The power of the variational approach in mechanics of solids and structures follows from its versatility: the approach is used both as a universal tool for describing physical relationships and as a basis for qualitative methods of analysis [1]. And there is yet another important advantage inherent in the variational approach the latter is a crystal clear,
pure and unsophisticated source of ideas that help build and establish numerical techniques for mechanics. This circumstance was realized thoroughly and became especially important after the advanced numerical techniques of structural mechanics, first of all the finite element method,
had become a helpful tool of the modern engineer. Certainly, it took some time after pioneering works by Turner, Clough and Melos until the finite element method was understood as a numerical technique for solving mathematical physics problems; nowadays no one would attempt to question an eminent role played by the variational approach in the process of this understanding. It is a combination of intuitive engineer thinking and a thoroughly developed mathematical theory of variational calculus which gave the finite element method an impulse so strong that its influence can still be felt.
It would be too rash to say that there are few publications or books on the subject matter discussed in this book. It suffices to list such names of prominent mathematicians and mechanicians as Leibenzon [2], Mikhlin [3], Washizu [4], Rectoris [5], Rozin [6] the ellipsis shows that this list could be continued. So, a person can be thought of as over much confident (even arrogant) to follow the listed authors and other recognized personalities, who furrowed up their way through the ocean of variational principles in mechanics long ago, and to make the venture of writing another book on the same subject. The words said by English physicist H.Bondy come into mind in this regard [7]: A book is a wonderful thing, but, honestly, there are too many books; so the readers have a hard time, and the authors maybe harder.
However, every book written is worth its readers audience. Some of the books (by Mikhlin or Rectoris) are intentionally oriented at mathematical aspects ofvariational solutions, while others (by Leibenzon, Washizu, Rozin) have a clear and pure mechanical accent.
Obviously, when an author is in process of writing a book like this one, there is a difficult issue that constantly crosses the way: who are the potential readers of the book and how to keep to their interests. K. Rektorys [5] is totally right by stating that it is quite a fancy matter how to make a book useful for both the mathematician and the engineer because: the said reader categories often have opposite opinions about a book like this, so they advance totally different requirements to it, which cannot be satisfied at the same time. For example, one can hardly accommodate oneself to the wish of the mathematician and provide a book written very concisely where the theory would be evolved at a quick pace.
This is a matter of choice, and the choice in this book is unambiguous: The book is oriented at people who took (or intend to take) their engineering degree and also have a certain awareness of mathematics generally, within the curriculum of the present mathematical education given to students of engineering at universities.
Here follows a short list of skills and knowledge that the reader of the book should possess. The reader is believed to have acquaintance with a standard set of solid mechanics subjects included in the curriculum on engineering at any university strength of materials, structural mechanics, basics of elasticity theory and to know something about basic notions of the calculus of variations. The concepts like a functional, Euler equations for one, principal and natural boundary conditions, the Lagrangian multiplier rule for a functional's point of stationarity when additional conditions are present, some others are assumed to be known to the reader and understood by him. The reader is also supposed to have
mastered the basics of linear algebra; as for the calculi, the Gauss Ostrogradski formula is used everywhere in several variations without additional explanation. Also, the author believes the reader will not have any difficulties with the differentiation of a function with respect to its vector argument; this operation can be met in the book a few times.
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