Royal Society Publishing

Jog my shape memory: dynamics as a challenge in mathematical materials science

Johannes Zimmer


Many complex phenomena in nature exhibit multiple scales. The challenge is to understand how the effects on one scale influence those on another. This review discusses some aspects of a multiscale analysis of martensitic materials as a prominent example of materials that exhibit nuanced structures and surprising implications on various scales. The emphasis is on dynamic issues. Some speculations are offered on future research directions.


1. What are martensites?

Everyone has seen phase transitions, i.e. a transformation of a (thermodynamic) system from one phase to another. Melting ice, freezing water or condensing vapour are some classic examples. Less well known, but equally important, are phase transitions occurring between different solid states. Around 1890, the German metallurgist Martens discovered something fairly unexpected, taking advantage of the strident progress in optical technology in the nineteenth century. He inspected steel under a microscope, discovering complex, needle-like structures consisting of different solid phases, which form distinctive patterns, invisible to the naked eye. Two examples are shown in figure 1. To date, such patterns (ranging from a few nanometres to a few micrometres) are commonly called microstructures. The microstructure strongly influences the mechanical properties of the material. For example, the pattern in hard steels is usually more coherent than that in inferior steels.

Figure 1

Examples of microstructure in martensitic materials. (a) Reconstructive transformation. The photograph shows martensite in 0.8% carbon steel. (b) Weak transformation. The photograph shows the formation of needles and wedges in NiTi.

In honour of Martens' discovery, solid-to-solid phase transitions, where the lattice structure changes abruptly at some critical temperature, are called martensitic phase transitions. Obvious questions arising in the context of martensites are: how does the microstructure form? How does it evolve? What is its influence on macroscopic properties? Our understanding of dynamic issues is still in its infancy. In this review article, some recent contributions are outlined and future challenges are sketched. To convey the main ideas, we ignore many complicated aspects of real microstructure, preferring to discuss idealized situations. Space limitations oblige us to omit a discussion of many valuable contributions. Fortunately, several excellent surveys offer a rather different emphasis, e.g. Roubíček (2000) and the monograph by Bhattacharya (2003). Sethna's web site provides a fascinating read. The state of the art in the passage from quantum chemistry to crystalline (microscopic) structures and their macroscopic limits is described by Le Bris & Lions (2005).

The mathematical treatment of microstructured materials dates from the pioneering work by Ball & James (1987, 1992), Ericksen (1980) and Chipot & Kinderlehrer (1988). There is an intricate theory based on the concept of global energy minimization. This has proved highly successful in explaining the various phenomena of elastic and plastic behaviour of materials with microstructures. A recent trend is to investigate dynamic aspects, developing a dynamic theory for situations where the material does not attain the ground state (the configuration with the lowest energy). Ball et al. (1991) and Friesecke & McLeod (1996) have made early contributions which highlight the importance of dynamical aspects, e.g. by acting as a selection criterion, preventing martensitic materials from forming arbitrarily fine microstructures. For a model proposed by Ball et al. (1991), Friesecke & McLeod (1996) study rigorously the dynamics of microstructure formation in a context where energy minimization predicts the formation of infinitely fine patterns. In experiments, one typically observes the pattern formation on a small but finite length-scale. This effect can also be seen in the dynamic model (Friesecke & McLeod 1996). Friesecke & McLeod (1996) prove that for suitable initial data (with strain fields having a transition layer; see §2 for the definition of strain), the limiting patterns for time going to infinity are not global minimizers, but relative ones. Relative minimizers are the minimizers in the class of patterns with same positions for the strain interfaces. Therefore, one can say that the dynamics selects a relative minimizer, rather than a global minimizer. This highlights a stark contrast between the dynamic and the static picture.

