The book offers a modern description of the mathematical methods required for solving a wide class of problems in the theory of elasticity, heat conductivity, thermo- and electroelasticity for composites of regular structure.

It will be of use for professionals working in the fields of mechanics of continuous media, composites, as well as for postgraduates and students specializing in the field of materials science.

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.

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.

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