Mathematical Methods in Electro-Magneto-Elasticity

538 Pages·2007·12.806 MB·English

by  Demosthenis I. Bardzokas, Michael L. Filshtinsky, Leonid A. Filshtinsky (auth.)

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Mathematical Methods
in Electro-Magneto-Elasticity
Demosthenis I. Bardzokas • Michael L. Filshtinsky •
Leonid A. Filshtinsky
Authors
D.I.Bardzokas
NationalTechnicalUniversityofAthens
SchoolofAppliedMathematics
andPhysicalSciences
DepartmentofMechanicsLaboratory
ofTestingandMaterials
Athens,Hellas,Greece
L.A.Filshtinsky
Dept.ofMathematicalPhysicsSumy
StateUniversity
Sumy,Ukraina
M.L.Filshtinsky†
LibraryofCongressControlNumber:2007923590
ISSN 1613-7736
ISBN 978-3-540-71030-1 SpringerBerlinHeidelbergNewYork
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publicationorpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawof
September9,1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.
ViolationsareliableforprosecutionundertheGermanCopyrightLaw.
SpringerisapartofSpringerScience+BusinessMedia
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(cid:2)c Springer-VerlagBerlinHeidelberg2007
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This book is dedicated to Michael Filshtinsky who had always a vision
to create. This creation was expressed both in science and music.
Preface
Thescienceofmechanicsofcoupledfieldsisadynamicresearcharea,studying
in a unified way continuum mechanics, heat transfer phenomena and electro-
magnetism, i.e. branches of science that are usually studied separately.
Foramorerigorousandaccuratedescriptionofthe influence ofstatic and
dynamic loadings, high temperatures and strong electromagnetic fields on
elastic media and constructions we need a single approach, i.e. one that has
thepotentialto integratealltheabovementionedphysicalfields.Theinvesti-
gationofthemechanicsofstressed-deformedmediasubjecttocoupledfieldsis
one of the most important tasks in today’s material science with major tech-
nological importance. Our scope is the rigorous qualitative and quantitative
analysis of the stressed-deformed states of materials and constructions and
their physical and mechanicalproperties within the phenomenologicaltheory
of electro-magneto-thermo elasticity. For that reason we derive the constitu-
tive equations that govern the behavior of the problem under study, we pose
the appropriate boundary conditions and finally we present and apply the
appropriate mathematical methods for their solution.
General relations of the mechanics of deformed bodies interchanging with
electromagnetic field are used and special attention is paid to the construc-
tion of the theory of shells and plates made from piezoelectric materials. We
present in detail art of the state mathematical methods for the solution of
a broad class of two-dimensional problem of electroelasticity for multicon-
nected bodies. We also examine several static and dynamic problems for
piecewise-homogeneouspiezoelectricplates,weakenedbycracksandopenings.
We mostly consider questions connected with the application of the method
of boundary integral equations for the investigation of the problem of de-
formation of electroelastic waves on heterogeneities of various types. Special
emphasisisgiveninthedefinitionofstrengthandbreakdowncharacteristicsof
bodies with defects. We state and solve some inverse problems of electroelas-
ticity about optimal, in some sense, equations by parameters of strength and
breakdown.
VIII Preface
Theinterplayofcoupledphysicalfieldsinanisotropicmediaintroducesad-
ditionaldifficultiesintotheanalysisofboundaryproblemsofelectro-magneto-
elasticity. The boundary conditions are not usually separated, which leads to
the necessity ofconsiderationof compoundboundaryproblemsof mathemat-
ical physics.
We believe that this monograph provides the theoretical background and
the tools for the systematic analysis of electromegneto-elastic media with a
broad spectrum of applications.
This book is the condensation of a long scientific collaboration of the
authors despite the fact that Dr. M. L. Filshtinsky, passed away unexpect-
edly 3 years ago. Michael L. Filshitinsky was born in 1961 in the town of
Novosibirsk. In 1989 he received his PhD in Mathematical Physics at the
Mechanic-Mathematical Faculty of the State University of Moscow
(Lomonosov). His Thesis was entitled: “Solution of two-dimensional dynami-
cal problems of elasticity and electroelasticity for cracked bodies”. The most
characteristic aspect of the work of M.L. Filshtinsky was the combination of
his natural talent and his hard work, for solving a vast number of novel and
real-worldproblems.Anotherinterestingcomponentofhispersonalitywashis
interestinmusiccomposition.M.L.Filshtinskyandushavepublishedtogether
many articles in international scientific journals and congress proceedings as
well as 5 monographs in Russian language [[190, 254]–[256]]
The present work is a small component of the knowledge of the great
SovietSchoolin the areaofCoupledFields Mechanicsandispartofaneffort
to bring out the rich Russian literature in the area.
At this point Prof. D.I. Bardzokas would also like to thank, with all his
heart,hisassociate,friendandstudentGeorgeI.Sfyrisaswellashiscolleague
and good friend Constantinos I. Siettos for the help they provided in writing
this monograph. Without their valuable help the publication of this work
would be impossible.
This book is for the specialists in Continuous Mechanics, Acoustics and
Defectoscopy, and also for advanced undergraduate and graduate –level stu-
dents in Applied Mathematics, Physics, Engineering Mechanics and Physical
Sciences.
D.I. Bardzokas
Professor NTUA, Section of Mechanics, Greece
L.A. Filshtinsky
Professor of Sumy State Univerisity, Ukraine
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 Halflayer 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