Introduction |

1 | General relations of the mechanics of piezoelectrics and electroconductive media in an electromagnetic field |

| 1.1. | Fundamental relations of electrodynamics |

| 1.2. | A piezoelectric effect in crystals and the equations of electroelasticity of piezoelectrics |

| 1.3. | Defining relations of the theory of magnetoelasticity |

| 1.4. | Energy of propagation of electromagnetoelastic waves |

2 | Coupled electromagnetic elastic waves in an unbounded medium |

| 2.1. | Propagation of electroelastic waves in an unbounded piezoelectric medium with finite conductivity |

| 2.2. | Propagation of plane waves in a piezoceramic medium of class $6mm$ |

| 2.3. | Plane magnetoelastic waves in an isotropic electroconductive medium |

3 | Surface Rayleigh waves in piezoceramic and magnetoelastic media |

| 3.1. | Excitation of Rayleigh waves in a piezoelectric space by two electrodes of opposite charge |

| 3.2. | Oscillation of a piezoelectric halfspace with a periodic system of electrodes |

| 3.3. | An anti-symmetric finite system of electrodes on the boundary of a piezoelectric |

| 3.4. | Excitation of axial symmetric Rayleight waves in a piezoelectric halfspace by end electrodes and circular electrodes |

| 3.5. | Rayleigh magnetoelastic surface waves in an elastic halfspace |

4 | Shear waves in piezoelectrics |

| 4.1. | Excitation of Gulyaev--Bluestein surface waves in a piezoelectric half-space with a finite anti-symmetric system of electrodes |

| 4.2. | Structure and characteristics of a shear wave in a hexagonal layer in a piezoelectric |

| 4.3. | Oscillations of a piezoelectric circular cylinder with a system of surface electrodes |

| 4.4. | Shear waves in acoustically connected halfspaces of a dielectric and piezoelectric |

5 | Harmonic Lamb waves in a piezoelectric strip |

| 5.1. | Excitation and characteristics of symmetric Lamb waves in a piezoelectric wave |

| 5.2. | Excitation and characteristics of anti-symmetric Lamb waves in a piezoelectric strip |

6 | Criteria of the breakdown of dielectrics and piezoelectrics |

| 6.1. | Energetic analysis of the breakdown of dielectrics |

| 6.2. | Invariant integrals, intensity factors of the density of charges and phenomenological criteria of the breakdown of dielectrics |

| 6.3. | Practical application of the criteria for the breakdown of dielectrics |

| 6.4. | The energetic analysis and phenomenologic criteria of an electromechanic breakdown of piezoelectrics |

Bibliography |

About the authors |

Discovered 100 years ago by French scientists Pierre and Jacques Curie the
piezoelectric effect is widely applied in different fields of science and
technology, but mainly in radio engineering, acoustics, non-destructive
testing, metrology, computer engineering and acoustoptics. The basis of this
phenomenon is the capability of some materials to deform under the influence
of external electric fields. The reversibility of this phenomenon is
well-known, which is manifested in the excitation of an electric field during
the deformation of a piezoelectric by external mechanic loads. The first
practice application of this phenomenon is connected with the name of Paul
Langevin, who constructed a sonar, where a sensing element made from quartz was
used as a radiator and receiver. From the middle of fifties of XX century after
creating artificial materials possessing a piezoeffect, the field
of application of piezoelectrics was broadened substantially and at present
a large quantity of different devices has been developed based on the
phenomenon of piezoelectric. Approximately that time started the investigation
of the problem of interaction of deformed media with electromagnetic fields and
up to now a series of models were developed, taking into account either aspects
of the interaction. In particular, models of piezoelectric and piezomagnetic
media were constructed, a piezoconductive medium, an electromagnetic conductive
medium and a series of other models. A more full representation of various
models of interaction, fields of their application and methods of calculations
of various devices may be found in
reviews [56], [79], [76], [75], [93], [80],
and also in special
monographs [32], [3], [8], [27],
[125], [128]--[65], [36], [82], [84], [89], [97], [102],
[106], [39], [64], and in references
in them.

The increased requirements in calculations precision of numerous
acoustoelectronics devices on the surface of acoustic waves, and also the
necessity of estimation of the influence of external electromagnetic fields on
their parameters stimulated intensive investigation in this field, which is
stated in numerous publications. Substantial part of this investigation was
fulfilled by using various approximation approaches. In connection with this
the mathematical investigation of wave processes in electromagnetic media on
the basis of a strict solution to the corresponding problems
of electroelasticity is quite actual. Recently much attention is paid to this
problem in the works of prominent scientists working in this field. Up to now
different types of investigations and concrete results in the sphere of static
and dynamic problems of electroelasticity appeared
in [4], [5], [22]--[26], [55], [77],
[70], [88], [99], [100], [107], [110], [47]--[60], [112], [1]--[126].

