# Mathematical Theory of Elastic Structures

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Book Preface

Elasticity theory is a classical discipline. The mathematical theory of elasticity in mechanics, especially the linearized theory, is quite mature, and is one of the foundations of several engineering sciences. In the last twenty years, there has been significant progress in several areas closely related to this classical field, this applies in particular to the following two areas.

First, progress has been made in numerical methods, especially the development of the finite element method. The finite element method, which was independently created and developed in different ways by scientists both in China and in the West, is a kind of systematic and modern numerical method for solving partial differential equations, especially elliptic equations. Experience has shown that the finite element method is efficient enough to solve problems in an extremely wide range of applications of elastic mechanics. In particular, the finite element method is very suitable for highly complicated problems. One of the authors (Feng) of this book had the good fortune to participate in the work of creating and establishing the theoretical basis of the finite element method. He thought in the early sixties that the method could be used to solve computational problems of solid mechanics by computers. Later practice justified and still continues to justify this point of view. The authors believe that it is now time to include the finite element method as an important part of the content of a textbook of modern elastic mechanics.

The second area is the development of composite elastic structural mechanics. In modern engineering practice we face not only the geometrically simple, elastic body, but also, more importantly, bodies composed of several elastic members, including those with different dimensions and with different properties, i.e., composite structures, such as aerospace structures, reactor structures, tall building structures, off-shore platform structures, underground structures.

The development of composite elastic mechanics has great significance in both practice and theory.

The development of composite elastic structural mechanics and the development of the finite element method, influence each other. To this date, however, the mathematical basis of composite structure theory is still not sufficiently rigorous and complete. At the same time, the theory of elliptic equations on composite manifolds developed in recent years by one of the authors (Feng), can be applied to place composite structure theory in a comparatively strict theoretical fra.mework, aiding the further development of both theory and application. Thus, it also makes it possible to include a mathematical theory of composite structures in this book on modern elastic structural mechanics.

In view of these considerations this book covers the following main topics: 1. The classical theory of linear elasticity 2. The mathematical theory of the composite elastic structures. Here the mathematical method proposed by the authors is presented. 3. The numerical method for solving elastic structural problems-the finite element method. The authors try to treat these three topics within the framework f a unified theory. It seems likely that the material covered in the book has not been published before. To this end, the whole book carries on a theoretical discussion on the mathematical basis of the principle of minimum potential energy, using displacement as the fundamental variable. The emphasis is on the accuracy and completeness of the mathematical formulation ofelastic structural problems. This is  another unique feature of this book. As is well known, the variational principle is one of the mathematical formulations of elasticity.

The variational principle can be used to deal with almost any elastic problem, although it is not the only possible method. As is also well known, although the principle of minimum potential energy based on displacement is not the only form of the variational principle of elasticity, it is sufficient for our needs and has the greatest generality. It is especially suitable for the problems with high complexity. We should point out that the finite element method, based on the mathematical form of the variational principle, especially the form of the variational principle based on displacement, has had great success in practice. Proceeding according to this guiding idea, we hope to achieve our goal rather economically.

The authors express their heartfelt thanks to Lin Qun, Wang Jin-xian, Van Chang-zhou, Fu Zi-zhi and others for their enthusiastic support and help during the process of writing, lecturing on and revising the book.