Mathematical Methods in Electro-Magneto-Elasticity:Mathematical Methods in Electro-Magneto-Elasticity (Lecture Notes in Applied and Computational Mechanics)
By Demosthenis I. Bardzokas, Michael L. Filshtinsky, Leonid A. Filshtinsky
Publisher: Springer
Number Of Pages: 530
Publication Date: 2007-06-27
ISBN-10 / ASIN: 3540710302
ISBN-13 / EAN: 9783540710301
Binding: Hardcover
The mechanics of Coupled Fields is a discipline at the edge of modern research connecting Continuum Mechanics with Solid State Physics. It integrates the Mechanics of Continuous Media, Heat Conductivity and the theory of Electromagnetism that are usually studied separately. For an accurate description of the influence of static and dynamic loadings, high temperatures and strong electromagnetic fields in elastic media and constructive installations, a new approach is required; an approach that has the potential to establish a synergism between the above mentioned fields. Throughout the book a vast number of problems are considered: two-dimensional problems of electro-magneto-elasticity as well as static and dynamical problems for piecewise homogenous compound piezoelectric plates weakened by cracks and openings. The boundary conditions, the constructive equations and the mathematical methods for their solution are thoroughly presented, so that the reader can get a clear quantitative and qualitative understanding of the phenomena taking place.
This book is for the specialists in Continuous Mechanics, Acoustics and Defectoscopy, and also for advanced undergraduate and graduate-level students in Applied Mathematics, Physics, Engineering Mechanics and Physical Sciences.
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Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1 Physical Fields in Solid Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.1 Heat Field. Heat Conduction Equation in Solid Bodies . . . . . . . 9
1.1.1 Heat Field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.1.2 Equilibrium Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.1.3 Heat Conduction Equation . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.1.4 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2 Electromagnetic Fields. Maxwell’s Equations . . . . . . . . . . . . . . . . 13
1.2.1 The Laws of Electrodynamics in Integral Forms . . . . . . . 13
1.2.2 Maxwell’s Equations in Differential Forms . . . . . . . . . . . . 15
1.2.3 Magnetization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.2.4 Electric Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.2.5 The Balance Equation of the Electromagnetic Field’s
Energy. Umov-Poynting Vector . . . . . . . . . . . . . . . . . . . . . . 18
1.2.6 Vector and Scalar Potentials of Electromagnetic Field . . 20
1.2.7 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.2.8 Stationary Electromagnetic Field. Electrostatic
Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.3 Stresses and Deformations. Hooke’s Law. . . . . . . . . . . . . . . . . . . . 24
1.3.1 State of Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.3.2 Equations of Equilibrium and Motion. Symmetry
of Stress Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.3.3 Deformed State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.3.4 Equations of Compatibility . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.3.5 Elastic Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.3.6 Hooke’s Law. Stress Potential . . . . . . . . . . . . . . . . . . . . . . . 32
1.3.7 Matrix Designations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.4 Singular Physical Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