From the multiple effects appearing as a result of interaction between an elastic
media and an electromagnetic field in the suggested monograph, the
statement of main results obtained in the field of investigation of harmonic
waves in piezoelectric magnitoelectric media is given. This choice may be
explained by the scientific interests of the authors on the one hand, and by
large practical application of harmonic waves in piezoelectric and
magnitoelectric media on the other hand.

The given monograph consists of 6 chapters, introduction and references and
arbitrarily may be divided into two parts. In the first part (the
Introduction and Chapter 1) a brief review of basic problems is included,
appearing in the process of calculation of volumic and surface waves in
magnetoelastic and electroelastic media and of the methods of calculation of devices on surface acoustic waves in piezoelectrics.

In the first chapter, which is of subsidiary character, the basic relations
of electrodynamics and the description of piezoelectric and magnetoelastic
effects, the statement of corresponding problems of excitation and
propagation of waves in various media, are given as well as, the main
energetic characteristics of wave fields. One of our purposes was to give
a brief but rather full statement of basic problems of electromagnetoelasticity, so that it would be possible to understand the
material stated in the following chapters without reading the initial
sources.

In the second chapter the results of investigation of harmonic waves in
unbounded electromagnetoelastic media are given, and unlike other monographs
on the subject, the investigation of volumic waves in piezoelectrics and
the investigation of volumic waves in piezoelectrics and magnetoelastic
media is built with due regard for electrodynamic effects and conductivity
of materials. Taking into account the mentioned
effects the real properties of materials are reflected more accurately and
the stated results allows us to analyze the influence of these quantities on the
basic parameters of the waves, and due to it to use more simple media models
in a certain wave band.

In the third, fourth and fifth chapters the problems
connected with excitation of surface Rayleigh waves, shear surface waves and
Lamb waves are considered. Despite the large number of works connected with
the investigation of excitation of harmonic surface waves in piezoelectrics
with the help of surface electrodes, only in few of them the solutions of boundary problems are built by using the equations of electroelasticity
without any additional simplification. The proposed method of investigation
is based on the reduction of the solution of the problems to a system of singular integral equations which is solved by the
Bubnov--Galerkin method and allows us to build efficient algorithms of solution
of the problems on computers and with the help of them to investigate the
cinematic and energetic characteristics of the excited waves. A rather
detailed investigation of the behaviour of electroelastic waves in the
vicinity of the electrode edges is given as well as; formulas of electrode
capacities and displacement current, which are important for application,
and at last the parametric investigation of intensity factor of the charge
density is represented, which allows to consider differently the solution of the important problem of electromechanic breakdown of piezoelectrics with
systems of electrodes. The sixth chapter of the book is devoted to the
solution of this problem.

We think that, investigation of the phenomenon of electromechanic breakdown of piezoelectrics, when the site of breakdown or puncture is the edge of electrodes, has theoretical and practical interest. The
necessity of electromechanic breakdown prediction of various devices on
surface waves, excited by electrodes, stimulated the work of the authors in
that direction, and its results are stated in the sixth chapter. The
proposed new approach to the solution of this problem, based on the
generalized methods of breakdown mechanics in case of piezoelectric
media with systems of electrodes. The suggested in this chapter criteria of puncture of dielectrics and electromechanic breakdown of piezoelectric are
based on the laws of conservation of energy and generalize the ideas and
methods of linear mechanics of breakdown. It should be noted that in case of vacuum puncture the suggested criteria give results which agree with the
experiment. The obtained in this chapter results indicate the efficiency of the proposed criteria of electromechanic breakdown of piezoelectrics and
dielectrics, and due to it approach of the solution of the problem of electromechanic breakdown of piezoelectrics is feasible. We expect that the material of this chapter will attract the attention of researchers and developers of devices
with surface electrodes to the problem of electromechanic breakdown of piezoelectrics.