We discuss some recent developments, with the following layout: two different categories of martensitic materials, characterized by weak and reconstructive transformations, are described in §1a,b, respectively. Typically, a symmetry breaking occurs for weak martensitic transformations at the transition point. Phase transformations which are not weak are called reconstructive. In §1, we mention the importance of this classification and some surprising consequences: weak martensitic may exhibit the so-called shape-memory effect (see figure 4); reconstructive phase transformations occur, for example, in steels and are irreversible. The rest of the paper is dedicated to dynamic aspects of martensitic materials and potential implications beyond martensites. We start the tour in §2 on the continuum scale, which leads to an investigation on the atomistic scale in §3. A discussion of the implication on the continuum scale in §3c completes the cycle. An excursion to dislocation models in §3d shows that similar ideas are applicable in a different context.

(a) Weak martensitic transformations and the shape memory effect

What are weak martensitic transformations, such as in NiTi? Let us start by inspecting the microstructure and the underlying phase transition, before discussing a surprising implication on the macroscopic scale. An example of microstructure arising in NiTi is shown in figure 1b. Weak martensitic transformations are characterized by rapid loss of symmetry of the crystal structure in one direction of the transformation (a slightly more general definition is given by Bhattacharya et al. (2004)); they are diffusionless rearrangements of the lattice structure. This is shown in figure 2 for the cubic-to-tetragonal transition. If no loads are applied, above a critical temperature, θc, the lattice forming the structure is stable in a cubic configuration. When a temperature drop below θc occurs, three different symmetry-related tetragonal lattices become stable and the cubic lattice becomes metastable (figure 3). The high-temperature, high-symmetry phase is commonly referred to as austenite, and the low-temperature, low-symmetry phase is called martensite. Figure 2 shows that there are three tetragonal variants of martensite for the cubic-to-tetragonal transformation.

Figure 2

Cubic-to-tetragonal transition. For this transition, the austenitic phase, stable above the critical temperature, is cubic, while the martensitic phase, stable below the critical temperature, is tetragonal. There are three variants of martensite.

Figure 3

Schematic plot of the energy landscape for the cubic-to-tetragonal transition at different temperatures. The vertical axes represent the density of the potential energy, while the horizontal plane symbolises the different crystalline states (in symmetric coordinates). (a) Above the critical temperature, the cubic lattice is stable, i.e. has the smallest potential energy. (b) At the critical temperature θc, the austenitic (cubic) and the martensitic (tetragonal) phases are stable.

Suppose a shape memory material is initially in the austenitic state, and we apply an external biaxial load. The material will first deform elastically as an ‘ordinary’ elastic material. Once a critical load is reached, it will start to transform under stress to martensite. This is called a stress-induced transformation. For suitable alignment of the load and the crystallographic axes, one will observe the formation of martensitic twins (owing to the coexistence of different variants of martensite at temperatures below the critical temperature). This procedure, called twinning, means that atoms on one side of the twin plane undergo a shear displacement relative to the atoms on the other side, in a direction which is parallel to the twin plane. It is often favourable (e.g. to accommodate boundary conditions) to form many interfaces between different martensitic variants. Thus, in this case, twinning occurs on a fine scale and a microstructure is generated. The formation of twin boundaries obviously increases the interfacial energy, but typically facilitates a very significant reduction of the elastic strain energy of the bulk.

It is not hard to see that the coexistence of several martensitic phases is at the basis of the intriguing shape memory effect, which occurs for example in Nitinol (figure 4): a deformation can be reversed upon heating. The transformation is reversible, and the cycle can be repeated many times. Bhattacharya (2003) offers an excellent discussion of how microstructures give rise to the shape memory effect.

Figure 4

The shape memory effect. Macroscopically, small deformations (depending on the material, typically up to 8%) disappear upon heating the material beyond the critical temperature, θc. When cooling down to temperatures below the critical temperature, the specimen remains macroscopically unchanged.