1.4.1 Analysis of the Singularities of Physical Fields . . . . . . . . 36
1.4.2 The Electric Field of a Charged Conductive Disk . . . . . . 39
X Contents
1.4.3 Mathematical Idealization of Cracks
in an Elastic Body. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
1.4.4 Criterion of Fracture of a Body with a Crack . . . . . . . . . 56
2 Basic Equations of the Linear Electroelasticity . . . . . . . . . . . . . 63
2.1 The Linear Theory of the Piezoelectricity. . . . . . . . . . . . . . . . . . . 63
2.2 Equations of State for Piezoelectric Ceramics . . . . . . . . . . . . . . . 70
2.3 Two-Dimensional Problems of Electroelasticity . . . . . . . . . . . . . . 72
2.4 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
2.5 Mechanics of Fracture of Piezoelectrics . . . . . . . . . . . . . . . . . . . . . 78
3 Static Problems of Electroelasticity
for Bimorphs with Stress Concentrators . . . . . . . . . . . . . . . . . . . 85
3.1 Complex Representations of Solutions in Two-Dimensions . . . . 85
3.2 A Bimorph with Cracks in One of the Pair Components . . . . . . 91
3.3 Bimorph with Openings in One of the Pair Components . . . . . . 100
3.4 A Composite Plate with a Crack Crossing the Interphase . . . . . 106
3.5 A Composite Plate with an Opening Crossing the Interphase . . 112
3.6 Green’s Function for a Composite Plate
with an Interphase Crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
3.7 A Case of an Inner Crack Reaching the Interphase . . . . . . . . . . . 125
4 Diffraction of a Shear Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
4.1 An Anisotropic Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
4.2 A Piezoceramic Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
4.3 A Piezoceramic Halfspace. Free Boundary and Rigid Fixture
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
4.4 A Halfspace with a Crack Reaching the Boundary . . . . . . . . . . . 156
4.5 Harmonic Excitation of a Halfspace by External Sources . . . . . . 158
4.6 Arbitrary with Time Excitation of a Halfspace . . . . . . . . . . . . . . 162
4.7 A Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
4.8 A Halflayer. Various Variants of Boundary Conditions . . . . . . . . 174
5 Scattering of a Shear Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
5.1 A Space and a Half-Space with Tunnel Openings . . . . . . . . . . . . 181
5.2 Impulse Excitation of a Half-Space with Openings . . . . . . . . . . . 190
5.3 Stress Concentration in a Layer with Openings . . . . . . . . . . . . . . 193
5.4 A Half layer with Openings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
5.5 A Space and a Halfspace with Cylindrical Inclusions.
Integrodifferential Equations of a Boundary Problem . . . . . . . . . 199
5.6 Interaction of Openings and Cracks in a Space . . . . . . . . . . . . . . 207
5.7 Fundamental Solution for a Composite Anisotropic Space . . . . . 216
5.8 An Anisotropic Bimorph with Tunnel Openings . . . . . . . . . . . . . 223
Contents XI
6 Mixed Dynamic Problems of Electroelasticity
for Piezoelectric Bodies with Surface Electrodes . . . . . . . . . . . 229
6.1 An Unbounded Medium with a Tunnel Opening. Direct
and Inverse Piezoelectric Effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
6.2 Interaction of Two Openings in an Unbounded Medium . . . . . . 237
6.3 Excitation of a Medium with an Opening
by an Electric Impulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
6.4 Excitation of Shear Waves in an Infinite Cylinder
with an Arbitrary System of Electrodes . . . . . . . . . . . . . . . . . . . . 245
6.5 A Hollow Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
6.6 A Halfspace with Tunnel Openings . . . . . . . . . . . . . . . . . . . . . . . . 255
6.7 A Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
6.8 Interaction of a Partially Electrodized Opening and Crack . . . . 280
6.9 An Opening Strengthened by a Rigid Stringer . . . . . . . . . . . . . . . 290
7 Harmonic Oscillations of Continuous Piezoceramic
Cylinders with Inner Defects (Antiplane Deformation) . . . . 301
7.1 A Cylinder Weakened by Tunnel Cracks (Direct Piezoeffect) . . 301
7.2 A Cylinder with a Thin Rigid Inclusion . . . . . . . . . . . . . . . . . . . . 309
7.3 A Cylinder with a Crack Excited by a System of Electrodes . . . 317
7.4 A Cylinder With an Inclusion Excited by a System
of Electrodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
8 Electroacoustic Waves in Piezoceramic Media
with Defects (Plane Deformation) . . . . . . . . . . . . . . . . . . . . . . . . . 333
8.1 Waves in a Homogeneous Medium . . . . . . . . . . . . . . . . . . . . . . . . . 333
8.2 General Representations of Coupled Fields in a Medium
of the Hexagonal Class of Symmetry . . . . . . . . . . . . . . . . . . . . . . . 338
8.3 An Unbounded Medium with Tunnel Cracks. Integral
Representations of Complex Potentials . . . . . . . . . . . . . . . . . . . . . 342
8.4 Integrodifferential Equations of a Boundary
Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344
8.5 Reducing to a Case of an Isotropic Medium. . . . . . . . . . . . . . . . . 348
8.6 Effect of Mutual Hardening of Cracks . . . . . . . . . . . . . . . . . . . . . . 353
8.7 An Inertial Effect in the Process of Impact Effect
on a Crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
8.8 A Matrix of Fundamental Solutions of Two-Dimensional
Equations of Electroelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358
8.9 An Unbounded Medium with Tunnel Openings . . . . . . . . . . . . . . 365
8.10 Oscillation of a Cylinder Under the Influence
