Book review

The history of viscoelasticity goes back in the middle of previous century. At that time became clear that the existing ideas and technics could not adequately describe the deformation and flow of the real materials. What were the first steps of a new science (further called as viscoelasticity). There were some periods of its successful development. The last one began at the early sixties.

The authors of this book have had two goals behind them. First of all they have tried to introduce the most of known ideas, to describe many of the essential results (not only modern but also those old ones which had and still have the great influence on the ideas involved). For these purposes are reviewed many books, surveys and articles. As a result is a quite satisfactory and complete list of references, many interesting notes and remarks and explanations of results and ideas. Besides are gathered some notations and propositions from real and complex analysis, differential and integral equations theories which make this monograph selfcontained. All the material in the book is unified due to choosen method of boundary value problem (BVP). In fact it is whole collection of closed to each other approaches. For plane models they are BVP for analytic functions and connected with them integral equations (singular integral equations are among them). Besides are involved different kind of integral transforms. Are discussed also some BVP for more comlicate (than Cauchy-Riemann type) partial differential equations (two and three dimensional). At last are presented some elements of free boundary method and so on.

The second goal of this book is to give this material (in fact - to present the theory and applications of Viscoelasticity) in such a form which can be useful and applicable for all who is using it - mathematicians (pure and applied), physicists and engineers. From this point of view we can see on the book as on the textbook, for which all the needed results are presented and introduced with all possible details and explanationary examples. Are given also a number of theoretical and practical exercises. Every chapter has a summary in which all the discussed problems are explane in a short and comprehensive form. In any case it helps to understand the given ideas more deeply and without technical details presented in the main text.

But the main attention is paid in the book to the construction, description and comparing many physical and mathematical phenomenal known as viscoelastic models. It makes this monograph the serious, systematic and background text on one of the quickly developing subject of applied mathematics and physics.

The book cosists of seven chapters, four appendices, list of references and subject index.

The main ideas leading to the investigation of viscoelasticity are gathered in the first chapter ("Fundamental Relationships"). On the discussion are different kind of models beginning from the classical ones which are describing this physical phenomena. But mainly the authors have tried to obtain the most fundamental equations for the viscoelastic medium in its different modifications. These equations are connected of course with the type of using models and so are quite differet.

The chapter begins with two notions known so far as in the classical elasticity theory - stress and strain. Are given some relations between them and their representations in the different coordinate systems.

Then are discussed physical considerations which lead to one-dimensional viscoelasticity model. All the equations of this model are presented in terms of relation function and greep function. On the discussion are different approaches and representations of the general viscoelasticity model. The end of the chapter is devoted to the construction of the three-dimensional viscoelastic model. In the isothermal case are obtained constitutive and dynamical equations. It is explaned the meaning of the non-inertial approximation for the model. Due to fact that in this case an approach in three-dimensional situation is essentially the same as in one-dimensional one, the most of details and considerations are also the same. Special attention is paid for isotropic case. At last is studied the phenomena of causality and its different analytic sense.

Summarizing all mentioned we can say that in the chapter 1 is observed the construction and discussion of the viscoelastic models and presented several its forms suitable for applications of methods given in the next chapters.

In a contrary to the previous one the chapter 2 ("General Theorems and Methods of Solution of Boundary Value Problems") is mainly devoted to depeloping an analytic machinery available for viscoelastic problems. First of all is presented the Classical Correspondence Principle which allows to give viscoelastic solutions in terms of elastic ones. Are discussed also some generalizations of this principle - for the problems involving time-dependent regions (Extended Correspondence Principle), for the case when regions over which certain types of boundary conditions prevail are contain in (or contain themselves) the regions where that type of boundary condition was given at all previous time (Generalized Partial Correspondence Principle).

Besides is presented so called viscoelastic Papkovich-Neuber solution. More precisely it is obtained an equation which provides an extension of the Papkovich-Neuber stress function of threedimensional elasticity to viscoelasticity.

At the end of the chapter is given an introduction to two more general problems. First of all it is a problem of contact between viscoelastic bodies. Its discussion is continued then in the next two chapters. Here are presented only general results and methods.

At last are studied so called aging materials or solution of problems involving non-isothermal conditions. The study of the corresponding equations under these two (or at least one) conditions is quite difficult even in the elastic case. Thus here is presented the only description of main approaches to the problem and given a short reference of known results.

The next two chapters are based on the complex analytic functions techniques developed for the discussed problems in the forties-sixties. This is connected with the well-known Kolosov-Muskhelishvili formulae.

In the third chapter ("Plane Non-inertial Contact Problems") is described a general metodology of these problems' solution. Mainly it is connected with the method of boundary value problems for analytic functions (Hilbert or Riemann-Hilbert-type problems) and the singular integral equations method. On the discussion are some of situations in which these methods can be applied.

From the technical point of view the chapter 4 ("Plane Noninertial Crack Problems") is a continuation of the previous one. The simple fracture mechanics problems are considered only for viscoelastic media. Besides it is choosen the only macroscopic approach. For alternative microscopic direction is done small description and reference introduction. The macroscopic approach allows the authors to use the advantages of complex analytic functions' theory and the boundary value problems. Here is presented more complicative variant of the latter problems than in the chapter 3. Mainly on the discussion are the problems of stationary crack (possibly growing). It should be stressed that at the study of cracks problems the exact nature of the corresponding boundary conditions is not always a priori known but has to be determined in the course of the solving the problems.

Contact problems in the space are discussed in the chapter 5 ("Three-dimensional Contact Problems"). This situation is more complicative firstly due to deeper dependence of the involving methods of the geometry of contact area, and secondly due to impossibility of using some plane approaches to solving the corresponding problems (the complex analytic functions' methods are not on the discussion here). Anyway some results presented in this chapter are obtained quite at the same manner as in previous chapters. Mostly this is connected with integral transforms and integral equations method. The authors have discussed the following questions: generalized Boussinesq formulae (which is extended for viscoelastic situation from the classical elastic one), the normal contact problem under varying load (in the case when a rigid identor is pressed into viscoelastic half-space), impact problems (an application of the previously developed technics to the problem of a rigid identor impacting once on a viscoelastio half-space and the losing contact with it) and at last hysteric function (in fact the approximation formula in the case of small viscoelasticity and small velocity).

In the small two last chapters (ch.6 "Thermoviscoelastic Boundary Value Problems", ch.7 "Plane Inertial Problems") are presented some results on the mentioned problems. As a matter of fact there are the only few of them in these situations.

In the case of temperature depending materials (see ch.5) is assumed that they are thermorheologically simple. The most of the equations even in this situation are quite complicative. Thus the main method of their solution is connected with Neumann series expansion for corresponding integral equations.

The only little progress in analytic method of solution there exists also for inertial problems. A few number of them are presented in the chapter 7.

At last in the appendices are given some tables and auxiliary results of general contents (Appendix I "Tables of Relevant Integrals and Other Formulae", Appendix II "Boundary Value Problems for Analytic Functions", Appendix III "Fourier Transforms", Appendix IV "Non-singular Integral Equations").

The book is an extremely interesting combination of deep monograph and simple-written textbook. It is addressed to the specialists in mechanical problems (first of all to those who is interesting in the viscoelastic phenomena) as well as to the specialists in real and complex analysis, partial differential and integral equations.