The shape memory effect has numerous applications in medical and engineering devices (Otsuka & Wayman 1998). These materials are candidates for artificial muscles (de Gennes & Okumura 2003). Since, always with the shape memory effect in mind, Nitinol enjoys an adaptivity and flexibility ordinary materials lack, it is regarded as an adaptive material. Lessons learnt from the shape memory effect can often be transferred to similar effects occurring in other adaptive materials, such as ferromagnetic shape memory alloys and ferroelectrics. Salje (1993) discusses martensitic phase changes in metals. Olson & Hartman (1982) observe martensitic phase transformations in biological systems, namely the tail sheath of the T4 bacteriophage virus. A relatively sound modelling on the crystalline level makes martensitic materials an ideal sand box for a mathematical analysis of multiscale phenomena. Before discussing mathematical models for the shape memory effect, we first contrast the martensitic transformations in shape memory alloys with those occurring in steels.

(b) Martensites in steel

Steel is another example of a material forming martensitic structures, and the microstructure plays a decisive role in the mechanical properties. Notably, martensitic transformations are induced in steel by quenching (rapid cooling), to enhance the alloy's strength (Olson & Owen 1992). An ongoing endeavour in casting and rolling of steel products is to influence the microstructure and optimize it in the production process. This industrially relevant problem is not quite completely understood. The challenges are close at hand: steel is usually rolled in mills in order to produce strips of a desired thickness and width. Typically, in a so-called hot rolling mill, the strip enters at a speed of 1 m s−1 at a temperature of 1100°C, and exits at 900°C with a speed of 3–15 m s−1. In the subsequent cooling section, the strip is cooled down with water to ca 500–600°C (M. Kurz & M. Metzger 2006, personal communication). The temperature gradient influences the microstructure, and the aim is to control a structure, at most, of a few micrometres at such high speeds and temperatures!

A stark contrast is exhibited between microstructures in steel and shape memory alloys. In the latter, the microstructure formed at low temperature disappears upon reheating, and the transformation is reversible (Otsuka & Wayman 1998). In the former, a microstructure generated by quenching remains essentially unchanged upon loading or heating, and the transformation is not reversible (Olson & Owen 1992).

Bhattacharya et al. (2004) recently accounted for this difference on the basis of the symmetry of phases involved in the transition. Specifically, they showed that two different classes of martensitic transformations are plausible. One class, which includes steels, sports ‘cheap’ plastic deformations (via lattice-invariant shears) in the sense that such a deformation is energetically not more expensive than a phase transition. The occurrence of plastic deformations then introduces an irreversibility. For the other class, the energetic price for lattice-invariant shears can be much higher than that for phase changes. Consequently, these transformations can be reversible. The common shape memory alloys belong to this class.

2. Macroscopic models for shape memory alloys

The question is how to model martensitic materials, and in particular, account for dynamic effects in shape memory materials. We begin the journey through mathematical models for shape memory alloys on the macroscopic level, before travelling to the microscopic level, and returning to the continuum (macroscopic world). Although we should pause at the mesoscopic scale to discuss grain boundaries and the polycrystalline nature of materials and their influence on macroscopic properties, we rather refer the reader to the literature (Bhattacharya & Kohn 1997; Kružík & Otto 2004; Chenchiah & Bhattacharya 2005).

Mathematical models for shape memory alloys are often expressed in terms of continuum mechanics. Let us assume that, at a fixed time, the material occupies a region Embedded Image. Then the key variables describing the alloy are the displacement field Embedded Image, (x, t)↦u(x, t), the velocity Embedded Image and the strain Embedded Image. Let Φ=Φ(F) with Embedded Image denote the elastic energy density, which is a function of the deformation gradient. The (Piola–Kirchhoff) stress tensor is Embedded Image.

In classical elastodynamics, the equations of motion are considered as the governing partial differential equations,Embedded Image(2.1)(Here and in the following, the dynamics are assumed to take place isothermally, since many challenges can already be cast in this setting. The temperature will be below θc, so that several variants are stable.)

(a) The elastic energy density

A key ingredient to equation (2.1) is the elastic energy density, Φ, since it determines the stress tensor, σ. What is the shape of Φ for a shape memory alloy? This question proves to be surprisingly tricky. In general, Φ=Φ(F), with F denoting the deformation, can be written as a function Embedded Image, to account for the invariance of the energy under a change of coordinates of an observer. Thus, the energy is a function of the six components of the (right) Cauchy–Green tensor FTF. For a martensitic material, the energy has to meet a symmetry condition imposed by the austenitic phase. Symmetric functions in six components with minimizers at prescribed positions, correct elastic moduli (second derivatives) and a phenomenologically correct dependence on the temperature are not easy to write down explicitly.

