Rods as Oriented Bodies

Rods or, more generally, thin bodies are conceptual abstractions of physical bodies having one or two dimensions much smaller than the third. The prototype of a thin body is a cylinder in which the diameter of the cross-section is much smaller than the length of its generators. If we have to determine the stresses and strains in an elastic cylinder under the action of given loads, the solution of equations of threedimensional elasticity appears as the canonical procedure. Sometimes, however, an alternative method of solution may be suggested by the slenderness of the cylinder. This leads us to attempt a solution which will take the form of a limit of the threedimensional solution when the diameter of the cylinder approaches zero, while the resultant of loads and the global stiffness of the body tend to finite limits. Obviously, the cylinder must not be conceived as a straight line which can only be lengthened, but account must also be taken of the bending and twisting of this line. This is achieved by regarding the cylinder in the limit as a line endowed with directions that may rotate independently of the deformation of the line to which they are attached. The directions, associated with the axis, are called directors and, a material curve, together with a collection of directors assigned to each particle, that is able to deform independently constitutes a Cosserat rod (Cosserat and Cosserat 1907).

In general, the present configuration of a Cosserat rod is specified by a vector function of the variable S (S1≤ S≤S2) and the time t:

This gives the position vectors of a point of a curve in three-dimensional Euclidean space at S, the arc length of the curve. The directors are a set of vectors dk(k= 1,…, K), which are also functions of (S, t).

We require that, under the change of frame

the vector p0 transforms according to the rule

and the directors transform according to the rule

This means that, while p0 behaves like a position vector under a change of frame, the dk terms behave like differences between position vectors.

Density, Motion, and Temperature

Mechanics studies the conditions for the motion and equilibrium of natural objects. In order to treat these in a sufficiently general form, mechanics introduces the notion of the body, which is a mathematical concept designed to give an abstract representation of the most important properties of these physical objects in order to describe their mechanical behavior. By the term “body” we mean a regularly open set $ in some topological space. The elements of a body are called particles, or substantial points to avoid confusion with the term “particle” as used in physics (Truesdell 1991, Ch. I, p.3). Bodies are available in their configurations, which are the regions that they may occupy in Euclidean space at some time. It is commonly assumed that configurations are regularly open sets in Euclidean space and there is a one-to-one mapping between the particles of the body and a possible configuration. It is often convenient to select one particular configuration K and to identify each particle of K by its position in this configuration, called the reference configuration. If, in the reference configuration, we choose Cartesian coordinates, the particle can be designated XA or X, Y, Z.

At the time t varies the particles change their positions in space, and consequently the position of a particle XA at the instant t is characterized by a vector function xi or x, y, z, of the form

which is called the motion of the body. The motion determines the shape of the body at each instant.

The inertia of a particle is determined by a scalar function p = p(XA, t), called the density; and the hotness of a particle is determined by another scalar function Ө = Ө(XA, t), called temperature.

Motion, density, and temperature are regarded as primitive concepts: that is, they need not be defined in terms of other known quantities, and must be measurable, at least in principle. The aim of mechanics is to determine the fields associated with Xi, p, and Ө. However, these quantities are not completely unrestricted, because both density and temperature must be positive-valued functions, and the functions // must be continuous and invertible so as to exclude the possibility of two particles assuming the same position.

The theory of water waves has been a source of intriguing – and often difficult – mathematical problems for at least 150 years. Virtually every classical mathematical technique appears somewhere within its confines; in addition, linear problems provide a useful exemplar for simple descriptions of wave propagation, with nonlinearity adding an important level of complexity. It is, perhaps, the most readily accessible branch of applied mathematics, which is the first step beyond classical particle mechanics. It embodies the equations of fluid mechanics, the concepts of wave propagation, and the critically important rôle of boundary conditions. Furthermore, the results of a calculation provide a description that can be tested whenever an expanse of water is to hand: a river or pond, the ocean, or simply the household bath or sink. Indeed, the driving force for many workers who study water waves is to obtain information that will help to tame this most beautiful, and sometimes destructive, aspect of nature. (Perhaps ‘to tame’ is far too bold an ambition: at least to try to make best use of our knowledge in the design of man-made structures.) Here, though, we shall – without apology – restrict our discussion to the many and varied aspects of water-wave theory that are essentially mathematical. Such studies provide an excellent vehicle for the introduction of the modern approach to applied mathematics: complete governing equations; nondimensionalisation and scaling; rational approximation; solution; interpretation. This will be the type of systematic approach that is adopted throughout this text.

