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.

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.

Interfaces in Materials

Much of materials science and engineering is played out at the level of a material's microstructure. From a more refined perspective, the quest to tune or create new microstructures may also be thought of as a series of exercises in interfacial control. Indeed, interfaces in materials are the seat of an extremely wide variety of processes ranging from the confinement of electrons to the fracture of polycrystals. The character of the interfaces that populate a given material leaves an indelible imprint on that material's properties. Whether we consider the susceptibility of a given material to chemical attackat its surfaces, the transport properties of complex semiconductor heterostructures, or the fracture of steels, ushered in by the debonding of internal interfaces, ultimately, interfaces will be seen as one of the critical elements yielding a particular material response. In some cases, the presence of such interfaces is desirable, in others, it is not. In either case, much effort attends the ambition of controlling such interfaces and the properties that they control.

Until this point in the book we have concerned ourselves with defects of reduced dimensionality. The present chapter aims to expand that coverage by turning to two-dimensional wall defects. The analysis of these defects will serve as our jumping off point for the consideration of microstructures within materials.

Thermal and Elastic Material Response

As yet, our discussion of modeling the energetics of crystalline solids has centered on the analysis of atomic configurations that are undisturbed by deformation, by the thermal vibrations that are inevitably present at finite temperatures or by defects. Before embarking on an analysis of the full-fledged defected solid, we begin with a consideration of geometric states that may be thought of as relatively gentle perturbations about the perfect crystal. In particular, the aim of the present chapter is to assess the energetics of both thermal vibrations and elastic deformations, with a treatment that is unified by the fact that in both cases the energy can be expanded about that of the perfect crystal which serves as the reference state.

Prior to examining the ways in which the methods of chap. 4 can be used to examine the energetics of both thermal vibrations and elastic deformations, we consider the significance of these excitations to the properties of materials at large. The methods of the present chapter feed into our quest to understand the properties of materials in several distinct ways. First, as will be described in the following paragraphs, a variety of important material properties involve the energetics of thermal vibration or elastic deformation directly, including the specific heat, the thermal conductivity, the thermal expansion and the elastic moduli themselves. However, in addition to the direct significance of the ideas to be presented here to the analysis of thermal properties, much of what we will have to say will form the methodological backdrop for our discussion of the thermodynamic ideas invoked to understand phase diagrams, diffusion and microstructural evolution.

Materials science as a formal discipline has quietly emerged as one of the central pillars of the physical sciences. Whether one interests oneself in the creation of lighter, more durable surfboards, or in the invention of new technologies to free up the traffic jam on the world wide web, in the end, these questions will always imply demands which must be met by new classes of materials.

Though the study of materials is largely rooted in the enlightened empiricism of traditional metallurgy, the advent of high-speed computers and the emergence of robust quantum simulations suggests new engineering strategies in which mechanism-based understanding might be hoped to lead to new materials. As a result of the maturation of theoretical materials science, it has become increasingly possible to identify a corpus of central results which serve as the basis for the analysis of materials. In addition, increasingly complex materials with structures at many different scales have led to the emergence of methods built around the explicit consideration of multiple scales simultaneously. As a result of these advances, this book attempts to take stock of the present capacity for modeling materials, with the word modeling used in the broadest sense.

The book is divided into four basic parts. It opens with an overview of some of the key issues concerning materials that one might hope to succeed in modeling. Special reference is made to the notion of material parameters as characterized by the broad variety of data about materials that may be found in any databook. Though my comments on material response will be rather generic, my efforts to model such response will be restricted almost exclusively to crystalline solids.

Defect Interactions and the Complexity of Real Material Behavior

In previous chapters, we have begun the attack on the problem of the observed thermomechanical properties of materials from a number of different perspectives. One of our key realizations has been the conspiratorial role of defects in governing the response of materials. Thus far, we have largely concentrated on the structure and energetics of individual defects with less attention given to the implications of these defects for material response. Our efforts have been primarily divided along dimensional lines, with point defects, line defects and interfacial defects each being treated as an end unto itself. The concern of the present chapter is to make evident the claim that such divisions are a theorist's artifice and that, in general terms, it is the synthetic response of all of these different types of defects in concert that gives rise to the observed complexity of materials.

We begin our discussion with the consideration of diffusion. In earlier chapters, we noted that diffusion is mediated by the presence of point defects. However, until now, we have not given any quantitative analysis of the lubricating effect of other defects in the context of diffusion. We have already seen in fig. 9.1 that bulk diffusion can be negligible in comparison with the diffusion that takes place along surfaces, grain boundaries and dislocations. As a particular realization of this class of problem, we consider the case of surface diffusion with special reference to some of the counterintuitive diffusion mechanisms that take place at surfaces.