Introduction

Prominent examples of long-lived large-scale vortices in geophysical flows are those such as the Great Red Spot (GRS) on Jupiter (Dowling, 1995; Marcus, 1993; Rogers, 1995). The emergence and persistence of such coherent structures at specific latitudes such as 22.4° S for the GRS in a background zonal shear flow that seems to violate all of the standard stability criteria (Dowling, 1995) are a genuine puzzle. Here we show how to utilize the equilibrium statistical theory with a suitable prior distribution, ESTP, introduced in Section 9.2 and discussed in Chapters 10, 11, and 12 to predict the actual coherent structures on Jupiter in a fashion which is consistent with the known observational record. As discussed in Section 9.2, the statistical theory is based on a few judiciously chosen conserved quantities for the inviscid dynamics such as energy and circulation and does not involve any detailed resolution of the fluid dynamics. The key ingredient of the ESTP is the prior distribution, Π(λ), for the one-point statistics of the potential vorticity which parameterizes the unresolved small-scale turbulent eddies that produce the large-scale coherent structures. Below, we show how the observational record of Jupiter from the recent Galileo mission suggests a special structure for such a prior distribution resulting from intense small-scale forcing. Recall from the studies discussed in Chapter 12 (DiBattista, Majda, and Grote, 2001) that ESTP can potentially describe the meta-stable large-scale coherent structures occurring from strong small-scale forcing, provided that the flux of energy to large scales is sufficiently weak.

What this monograph is about

Certain crystalline materials can exist in more than one solid phase, where a phase is identified by a distinct crystal structure. Typically, one phase is preferred under certain conditions of stress and temperature, while another is favored under different conditions. As the stress or temperature varies, the material may therefore transform abruptly, from one phase to another, leading to a discontinuous change in the properties of the body. Examples of such materials include the shape-memory alloy NiTi, the ferroelectric alloy BaTiO3, the ferromagnetic alloy FeNi and the high-temperature superconducting ceramic alloy ErRh4B4. In each of these examples the transition occurs without diffusion and one speaks of the transformation as being martensitic (or displacive).

Alloys such as Au–47.5%Cd and Cu–15.3%Sn are known to have a cubic lattice at high temperatures and an orthorhombic lattice at low temperatures. Therefore, if such a material is subjected to thermal cycling, it will transform between these two phases. Similarly, alloys such as Ni–36%Al and Fe–7%Al–2%C transform between a high-temperature cubic phase and a low-temperature tetragonal phase, whereas near-equiatomic NiTi has a high-temperature cubic phase and low-temperature monoclinic phase.

If a stress-free single crystal of such a two-phase material is slowly cooled from a sufficiently high temperature, it starts out in the high-temperature phase and at first, merely undergoes a thermal contraction.

Introduction

We next turn to the dynamics of the two-phase nonlinearly elastic materials introduced in Chapter 2. As in the theory of mixed-phase equilibria and quasistatic processes for such materials set out in Chapter 3, the notion of driving force plays a central role in the analysis when inertial effects are taken into account. The indeterminacy exhibited in Chapter 3 by even the simplest static or quasistatic problems for two-phase materials manifests itself again in the present much richer dynamical context. Moreover, the continuum-mechanical interpretations of nucleation and kinetics again serve to restore the uniqueness of solutions to the dynamic problems to be considered here. As in the preceding chapters, thermal effects are omitted; they will be included in the more general settings of later chapters.

The main vehicle for our study of one-dimensional dynamics of two-phase materials is the impact problem. There is an enormous body of experimental literature pertaining to the response of solids to shock or impact loading, much of it motivated by questions concerning the behavior of materials at extremely high pressures, as occurs, for example, deep in the earth. The reader will find some guidance to the experimental literature in this field of the dynamic behavior of materials in the books by Graham [11] and Meyers [25].