Before we commence our presentation of the theory of water waves, we require a firm and precise base from which to start. This must be, at the very least, a statement of the relevant governing equations and boundary conditions. However, it is more satisfactory, we believe, to provide some background to these equations, albeit within the confines of an introductory and relatively brief chapter. The intention is therefore to present a derivation of the equations for inviscid fluid mechanics (Euler's equation and the equation of mass conservation) and a few of their properties. (The corresponding equations for a viscous fluid – primarily the Navier–Stokes equation – appear in Appendix A.) Coupled to these general equations is the set of boundary (and initial) conditions which select the water-wave problem from all other possible solutions of the equations. Of particular importance, as we shall see, are the conditions that define and describe the surface of the fluid; these include the kinematic condition and the rôles of pressure and surface tension. Some rather general consequences of coupling the equations and boundary conditions will also be mentioned.

Once we have available the complete prescription of the water-wave problem, based on a particular model (such as for inviscid flow), we may ‘normalise’ in any manner that is appropriate.

The Equations of Linear Theory

A planar plate, like a planar membrane, is a solid layer bounded by two parallel planes and by a cylinder that intercepts the planes at right angles. The plane midway between the faces is called the “middle plane.” We can draw any cylindrical surface to cut the middle plane in a curve s. The particular cylinder that coincides with the lateral surface of the plate is called the “edge.” We choose rectangular axes x and y placed in the middle plane and draw the z axis at right angles to this plane, so that the upper and lower plane faces have the equations z = h and z = —h, respectively, where 2h represents the thickness of the plate. We draw the outer normal v to s, lying in the middle plane, the unit vector s tangentially to s, and the unit vector n normal to the middle plane (Love 1927, Art. 296). These three unit vectors are oriented such that they form a right-handed orthonormal basis (v, s, n).

Let δs be an arc element of the curve s, and let us suppose that the generators of, drawn through the extremities of δs, mark out on a piece of cylindrical surface of base δs and height 2h, called the area of this element δA. The tractions across the area δA are statically equivalent to a force applied at the centroid of δA together with a couple. We resolve, as we did for membranes, this force into three components 8T, δS, and δN directed along v, s, and n, respectively, and resolve the couple into three components δH, δG, and δK directed along the same vectors. We then take the limits of the ratios δT/δs,…, δK/δs as δs tends to zero. We postulate that the limit of δK/δs is zero, but all other limits are finite The limits of δT/δs, δS/δs, and 8N/ds are denoted by T S, and N, respectively, and those of SH/δs, and δG/δs by H and G. T is a tension, S and N are shearing forces tangential and normal to the middle plane, G is a bending couple, and H is a torsional couple.

What does it mean to solve a problem in classical elasticity? The question may appear trivial, but, if we ask scholars working in the field, we receive surprisingly different answers. Let us assume, in order to make the subject more explicit, that the problem concerns the impact between two elastic bodies. For an experimental physicist solving the problem means interpreting those crucial experiments that make it possible to decide which are the important variables in the phenomenon: in this specific case, the densities and elastic moduli of the materials are important, but the temperature and atmospheric pressure, for instance, are not. A theoretical physicist will say instead that the solution consists in formulating the general equations of the problem, having inserted all the significant variables. For a mathematician it will be obvious that solving the problem means finding an existence, uniqueness, and possibly regularity, theorem for the equations of elastic impact. Yet another answer will be given by an engineer, who will require an explicit formula giving the stress components within the two bodies at each point and at each instant.

Confronted with such a variety of answers, a typical student feels disoriented, being immediately aware of the basic ambiguity in the way in which the question itself has been posed. Ludwig Wittgenstein would say in explanation that the confusion arises from the vague use of the verb “to solve.” For the same word has been used in different contexts with different meanings.

Though dissimilar, the four answers have a common characteristic. They represent four attempts at describing the same phenomenon by abstraction, setting aside the unessential details, with the purpose not merely of illustrating but also of predicting. We say that they propose four models for the elastic impact. At this point we immediately ask whether there is a rational criterion for deciding which model is preferable, provided that all satisfy the three necessary requisites of being realistic, logically coherent, and simple. It is evident that a model must not be in obvious conflict with the physical data, nor must it be self-contradictory or too complicated. However, unfortunately, there is no incontrovertible way of establishing that one model is better than another. Setting up a model means creating conceptual conditions suitable for posing a particular question about the problem.

