Introduction

This long and technical chapter aims at providing some basic connections between the mathematical theory of the Navier–Stokes equations (NSE) and the conventional theory of turbulence. As stated earlier, the conventional theory of turbulence (including the famous Kolmogorov spectrum law) is based principally on physical and scaling arguments, with little reference to the NSE. We believe that it is instructive to connect turbulence more precisely with the Navier–Stokes equations.

It is commonly accepted that turbulent flows are necessarily statistical in nature. Indeed, if a flow is turbulent, then all physical quantities are rapidly varying in space and time and we cannot determine the actual instantaneous values of these quantities. Instead, one usually measures the moments, or some averaged values of physical quantities; that is, only a statistical description of the flow is available. The first task in this chapter is to establish, in a more precise way, the time evolution of the probability distribution functions associated with the fluid flow – that is, the statistical solutions of the Navier–Stokes equations. Although the discussion is relevant to deterministic data (initial values of the velocities and volume forces), we extend our discussion to the case of random data; however, we will not examine the more involved case of very irregular forcing (such as white or colored volume forces), since deterministic or moderately irregular stochastic data suffice, in practice, to generate complex turbulent flows.

Introduction

The purpose of this chapter is to recall some elements of the classical mathematical theory of the Navier–Stokes equations (NSE). We try also to explain the physical background of this theory for the physics-oriented reader.

As they stand, the Navier–Stokes equations are presumed to embody all of the physics inherent in the given incompressible, viscous fluid flow. Unfortunately, this does not automatically guarantee that the solutions to those equations satisfy the given physics. In fact, it is not even guaranteed a priori that a satisfactory solution exists. This chapter addresses the means for specifying function spaces – that is, the ensembles of functions consistent with the physics of the situation (such as incompressibility, boundedness of energy and enstrophy, as well as the prescribed boundary conditions) – that can serve as solutions to the Navier–Stokes equations. An important point is made that the kinematic pressure, p, is determined uniquely by the velocity field up to an additive constant. Hence, one cannot specify independently the initial boundary conditions for the pressure. This observation leads naturally to a representation of the NSE by an abstract differential equation in a corresponding function space for the velocity field.

Two types of boundary conditions are considered: no-slip, which are relevant to flows in domains bounded by solid impermeable walls; and space-periodic boundary conditions, which serve to study some idealized flows (including homogeneous flows) far away from real boundaries.

Introduction

In this chapter we first briefly recall, in Section 1, the derivation of the Navier–Stokes equations (NSE) starting from the basic conservation principles in mechanics: conservation of mass and momentum. Section 2 contains some general remarks on turbulence, and it alludes to some developments not presented in the book. For the benefit of the mathematically oriented reader (and perhaps others), Section 3 provides a fairly detailed account of the Kolmogorov theory of turbulence, which underlies many parts of Chapters III–V. For the physics-oriented reader, Section 4 gives an intuitive introduction to the mathematical perspective and the necessary tools. A more rigorous presentation appears in the first half of Chapter II and thereafter as needed. For each of the aspects that we develop, the present chapter should prove more useful for the nonspecialist than for the specialist.

Viscous Fluids. The Navier–Stokes Equations

Fluids obey the general laws of continuum mechanics: conservation of mass, energy, and linear momentum. They can be written as mathematical equations once a representation for the state of a fluid is chosen. In the context of mathematics, there are two classical representations. One is the so-called Lagrangian representation, where the state of a fluid “particle” at a given time is described with reference to its initial position.

In this chapter we discuss applications of the perturbation approach to stability analysis of elastic bodies subjected to large deformations. Various ideas commonly used in the perturbation approach are explained by using simple examples. Two types of bifurcations are distinguished: bifurcations at a non-zero critical mode number and bifurcations at a zero critical mode number. For each type we first explain with the aid of a model problem how stability analysis can be carried out and then explain how the analysis could be extended to problems in Finite Elasticity. Although the present analysis focuses on the perturbation approach, the dynamical systems approach is also discussed briefly and references are made to the literature where more details can be found. In the final section, we carry out a detailed analysis for the necking instability of an incompressible elastic plate under stretching.