Introduction

In the preceding chapter we determined the kinetics of a certain phase transformation using experiments that involved fast loading in which inertia was important. In the present chapter we determine the kinetics of a different transformation using data from quasistatic experiments. The transformation studied here is a twinning deformation, not a phase transformation, a twin boundary being an interface that separates two variants of martensite; see Example 1 in Section 12.2. The change in lattice orientation across a twin boundary makes it analogous, in certain ways, to a phase boundary, and in particular, the motion of a twin boundary is governed by a kinetic relation.

As we have seen, the simplest form of kinetic relation governing the isothermal motion of an interface relates the driving force on it to its normal velocity of propagation: Vn = Φ(f). Since the kinetic response function Φ here is a function of a single scalar independent variable, one set of experiments, say uniaxial tension tests, completely determines Φ; and the function Φ thus determined characterizes all motions of this interface such as, say, in biaxial conditions. If the deformation field is inhomogeneous, and the phase or twin boundary is curved, one would use this same kinetic relation locally, at each point along the interface, relating the driving force at that point to the normal velocity of propagation of that point.

Introduction

In this chapter, we assemble the basic field equations and jump conditions for a one-dimensional, purely mechanical theory of nonlinear elasticity; although thermal effects will be omitted, inertia will be taken into account. The theory presented here is general enough to describe nonlinearly elastic materials that, under suitable conditions of stress, are capable of existing in either of two phases. As we shall see, a key feature of this theory is that the potential energy of the material, as a function of strain at a fixed stress, has two local minima. The associated constitutive relation between stress and strain will then necessarily be nonmonotonic, possessing a maximum and a minimum connected by an unstable regime in which stress declines with increasing strain.

Experiments that provide the motivation for the theory about to be developed fall into two categories. The first of these involves slow tensile loading and unloading of slender bars or wires composed of materials such as shape-memory alloys. The model to be constructed to describe experiments of this kind is one of uniaxial stress in a one-dimensional nonlinearly elastic continuum, and the processes to be studied for this model are quasistatic. The stress-induced phase transitions in such experiments occur in tension, so the two minima in the potential energy density occur at positive – or extensional – values of strain, as do the extrema in the stress– strain relation.

This monograph threads together a series of research studies carried out by the authors over a period of some fifteen years or so. It is concerned with the development and application of continuum-mechanical models that describe the macroscopic response of materials capable of undergoing stress- or temperature-induced transitions between two solid phases.

Roughly speaking, there are two types of physical settings that provide the motivation for this kind of modeling. One is that associated with slow mechanical or thermal loading of alloys such as nickel–titanium or copper–aluminum–nickel that exhibit the shape-memory effect. The second arises from high-speed impact experiments in which metallic or ceramic targets are struck by moving projectiles; the objective of such studies – often of interest in geophysics – is usually to determine the response of the impacted material to very high pressures. Phase transitions are an essential feature of the shape-memory effect, and they frequently occur in high-speed impact experiments on solids. Those aspects of the theory presented here that are purely phenomenological may well have broader relevance, in the sense that they may be applicable to materials that transform between two “states,” for example, the ordered and disordered states of a polymer.

Our development focuses on the evolution of the phase transitions modeled here, which may be either dynamic or quasistatic. Such evolution is controlled by a “kinetic relation,” which, in the framework of classical thermomechanics, represents information supplementary to the usual balance principles and constitutive laws of conventional theory.

Introduction

In Chapters 3 and 4 we used particular one-dimensional initial–boundary value problems to demonstrate that, because of a massive lack of uniqueness that exists otherwise, elasticity theory must be supplemented with a kinetic law and nucleation condition if it is to be used to model the emergence and evolution of multiphase configurations. As shown there, not only is there a need for such information, there is also room for it in the theory. A second motivation for a kinetic law, also presented in Chapter 3, arose by casting the quasistatic problem considered there in the framework of standard internal-variable theory; the evolution law characterizing the rate of change of the internal variable in that theory is then the kinetic law.

In the present chapter we provide a third approach to the notion of kinetics, this one from a thermodynamic point of view. In addition to providing a motivation, the discussion here allows us to describe the kinetic law within a general three-dimensional thermoelastic setting.

