) 369

References 375

Chapter 13 Numerical Method for Analyzing Fracture ofPiezoelectric and Ferroelectric Materials 377

13.1 Generalized variation principle 380

13.1.1 Generalized variation principle of linear elastic mechanics 380

13.1.2 Variation principle of electromechanical coupling problem 382

13.2 Finite element method for piezoelectric material fracture 384

13.2.1 Basic format of finite element for piezoelectric fracture 384

13.2.2 Calculation example:the electromechanical field around the circular hole in an infmite piezoelectric matrix 387

13.2.3 Calculation example:model ofpiezoelectric material with two-sided notches 389

13.3 Meshless method for piezoelectric material fracture 391

13.3.1 Basic format of electromechanical coupling meshless method 391

13.3.2 Some problems about electromechanical coupling meshless method 393

13.3.3 Numerical example 397

13.4 Nonlinear finite element analysis of ferroelectric material fracture 397

13.4.1 Solution of field quantity with given electric domain distribution 398

13.4.2 New electric domain distribution and finite element iterative process determined by field quantity 404

13.4.3 Calculation example:Ferroelectric crystal containing insulating circular hole plus vertical electric field 406

13.4.4 Calculation example:Ferroelectric crystal containing insulating crack plus electric field(E=0.72Ec)perpendicular to crack surface 411

References 415

Appendix The Material Constants ofPiezoelectric Ceramics 417