Consequently, only a handful of specific examples of energy densities can be found in the literature. A prominent one is given by R. D. James (1988, unpublished notes) and Ericksen (1986) for the cubic-to-tetragonal transformation in InTl. Vedantam (2000) uses irreducible polynomials of the strain tensor to derive an elastic energy density for CuAlNi. A different line of thought introduces symmetry-adapted nonlinear coordinates in conjunction with suitable splines (Zimmer 2004; Hormann & Zimmer submitted); the additional degrees of freedom make it possible to fit elastic moduli, which cannot be fitted with invariant polynomials of lowest order.

There seems to be no general framework to derive macroscopic energy densities from ab initio calculations. However, Vedantam (2000) specializes his macroscopic energy density to a shear deformation on a twin plane along the twinning direction and obtains a remarkable agreement with an energy calculated from molecular statistics. The approach to combine symmetry-adapted coordinates and splines may provide a general method to accommodate the wealth of data obtained by ab initio calculations.

(b) Ill-posed equations of motion

Unfortunately, the problems only start once we have chosen an energy density and try to solve an initial-boundary value problem for equation (2.1). To see this, let us consider the simplest situation, a one-dimensional bar under a tensile force. We also ignore effects of inertia and consider the equilibrium situation. It is not hard to see that the equilibrium configuration is generically not uniquely determined; any of the infinitely many solutions involving a mixture of stable variants may be the response of the material to the applied force.

It is useful to reformulate this insight in mathematical terms. Since Φ is non-convex for low temperatures, the stress tensor will be non-monotone. It is immediate that the hyperbolicity condition σ′(ux)>0 is thus violated. This implies that equation (2.1) changes type.

Young measures have been introduced as generalized solutions (Ball & James 1987). For details, we refer the reader to lecture notes by Bethuel et al. (1999).

(c) Rankine–Hugoniot conditions

The ill-posedness of equation (2.1) arises as a consequence of the coexistence of two phases. Thus, let us investigate the situation more closely at an interface between the two coexistent phases. For simplicity, we think of the interface as straight. It is convenient to take unstressed austenite as the reference configuration; then two martensitic variants meeting at the interface are generated by different deformations, since they are characterized by a distinct crystallographic lattice and orientation. Thus, at the interface, the strain will be discontinuous.

This simple observation is already the key, as becomes apparent once we apply a force in this Gedanken experiment to ‘jog the shape memory’, i.e. make the phase boundary move. Assume that the force favours one of the two variants at the interface. Then, the interface propagates towards the phase with higher energy, since this is transformed into the low-energy phase. It is not hard to see that a discontinuity such as the interface cannot move arbitrarily; the RankineHugoniot conditions state that if the interface moves with the velocity, Embedded Image, and either the strain, ux, or the velocity, Embedded Image, are discontinuous at the interface, the following relations must hold:Embedded Imagewhere, for a function f(x, t), 〚f〛 stands for f(s(t)+, t)−f(s(t)−, t), i.e. the difference of the limiting values from the right and the left.

A moving interface can be seen to dissipate energy, and the amount of dissipation is measured by the configurational force (or driving force). To define it, we let Embedded Image denote the average stress across the discontinuity. Then, the configurational force isEmbedded ImageThe rate of the energy dissipation is then given by Embedded Image. The entropy inequality requires that Embedded Image.

There are two kinds of moving interfaces, subsonic or supersonic with respect to the sound speed of the surrounding elastic medium, which is here the same for both the phases. Supersonic phase boundaries satisfy the Lax condition for hyperbolic PDEs, while subsonic phase boundaries violate this condition, and it can be seen that this leads to an ill-posed initial problem for equation (2.1). Thus, unless otherwise indicated, we focus on the subsonic case, since it poses particular challenges.

