Introduction

In Chapter 4, we derived various approximate models for thin or slender elastic configurations such as rods and plates. These models were obtained using net force and moment balances combined with ad hoc constitutive relations, for example between the bending moments and the curvatures. In this chapter, we show how such models may be derived systematically from the underlying continuum equations and boundary conditions. We concentrate on a few canonical models for plates, beams, rods and shells. Each of these models is important in its own right, and their derivation illustrates the tools that are widely useful for analysing more general thin structures.

The basic idea is to exploit the slenderness of the geometry so as to simplify the equations of elasticity asymptotically. This process is made systematic by first non-dimensionalising the equations, so that all the variables are dimensionless and of order one. This highlights the small slenderness parameter ε = h/L, where h is a typical thickness and L a typical length of the elastic body. A simplified system of equations is then obtained by carefully taking the limit ε → 0. Typically, the solution is sought as an asymptotic expansion in powers of the small parameter ε, and the techniques demonstrated here fall within the general theories of asymptotic expansions and perturbation methods. Kevorkian & Cole (1981), Hinch (1991) and Bender & Orszag (1978) provide very good general expositions of these methods.

Understanding of turbulence is one of the most challenging problems of modern mathematical and theoretical physics. Turbulence can be described as a chaotic, highly non-equilibrium state of a non-linear physical system. Defined this way, turbulence embraces a broader class of examples than the chaotic Navier-Stokes flow, – a system for which the concept of turbulence was originally introduced and developed. In particular, turbulence appears as far-from-equilibrium states in plasmas, solids, Bose-Einstein condensates and even in nonlinear optics. The characteristic feature of turbulence is the presence of significant energy exchange between many degrees of freedom, which renders most attempts of perturbative treatment of the problem useless. Still mathematicians, physicists and engineers invest a large effort in understanding of turbulence due to the unprecedented importance of turbulence both for theoretical and applied science.

Thanks to fundamental works of Richardson, Taylor, Kolmogorov and Obukhov, we have a phenomenology of turbulent cascades leading to the famous Kolmogorov spectrum. This theory has been successfully applied to a wide range of turbulent systems. What is still lacking, however, is a fundamental theory of turbulence, which would allow both finding a rigorous mathematical foundation for the Kolmogorov theory and understanding of its limitations.

Introduction

This chapter concerns steady state problems in linear elasticity. This topic may appear to be the simplest in the whole of solid mechanics, but we will find that it offers many interesting mathematical challenges. Moreover, the material presented in this chapter will provide crucial underpinning to the more general theories of later chapters.

We will begin by listing some very simple explicit solutions which give valuable intuition concerning the role of the elastic moduli introduced in Chapter 1. Our first application of practical importance is elastic torsion, which concerns the twisting of an elastic bar. This leads to a class of exact solutions of the Navier equation in terms of solutions of Laplace's equation in two dimensions. However, as distinct from the use of Laplace's equation in, say, hydrodynamics or electromagnetism, the dependent variable is the displacement, which has a direct physical interpretation, rather than a potential, which does not. This means we have to be especially careful to ensure that the solution is single-valued in situations involving multiply-connected bars.

These remarks remain important when we move on to another class of two-dimensional problems called plane strain problems. These have even more general practical relevance but involve the biharmonic equation. This equation, which will be seen to be ubiquitous in linear elastostatics, poses significant extra difficulties as compared to Laplace's equation. In particular, we will find that it is much more difficult to construct explicit solutions using, for example, the method of separation of variables.

Introduction

This chapter concerns simple unsteady problems in linear elasticity. As noted in Section 1.10, the unsteady Navier equation (1.7.8) bears some similarity to the familiar scalar wave equation governing small transverse displacements of an elastic string or membrane. We therefore start by reviewing the main properties of this equation and some useful solution techniques. This allows us to introduce, in a simple context, important concepts such as normal modes, plane waves and characteristics that underpin most problems in linear elastodynamics.

In contrast with the classical scalar wave equation, the Navier equation is a vector wave equation, and this introduces many interesting new properties. The first that we will encounter is that the Navier equation in an infinite medium supports two distinct kinds of plane waves which propagate at two different speeds. These are known as P-waves and S-waves, and correspond to compressional and shearing oscillations of the medium respectively. Considered individually, both P- and S-waves behave very much like waves as modelled by the scalar wave equation (1.10.9). In practice, though, they very rarely exist in isolation since any boundaries present inevitably convert P-waves into S-waves and vice versa. We will illustrate this phenomenon of mode conversion in Section 3.2.5 by considering the reflection of waves at a plane boundary.

In two-dimensional and axisymmetric problems, we found in Chapter 2 that the steady Navier equation may be transformed into a single biharmonic equation by introducing a suitable stress function. In Sections 3.3 and 3.4 we will find that the same approach often works even for unsteady problems, and pays dividends when we come to analyse normal modes in cylinders and spheres.

