Preface |

1 | Introduction. Models of composite materials |

| 1.1. | Composite materials |

| 1.2. | Mathematical models of composite materials |

| 1.3. | Properties of composite materials |

2 | Equations of the mathematical physics with quickly changing coefficients |

| 2.1. | Homogenization |

| 2.2. | Effective constant |

| 2.3. | Local fields |

| 2.4. | A medium with a periodic structure |

3 | The asymptotic method of averaging |

| 3.1. | The method of two-scaled expansions |

| 3.2. | Solution of the ordinary differential equation with quickly oscillating coefficients (zero approximation) |

| 3.3. | Solution of the ordinary differential equation with quickly oscillating coefficients (asymptotic expansion) |

4 | Averaging of the non-stationary equation of heat conduction for composite materials of a periodic structure |

| 4.1. | The problem of heat conduction for a composite material of a periodic structure |

| 4.2. | Calculation of temperature fields in a heterogeneous medium with a periodic structure |

5 | Averaging of the equations of electrodynamics in a periodic medium |

| 5.1. | Averaging of Maxwell's equations for a continuous medium with a periodic structure |

| 5.2. | Equations of electrodynamics for high-frequency electromagnetic field in a continuous medium with a periodic structure |

| 5.3. | Averaging of Maxwell's equations for high-frequency electromagnetic field in a periodic medium |

6 | Heating of the composite periodic structure in a high-frequency electromagnetic field. Solution of the problem of heating of a laminated composite |

| 6.1. | Statement of the problem of a composite heating in an electromagnetic field |

| 6.2. | Heating of a laminated composite of a periodic structure in a high-frequency electromagnetic field |

| 6.3. | Solution of the problem of heating of a laminated plate in a high-frequency electromagnetic field |

7 | Some information about the theory of periodic functions |

| 7.1. | Meromorphic functions |

| 7.2. | Elliptic functions |

| 7.3. | Weierstrasse's function |

| 7.4. | Quasiperiodic Weierstrass functions |

| 7.5. | Expansion of Weierstrass functions in Laurent series |

| 7.6. | Construction of two-periodic solutions of Laplace equation |

| 7.7. | Construction of two-periodic solutions of Poisson equation |

8 | Heating of a fibrous composite of a periodic structure in a high-frequency electromagnetic field |

| 8.1. | Statement of the problem of heating a fibrous composite in an electromagnetic field |

| 8.2. | Solution of the problem of electrodynamics for a fibrous one-way directed composite |

| 8.3. | Calculation of effective dielectric permeabilities |

| 8.4. | Calculations of the specific density of the heat release sources |

| 8.5. | Solution of the problem of heat conduction for a fibrous one-way directed composite |

| 8.6. | Evaluation of the local heterogenities of a temperature field under macro homogeneous heating |

| 8.7. | Calculation of temperature fields in a fibrous plate under heating, in a high-frequency electromagnetic field |

9 | About the propagation of acoustic waves in a fibrous material filled with liquid |

| 9.1. | Acoustic waves of an infinitesimal amplitude in an ideal medium |

| 9.2. | Averaging of the equations of acoustics of an ideal liquid in a periodic medium |

| 9.3. | Calculation of the velocity of waves in a fibrous one-way directed material with regular laying of fibres |

10 | Motion of viscous liquid in a porous medium with a periodic structure |

| 10.1. | Equations of motion of viscous liquid |

| 10.2. | Averaging of the Stokes equations in a porous body with a periodic structure |

| 10.3. | Calculation of the tensor of coefficients of filtration in a porous body with an orthogonal system of capillaries |

| 10.4. | Averaging of the equations of acoustics of viscous liquid in a periodic medium |

| 10.5. | Calculation of wave processes in a porous body with an orthogonal system of capillaries |

| 10.6. | Calculation of wave processes in a porous body with a fibrous structure |

11 | The theory of elasticity of composite materials with periodic structures |

| 11.1. | Averaging of the equations of linear problems in the theory of elasticity of composite materials with periodic structures |

| 11.2. | Effective elastic constants and the criteria of failure of laminated composite materials with periodic structures |

12 | Coupled fields in composite materials with periodic structures |

| 12.1. | Averaging of non-stationary equations of thermoelasticity for composite materials with periodic structures |

| 12.2. | Basic equations of the linear theory of electroelasticity |

| 12.3. | Averaging of the equations of electroelasticity for composite materials with periodic structures |

