Fracture Mechanics of Piezoelectric and Ferroelectric Solids (eBook)

Cover 1
Title Page 3
Copyright Page 4
Foreword 8
Preface 10
Table of Contents 13
Chapter 1 Introduction 19
1.1 Background of the research on fracture mechanics of piezoelectric/ferroelectric materials 19
1.2 Development course and trend 21
1.3 Framework of the book and content arrangements 22
References 24
Chapter 2 Physical and Material Properties of Dielectrics 26
2.1 Basic concepts of piezoelectric/ferroelectric materials 26
2.2 Crystal structure of dielectrics 29
2.3 Properties of electric polarization and piezoelectricity 33
2.3.1 Microscopic mechanism of polarization 34
2.3.2 Physical description of electric polarization 34
2.3.3 Dielectric constant tensor of crystal and its symmetry 37
2.4 Domain switch of ferroelectrics 38
2.4.1 Electric domain and domain structure 38
2.4.2 Switching of electric domain and principles for domain switch 43
2.4.2.1 Switching of electric domain 43
2.4.2.2 Principles for domain switching 46
References 48
Chapter 3 Fracture of Piezoelectric/Ferroelectric Materials — Experiments and Results 50
3.1 Experimental approaches and techniques under an 52
3.1.1 High-voltage power supply 52
3.1.2 High voltage insulation 52
3.1.3 Moire interferometry 57
3.1.4 Digital speckle correlation method 59
3.1.5 Method of polarized microscope 60
3.1.6 Experimental facilities 61
3.2 Anisotropy of fracture toughness 62
3.3 Electric field effect on fracture toughness 63
3.4 Fracture behavior of ferroelectric nano-composites 69
3.5 Measurement of strain field near electrode in doublelayer structure of piezoelectric ceramics 72
3.6 Observation of crack types near electrode tip 75
3.7 Experimental results and analysis related to ferroelectric single crystal out-of-plane polarized 77
3.7.1 Restorable domain switch at crack tip driven by low electric field 78
3.7.2 Cyclic domain switch driven by cyclic electric field 81
3.7.3 Electric crack propagation and evolution of crack tip electric domain 82
3.8 Experimental results and analysis concerning in-plane polarized ferroelectric single crytal 84
3.8.1 Response of specimen under a positive electric field 84
3.8.2 Crack tip domain switch under low negative electric field 85
3.8.3 Domain switching zone near crack tip under negative field 86
3.8.4 Evolution of electric domain near crack tip under alternating electric field 89
References 92
Chapter 4 Basic Equations of Piezoelectric Materials 94
4.1 Basic equations 94
4.1.1 Piezoelectric equations 94
4.1.2 Gradient equations and balance equations 100
4.2 Constraint relations between various electroelastic constants 101
4.3 Electroelastic constants of piezoelectric materials 102
4.3.1 Coordinate transformation between vector and tensor of the second order 102
4.3.2 Coordinate transformation of electroelastic constants 103
4.3.3 Electroelastic constant matrixes of piezoelectric crystals vested in 20 kinds of point groups 105
4.4 Governing differential equations and boundary conditions of electromechanical coupling problems 109
4.4.1 Governing differential equations of electromechanical coupling problems 109
4.4.2 Boundary conditions of electromechanical coupling 112
References 112
Chapter 5 General Solutions to Electromechanical Coupling Problems of Piezoelectric Materials 113
5.1 Extended Stroh formalism for piezoelectricity 113
5.1.1 Extended Stroh formalism 114
5.1.2 Mathematical properties and important relations of Stroh formalism 118
5.1.2.1 Standard eigenvalue problem 118
5.1.2.2 Orthogonality relations and closure relations 119
5.1.2.3 S-H-L matrices and their integral formula 120
5.1.2.4 Admittance and impedance matrices 123
5.2 Lekhniskii formalism for piezoelectricity 123
5.3 General solutions to two-dimensional problems of transversely isotropic piezoelectric materials 128
5.3.1 The general solutions to the anti-plane problems of transversely isotropic piezoelectric materials 128
5.3.2 The general solutions to the in-plane problems of transversely isotropic piezoelectric materials—Stroh method 129