Although solid mechanics is a vitally important branch of applied mechanics, it is often less popular, at least among students, than its close relative, fluid mechanics. Several reasons can be advanced for this disparity, such as the prevalence of tensors in models for solids or the especial difficulty of handling nonlinearity. Perhaps the most daunting prospect for the student is the multitude of different behaviours that can occur and cause elementary theories of elasticity to become irrelevant in practice. Examples include fracture, buckling and plasticity, and these pose intellectual challenges in solid mechanics that are every bit as fascinating as concepts like flight, shock waves and turbulence in fluid dynamics. Our principal objective in this book is to demonstrate this fact to undergraduate and beginning graduate students.

We aim to give the subject as wide an accessibility as possible to mathematically- minded students and to emphasise the interesting mathematical issues that it raises. We do this by relating the theory to practical applications where surprising phenomena occur and where innovative mathematical methods are needed.

Our layout is essentially pragmatic. Although more advanced texts in solid mechanics often begin with quite general theories founded on basic mechanical and thermodynamic principles, we start from the very simplest models, based on elementary observations in engineering and physics, and build our way towards models that are the basis for current applied research in solid mechanics. Hence, we begin by deriving the basic Navier equations of linear elasticity, before illustrating the mathematical techniques that allow these equations to be solved in many different practically relevant situations, both static and dynamic.

The emphasis of this short course is on fundamental properties of developed turbulence, weak and strong. We shall be focused on the degree of universality and symmetries of the turbulent state. We shall see, in particular, which symmetries remain broken even when the symmetry-breaking factor goes to zero, and which symmetries, on the contrary, emerge in the state of developed turbulence.

Introduction

Turba is Latin for crowd and “turbulence” initially meant the disordered movements of large groups of people. Leonardo da Vinci was probably the first to apply the term to the random motion of fluids. In 20th century, the notion has been generalized to embrace far-from-equilibrium states in solids and plasma. We now define turbulence as a state of a physical system with many interacting degrees of freedom deviated far from equilibrium. This state is irregular both in time and in space and is accompanied by dissipation.

We consider here developed turbulence when the scale of the externally excited motions deviate substantially from the scales of the effectively dissipated ones. When fluid motion is excited on the scale L with the typical velocity V, developed turbulence takes place when the Reynolds number is large: Re = V L/v ≫ 1. Here v is the kinematic viscosity. At large Re, flow perturbations produced at the scale L have their viscous dissipation small compared to the nonlinear effects. Nonlinearity produces motions of smaller and smaller scales until viscous dissipation stops this at a scale much smaller than L so that there is a wide (so-called inertial) interval of scales where viscosity is negligible and nonlinearity dominates.

Introduction

Fracture describes the behaviour of thin cracks, like the one illustrated in Figure 7.1, in an otherwise elastic material. The crack itself is a thin void, whose faces are usually assumed to be stress-free. When a solid containing a crack is stressed, we will find that the stress is localised near the crack tip, becoming singular if the tip is sharp. We will see that the strength of the singularity can be characterised by a stress intensity factor, which depends on the size of the crack and on the applied stress. This factor determines the likelihood that a crack will grow and, therefore, that the solid will fail.

On the other hand, contact refers to the class of problems in which two elastic solid bodies are brought into contact with each other, as illustrated in Figure 7.2. When the material properties are the same in the two bodies, the geometrical configuration near the edge of the contact region is apparently similar to that of fracture, with voids now outside the contact set, that is, the set of points at which the two solids are in contact. The mathematical setup of such problems does therefore have some similarity with fracture, but, in the steady state, there is one crucial difference. Whereas much of the study of fracture concerns cracks of prescribed length, in contact problems the contact set itself is often unknown in advance. They are thus known as free boundary problems, that is problems whose geometry must be determined along with the solution.

Introduction

So far in this book we have been considering linear elasticity only for very simple geometries such as cylinders, spheres and half-spaces. In this chapter, we will consider more general solids under the restriction that they are thin and the equations of elasticity can consequently be simplified. A familiar example that we have already encountered is the wave equation governing the transverse displacements of a thin elastic string, and we will revisit this model below in Section 4.3.

A string is characterised by its inability to withstand any appreciable shear stress, so its only internal force is a tension acting in the tangential direction. Similarly, a membrane is a thin, nearly two-dimensional structure, such as the skin of a drum, which supports only in-plane tensions. A bar, on the other hand, is a nearly one-dimensional solid that can be subject to either tension or compression. However, many thin elastic bodies also have an appreciable bending stiffness and therefore admit internal shear stress as well as tension. A familiar example is a flexible ruler, which clearly resists bending while deforming transversely in two dimensions, and is known as a beam. A thin, nearly one-dimensional object which can bend in both transverse directions, such as a curtain rod or a strand of hair, will be referred to as a rod. On the other hand, a nearly planar elastic structure with significant bending stiffness, for example a pane of glass or a stiff piece of paper, is called a plate. Finally, a shell is a thin, nearly two-dimensional elastic body which is not initially planar, for example a ping-pong ball or the curved panel of a car.