13 | Averaging of the equations of physical processes for bodies with a wavy boundary |

| 13.1. | Solution of the two-dimensional problem of heat conductivity for a body with a wavy boundary |

| 13.2. | Calculation of temperature fields in bodies with wavy boundaries using the method of averaging |

| 13.3. | Averaging of the three-dimensional equations of the theory of elasticity for an anisotropic plate of variable thickness with a periodic structure |

14 | Special integral transformations for solution of the problems of the mathematical physics in periodic media |

| 14.1. | New generalized integral transformations in axisymmetric boundary-value problems in the mechanics of composites |

| 14.2. | Torsion of a composite cylindrical shaft of finite length |

| 14.3. | Heat conductivity in a multi-layer wedge-form composite |

| 14.4. | A generalized integral transformation of Kontorovich--Lebedev type used for the solution of boundary problems of the theory of elasticity |

| 14.5. | About generalized integral transformation of Kontorovich--Lebedev type and its application for solving boundary problems of elasticity and heat conductivity |

Bibliography |

About the authors |

Introduction. Models of composite materials

### Composite materials

*And the same day Pharaoh ordered the intendents of the people
and supervisors: "Do not give them the straw to make bricks as the
day before and day before yesterday. Let them collect the straw
themselves. And make them produce the same number of bricks as they
did yesterday and the day before yesterday, and do not decrease the
number, because they are idle and therefore they shout. Let us go and
make a sacrifice to our god".* (Exodus, Chapter 5, vlrs. 6-8).

That was the answer of the Egypt Pharao to Moses when he demanded to
set free the Hebrews -- he stopped the delivery of the straw which was
added to the brick earth in order to increase the strength of the
produced bricks. So, the Bible testifies the wide use of artificial composite building materials in ancient Egypt. The history
of composite materials is lost in the depth of centuries, and it should not surprise us because man was always
surrounded by composite materials, such as wood, caulescents and
leaves of plants, shells and also bones, muscles and blood vessels of animals and of men. And, as the modern calculations show, the
biological composites are created in different ratios optimally and
man will always imitate it.

Nowadays we have an overturn in construction due to the appearance of such composite materials as concrete and reinforced concrete. The modern
aircraft and space technology cannot exist without wide use of such
composite materials as glass-reinforced plastic, boron plastic, carbon
composite materials and metal composites. Their production increases,
the manufacturing cost decreases, and thus, the use of the composites
in transport, consume and health service becomes perspective. The new
century will become the century of composites.

### Mathematical models of composite materials

What is a composite material from the point of view of modern
science? First of all before we answer this question we should
remember that in the title of the book we refer to mathematical
modeling. These words underline that in any theory we do not consider
the physical object itself, but a certain mathematical
model, which describes the behaviour of the real object more or less
precisely.

From the XIX^{th} century two approaches to the
consideration of properties of solid bodies are known: molecular approach
of Louis Navier and continuous approach of Ogusten Cauchy, named after the two
famous French physicists. The first approach was based on the
consideration that a body is a system of interacting molecules, which
brought us to rather strict crystallophysical theories. The second
approach consists of substituting a real body by an imaginary
continuous medium, infinitely filling the space. In order to describe
the behaviour of the continuous medium we should postulate the
defining equations. The obtained model is considered to be useful for
calculating the processes in some real bodies, if the results of the calculations correspond to the results of macroscopic experiments
rather precisely. Just in the frames of such approach, called the
phenomenological and consisting of fundamentals of the
continuous medium mechanics, we will carry on our statement in this book.

It is always important to remember the hypotheses laid into the
basis of the model and the limits of its application. So, the
hypothesis of a continuous medium loses its validity if we talk about
objects of molecular sizes, e.g., about the tip of a crack. In an
elastic body the continuous model predicts unreal infinite stresses
at the tip of a crack under any infinitesimal loadings. But it means
that we should add the description of an end zone of a crack the to
model. The built up model will have the right to exist if it
describes correctly what is necessary. In the given case it should
describe the limiting loadings, the velocity of propagation of a
crack and durability of the body with a crack.

