A fundamental aspect of engineering is the desire to design artifacts that exploit materials to a maximum in terms of performance under working conditions and efficiency of manufacture. Such an activity demands an increasing understanding of the behavior of the artifact in its working environment together with an understanding of the mechanical processes occurring during manufacture.

To be able to achieve these goals it is likely that the engineer will need to consider the nonlinear characteristics associated possibly with the manufacturing process but certainly with the response to working load. Currently, analysis is most likely to involve a computer simulation of the behavior. Because of the availability of commercial finite element computer software, the opportunity for such nonlinear analysis is becoming increasingly realized.

Such a situation has an immediate educational implication because, for computer programs to be used sensibly and for the results to be interpreted wisely, it is essential that the users have some familiarity with the fundamentals of nonlinear continuum mechanics, nonlinear finite element formulations, and the solution techniques employed by the software. This book seeks to address this problem by providing a unified introduction to these three topics.

The style and content of the book obviously reflect the attributes and abilities of the authors. The authors have lectured on this material for a number of years to postgraduate classes, and the book has emerged from these courses. We hope that our complementary approaches to the topic will be in tune with the variety of backgrounds expected of our readers and, ultimately, that the book will provide a measure of enjoyment brought about by a greater understanding of what we regard as a fascinating subject.

The original edition of this book, titled Nonlinear Continuum Mechanics for Finite Element Analysis, published in 1997, contained a chapter on a FORTRAN program implementation of the material in the text, this being freely available at www.flagshyp.com. In 2008 a second edition included new chapters onelasto-plastic behavior of trusses and solids and retained the FORTRAN implementation. It was envisioned that an expanded third edition could include dynamics, although this would involve substantial additional material not suitable to the needs of all readers.

The measurement of lengths and areas has a long history dating back to the ancient cultures of Egypt and Greece. It has led to the development of measure theory as an area of modern mathematics, dealing with systematic approaches for measuring complicated objects based on available measurements for simpler objects. The aim of the present chapter is to provide the basic tools for acquiring working knowledge of the Lebesgue integral and its generalisations to abstract measure and integration theory. Rather than providing in this survey full proofs of the main theorems, we motivate/explain the main building blocks of the theory and we illustrate the flexibility of this powerful concept. For the results that are used but not proved we provide adequate references.

Historical considerations

A full appreciation of measure theory requires, we believe, some insight into the genesis of the subject. For this we relied on the material provided in Kupka (1986).

Ancient measure theory

In ancient Egypt, several hundreds of years B.C., the relatively flat ground was basically subdivided into rectangular plots whose area could be expressed in whole numbers of square cubits (the unit of length used in the time of the pharaohs). With regard to measure theory, the vastly greater refinement of Greek mathematics over that of the Egyptians is to a large extent attributed to the necessity of the ancient Greeks to ascertain, for agricultural needs, areas in an irregularly shaped hilly/mountaineous terrain. To obtain approximations of areas of more complicated regions, precise enough to satisfy the practical requirements of the time, the Greek mathematicians used the “paving stone technique”: the given region was paved as exactly as possible with variously chosen rectangular stones of known area, so that the unknown area is well approximated by the sum of the known areas of the individual nonoverlapping stones. The greater sophistication of Greek mathematicians came about because of their subsequent preference of triangles over rectangles. This is advantageous because any region bounded by straight lines can always be paved exactly by finitely many triangles and such regions provide accurate information about certain areas bounded by curves.

This being a textbook, it would seem appropriate to offer to the aspiring mathematician/prospective graduate student, from the author's perspective, some subjective points of view that could be worth pondering at the beginning of the journey towards advanced mathematics.

There is a strong parallel between mountain climbing and mathematics: both are very hard work, and both give their enthusiasts lots of pleasure in what they achieve, along with a view of the world that most people don't get. Both activities expose the usual fallacy of only looking at the immediate, obvious risks, and not taking a long-term view. Also, just like after completing the struggle to find a route to the summit, other possible routes may be discerned for the descent or for subsequent ascents; so in mathematics, once an approach is found, other mathematicians can usually find an alternative that is often much better and/or shorter. Indeed, one can push the comparison further. Learning about an unknown but well-established subject is to some extent similar to taking a trip to a vast unknown area that is considered to be a tourist attraction. There are guidebooks, maps and route signs. It is advisable to start by taking the main road and having a look at some of the recommended attractions. Once one is familiar with the basic layout, a favourable first impression might entice one to look more closely. Often old, practically forgotten trails have much to offer. In following the lure to explore uncharted territory by venturing off the trail one should, however, be wary of inadequate preparation. In the mathematical context, the more you know about the background and the more techniques you master, the vaster are the opportunities that you can try out. In particular, a proof is the outcome of the interaction between creative imagination and critical reasoning; further, rigorous formal proofs become really important in the advent of a crisis, e.g. a counter-intuitive behaviour, or when a paradox of some kind arises. For this reason, one ought to master the basic tools that are available, seeking also to get acquainted with recent technical advances.

