Numerous applications of rod structures in civil engineering, aircraft and spacecraft confirm the importance of the topic. On the other hand the majority of books on structural mechanics use some simplifying hypotheses; these hypotheses do not allow to consider some important effects, for instance the boundary layer effects near the points of junction of rods. So the question concerning the limits of applicability of structural mechanics hypotheses and the possibilities of their refinement arise. In this connection the asymptotic analysis of equations of mathematical physics, the equations of elasticity in rod structures (without these hypotheses and simplifying assumptions being imposed) is undertaken in the present book. Moreover, a lot of modern structures are made of composite materials and therefore the material of the rods is not homogeneous. This inhomogeneity of the material can generate some unexpected effects. These effects are analysed in this book. The methods of multi-scale modelling are presented by the homogenization, multi-level asymptotic analysis and the domain decomposition. These methods give an access to a new class of hybrid models combining macroscopic description with "microscopic zooms".

In this paper we shall discuss the construction of formal short-wave asymp totic solutions of problems of mathematical physics. The topic is very broad. It can somewhat conveniently be divided into three parts: 1. Finding the short-wave asymptotics of a rather narrow class of problems, which admit a solution in an explicit form, via formulas that represent this solution. 2. Finding formal asymptotic solutions of equations that describe wave processes by basing them on some ansatz or other. We explain what 2 means. Giving an ansatz is knowing how to give a formula for the desired asymptotic solution in the form of a series or some expression containing a series, where the analytic nature of the terms of these series is indicated up to functions and coefficients that are undetermined at the first stage of consideration. The second stage is to determine these functions and coefficients using a direct substitution of the ansatz in the equation, the boundary conditions and the initial conditions. Sometimes it is necessary to use different ansiitze in different domains, and in the overlapping parts of these domains the formal asymptotic solutions must be asymptotically equivalent (the method of matched asymptotic expansions). The basis for success in the search for formal asymptotic solutions is a suitable choice of ansiitze. The study of the asymptotics of explicit solutions of special model problems allows us to "surmise" what the correct ansiitze are for the general solution."

'Et moi, .... si j'avait su comment en revenir, One service mathematics has rendered the je n'y semis point all,,: human race. It has put common sense back Jules Verne where it belongs, on the topmost shelf next to the dusty canister labelled 'discarded non The series is divergent: therefore we may be sense'. able to do something with it. Eric T. Bell O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non !inearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.

In this paper we shall discuss the construction of formal short-wave asymp totic solutions of problems of mathematical physics. The topic is very broad. It can somewhat conveniently be divided into three parts: 1. Finding the short-wave asymptotics of a rather narrow class of problems, which admit a solution in an explicit form, via formulas that represent this solution. 2. Finding formal asymptotic solutions of equations that describe wave processes by basing them on some ansatz or other. We explain what 2 means. Giving an ansatz is knowing how to give a formula for the desired asymptotic solution in the form of a series or some expression containing a series, where the analytic nature of the terms of these series is indicated up to functions and coefficients that are undetermined at the first stage of consideration. The second stage is to determine these functions and coefficients using a direct substitution of the ansatz in the equation, the boundary conditions and the initial conditions. Sometimes it is necessary to use different ansiitze in different domains, and in the overlapping parts of these domains the formal asymptotic solutions must be asymptotically equivalent (the method of matched asymptotic expansions). The basis for success in the search for formal asymptotic solutions is a suitable choice of ansiitze. The study of the asymptotics of explicit solutions of special model problems allows us to "surmise" what the correct ansiitze are for the general solution."

'Et moi, .... si j'avait su comment en revenir, One service mathematics has rendered the je n'y semis point all,,: human race. It has put common sense back Jules Verne where it belongs, on the topmost shelf next to the dusty canister labelled 'discarded non The series is divergent: therefore we may be sense'. able to do something with it. Eric T. Bell O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non !inearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.

Numerous applications of rod structures in civil engineering, aircraft and spacecraft confirm the importance of the topic. On the other hand the majority of books on structural mechanics use some simplifying hypotheses; these hypotheses do not allow to consider some important effects, for instance the boundary layer effects near the points of junction of rods. So the question concerning the limits of applicability of structural mechanics hypotheses and the possibilities of their refinement arise. In this connection the asymptotic analysis of equations of mathematical physics, the equations of elasticity in rod structures (without these hypotheses and simplifying assumptions being imposed) is undertaken in the present book. Moreover, a lot of modern structures are made of composite materials and therefore the material of the rods is not homogeneous. This inhomogeneity of the material can generate some unexpected effects. These effects are analysed in this book. The methods of multi-scale modelling are presented by the homogenization, multi-level asymptotic analysis and the domain decomposition. These methods give an access to a new class of hybrid models combining macroscopic description with "microscopic zooms."