A medium is usually called homogeneous, when similar
dimensions of it have similar properties. It is obvious that an
amorphous material (for example, glass) is homogeneous in the frames
of a continuous approach. However, the technological alloys are
polycrystallic, and here we have to answer the question whether it is
possible to simulate as a homogeneous continuous medium. The answer
to this question is not simple, because it depends on the problem we try
to solve. If it is necessary to evaluate the possibility of microcracks formation at grain joints we should solve the problem for a
heterogeneous body consisting of several crystal grains of various
orientation. But when calculating the deflection of the bar under the
influence of a definite force we may consider the bar homogeneous.
The mistake from substituting the real material with a homogeneous
continuous medium should not be essential, if the thickness of the
bar is measured in centimeters, but the dimensions of the grains are
the hundredth fractions of a millimeter.

Similar problems also appear in the process of designing
elements of constructions from composite materials, e.g. from
glass-reinforced plastic. Solving the problem of influence of the
inner pressure on a tube made from glass-reinforced plastic we may
successfully use a model of an equivalent homogeneous continuous
medium. The analysis of delamination between fibre and epoxy or
between fibre breaks is carried out by solving a problem of a single
or several fibres submerged into homogeneous medium. Such methods
remind us the investigations with the help of a microscope with various
magnification and are called "the microscope principles".

Thus, let us go back to the definition of composite materials
from the point of view of mathematical description of the physical
processes in them. Normally composite materials or composites are
considered to be multiphase materials consisting of two or more
number of components. Their components keep their individuality and
between the components interphases exist. One of the components,
filling the space as a binder is called a matrix or a binder. Other
components occupying isolated areas are called inclusions (of reinforced material or reinforcement). Usually on the one hand the
dimensions of the inclusions and the distances between them are
larger in comparison with molecular ones, and on the other hand are
smaller in comparison with the characteristic dimensions of the
problem. Such a composite is homogeneous in macroscopic scale (the
dimensions of the investigated body) but is heterogeneous in
microscopic scale (the dimensions of the inclusions and the distances
between them). If all the dimensions of an inclusion have the same
order, it may be called a grain or a dispersed particle, and the
composite may be called dispersed or granulated. Very
extended inclusions are called fibres and the composite is called fibrous composite. If
the inclusions are parallel cylinders, the material is called a
fibrous one-way directed composite. Laminated fibrous composites
consist of one-way layers with various orientation of the fibres.

### Properties of composite materials

A characteristic property of a composite material is the ability
to unify the useful properties of separate components, so the
manifestation of new properties differs from the properties of the
components. In many cases the composite materials are developed and
created together with the construction. First of all that refers to
fibrous one-way directed and winding laminated fibrous materials. The
materials and constructions from them are produced simultaneously by
the method of continuous winding. The technological conditions of the
process of winding determine the possible designs of the products,
and the material is formed so that it could respond to the real
stresses efficiently.

Nowaday composite materials may be produced with various physical
and mechanical properties, and that means that the constructions from
composite materials may be optimized. Experimental definition of the
properties of composite materials with all kinds of schemes of reinforcing requires a rather big volume of expensive investigations.
Therefore, building theoretical models of composite
materials, allows for determination not only of averaged
characteristics, but also description of the singularities of the
processes taking place in such media.

**Bardzokas Demosthenis Ioannis** is a professor of National Technological
University of Athens (NTUA). He was born in Tashkent in 1952 in the 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
a well-known scientist, academician P.S.Teokaris. 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.).

**Zobnin Alexander Igorevich** has a doctorate diploma on physico-mathematical
science, and is a senior lecturer. He was born in 1948 in Kaunas (Lithuania). In 1971
he graduated the mechanico-mathematical faculty of Moscow State University
(department of plasticity theory).

In 1975 after finishing the postgraduate courses of mechanico-mathematical
faculty of Moscow State University and defended the thesis "Several
problems of the mechanics of destruction" (superviser of studies
academician of Acad. Sci. of USSR U.N.Rabotnov).

He worked at Central Research and Design Institute of Constructive
metalloconstructions. Since 1978 he has become a senior lecturer of Moscow
State University of Engineering Ecology, department of higher mathematics.

He published about 30 scientific works concerning different fields of the
mechanics of a deformed solid body (mechanics of destruction, mechanics of connected fields in a deformed solid body, mechanics of composite materials
of periodic structure).

A.I.Zobnin, D.I.Bardzokas