In Section 8.2 we present the thermodynamic formalism of irreversible processes in a thermoelastic body. Based on this, we introduce the notion of a thermodynamic driving force and the flux conjugate to it, and the notion of a kinetic relation then follows naturally. In Section 8.3 we present some phenomenological examples of kinetic relations, while in Section 8.4 we describe examples of kinetic relations based on various underlying transformation mechanisms. Some remarks on the nucleation condition are made in Section 8.5.

Introduction

In the present chapter, we consider a continuum in which there are three-dimensional, thermomechanical fields involving moving surfaces of discontinuity in strain, particle velocity, and temperature. Our objective is to set out the theory of driving force acting on such surfaces without specifying any particular constitutive law. The theory should be applicable to physical settings ranging from the quasistatic response of solids under slow thermal or mechanical loading in which heat conduction is present, to fast adiabatic processes in which temperature need not be continuous and inertia must be taken into account. The theory must be general enough to accommodate such disparate settings.

We begin by stating the fundamental balance laws for momentum and energy in their global form. These laws are then localized where the fields are smooth, leading to the basic field equations of the theory. Localization of the global laws at points where the thermomechanical fields suffer jump discontinuities provides the jump conditions appropriate to such discontinuities.

Without making constitutive assumptions, we then introduce the notion of driving force. The driving force arises through consideration of the entropy production rate associated with the thermomechanical fields under study; it leads to a succinct statement of the implication of the second law of thermodynamics for moving surfaces of discontinuity.

The theory is developed for three space dimensions in Lagrangian, or material, form, according to which one follows the evolution of fields attached to a given particle of the continuum.

Introduction

In Chapter 4 we studied one-dimensional models of dynamic phase transitions in the purely mechanical theory of elastic materials. Our objective here is to extend the ideas of that chapter to the dynamics of two-phase thermoelastic materials. Much of the analysis will be directed to the materials of Mie–Grüneisen type introduced in Chapter 9, with special emphasis on the trilinear thermoelastic material. As in Chapter 4, we shall study an impact-induced phase transition that occurs in compression, rather than in tension; this will require some minor modifications of the details of the constitutive models presented in Chapter 9.

The subject of nonlinear wave propagation in solids has an enormous literature encompassing both experimental and theoretical work. For a sample of background references representing a variety of viewpoints in this field, the reader might consult the classic work on gas dynamics of Courant and Friedrichs [4], the extensive review article of Menikoff and Plohr [7], the discussion by Ahrens [2] of the experimental determination of the “equation of state” of condensed materials, the work of Swegle [9] on phase transitions in materials of geologic interest, and the theory of shock waves in thermoelastic materials presented by Dunn and Fosdick [5].

In the next section, we set out the basic field equations and jump conditions of the dynamical theory of thermoelasticity when the kinematics are those of uniaxial strain and the processes are adiabatic.

The purely mechanical quasistatic response of one-dimensional, two-phase elastic bars was discussed in Chapter 3. In the present chapter, we shall generalize that discussion to incorporate thermal effects. After setting out some preliminaries in Section 10.1, in Section 10.2 we describe the thermomechanical equilibrium states of a two-phase material. Quasistatic processes, taken to be one-parameter families of equilibrium states, are studied in Section 10.3. We specialize the discussion to a trilinear thermoelastic material in Section 10.4 and then evaluate the response of the bar to some specific loading programs: in Sections 10.5, 10.6, and 10.7 we consider stress cycles at constant temperature, temperature cycles at constant stress, and the shape-memory cycle respectively; qualitative comparisons with some experiments are also made in these sections. In Section 10.8 we describe an experimental result of Shaw and Kyriakides [15, 16] and compare the theoretical predictions of our model with it. Finally in Section 10.9 we comment on processes that are slow in the sense that inertial effects can be neglected but are not quasistatic in the preceding sense of being one-parameter families of equilibrium states.

Preliminaries

We begin by setting out the one-dimensional version of the theory of thermoelasticity given in Chapter 7. Consider a tensile bar that occupies the interval [0, L] of the x-axis in a reference configuration.