In Chapter 2 we presented some classical ideas in the theory of water waves. One particular concept that we introduced was the phenomenon of a balance between nonlinearity and dispersion, leading to the existence of the solitary wave, for example. Further, under suitable assumptions, this wave can be approximated by the sech2 function, which is an exact solution of the Korteweg–de Vries (KdV) equation; see Section 2.9.1. We shall now use this result as the starting point for a discussion of the equations, and of the properties of corresponding solutions, that arise when we invoke the assumptions of small amplitude and long wave-length. In the modern theories of nonlinear wave propagation – and certainly not restricted only to water waves – this has proved to be an exceptionally fruitful area of study.

The results that have been obtained, and the mathematical techniques that have been developed, have led to altogether novel, important and deep concepts in the theory of wave propagation. Starting from the general method of solution for the initial value problem for the KdV equation, a vast arena of equations, solutions and mathematical ideas has evolved. At the heart of this panoply is the soliton, which has caused much excitement in the mathematical and physical communities over the last 30 years or so. It is our intention to describe some of these results, and their relevance to the theory of water waves, where, indeed, they first arose.

The study of problems in water-wave theory, particularly under the umbrella of the linear approximation, goes back over 150 years. In the intervening time, many different problems – and extensions of standard problems – have been discussed by many authors. In a text such as ours, it is necessary to make a selection from this body of classical work; we cannot hope to describe all the various problems, nor all the subtle variants of standard problems. Our intention is, of course, to include the simplest and most fundamental results (such as, for example, the speed of waves over constant depth and the description of particle paths), but otherwise we choose those topics which contain some interesting and relevant mathematics. However, since we shall not present all that some readers might, perhaps, expect or prefer, we endeavour to remedy this by introducing additional examples through the exercises. The sufficiently dedicated reader is therefore directed to the exercises, particularly if a broader spectrum of water-wave theory is desired.

The material here is presented under two separate headings. The first is linear problems, where, apart from the elementary aspects mentioned above, we single out those topics that are attractive and which will prove relevant to some of our later discussions. Thus we describe waves on sloping beaches, as well as the phenomenon of edge waves.

In the earlier chapters we have described the mathematical background – and the mathematical details – of many classical linear and nonlinear water-wave phenomena. In addition, in the later chapters, we have presented many of the important and modern ideas that connect various aspects of soliton theory with the mathematical theory of water waves. However, much that is significant in the practical application of theories to real water waves – turbulence, random depth variations, wind shear, and much else – has been omitted. There are two reasons for this: first, most of these features are quite beyond the scope of an introductory text, and, second, the modelling of these types of phenomena follows a less systematic and well-understood path. Of course, that is not meant to imply that these approaches are unimportant; such studies have received much attention, and with good reason since they are essential in the design of man-made structures and in our endeavours to control nature.

What we have attempted here, in a manner that we hope makes the mathematical ideas transparent, is a description of some of the current approaches to the theory of water waves. To this end we have moved from the simplest models of wave propagation over stationary water of constant depth (sometimes including the effects of surface tension), to more involved problems (for example, with ‘shear’ or variable depth), but then only for gravity waves.

Geometrical Properties of Rods

From the beginnings of human civilization, rods have been used in making rudimentary tools and weapons. In the theory of the elasticity, rods have been used to model a range of physical structures such as beams, columns, and shafts, the characteristics of which is slenderness that is of having a longitudinal dimension much greater than either of the transverse dimensions. The term “bar” is sometimes used synonymously with rod, although in general speech it need not imply that one dimension is very much greater than others. Occasionally, when modeling a physical object, it is possible to identify it immediately as a rod; for example, the mast of a boat. On other occasions a physical structure can be regarded as an assembly of rods; for example, certain buildings or bridges. Rods are important because they offer the simplest but nontrivial model to which the theory of elasticity can be applied and verified experimentally. The earliest theories of rods formulated at the beginning of the seventeenth century arose as a consequence of the research of Beeckman and Mersenne (Truesdell 1960, Prologue, Sec. 3.4), but the most significant progress made in the theory was due to James Bernoulli, Euler, Saint-Venant, Kirchhoff, and Michell. Implicit in using rod models is the facility of being able to work with a single independent variable representing the arc measured along a certain curve. In slender bodies this avoids the mathematical obstacles that arise from having the three independent variables of general continuum theory.

This then requires the construction of a rational scheme for approximating the system of field equations of general continua by a system containing one spatial independent variable and, ultimately, the time. However, in order to be acceptable, the approximation procedure must explain at each stage the relationship between the approximate solutions and those of the original three-dimensional problem; in addition, it must contain, at least in principle, the method of estimating the approximation error; finally, the approximate quantities must have a significant mechanical interpretation in the form of stress averages over certain sections of the body.