This book is an introduction to the theories of bifurcation and chaos. It treats the solution of nonlinear equations, especially difference and ordinary differential equations, as a parameter varies. This is a fascinating subject of great power and depth, which reveals many surprises. It requires the use of diverse parts of mathematics – analytic, geometrical, numerical and probabilistic ideas – as well as computation. It covers fashionable topics such as symmetry breaking, singularity theory (which used to be commonly called catastrophe theory), pattern selection, chaos, predictability, fractals and Mandelbrot sets. But it is more than a fashionable subject, because it is a fundamental part of the theory of difference and differential equations and so destined to endure. Also the theory of nonlinear systems is applied to diverse and countless problems in all the natural and social sciences, and touches on some problems of philosophy.

The writing of the book evolved with lecture courses I have given to final-year undergraduates at the University of Bristol and to graduates at the University of Washington and Florida State University in the USA over the last decade. I hope that others will enjoy this book as our students have enjoyed the courses.

Most of the equations treated in traditional mathematics courses at university are linear. These linear algebraic, ordinary differential, partial differential and integral equations are solved by various powerful methods, which essentially depend upon the principle of superposition.

In setting down a general continuum mechanics description of finite deformation processes in metal crystals that takes into account both lattice straining and gross crystallographic slip, it is useful to begin with an assessment of the minimum physical scale at which such a description has meaning. The following discussion, based upon similar discussions in Havner (1973a,b; 1982a), is pertinent to the determination of that minimum scale.

As seen by an observer resolving distances to 10−3 mm, the deformation of a crystal grain (of typical dimensions 10−3−10−2 cm within a finitely strained polycrystalline metal) may be considered relatively smooth. At this level of observation, which for convenience we shall call microscopic, one can just distinguish between slip lines on crystal faces after extensive distortion of a specimen. In contrast, a submicroscopic observer resolving distances to 10−5 mm (the order of 100 atomic spacings) is aware of highly discontinuous displacements within crystals. The microscopic observer's slip lines appear to the submicroscopic observer as slip bands of order 10−4 mm thickness, containing numerous glide lamellae between which amounts of slip as great as 103 lattice spacings have occurred, as first reported by Heidenreich (1949); hence a continuum perspective at this second level would seem untenable. Accordingly, we adopt a continuum model in which a material “point” has physical dimensions of order 10−3 mm. This is greater than 103 lattice spacings yet at least an order of magnitude smaller than typical grain sizes in polycrystalline metals.

In this chapter we return to the general theoretical framework of Chapter 3 and extend it to the analysis of characteristics of overall response of macroscopically uniform polycrystalline solids. The objective is the presentation of a rigorous theoretical connection between single-crystal elastoplasticity and macroscopic crystalline aggregate behavior. The development is based upon the original analysis of Hill (1972) and other basic contributions in Hill & Rice (1973), Havner (1974, 1982a, 1986), and Hill (1984, 1985). Central to an understanding of the crystal-to-aggregate transition is the well-known “averaging theorem” introduced by Bishop & Hill (1951a) but only given its final form and initial proof at finite strain in Hill's (1972) seminal work.

Crystalline Aggregate Model: The Averaging Theorem

At the beginning of Chapter 3, the scale of a crystal material point in a continuum model was defined to have linear dimension of order 10−3 mm: greater than 103 lattice spacings but at least an order of magnitude smaller than normal grain sizes in polycrystalline metals. Consider now the choice of physical size of a representative “macroelement” that defines a continuum point at the level of ordinary stress and strain analysis (that is, in structural and mechanical components or materials-forming operations.)

The wall thickness of thin-walled metal tubes used in combined stress tests (say, axial loading and torsion) often is in the range 1−2 mm and 10 to 30 grains (see, for example, Mair & Pugh (1964) or Ronay (1968)).

Turning from the rigorous theoretical analysis of Chapter 6 to the subject of (and literature on) the calculation of approximate polycrystalline aggregate models at finite strain, one can identify three prominent themes: the prediction of (i) macroscopic axial-stress–strain curves, (ii) macroscopic yield loci, and (iii) the evolution of textures (that is, the development of preferred crystal orientations in initially statistically isotropic aggregates). The topic of polycrystal calculations is vast and complex, warranting a monograph on its own (and by other hands). In this closing chapter of the present work I primarily shall review selected papers (acknowledging others) from among those contributions that are particularly significant or noteworthy in the more than 50 years' history of the subject.

The Classic Theories of Taylor, Bishop, and Hill

Near the beginning of G. I. Taylor's (1938a) May Lecture to the Institute of Metals is the following splendid sentence. “I must begin by making the confession that I am not a metallurgist; I may say, however, that I have had the advantage of help from, and collaboration with, members of your Institute, whose names are a sure guarantee that the metals I have used were all right, even if my theories about them are all wrong.” More than anything else this statement reflects Taylor's irrepressible humor, for of course his theories were not “all wrong.”