Introduction

This chapter is concerned with nonlinear stability analysis of elastic bodies subjected to large elastic deformations. A typical problem we have in mind is the stability of a cylindrical rubber tube that is compressed either by an external pressure or by forces at the two flat ends. In general terms, we consider an elastic body which has an undeformed configuration Br in a three-dimensional Euclidean point space. This elastic body is then subjected to some external forces. It is now customary to refer to such an elastic body as pre-stressed in the Finite Elasticity literature (the pre-stress considered in the present context is not therefore that induced in a manufacturing process).

This chapter is an overview of a theory of a class of nonlinear elastic materials for which the deformation is subject to an internal material constraint described in experiments by James F. Bell on the finite plastic deformation of a variety of annealed metals. Research by Bell and his associates published since about 1979 is reviewed, and Bell's empirically deduced rules and laboratory data are compared with analytical results obtained within the context of nonlinear elasticity theory. First, Bell's empirical characterization of the constrained response of polycrystalline annealed metals in finite plastic strain is sketched. A few kinematical consequences of Bell's constraint, an outline of the constitutive theory developed to characterize the isoteopic, nonlinearly elastic response of Bell materials, and theoretical results that lead to Bell's empirical parabolic laws within the structure of isotropic, elastic and hyperelastic Bell constrained materials are presented. The study concludes with discussion of Bell's empirically based incremental theory of plasticity.

Introduction

It is common in technical writing to begin with a sketch of related research assembled to set the stage for the work ahead. But I'm not going to follow the usual path. There is more to this account than just its technical side - teachers and students, colleagues and associates, family and friends, places and events, life and death - the ingredients of the human side of the story. A reader who feels no interest in this sort of personal, anecdotal retrospection, however, will find immediate relief and surely suffer no loss in skipping ahead to Section 2.3 where Bell's important experiments and his internal material constraint are introduced. We'll return to this shortly.

A deformation or a motion is said to be a universal solution if it satisfies the balance equations with zero body force for all materials in a given class, and is supported in equilibrium by suitable surface tractions alone. On the other hand for a given deformation or motion, a local universal relation is an equation relating the stress components and the position vector which holds at any point of the body and which is the same for any material in a given class. Universal results of various kinds are fundamental aspects of the theory of finite elasticity and they are very useful in directing and warning experimentalists in their exploration of the constitutive properties of real materials. The aim of this chapter is to review universal solutions and local universal relations for isotropic nonlinearly elastic materials.

Introduction

One of the main problems encountered in the applications of the mechanics of continua is the complete and accurate determination of the constitutive relations necessary for the mathematical description of the behavior of real materials.

After the Second World War the enormous work on the foundations of the mechanics of continua, which began with the 1947 paper of Rivlin on the torsion of a rubber cylinder published in the Journal of Applied Physics, has allowed important insights on the above mentioned problem (Truesdell and Noll, 1965).

In this chapter we provide an introductory exposition of singularity theory and its application to nonlinear bifurcation analysis in elasticity. Basic concepts and methods are discussed with simple mathematics. Several examples of bifurcation analysis in nonlinear elasticity are presented in order to demonstrate the solution procedures.

Introduction

Singularity theory is a useful mathematical tool for studying bifurcation solutions. By reducing a singular function to a simple normal form, the properties of multiple solutions of a bifurcation equation can be determined from a finite number of derivatives of the singular function. Some basic ideas of singularity theory were first conjectured by R. Thorn, and were then formally developed and rigorously justified by J. Mather (1968, 1969a, b). The subject was extended further by V. I. Arnold (1976, 1981). In two volumes of monographs, M. Golubitsky and D. G. Schaeffer (1985), and M. Golubitsky, I. Stewart and D. G. Schaeffer (1988) systematized the development of singularity theory, and combined it with group theory in treating bifurcation problems with symmetry. Their work establishes singularity theory as a comprehensive mathematical theory for nonlinear bifurcation analysis.

