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.”

The scientific study of finite distortion of cubic metal crystals was formally inaugurated by G. I. Taylor's 1923 Bakerian Lecture to the Royal Society (Taylor & Elam 1923), and the development of a rational mechanics of finite plastic deformation of crystalline solids may be said to have begun. The remarkable feature of this pioneering work (and of all Taylor's subsequent experimental investigations of f.c.c. and b.c.c. crystals) is that, in addition to the use of X-ray analysis to determine the changing orientation of the crystal atomic lattice, external measurements sufficient to completely define the uniform distortion of the crystal specimen were made at each stage of the test. All material directions that remained unchanged in length were then established by exact geometric analysis.

Taylor's approach is to be distinguished from that of Mark, Polanyi & Schmid (1923), whose experimental study of crystals of hexagonal structure is of comparable historical importance. They pulled single crystal wires of zinc and assumed that slip lines (or bands) on the specimen surface were traces of a family of planes of single slip, which were then shown to coincide with a crystal plane. Taylor (1926), discussing such methods in general, rather amusingly remarked: “They depend in fact on knowing the form which the answer will take before starting to solve the problem.”

Taylor & Elam (1923) firmly established two experimental laws of fundamental and lasting value for the foundations of crystal mechanics that seven decades of experimentation on f.c.c. crystals have only served to reinforce.

As remarked in Chapter 1, v. Goler & Sachs (1927) derived equations for equal double slip in f.c.c. crystals in tension with the loading axis on a symmetry line, and Taylor (1927a) derived comparable equations in compression (and also gave the equation of the unstretched cone for f.c.c. crystals). Equations applicable to any crystal class for these same symmetry conditions and for both tension and compression were developed by Bowen & Christian (1965), who presented formulas for various specific combinations of slip systems in f.c.c. and b.c.c. crystals. A general equation for the deformation gradient in (proportional) double slip of arbitrary relative amounts was first given in the work of Chin, Thurston, & Nesbitt (1966) mentioned previously. They carried the analysis no further, however, and applied the equation only to cases of equal, symmetric double slip in f.c.c. crystals.

Apparently the first explicit equations for rotation and stretch of a crystal material line in arbitrary (proportional) double slip in f.c.c. and b.c.c. crystals were developed by Shalaby & Havner (1978) (independently of the work of Chin, Thurston, & Nesbitt (1966)). The equations were illustrated for various nonsymmetric axis positions and relative amounts of slip. General equations for material line and areal vectors and both the finite deformation gradient and its inverse in arbitrary (proportional) double slip were derived in Havner (1979). Here we shall follow this last approach to the analysis of double slip in crystals.

Rodney Hill wrote his preface to The Mathematical Theory of Plasticity 41 years ago this month. As a reader of the present monograph likely knows, that classic work dealt with the macroscopic theory of metal plasticity and its applications as the subject stood at mid-century; and Hill only briefly (albeit superbly) discussed in his introductory chapter the physical background of the plastic properties of crystals and polycrystalline aggregates. The same year, however, saw publication of the English translation of an earlier (1935) classic specifically concerned with that background, Kristallplastizität by E. Schmid and W. Boas. Not entirely coincidentally, both Rodney Hill and this translation were associated with the Cavendish Laboratory, Cambridge, during the period immediately following World War II.

Each of these books when first published was in many respects a treatise on its respective subject, but there was no contemporary work which integrated these fields. Today, I doubt a comprehensive treatise could be written on all that has transpired both in the development of mathematical theory and in the experimental study of plastic behavior of crystalline materials during this century (or even since 1950). Accordingly, in planning and carrying out the writing of the present work, I decided to restrict its scope to those aspects of the broad subject of crystalline plasticity that have particularly interested me and that I have contributed to or at least seriously studied during the years since 1968.

The following notes have been the basis for an introductory course in atmospheric dynamics which has been taught at Harvard and M.I.T. for the past seven years. The individual chapters were initially intended to correspond to ninety-minute lectures, but, as a result of innumerable changes based on practical demands, the author's predilections, and so forth, this is no longer the case. Some chapters have been reduced while others have been greatly expanded.

Many of the topics covered in these notes may seem somewhat advanced for an introductory course. There are several reasons why they have been included (and why other more traditional topics have been neglected). First, I feel that many topics are considered ‘advanced’ or ‘elementary’ for historical reasons and not because they are particularly difficult or easy. The topics I have included do not call on especially advanced mathematical skills; they are, moreover, topics which I believe to be basic to the contemporary study of atmospheric dynamics (wave-mean flow interaction, for example). Second, the students who have taken this course at Harvard and M.I.T. have usually had good backgrounds in undergraduate physics and applied mathematics. In many cases, moreover, the students have had some earlier introduction to fluid mechanics. That said, the material in these notes is in many instances conceptually demanding, and students should not feel discouraged if they have difficulty following it. Some topics may require considerable effort.

Because of the background of most of the students, and, in particular, because most of the students have already been introduced to the equations of motion, I have adopted a somewhat unusual approach to the derivation of the equations in Chapter 6.

These notes end (as do many courses) rather abruptly. I hope to leave the reader with the sense that he or she has learned a lot. But, I would hardly wish to disguise the fact that we have barely begun the exploration of atmospheric dynamics. The nonlinear evolution and possible equilibration of instabilities which should give us the wave and flux magnitudes has only been touched on – both in these notes and in current research. A major current approach to questions of the general circulation – namely, the use of large numerical computer simulations – has not even been discussed. Areas whose impact on large-scale dynamics is almost certainly major – like boundary layer turbulence and convective cloud activity – have likewise been only peripherally dealt with in these notes. Although we have come quite far in improving our understanding of many of the phenomena and features described in Chapter 5, we are still far from a satisfactory state, and, as we have earlier noted, there exists a world of important and challenging phenomema whose scales are smaller than those discussed in Chapter 5: hurricanes, fronts, thunderstorms, squalls, to name a few. Even those topics that we have dealt with in some detail have hardly been dealt with in any measure of completeness. Whole books (in most cases several) and countless articles have been devoted specifically to instability, wave theory, the general circulation, and even tides.

The shear scope of problems which fall under the general rubric of atmospheric dynamics is so great as to lead, unfortunately, but inevitably, to a high degree of specialization.