INTRODUCTION

Nearby, but below, the order-disorder phase-transition temperature, the dynamical equations that govern elastic crystals with a microstructure exhibit all the necessary, if not sufficient, properties (essentially, nonlinearity and dispersion with a possible compensation between the two effects) to allow for the propagation of so-called solitary waves. Whether these are true solitons or not is a question that can be answered only in each case through analysis and/or numerical simulations. These waves, however, are supposed to represent domain walls in motion. The latter are layers of relatively small thickness which carry a strong nonuniformity in a relevant parameter between two adjacent phases or degenerate ground states (see, e.g., Maugin and Pouget, 1986).

This volume contains most of the invited lectures delivered as part of the Symposium on “Non-classical Continuum Mechanics, Abstract Techniques and Applications” held in the University of Durham from 2 – 12th July, 1986. The meeting was under the auspices of the London Mathematical Society with financial support generously provided by the Science and Engineering Research Council of Great Britain. To these organisations and also to the University of Durham (including staff of the Department of Mathematics and of Grey College) grateful thanks are extended on behalf of all participants and members of the Organising Committee. Sincere thanks are also due to Mrs P. Hampton (Heriot-Watt University) and Mrs S. Cuttle (University of Durham) who admirably coped with the secretarial and administrative burdens.

One objective of the Symposium was to provide the opportunity for interaction between two broad trends now discernible in continuum mechanics which respectively emphasise applications and rigorous mathematical analysis. The mutual enrichment of these two developments requires frequent exchange of information and thanks to the right note struck in the formal addresses this certainly occurred at the Durham meeting. Indeed, all contributors are to be commended for ensuring that their presentations were readily accessible to both groups thus helping to further the aims of the organisers. This in turn has resulted in the present volume containing articles which besides being accounts of recent progress should also be of wide interest to the specialist and non-specialist alike.

INTRODUCTION

Twinning, phase transitions and memory shape effects are frequently observed in solid materials with crystalline structure (cf. [3], [19]). As remarked by Wayman & Shimizu [19], it is of great theoretical and technological interest to investigate and predict the onset of such phenomena and to analyse equilibria and stability for this type of material.

The mathematical approach to these problems has recently been addressed by several authors, including Chipot & Kinderlehrer [4], Ericksen [7], [8], [9], Fonseca [10], [11], [12], Fonseca & Tartar [13], James [13], Kinderlehrer [15], Parry [17] and Pitteri [18], within the framework of a continuum theory based on nonlinear thermoelasticity proposed by Ericksen [7], [8]. The thermoelastic behaviour is quite noticeable in certain alloys for some ranges of temperature. As an example, Basinski & Christian [3] found that at room temperature Indium-Thallium alloys are perfectly plastic but for temperatures above 100°C or below 0°C they exhibit rubber like behaviour.

We summarize briefly the notion of an elastic crystal as described by Ericksen [9] and Kinderlehrer [15] (see also Ericksen [7], [8], and Fonseca [10]). In terms of molecular theory, a (pure) crystal is a countable set of identical atoms arranged in a periodic manner. By fixing a cartesian coordinate system with origin at one of the atoms and orthonormal basis [e1, e2, e3]l the position vectors of the atoms of the lattice relative to some choice of linearly independent lattice vectors {a1, a2, a3} may be given by where 114 mi ∈ Z, i =; 1, 2, 3, and A is the matrix with columns a1, a2, a3.

INTRODUCTION

When a granular material is sheared at a sufficiently high rate, the shear stress and the normal stress required to maintain its motion are observed to vary with the square of the shear rate (Bagnold (1954), Savage (1972), Savage & McKeown (1983), Savage & Sayed (1984), Hanes & Inman (1985a)). The interpretation of these observations is that, at high shear rates, the dominant mechanism of momentum transfer is collisions between grains, with the interstitial liquid or gas playing a relatively minor role.

There is an obvious analogy between the colliding macroscopic grains of a sheared granular material and the agitated molecules of a dense gas. There are also several important differences: collisions between grains are inevitably inelastic; typical grain flows involve spatial variations over far fewer grains than their molecular counterparts; and, consequently, the influence of boundaries is far more pervasive.