The purpose of this chapter is to give a brief exposition of singularity theory for researchers in elasticity. The emphasis is on providing a working knowledge of the theory to the reader with minimal mathematical prerequisites. It can also serve as a handy reference source of basic techniques and useful formulae in bifurcation analysis.

In this chapter we provide a brief overview of the main ingredients of the nonlinear theory of elasticity in order to establish the basic background material as a reference source for the other, more specialized, chapters in this volume.

Introduction

In this introductory chapter we summarize the basic equations of nonlinear elasticity theory as a point of departure and as a reference source for the other articles in this volume which are concerned with more specific topics.

There are several texts and monographs which deal with the subject of nonlinear elasticity in some detail and from different standpoints. The most important of these are, in chronological order of the publication of the first edition, Green and Zerna (1954, 1968, 1992), Green and Adkins (1960, 1970), Truesdell and Noll (1965), Wang and Truesdell (1973), Chadwick (1976, 1999), Marsden and Hughes (1983, 1994), Ogden (1984a, 1997), Ciarlet (1988) and Antman (1995). See also the textbook by Holzapfel (2000), which deals with viscoelasticity and other aspects of nonlinear solid mechanics as well as containing an extensive treatment of nonlinear elasticity. These books may be referred to for more detailed study. Subsequently in this chapter we shall refer to the most recent editions of these works. The review articles by Spencer (1970) and Beatty (1987) are also valuable sources of reference.

Section 1.2 of this chapter is concerned with laying down the basic equations of elastostatics and it includes a summary of the relevant geometry of deformation and strain, an account of stress and stress tensors, the equilibrium equations and boundary conditions and an introduction to the formulation of constitutive laws for elastic materials, with discussion of the important notions of objectivity and material symmetry.

In this chapter we describe how certain features of the nonlinear inelastic ehaviour of solids can be described using a theory of pseudo-elasticity. ecifically, the quasi-static stress softening response of a material can be described by allowing the strain-energy function to change, either continuously or discontinuously, as the deformation process proceeds. In particular, the strain energy may be different on loading and unloading, residual strains may be generated and the energy dissipated in a loading/unloading cycle may be calculated explicitly. The resulting overall material response is not elastic, but at each stage of the deformation the governing equilibrium equations are those appropriate for an elastic material. The theory is described in some detail for the continuous case and then examined for an isotropic material with reference to homogeneous biaxial deformation and its simple tension, equibiaxial tension and plane strain specializations. A specific model is then examined in order to illustrate the (Mullins) stress softening effect in rubberlike materials. Two representative problems involving non-homogeneous deformation are then discussed. The chapter finishes with a brief outline of the theory for the situation in which the stress (and possibly also the strain) is discontinuous.

Introduction

For the most part the chapters in this volume are concerned with elasticity per se. However, there are some circumstances where elasticity theory can be used to describe certain inelastic behaviour. An important example is deformation theory plasticity, in which nonlinear elasticity theory is used to describe loading up to the point where a material yields and plastic deformation is initiated.

In this chapter we give a simple account of the theory of isotropic nonlinear elastic membranes. Firstly we look at both two-dimensional and three-dimensional theories and highlight some of the differences. A number of examples are then used to illustrate the application of various aspects of the theory. These include basic finite deformations, bifurcation problems, wrinkling, cavitation and existence problems.

Introduction

The aim of this chapter is to give a simple basic account of the theory of isotropic hyperelastic membranes and to illustrate the application of the theory through a number of examples. We do not aim to supply an exhaustive list of all relevant references, but, conversely, we give only a few selected references which should nevertheless provide a suitable starting point for a literature search.

The basic equations of motion can be formulated in two distinct ways; either by starting from the three-dimensional theory as outlined in Chapter 1 of this volume and then making assumptions and approximations appropriate to a very thin sheet; or from first principles by forming a theory of twodimensional sheets. The former approach leads to what might be called the three-dimensional theory and can be found in Green and Adkins (1970), for example. A clear derivation of the two-dimensional theory can be found in the paper of Steigmann (1990). Since there are two different theories attempting to model the same physical entities it is natural to compare and contrast these two theories.