For subsonic waves, how can physically relevant solutions be singled out to make the problem well posed? Abeyaratne & Knowles (1991) introduce a kinetic relation as additional constitutive information to determine the macroscopic response of the body. The kinetic relation determines a functional dependence of the configurational force on a phase boundary and a velocity,Embedded ImageThe motivation behind this line of thought is that, according to the principles of irreversible thermodynamics, the propagation speed of the interface is determined by the configurational force, even though the laws of thermodynamics only impose restrictions on this relationship without specifying the relationship explicitly (Abeyaratne & Vedantam 2003).

(d) Which kinetic relation?

It is not obvious how to derive kinetic relations, and different approaches have been taken. One approach is to augment equation (2.1) by regularizing terms and derive kinetic relations from the augmented model. This is a successful strategy, but the physical meaning of the regularization is not immediate in the case of shape memory alloys. The complex nature of martensitic phase transitions, exhibiting a very rich phenomenology on various scales, casts some doubts on the possibility of choosing a regularization on physical grounds, be it by a penalization of interfaces (capillarity) or by artificial viscosity.

Alternatively, it is reasonable to model kinetic relations phenomenologically. Abeyaratne & Knowles (1997) calculated kinetic relations for an austenite–martensite interface in CuAlNi and found dry-friction type of behaviour.

Here, a different line of thinking is presented. The key idea is to resort to the atomistic level. Namely, underlying the macroscopic kinetic relation is the dynamics of the lattice as a result of the crystallographic transformation of one variant into another. Thus, let us replace equation (2.1) with a discretized counterpart. We obtain a model of discrete elasticity, where a one-dimensional body is simply regarded as a chain of atoms, linked by springs which represent chemical bonds. Lattice models automatically generate a kinetic relation as a consequence of radiative damping (Kevrekidis & Weinstein 2000).

3. Lattice models

We consider a two-sided infinite chain of atoms and assume nearest neighbour interactions (figure 5). The atoms are identical and, in normalized variables considered below, they have unit mass. The springs are identical, but will be nonlinear. For each spring, the elastic energy is a function Embedded Image, which takes the discrete strain as the argument.

Figure 5

The model of discrete elasticity considered in §3. The atoms are coupled to their neighbours by nonlinear springs.

The scaling is such that the reference interparticle distance is one. For atom Embedded Image, the displacement is Embedded Image. The discrete strain is given by the difference of the displacements, say uk+1(t)−uk(t). Newton's law then gives the equations of motionEmbedded Image(3.1)for every Embedded Image.

(a) A 50-year-old challenge: anharmonic interaction potential

The study of equation (3.1) has a lively history, which can be traced back at least to a famous computer experiment by Fermi et al. (1955) in Los Alamos. They considered a chain of 64 identical atoms, interacting with their nearest neighbours as described by equation (3.1), with an anharmonic potential, Φ, with a sinusoidal initial condition. The anticipation was that equipartition of the energy would occur, meaning that the lattice would reach a thermal equilibrium, and that averaged over time, the initial energy would be equipartitioned among all Fourier modes. Astonishingly, the discovery was that the system did not approach thermal equilibrium, but behaved essentially quasi-periodically; thermalization was only found for an initial energy exceeding a certain threshold. Significant progress has been made in the mathematical analysis of the Fermi–Pasta–Ulam (FPU) lattice. Friesecke & Wattis (1994) use a variational approach to obtain localized solutions, and Friesecke & Pego (1999) give, in a series of impressive papers, a very detailed description of solitary waves in the FPU lattice. Iooss (2000) uses a centre-manifold approach to find all the small bounded travelling waves for generic elastic energies near the first critical wave velocity. These results either assume the convexity of the elastic energy or prove results for small amplitude solutions, where it is an open question whether the confinement to a convex region within the potential can be overcome.

(b) Travelling phase transitions on the atomistic scale

For phase transitions, Φ is non-convex and thus the springs are bistable. This non-convexity makes the mathematical analysis even harder, and our current understanding of this situation is very fragmentary.