**Kudryavtsev Boris Alexandrovich** -- was born in 1937 near Ribinsk of Yaroslavl region. After school he entered Moscow aviation institute, which
he finished in 1959, and he was accepted at Central Research Institute of Machine Building, where he worked for 10 years, studying the problems of providing the strength of products of rocket and space technology and
concentrating his attention on the problems of development of the methods of calculation of thinwalled shells and plates, including the problems of studying of the features of stress-deformed state in the vicinity of heterogeneous inclusions, cavities and cracks, and also the practical
problems including a series of problems of breakdown mechanics. In the
beginning of 70s B.A.Kudryavtsev was invited to the Department of Higher
Mathematics of Moscow Institute of Chemical Engineering Industry, where he
worked for 20 years starting from a lecturer and becoming a chief of the
Department. During these years he organized an All-Union Seminar on
mechanics of hard deformed body, where the problems connected with the
investigation of interaction of a hard deformed body with electromagnetic
fields were considered. In 1983--1984 B.A.Kudryavtsev wrote and defended
a thesis for Doctorate degree in Leningrad State University, which connected
the problems of interaction of an elastic medium with an electromagnetic
field including the questions of interaction of bodies with cracks with
magnetic and electric fields. During the work in the Seminar he supervised
about 20 PhD's and 3 theses for Doctorate degrees in Science. The Seminar
was very popular in the USSR and cooperated with related Seminars, which
where led by such scientists as I.I.Vorovich (Rostov-on-Don), A.N.Guz
(Institute of Mechanics, Kiev), A.F.Ulitko (Kiev University), A.A.Iliushin
(Moscow University), A.S.Kosmodamiansky (Donetsk University), S.A.Ambartsumyan (Institute of Mechanics, Erevan).

B.A.Kudryavtsev is the author of 130 scientific articles and the books
"Electromagnetoelasticity of piezoelectric and electroconductive bodies"
(with V.Z.Parton as a co-author). He was also the supervisor of many
scientific and research works which where done in the Department by orders
of commercial institutes.

The book suggested to the reader was planned to be written in the beginning
of 90s by B.A.Kudryavtsev, and the initial variant was almost ready during
the life of our teacher. However, in 1994 B.A.Kudryavtsev died suddenly.
We, his pupils, hope that publishing of this book, the ideologist of which
he was, and in which the new works which appeared
during the last decade are also considered, will be deserving the memory
of B.A.Kudryavtsev -- a great scientist and teacher.

*D.I.Bardzokas (Athens),
N.A.Senik (Moscow)*

**Bardzokas Demosthenis Ioannis** -- is a professor at National Technological
University of Athens (NTUA). He was born in Tashkent in 1952 in a family
of Greek political refugees. After finishing secondary school in 1970 he
entered Tashkent State University, mechanicomathematical faculty and
graduated it in 1975.

After the fall of the dictratorship in Greece all his family returns back
home. In 1976 he became a research worker of National Technological
University of Athens, department of mechanics, the head of which was
well-known scientist, academician P.S.Teocaris. Under his supervision he
defended the thesis "Investigation of plane problems of strengthening
bodies with cracks and plane contact problems of elastic bodies by the
method of the theory of functions of complex variables". From 1987 to 1990
he worked on probation in Moscow Institute of Chemical Engineering under the
supervision of V.Z.Parton and B.A.Kudryavtsev.

At present he is a professor of the department of mechanics, faculty of applied mathematics and physical sciences of National Technological
University of Athens. He has published more than 100 works concerning various
fields of the mechanics of continuous media (mechanics of destruction,
elasticity, heat conductivity, electroelasticity, mechanics of composite
materials, theory of waves, etc.).

**Senik Nikolai Alexandrovich** -- was born near Zaporozie (Ukraine) in
1955. He graduated the mechanicomathematical faculty of Moscow State
University in 1978. In 1981 he started his postgraduate studies in the
Department of higher mathematics of Moscow Institute of Chemical Engineering
Industry, where B.A.Kudryavtsev was lecturing at that time. In 1984 he
defended a thesis connected with the theory of thinwalled piezoelectric
shells for his Phd of physical and mathematical sciences. From 1985 he works
as a senior research worker at Scientific and Research Institute of Electromechanics. Simultaneously he took active part in the Seminar of the
mechanics of hard deformed body in Department of higher mathematics of Moscow Institute of Chemical Engineering Industry. In 1992 he defended
a thesis for a doctorate degree of physical and mathematical sciences at
Moscow State University. In 1994 he became a chief of Scientific and
research laboratory of developing of perspective cosmic apparatus and also
of providing strength parameters and heat regimes of space apparatus of series "Meteor" in the same institute. In 2001 he became a deputy
chief constructor of space apparatus "Vulkan". At present he is a chief of Department of perspective space developments at Scientific and Research
Institute of Electromechanics.