This chapter studies nonlinear dispersive waves in a Mooney-Rivlin elastic rod. We first derive an approximate one-dimensional rod equation, and then show that traveling wave solutions are determined by a dynamical system of ordinary differential equations. A distinct feature of this dynamical system is that the vector field is discontinuous at a point. The technique of phase planes is used to study this singularity (there is a vertical singular line in the phase plane). By considering the relative positions of equilibrium points, we establish the existence conditions under which a phase plane contains physically acceptable solutions. In total, we find ten types of traveling waves. Some of the waves have certain distinguished eatures. For instance, we may have solitary cusp waves which are localized with a discontinuity in the shear strain at the wave peak. Analytical expressions for most of these types of traveling waves are obtained and graphical results are presented. The physical existence conditions for these waves are discussed in detail.

Introduction

Traveling waves in rods have been the subject of many studies. The study of plane flexure waves has formed one focus. See, e.g., Coleman and Dill (1992), and Coleman et al. (1995). Another focus is the study of nonlinear axisymmetric waves that propagate axial-radial deformation in circular cylindrical rods composed of a homogeneous isotropic material. This chapter is concerned with the latter aspect for incompressible Mooney-Rivlin materials. We mention in particular three related works by Wright (1982, 1985) and Coleman and Newman (1990).

For homogeneous isotropic incompressible nonlinearly elastic solids in equilibrium, the simplified kinematics arising from the constraint of no volume change has facilitated the analytic solution of a wide variety of boundary-value problems. For compressible materials, the situation is quite different. Firstly, the absence of the isochoric constraint leads to more complicated kinematics. Secondly, since the only controllable deformations are the homogeneous deformations, the discussion of inhomogeneous deformations has to be confined to a particular strain-energy function or class of strain-energy functions. Nevertheless, in recent years, substantial progress has been made in the development of analytic forms for the deformation and in the solution of boundary value problems. The purpose of this Chapter is to review some of these recent developments.

Introduction

For homogeneous isotropic incompressible materials in equilibrium, the simplified kinematics arising from the constraint of no volume change has facilitated the analytic solution of a wide variety of boundary-value problems, see, e.g., Ogden (1982, 1984), Antman (1995), and Chapter 1 of the present volume. Most well-known among these are the controllable or universal deformations, namely, those inhomogeneous deformations which are independent of material properties and thus can be sustained in all incompressible materials in the absence of body forces. For homogeneous isotropic compressible materials, Ericksen (1955) established that the only controllable deformations are homogeneous deformations Thus, inhomogeneous deformations for compressible materials necessarily have to be discussed in the context of a particular strain-energy function or class of strain-energy functions.

This chapter provides a brief introduction to the following basic ideas pertaining to thermoelastic phase transitions: the lattice theory of martensite, phase boundaries, energy minimization, Weierstrass-Erdmann corner conditions, phase equilibrium, nonequilibrium processes, hysteresis, the notion of driving force, dynamic phase transitions, nonuniqueness, kinetic law, nucleation condition, and microstructure.

Introduction

This chapter provides an introduction to some basic ideas associated with the modeling of solid-solid phase transitions within the continuum theory of finite thermoelasticity. No attempt is made to be complete, either in terms of our selection of topics or in the depth of coverage. Our goal is simply to give the reader a flavor for some selected ideas.

This subject requires an intimate mix of continuum and lattice theories, and in order to describe it satisfactorily one has to draw on tools from crystallography, lattice dynamics, thermodynamics, continuum mechanics and functional analysis. This provides for a remarkably rich subject which in turn has prompted analyses from various distinct points of view. The free-energy function has multiple local minima, each minimum being identified with a distinct phase, and each phase being characterized by its own lattice Crystallography plays a key role in characterizing the lattice structure and material symmetry, and restricts deformations through geometric compatibility. The thermodynamics of irreversible processes provides the framework for describing evolutionary processes. Lattice dynamics describes the mechanism by which the material transforms from one phase to the other. And eventually all of this needs to be described at the continuum scale.