Exploiting the analogy, methods from the kinetic theory of dense gases have been adopted and extended to derive balance laws, constitutive relations, and boundary conditions for systems of macroscopic, dissipative, grains. Jenkins (1987) provides a brief review of this activity.

Introduction

The study of pulse propagation in one dimensional random media arises in many applied contexts. While reflection and transmission of monochromatic waves was studied extensively some time ago [1–6 and references therein], new and perhaps surprising results emerge in the study of pulses that cannot be understood simply from the single frequency analysis by Fourier synthesis. The numerical study of Richards and Menke [7] drew our attention to these questions and led to [8] and [9]. Here we extend and simplify the analysis of [8] and give several new results. The computations are at a formal level comparable to the one in [8].

In [8] we analyzed the reflection of a pulse that is broad compared to the size of the inhomogeneities of the random medium. The random functions characterizing the medium properties were statistically homogeneous. We gave a rather complete description of the reflected signal process in a well defined asymptotic limit in which it has a canonical structure. We introduced the notion of a windowed process and showed that the canonical reflection process is windowed and Gaussian. We found a scaling law for the power spectral density but not its explicit form. All this was subjected to extensive numerical simulations in [9] where an intrinsic scaling, localization length scaling, was introduced that makes comparison to the theory much more reliable. This intrinsic scaling idea is not fully understood theoretically but seems to be very promising.

In this paper we extend the analysis to random media that are not statistically homogeneous. The incident pulse is now broad compared to the size of the inhomogeneities but short compared to the scale of variation of the mean properties.

INTRODUCTION

An outline is given of the mathematical modelling of some problems which occur in various situations in the oil service industry. In each case the objective is to use a mathematical model as a means of interpreting properties of the formation either during drilling or by well tests after drilling has been completed.

We begin with a brief outline of oil well testing. The main aim of such tests is the recovery of a fluid sample and an estimation of the flow capacity of the formation. We start with a description of a standard procedure for single phase flow (Homer's method (1)) and then discuss the complications introduced when the flow equations are coupled with the temperature. A description is also given of some two phase flow problems.

A second example is that of a mathematical model to be used for interpreting measurements while drilling. This consists of studying the axial vibrations generated by a roller cone bit during the drilling process. Through the model the axial force and displacement histories are calculated at the bit and related to the force and acceleration measured at an MWD (measurements while drilling) site just above the drill bit (possibly as far away as 60 ft.) or at the surface. It is, of course, important to identify the relation between the measured quantities and the values of these quantities at the bit. The objective of this interpretation is to determine formation and bit properties as drilling proceeds so as to facilitate the drilling process.

INTRODUCTION

This paper discusses certain types of stability questions that have been largely ignored in the literature, i.e. continuous dependence on geometry and continuous dependence on modeling. Although we shall consider these questions primarily in the context of ill-posed problems we shall briefly indicate some difficulties that might arise under geometric and/or modeling perturbations in well posed problems.

In setting up and analyzing a mathematical model of any physical process it is inevitable that a number of different types of errors will be introduced e.g. errors in measuring data, errors in determining coefficients, etc. There will also be errors made in characterizing the geometry and in formulating the mathematical model. In most standard problems the errors made will induce little error in the solution itself, but for ill-posed problems in partial differential equations this is no longer true.

Throughout this paper we shall assume that a “solution” to the problem under consideration exists in some accepted sense, but in the case of ill-posed problems such a “solution” will invariably fail to depend continuously on the data and geometry. We must appropriately constrain the solution in order to recover the continuous dependence (see [8]); however, appropriate restrictions are often difficult to determine. In the first place any such constraint must be both mathematically and physically realizable. At the same time a given constraint must simultaneously stabilize against all possible errors that may be made in setting up the mathematical model of the physical problem. Since a constraint restriction has the effect of making an otherwise linear problem nonlinear, one must use care in treating the various errors separately and superposing the effects.