Introduction

In everyday life we regularly encounter physical phenomena that apparently vary continuously in space and time. Examples are the bending of a paper clip, the flow of water or the propagation of sound or light waves. Such phenomena can be described mathematically, to lowest order, by a continuum model, and this book will be concerned with that class of continuum models that describes solids. Hence, at least to begin with, we will avoid all consideration of the “atomistic” structure of solids, even though these ideas lead to great practical insight and also to some beautiful mathematics. When we refer to a solid “particle”, we will be thinking of a very small region of matter but one whose dimension is nonetheless much greater than an atomic spacing.

For our purposes, the diagnostic feature of a solid is the way in which it responds to an applied system of forces and moments. There is no hard-andfast rule about this but, for most of this book, we will say that a continuum is a solid when the response consists of displacements distributed through the material. In other words, the material starts at some reference state, from which it is displaced by a distance that depends on the applied forces. This is in contrast with a fluid, which has no special rest state and responds to forces via a velocity distribution. Our modelling philosophy is straightforward. We take the most fundamental pieces of experimental evidence, for example Hooke's law, and use mathematical ideas to combine this evidence with the basic laws of mechanics to construct a model that describes the elastic deformation of a continuous solid.

Introduction

We will now mention a wide range of important solid mechanics phenomena that warrant discussion even in a mathematically-oriented book but have been ignored or scarcely mentioned so far.

A phenomenon of interest in industries ranging from food to glass is that of viscoelasticity. Here the molecular structure of the material is such that it flows under any applied stress, no matter how small. As we will see in Section 9.2, viscoelastic materials are quite unlike plastic materials. Their behaviour depends critically on the time-scale of the observer; for example when a ball of “silly putty” or suitably dilute custard impacts a wall, it rebounds almost elastically but, if left on a table, it will slowly spread horizontally under the action of gravity.

Yet another attribute of solids is well-known to be of great practical importance in the kitchen, when glass can be observed to break in hot water, and in the railway industry, where track can distort in high summer. This is thermoelasticity, which describes the response of elastic solids to temperature variations. To model this response ab initio, even at a macroscopic scale, requires more thermodynamics than is appropriate for this text, and in Section 9.3 we will only present the simplest ad hoc model that can give useful realistic results.

Even the above list of diverse phenomena only relates to solids that are fairly homogeneous on a macroscopic scale. Hence this book would certainly not be a fair introduction to the mathematics of solid mechanics without some discussion of the increasingly important properties of composites.

Introduction

In Chapters 2 and 3, we have analysed solutions of the steady and unsteady Navier equations, which were derived in Chapter 1 under the two assumptions that underpin linear elasticity.

First, we assumed that we could discard the nonlinear terms in the relation (1.4.5) between strain and displacement. Geometrically nonlinear elasticity concerns large deformations in which these terms are not negligible, so the strain is a nonlinear function of the displacement gradients. This inevitably leads to the further complication that the Lagrangian and Eulerian variables may no longer be approximated as equal.

The second assumption behind linear elasticity is that the stress is a linear function of the strain. This is a reasonable approximation for small strains, but it does not work well for materials such as rubber, which can suffer large strain and still remain elastic (see Treloar, 2005). Models for such materials require mechanically nonlinear elasticity, in which the stress is a nonlinear function of the strain.

As indicated in Section 1.6, the fundamental difficulty to be confronted is that the balance of stresses is performed in the deformed state, while the constitutive relation must be imposed relative to the reference configuration. As a first step in addressing this difficulty, we will revisit the concepts of stress and strain. We will show how they may both be expressed in a Lagrangian frame of reference, allowing a self-consistent constitutive law to be imposed between them. For mechanically nonlinear materials, such laws are far less easy to specify than (1.7.6), and we will see that great care has to be taken to avoid models that allow unphysical behaviour.

Introduction

Transport of matter, pollution or chemical and biological agents by turbulent flows is an important phenomenon with multiple applications from cosmology and astrophysics to meteorology, environmental studies, biology, chemistry and engineering. This is a series of lecture notes reviewing some aspects of the theoretical work on simple models of turbulent transport done over the last decade and partly described already, often with much more details, in [Fal01]. Other approaches to modeling turbulent transport, with the stress on turbulent diffusion, may be found in the review [Maj99]. More practical issues related to the influence of turbulence on the chemical or biological activity were addressed in [Tel05] from somewhat similar point of view.

The aim of simple models of turbulent transport that we shall discuss here is to explain or discover general phenomena and robust behaviors rather than to provide a detailed quantitative description. We shall study exclusively the passive transport approximation which is relevant for small concentrations of transported matter so that its back-reaction on the flow itself may be ignored. From the mathematical point of view, passive transport may be considered as a problem in random dynamical systems. Some of the techniques developed over years in the dynamical systems theory are indeed useful in analyzing transport phenomena, especially for flows with moderate Reynolds numbers.