5.3.3 The general solutions to the in-plane problems of transversely isotropic piezoelectric materials—Lekhniskii method 132
5.4 General solutions to three-dimensional problems of transversely isotropic piezoelectric materials 135
References 139
Chapter 6 Fracture Mechanics of Homogeneous Piezoelectric Materials 141
6.1 Anti-plane fracture problem 143
6.2 In-plane fracture problem 146
6.3 Three dimensional fracture problem 151
6.3.1 Description of problem 152
6.3.2 Derivation of electroelastic fields 154
6.4 Electromechanical coupling problem for a dielectric elliptic hole 158
6.4.1 Anti-plane problem of transversely isotropic piezoelctric material containing dielectric ellipic holes 158
6.4.2 Generalized plane problems of piezoelectric materials containing a dielectric elliptic hole 165
6.5 Influence on crack tip field imposed by electric boundary conditions along the crack surface 174
References 174
Chapter 7 Interface Fracture Mechanics of Piezoelectric Materials 177
7.1 Interfacial cracks in piezoelectric materials under uniform electromechanical loads 179
7.1.1 Tip field of interfacial crack 179
7.1.2 Full field solutions for an impermeable interfacial crack 183
7.2 Effect of material properties on interfacial crack tip field 186
7.3 Green’s functions for piezoelectric materials with an interfacial crack 188
7.3.1 Brief review of Green’s functions for piezoelectric materials 188
7.3.2 Green’s functions for anti-plane interfacial cracks 190
References 195
Chapter 8 Dynamic Fracture Mechanics of Piezoelectric Materials 198
8.1 Scattering of elastic waves in a cracked piezoelectrics 200
8.1.1 Basic concepts concerning propagation of elastic wave in a piezoelectrics 200
8.1.2 Dominant research work on elastic wave scattering caused by cracks in piezoelectrics 203
8.1.3 Scattering of Love wave caused by interficial cracks in layered elastic half-space of piezoelectrics 205
8.2 Moving cracks in piezoelectric medium 212
8.2.1 Anti-plane problems of moving interficial cracks 213
8.2.2 The plane problem of moving cracks 218
8.3 Transient response of a cracked piezoelectrics to electromechanical impact load 225
8.3.1 Anti-plane problems of cracked piezoelectrics under impact electromechanical loads 226
8.3.2 Transient response of crack mode-III in strip-shaped piezoelectric medium 231
8.3.3 In-plane problems of cracked piezoelectrics under the action of impact electromechanical loads 232
8.4 Dynamic crack propagation in piezoelectric materials 237
8.4.1 Dynamic propagation of conducting crack mode-III 238
8.4.2 Dynamic propagation of dielectric crack mode-III 244
References 248
Chapter 9 Nonlinear Fracture Mechanics of Ferroelectric Materials 250
9.1 Nonlinear fracture mechanical model 251
9.1.1 Electrostriction model 251
9.1.2 Dugdale model (strip saturation mode) 259
9.2 Domain switching toughening model 263
9.2.1 Decoupled isotropy model 264
9.2.1.1 Electromechanical atmosphere of domain switching zone 264
9.2.1.2 Domain switching zone 265
9.2.1.3 Domain switching toughening 266
9.2.2 Anisotropy model for electromechanical coupling 267
9.2.2.1 Electromechanical coupling field near crack tip 267
9.2.2.2 Domain switching zone 269
9.2.2.3 Domain switching toughening 275
9.3 Nonlinear crack opening displacement model 277
9.3.1 Definition of crack opening displacement 278
9.3.2 Crack opening displacement ?9 caused by piezoelectric effect 280
9.3.3 Effect ?? of domain switching on crack opening displacement 281
9.4 Interaction between crack tip domain switching of BaTiO3 single crystal and crack growth under electromechanical load 287
9.4.1 Experiment principle and technology 288
9.4.2 Experimental phenomena 288
9.4.3 Analysis of domain switching zone 291
9.4.3.1 Theoretical domain switching zone 292
9.4.3.1.1 Domain switching criterion 292
9.4.3.1.2 Stress field of crack tip 292
9.4.3.1.3 Domain switching strain 295
9.4.3.1.4 Theoretical crack tip domain switching 296
9.4.3.2 Analysis of experimental results of domain switching zone 297
9.4.3.2.1 Orientation of 90 a-a domain boundary 297
9.4.3.2.2 Domain structure of area A 298