of Pulsating Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
9 Magnetoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
9.1 Magnetic Field and its Properties. . . . . . . . . . . . . . . . . . . . . . . . . . 373
9.1.1 Action of the Magnetic Field on the Moving
Electric Charges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374
XII Contents
9.2 The Magnetic Properties of the Substance . . . . . . . . . . . . . . . . . . 375
9.2.1 Action of External Magnetic Field on the Substance . . . 375
9.2.2 Classification of Substances According to Magnetic
Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376
9.3 General Relations of the Magneto-Elasticity
of Electro-Conductive Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382
9.4 Linear Magneto-Elasticity of Diamagnetic Materials . . . . . . . . . 384
9.5 Equations of Magneto-Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
10 Induced Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
10.1 Initial Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
10.2 The Current-Conducting Medium with Tunnel Cracks
in a Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390
10.2.1 Antiplane Deformation of the Infinite Ideal Conductor
with Tunnel Cracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
10.3 Half – Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396
10.4 Layer and Semi – Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402
10.5 Stress Concentration in the Opening in the Conducting
Half-Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408
10.5.1 Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411
10.6 Interaction of Crack and Opening in the Current-Conducting
Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411
10.7 Diffraction of Shear Magneto-Elastic Wave in the Inclusions . . 414
10.8 Fundamental Solution of Two – Dimensional Equations . . . . . . 420
10.9 Diffraction of Magneto-Elastic Waves on the Opening
in a Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425
10.9.1 Statement of Boundary-Value Problem. Integral
Representations of the Solutions . . . . . . . . . . . . . . . . . . . . . 425
10.9.2 Integral Equations of the Boundary Value Problem . . . . 431
10.9.3 Dynamic Intensity of Conductor in the Opening . . . . . . . 432
11 Influence of Magnetizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441
11.1 Initial Relations of Linear Magneto-Elasticity
of Ferromagnetic Materials. Complex Representations
of the Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441
11.2 Ferromagnetic Medium, Weakened by the Tunnel Cracks . . . . . 444
11.3 Generalized Kirsh Problem for Ferromagnetic Medium
with Cavity in Strong Magnetic Field . . . . . . . . . . . . . . . . . . . . . . 454
11.3.1 Statement of the Problem. Complex Representation
of the Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454
11.3.2 Stress Concentration in Circular Opening. . . . . . . . . . . . . 458
11.3.3 Opening of Arbitrary Configuration in Ferromagnetic
Medium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461
Contents XIII
12 Optimal Control of Physical Fields
in Piezoelectric Bodies with Defects . . . . . . . . . . . . . . . . . . . . . . . 465
12.1 Optimization of Fracture Characteristics of Anisotropic
Semi-Infinite Plate with Cracks . . . . . . . . . . . . . . . . . . . . . . . . . . . 465
12.1.1 Direct Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465
12.1.2 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466
12.2 Statement of Certain Optimization Problems. . . . . . . . . . . . . . . . 472
12.3 Control of the Parameters of Fracture in Piezoceramic
Half-Plane with Cracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476
12.4 Control of the Stress Intensity Factor in a Bimorph
with an Interphase Crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479
12.5 On the Application of the General Problem
of Moments to Certain Optimization Problems
of the Theory of Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482
12.5.1 Statement of the Problems . . . . . . . . . . . . . . . . . . . . . . . . . 482
12.5.2 Approximational Approach . . . . . . . . . . . . . . . . . . . . . . . . . 484
12.6 Pulse Boundary Control of the Stressed State
of the Half-Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490
12.6.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 490
12.6.2 Control of Elastic State . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492
12.6.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494
12.7 Boundary Control of the Stress Intensity Factors
in a Halfspace with a Crack. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497
12.8 Control of Electric Charges on the Electrodes
in a Layer with a Partially Electrodized Opening . . . . . . . . . . . . 499
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517
Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509
Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511
B.1 The Approximate Solution of Singular Integrodifferential
Equations Prescribed on Smooth Disconnected Contours . . . . . 511
B.2 Solution of the Singular Integral Equations
Given on Smooth Connected Contours . . . . . . . . . . . . . . . . . . . . . 514
B.3 Numerical Solution of the Singular Integrodifferential
Equations Given on Connected Contours . . . . . . . . . . . . .
Mathematical Methods in Electro-Magneto-Elasticity.part1
ص
[ 本帖最后由 drjiachen 于 2008-12-24 10:03 编辑 ]
Mathematical Methods in Electro-Magneto-Elasticity.part2-3
[ 本帖最后由 drjiachen 于 2008-12-24 10:04 编辑 ]
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