To identify the kinetic relation, it is natural to seek travelling waves as solutions to equation (3.1). A travelling wave is a special solution of the formEmbedded Image(3.2)We insert this ansatz in equation (3.1) and obtainEmbedded Image(3.3)

Does equation (3.3) have a solution? Truskinovsky & Vainchtein (2005) consider the case where the elastic energy is piecewise quadratic (and each particle may interact with l neighbours, with a quadratic energy for every interaction except for the nearest neighbours). An explicit solution can be computed which has a vanishing strain at 0 and two different strains as ξ→±∞; the atoms at the left of the interface are in one phase, while those on the right are in the other phase.

For a general elastic energy, the picture is largely open. Schwetlick & Zimmer (in press) prove the existence of supersonic solitary waves, where the strain is asymptotically for ξ→±∞ in one state (the left well of the elastic energy, say), and in between makes an excursion towards the right well. The proof relies on a mountain pass argument and assumes a sufficiently high wave speed c. A qualitative description of the solution is not known. In particular, it remains open whether the homoclinic excursion in the strains reaches the second well. Another challenging open problem concerns the stability of these solitary waves.

(c) Return journey to the macroscopic world

For a piecewise linear stress–strain relation, specifically for a nearest neighbour interaction potential, which is the pointwise minimum of two quadratic wells, one can compute more, and use the microscopic information on the journey back to the macroscopic world. We recall that the microscopic model (3.1) is Hamiltonian, but the kinetic relation is to specify a macroscopic dissipation. Truskinovsky & Vainchtein (2005) construct explicit (microscopic) travelling wave solutions (with piecewise quadratic interaction potential) and show that the radiation of lattice waves carrying energy away from the propagating front results in a macroscopic dissipation. There seems to be no rigorous proof of stability for the travelling waves, but numerical simulations (Truskinovsky & Vainchtein 2005) indicate that at least some of the travelling waves are stable. For the same setting, Truskinovsky & Vainchtein (2006) give various dispersive quasicontinuum approximations, based either on Taylor or rational (Padé) approximations, and show that the associated kinetic relation agrees well with its microscopic prediction.

Abeyaratne & Vedantam (2003) derive a kinetic relation for twin boundary motion from a lattice model. They consider a ledge propagating transversely along the twin boundary. Abeyaratne & Vedantam (2003) determine the forward motion of the twin, and pass to the continuum using a higher order expansion. The lattice model resembles the Frenkel–Kontorova model discussed later in §3d. Remarkably, in the continuum limit, the solution exhibits an oscillatory behaviour behind the travelling wave, rather than converging asymptotically to a fixed displacement.

(d) An excursion to dislocations

The analysis of travelling waves and kinetic relations for phase transitions bears many similarities with the analogous study of dislocation dynamics. A simple model of the latter has been given by Frenkel & Kontorova (1939). The governing equation isEmbedded Image(3.4)which differs from equation (3.1) in the sense that the elastic energy is here assumed to be harmonic, and there is an additional on-site potential V. For the Frenkel–Kontorova model, V is periodic, V(u)=−cos(u). Again, uk denotes the displacement of particle k; the coupling constant γ is strictly positive.

In the mathematics literature, equation (3.4) is often referred to as the discrete KleinGordon equation. Then, V is taken to be anharmonic and analytic with V(0)=0 and V″(0)>0. Here, the centre-manifold approach yields ‘nanopterons’ (solitary waves superposed on exponentially small oscillatory tails), as shown by Iooss & Kirchgässner (2000). Of particular interest here are travelling breathers, i.e. spatially localized solutions which are time-periodic in a system of reference moving at constant velocity. Numerical calculations here as for the FPU chain are a delicate task. Duncan et al. (1993) report that they could only find travelling waves superposed to a tail, which appears periodic with a small amplitude, but no travelling waves with the property u(z)→0 mod 2π as z→±∞. It may be that this background radiation is necessary to enable a lossless propagation of the wave. Likewise, it is possible that the exponentially small tails are essential for the propagation of a wave which is otherwise solitary. This phenomenon seems not to be understood.