The Fourier series representation of a function is the function space counterpart of the decomposition of an n-dimensional vector into components with respect to an orthonormal basis for ℝn or Cn. To deal with the underlying infinite-dimensional setting some acquaintance with functional analysis is required. The present chapter aims to present the basic functional analytic framework. We introduce some powerful tools that will be used in Chapter 4 to gain insight into the behaviour of Fourier series.

An overall perspective

Despite Fourier's optimistic program for representing an arbitrary function by a trigonometric series, the convergence issue for Fourier series is a delicate matter. The challenge is twofold: with regard to the appropriate choice of functions, as well as concerning the suitable notion of convergence (with links between these two issues). Two results illustrate the intricate nature of trigonometric series. In 1872 Weierstrass used a trigonometric series to provide an example of a continuous but nowhere differentiable function (see Exercise 34 of Chapter 2). On the other hand, it was for a long time supposed that every function f which is periodic and continuous possesses a Fourier series which converges at every point to the function. In 1873 du Bois-Reymond exhibited a continuous periodic function with a divergent Fourier series at a point. Nevertheless, the intuition of Fourier, that for a large class of functions one can define the Fourier coefficients and recover the function from the knowledge of its Fourier coefficients, was essentially correct. Riemann's integral, introduced to deal with trigonometric series, showed great promise at first, but proved in the end to be insufficient to cope with the complexity of the problem. The surprising examples mentioned above shattered the confidence of mathematicians that Fourier series represent a convenient tool, so that towards the end of the nineteenth century, the subject of Fourier series appeared to be intractable and research in this direction reached a standstill. The resurgence of the topic at the beginning of the twentieth century was enabled by Lebesgue's theory of integration. It turns out that square Lebesgue integrable functions represent the class of functions with which Fourier series are most naturally associated. This specific setting was the main source of basic concepts in functional analysis, such as completion, separability and orthogonality.

Fourier analysis is a central area of modern mathematics, comprising deep results that rely on advanced principles, as well as numerous aspects that require manipulative ingenuity. The power of the theory is illustrated by its wide applicability. Ideas originating in Fourier analysis permeate many essential developments of modern mathematics, bridging analysis with algebra and providing effective tools for an astonishing variety of applications. We list here an alphabetical sample of subjects, illustrating either areas of mathematics with strong links to Fourier analysis or real world applications of Fourier analysis, that are covered briefly in this textbook: acoustics, complex analysis, functional analysis/operator theory, group theory/representation theory, heat flow, hydrodynamics, image processing, medical imaging, number theory, optics and astronomy, partial differential equations, probability and statistics, quantum mechanics, signal processing.

While formal approaches to Fourier analysis can be informative, to appreciate the subject fully and to strengthen the ability to use it in other contexts, one has to acquire a certain mathematical sophistication that draws on measure theory and functional analysis. Lebesgue's integral and the concepts of Hilbert and Banach spaces are intimately connected to Fourier analysis, providing not only an adequate setting but also being useful in obtaining fundamental results, often with surprisingly little effort. A detailed presentation of measure theory and functional analysis would be out of place in an introductory textbook, but ignoring these topics would amount to a lamentable attempt to run before we have learned to walk. For this reason we outline in Chapters 2 and 3 the principal facts about the Lebesgue integral and Hilbert/Banach spaces, as needed later, emphasising and illustrating the relevant conceptual ideas. This should provide some essential intuition that must, nevertheless, be adequately backed up by analytic rigour, so that we present at least sketchy proofs, avoiding only the proofs that demand an advanced degree of technical versatility. The reader may take on faith the results stated without proof, but detailed references for further study are provided. This material offers, to the interested reader, a basis for a solid grounding in these aspects and has been “class-tested” to groups of graduate students at Lund University and at the University of Vienna (during the academic years 2002–2004 and 2014–2015, respectively). However, the material in Chapters 2 and 3 is not an integral part of a standard course.