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.

The purpose of this chapter is to focus on a variety of exact results applying to the perfectly elastic incompressible Varga materials. For these materials it is shown that the governing equations for plane strain, plane stress and axially symmetric deformations, admit certain first order integrals, which together with the constraint of incompressibility, give rise to various second order problems. These second order problems are much easier to solve than the full fourth order systems, and indeed some of these lower order problems admit elegant general solutions. Accordingly, the Varga elastic materials give rise to numerous exact deformations, which include the controllable deformations known to apply to all perfectly elastic incompressible materials. However, in addition to the standard deformations, there are many exact solutions for which the corresponding physical problem is not immediately apparent. Indeed, many of the simple exact solutions display unusual and unexpected behaviour, which possibly reflects non-physical behaviour of the Varga elastic materials for extremely large strains. Alternatively, these exact results may well mirror the full consequences of nonlinear theory. This chapter summarizes a number of recent developments.

Introduction

Natural and synthetic rubbers can accurately be modelled as homogeneous, isotropic, incompressible and hyperelastic materials, and which are sometimes referred to as perfectly elastic materials. The governing partial differential equations tend to be highly nonlinear and as a consequence the determination of exact analytical solutions is not a trivial matter.

Liquid metals freeze in much the same way as water. First, snowflake-like crystals form, and as these multiply and grow a solid emerges. However, this solid can be far from homogeneous. Just as a chef preparing icecream has to beat and stir the partially solidified cream to break up the crystals and release any trapped gas, so many steelmakers have to stir partially solidified ingots to ensure a fine-grained, homogeneous product. The preferred method of stirring is electromagnetic, and has been dubbed the ‘electromagnetic teaspoon’. We shall describe this process shortly.

First, however, it is necessary to say a little about commercial casting processes.

Casting, Stirring and Metallurgy

(Leonardo da Vinci on the extraction and casting of metals)

Man has been casting metals for quite some time. Iron blades, perhaps 5000 years old, have been found in Egyptian pyramids, and by 1000 BC we find Homer mentioning the working and hardening of steel blades. Until relatively recently, all metal was cast by a batch process involving pouring the melt into closed moulds. However, today the bulk of aluminium and steel is cast in a continuous fashion, as indicated in Figure 8.1.

In his 1964 lectures on physics, R P Feynman noted that:

In this chapter we build the dam and write down the equations. Later, particularly in Chapter 7 where we discuss turbulence, we shall see how the dam bursts open.

Fluid Flow in the Absence of Lorentz Forces

In the first seven sections of this chapter we leave aside MHD and focus on fluid mechanics in the absence of the Lorentz force. We return to MHD in Section 3.8. Readers who have studied fluid mechanics before may be familiar with much of the material in Sections 3.1 to 3.7, and may wish to proceed directly to Section 3.8. The first seven sections provide a self-contained introduction to the subject, with particular emphasis on vortex dynamics, which is so important in the study of MHD.

Elementary Concepts

Different categories of fluid flow

The beginner in fluid mechanics is often bewildered by the many diverse categories of fluid flow which appear in the text books. There are entire books dedicated to such subjects as potential flow, boundary layers, turbulence, vortex dynamics and so on.

When Rm is high there is a strong influence of u on B, and so we obtain a two-way coupling between the velocity and magnetic fields. The tendency for B to be advected by u, which follows directly from Faraday's law of induction, results in a completely new phenomenon, the Alfvèn wave. It also underpins existing explanations for the origin of the earth's magnetic field and of the solar field. We discuss both of these topics below. First, however, it may be useful to comment on the organisation of this chapter.

The subject of high-Rm MHD is vast, and clearly we cannot begin to give a comprehensive coverage in only one chapter. There are many aspects to this subject, each of which could, and indeed has, filled text-books and monographs. Our aim here is merely to provide the beginner with a glimpse of some of the issues involved, offering a stepping-stone to more serious study. The subject naturally falls into three or four main categories. There is the ability of magnetic fields to support inertial waves, both Alfvèn waves and magnetostrophic waves.

Magnetohydrodynamics (MHD for short) is the study of the interaction between magnetic fields and moving, conducting fluids. In the following seven chapters we set out the fundamental laws of MHD. The discussion is restricted to incompressible flows, and we have given particular emphasis to the elucidation of physical principles rather than detailed mathematical solutions to particular problems.

We presuppose little or no background in fluid mechanics or electromagnetism, but rather develop these topics from first principles. Nor do we assume any knowledge of tensors, the use of which we restrict (more or less) to Chapter 7, in which an introduction to tensor notation is provided. We do, however, make extensive use of vector analysis and the reader is assumed to be fluent in vector calculus.

The subjects covered in Part A are:

1. qualitative overview of MHD

2. governing equations of electrodynamics

3. governing equations of fluid mechanics

4. kinematics of MHD: advection and diffusion of a magnetic field

5. at low magnetic Reynolds' number

6. at high magnetic Reynolds' number

7. turbulence at low and high magnetic Reynolds' numbers

One point is worth emphasising from the outset. The governing equations of MHD consist simply of Newton's laws of motion and the pre-Maxwell form of the laws of electrodynamics. The reader is likely to be familiar with elements of both sets of laws and many of the phenomena associated with them.