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.

Introduction

The developments in the preceding chapter assumed nothing about the constitutive response of the continuum; we now restrict attention to a special class of constitutive laws – the so-called thermoelastic materials introduced previously in Chapter 5. Our discussion in Chapter 5 was focused entirely on the energy wells of the characterizing energy potential. Here we discuss thermoelastic materials and nonlinear thermoelasticity in more detail.

In Section 7.2 we state the constitutive law of nonlinear thermoelasticity, in which stress and specific entropy are specified as functions of deformation gradient and absolute temperature through the Helmholtz free energy potential. An equivalent alternate form of the constitutive law, in which stress and temperature are given in terms of deformation gradient and specific entropy by means of the internal energy potential, is also discussed. The expression for the driving force is then specialized to this setting. Next we state the heat conduction law, and in Section 7.2.3, we write out the full theory in the form of four scalar partial differential equations involving the three components of displacement and temperature. The accompanying jump conditions are also laid out. In the final subsection we specialize the results to a state of thermomechanical equilibrium.

Introduction

In the present chapter we study the equilibrium and quasistatic response of a thin bar composed of a material modeled by the stress–strain curve shown in Figure 2.2 of the preceding chapter. There is an extensive experimental literature devoted to tensile loading and unloading of bars made of materials that are capable of undergoing displacive phase transitions; often the materials studied are technologically important shape-memory alloys such as nickel–titanium. For a small sample of this literature, the reader might consult the papers of Krishnan and Brown [13], Nakanishi [17], Shaw and Kyriakides [19], and Lin et al. [14], as well as the references cited there. The loading in such experiments is slow, in the sense that inertia is insignificant. The objective is typically the determination of the relation between the applied stress and the overall elongation of the bar, though in some studies, such as that of Shaw and Kyriakides [19], local strain and temperature measurements are made as well. The stress–elongation relation that is observed in such experiments exhibits hysteresis, the phase transition being the primary mechanism responsible for such dissipative behavior. For a given material, the size and other qualitative features of the hysteresis loops depend on the loading rate and the temperature at which the test takes place. In the model to be discussed in this chapter, thermal effects are omitted; they will be accounted for in later chapters.