I present a development of the modern theories of elastic shells, regarded as mathematical surfaces endowed with kinematical and constitutive structures deemed sufficient to represent many of the features of the response of thin shell-like bodies. The emphasis is on Cosserat theory, specialized to obtain a model of the Kirchhoff-Love type through the introduction of appropriate constraints. Noll's concept of material symmetry, adapted to surface theory by Cohen and Murdoch, is used to derive new constitutive equations for elastic surfaces having hemitropic, isotropic and unimodular symmetries. The last of these furnishes a model for fluid films with local bending resistance, which may be used to describe the response of certain fluid microstructures and biological cell membranes.

Introduction

I use the nonlinear Kirchhoff-Love theory of shells to describe the mechanics of a number of phenomena including elastic surface-substrate interactions and the equilibria of fluid-film microstructures. The Kirchhoff-Love shell may be interpreted as a one-director Cosserat surface (Naghdi 1972) with the director field constrained to coincide with the local orientation field.

The phenomenology of surfactant fluid-film microstructures interspersed in bulk fluids poses significant challenges to continuum theory. By using simple models of elastic surfaces, chemical physicists have been partially successful in describing the qualitative features of the large variety of equilibrium structures observed (Kellay et al. 1994, Gelbart et al. 1994). The basic constituent of such a surface is a polar molecule composed of hydrophilic head groups attached to hydrophobic tail groups.

The subject of Finite Elasticity (or Nonlinear Elasticity), although many of its ingredients were available much earlier, really came into its own as a discipline distinct from the classical theory of linear elasticity as a result of the important developments in the theory from the late 1940s associated with Rivlin and the collateral developments in general Continuum Mechanics associated with the Truesdell school during the 1950s and 1960s. Much of the impetus for the theoretical developments in Finite Elasticity came from the rubber industry because of the importance of (natural) rubber in many engineering components, not least car tyres and bridge and engine mountings. This impetus is maintained today with an ever increasing use of rubber (natural and synthetic) and other polymeric materials in a broader and broader range of engineering products. The importance of gaining a sound theoretically-based understanding of the thermomechanical behaviour of rubber was only too graphically illustrated by the role of the rubber O-ring seals in the Challenger shuttle disaster. This extreme example serves to underline the need for detailed characterization of the mechanical properties of different rubber like materials, and this requires not just appropriate experimental data but also the rigorous theoretical framework for analyzing those data. This involves both elasticity theory per se and extensions of the theory to account for inelastic effects.

Over the last few years the applications of the theory have extended beyond the traditional regime of rubber mechanics and they now embrace other materials capable of large elastic strains.

In this chapter, we deal with the theory of finite strain in the context of nonlinear elasticity. As a body is subjected to a finite deformation, the angle between a pair of material line elements through a typical point is changed. The change in angle is called the “shear” of this pair of material line elements. Here we consider the shear of all pairs of material line elements under arbitrary deformation. Two main problems are addressed and solved. The first is the determination of all “unsheared pairs”, that is all pairs of material line elements which are unsheared in a given deformation. The second is the determination of those pairs of material line elements which suffer the maximum shear.

Also, triads of material line elements are considered. It is seen that, for an arbitrary finite deformation, there is an infinity of oblique triads which are unsheared in this deformation and it is seen how they are constructed from unsheared pairs.

Finally, for the sake of completeness, angles between intersecting material surfaces are considered. They are also changed as a result of the deformation. This change in angle is called the “planar shear” of a pair of material planar elements. A duality between the results for shear and for planar shear is exhibited.

6.1 Introduction

At a typical particle P in a body, material line elements are generally translated, rotated and stretched as a result of a deformation, so that angles between intersecting material lines are generally changed.