9.4.3.2.3 Domain structures of areas B and C 298
9.4.4 Ferroelastic domain switching toughening 300
9.4.4.1 Domain switching tail region model 300
9.4.4.2 Theoretical R curve 302
References 304
Chapter 10 Fracture Criteria 308
10.1 Stress intensity factor criterion 309
10.2 Energy release rate criterion 309
10.2.1 Total energy release rate criterion 309
10.2.2 Mechanical strain energy release rate criterion 312
10.3 Energy density factor criterion 316
10.4 Further discussion on stress intensity factor criterion 320
10.5 COD criterion 323
References 325
Chapter 11 Electro-elastic Concentrations Induced by Electrodes in Piezoelectric Materials 327
11.1 Electroelastic field near surface electrodes 328
11.1.1 Electroelastic field near stripe-shaped surface electrodes 328
11.1.2 Electroelastic field near circular surface electrodes 336
11.2 Electroelastic field near interface electrode 342
11.2.1 General solution to the interface electrode of anisotropic piezoelectric bi-materials 343
11.2.2 Electroelastic field near the interface electrode in transversely isotropic piezoelectric bi-materials 346
11.3 Electroelastic field in piezoelectric ceramic-electrode layered structures 348
11.3.1 Laminated structure model, experimental set-up and finite element calculation model 348
11.3.2 Numerical calculation and experimentally measured results 351
References 354
Chapter 12 Electric-Induced Fatigue Fracture 356
12.1 Experimental observation and results 357
12.1.1 Electrically induced fatigue experiment by Cao and Evans (1994) 357
12.1.2 Electrically induced fatigue experiment of samples containing penetrating cracks 359
12.1.2.1 Design and preparation of samples and experiment apparatus 359
12.1.2.2 Experimental phenomena 360
12.1.2.3 Experimental results and analysis 363
12.2 Phenomenological model 369
12.2.1 Model I 369
12.2.2 Model II 373
12.3 Domain switching model 374
12.3.1 Electrically induced fatigue investigated by means of crack tip intensity factor 374
12.3.1.1 Gradual analysis of model 374
12.3.1.2 Results of numerical calculation 378
12.3.2 Investigation of electrically induced fatigue by means of crack opening displacement (COD) 382
12.3.2.1 Gradual analysis of the model 382
12.3.2.2 Numerical calculation result 385
References 388
Chapter 13 Numerical Method for Analyzing Fracture of Piezoelectric and Ferroelectric Materials 390
13.1 Generalized variation principle 393
13.1.1 Generalized variation principle of linear elastic mechanics 393
13.1.1.1 The minimum potential energy principle 393
13.1.1.2 Minimum complementary energy principle 394
13.1.2 Variation principle of electromechanical coupling problem 395
13.2 Finite element method for piezoelectric material fracture 397
13.2.1 Basic format of finite element for piezoelectric fracture 397
13.2.2 Calculation example: the electromechanical field around the circular hole in an infinite piezoelectric matrix 400
13.2.3 Calculation example: model of piezoelectric material with two-sided notches 402
13.3 Meshless method for piezoelectric material fracture 404
13.3.1 Basic format of electromechanical coupling meshless method 404
13.3.2 Some problems about electromechanical coupling meshless method 406
13.3.2.1 Effect of weight function and nodes on the selection of radius 406
13.3.2.2 Treatment of crack problem 406
13.3.2.3 Selection of integral domain scheme 408
13.3.3 Numerical example 410
13.4 Nonlinear finite element analysis of ferroelectric material fracture 410
13.4.1 Solution of field quantity with given electric domain distribution 411
13.4.1.1 Plane strain case 414
13.4.1.2 Plane stress case 415
13.4.2 New electric domain distribution and finite element iterative process determined by field quantity 417
13.4.3 Calculation example: Ferroelectric crystal containing insulating circular hole plus vertical electric field 419
13.4.4 Calculation example: Ferroelectric crystal containing insulating crack plus electric field (E=0.72Ec) perpendicular to crack surface 424
References 428
Appendix The Material Constants of Piezoelectric Ceramics 430