Currently, there is no rigorous framework for the passage from the atomistic scale to the continuum for dislocation dynamics. Carpio et al. (2001) discuss some continuum models for dislocation densities and highlight open problems. Again, some models are ill-posed, and thus some challenges resemble those we discussed earlier for martensites. The forthcoming book by W. Cai & V. Bulatov (computer simulations of dislocations) offers a detailed discussion of dislocation dynamics on different scales.

(e) Modulation and macroscopic models

Modulation theory gives a formal way of deriving a thermodynamic description for lattice models. Here, the goal is to obtain macroscopic equations describing temperature effects due to microscopic oscillations, rather than to find kinetic relations. We thus leave the world of phase transitions and dislocations and consider lattice models with convex interaction potentials. Let us assume that the microscopic dynamics is described by equation (3.1) (thus, t is the microscopic time and k is the microscopic particle index). From this information, we would like to infer the evolution of macroscopic thermodynamic fields, such as momentum, energy and entropy.

A powerful, yet formal, procedure for this micro–macro transition is as follows. On the microscopic level, temperature corresponds to oscillatory behaviour of the atoms. Thus, it is reasonable to consider periodic travelling waves which may mimic these oscillations. Let the travelling wave be given as Embedded Image for Embedded Image (compare equation (3.2)). The parameters are mean distance, r, mean velocity, v, wave-number, j, and frequency, ω. We define a macroscopic time τϵt and a macroscopic particle index yϵk (this is the hyperbolic scaling; other scalings yield the Korteweg–deVries equation (Friesecke & Pego 1999) and the nonlinear Schrödinger equation (Giannoulis & Mielke 2006) as governing macroscopic evolution equations). The main idea of modulation theory is to consider approximate travelling wave solutions by allowing the travelling wave parameter to vary on the macroscopic scale. The travelling wave parameters become fields in the macroscopic variables τ and y. Dreyer et al. (2006) describe this procedure and obtain four conservation laws for the four parameters r, v, j and ω, known at Whitham's equations. We refer the reader to the literature (Dreyer et al. 2006) for the precise procedure and further references.

Giannoulis & Mielke (2006) proved the validity of the nonlinear Schrödinger equation as an approximation of a nonlinear version of the Klein–Gordon equation (where the interaction potential is allowed to be nonlinear) with a nonlinear on-site potential for large, but finite time-intervals. For the FPU lattice, the validation of a modulation equation seems to be open.

4. Looking ahead

Several open problems for martensitic materials have been highlighted in the previous discussion, and they are challenging mathematical questions in their own right. On the atomistic scale, key challenges are to remove a smallness assumption, for example, on the amplitude of a wave, and thus gain a qualitative knowledge of lattice waves in phase-transforming materials. Furthermore, the rigorous passage from atomistic to continuum is widely open.

Implications of a deeper mathematical appreciation are likely to go beyond shape memory alloys. In general, our understanding of the micro–macro transition is very poor. For example, does the motion of defects give rise to friction, a phenomenon we encounter every day but do not understand well at all? If so, can we develop mathematical ways of deriving macroscopic descriptions for, say, defect densities and their motion, rather than postulating such laws phenomenologically?

There are other examples of defect dynamics which we would like to understand better, among them are crack tips, and interstitials and vacancies. Defect phenomena on the atomistic scale are of direct relevance to materials science, solid-state physics and physical chemistry. For many applications, such as fuel cells, these transport phenomena need to be understood. The movement of a martensitic phase boundary may be seen as a relatively simple problem with a moving singularity, and further advancement in this field is likely to spark insights in the aforementioned disciplines.


The author is particularly grateful for the assistance received from Frank Hammett of the Department of Mechanical Engineering at the University of Bath in the preparation of micrographs shown in figure 1. Sincere thanks are expressed to Hans-Georg Zimmer for drawing figures 2–4. The author gratefully acknowledges the financial support of the EPSRC through an Advanced Research Fellowship (GR/S99037/1).


  • One contribution of 23 to a Triennial Issue ‘Mathematics and physics’.


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