Shape Memory Wires in R3

We propose a new model describing the dynamics of wire made of shape memory alloys, by combining an elastic curve theory and the Ginzburg-Landau theory. The wire is assumed to be a closed curve and is not to be stretched with deformation. The derived system of nonlinear partial differential equations consists of a thermoelastic system and a geometric evolution equation under the inextensible condition. We also show that the system has dual variation structure as well as a straight material case. The structure implies stability of infinitesimally stable stationary state in the Lyapunov sense.


Introduction
Shape memory effect arises from the phase transition of lattice structure. Although there are many models for shape memory alloys, one of the classical model is proposed by Falk, which we call the Falk model. Falk applied the Ginzburg-Landau theory for phase transition to shape memory alloys by regarding shear strain E as an order parameter (see e.g., [1]). That is, the Helmholz free energy density proposed by Falk is given by f ðE, θ, ∂ 2 x uÞ :¼ where u and θ are displacement and absolute temperature, respectively, and a positive constant θ c denotes the critical temperature of the phase transition. We call the first term j∂ 2 x uj 2 2 curvature energy density, the second term f(E, θ) nonlinear elastic energy density and the third term f 0 (θ) thermal energy density. In other words, Falk represented the phase transition by using the form of the nonlinear elastic energy density. We also remark that for simplicity, all physical constants without the critical temperature are normalized by unity. The Falk model was proposed for straight materials. In the model, the material is built up by a stack of layers parallel to the so-called habit plane (see [2]) and assumed that the displacement u in that direction to depend only on a coordinate x perpendicular to the habit plane, that is, the variable x runs in the stacking direction. Then, the material conserves its volume.
From the point of the small deformation theory, we may use a linearized approximation relation E = ∂ x u. Moreover, we take f 0 as the following typical form: f 0 ðθÞ :¼ θ À θlogθ: Then following a standard procedure of derivation of thermoelastic system (see e.g., [3]), we can derive the system of nonlinear partial differential equations called the Falk model: Here, unknowns are displacement u and absolute temperature θ, and ∂ x and ∂ t represent partial differential operator with respect to x and t, respectively. The model is well known as one of classical models describing shape memory alloys. For the other models, we refer Fremond [4], Fremond-Miyazaki [5] and reference therein. The Falk model (Eq. (2)) has been studied actively in the mathematical literature. In the isothermal case, well-posedness, stability of solitary-wave solution, existence of steady state, travelling wave solution and invariant measure have been investigated by Fang-Grillakis [6], Falk-Laedke-Spatschek [7], Friedman-Sprekels [8], Garcke [9] and Tsutsumi-Yoshikawa [10], respectively. For the full system (Eq. (1)), the well-posedness results are found in, for example, [11][12][13] and so on, and numerical results are found in Hoffmann-Zou [14], Niezgodka-Sprekels [15] by finite element method and in Matus-Melnik-Wang-Rybak [16] and Yoshikawa [17,18] by the finite difference method. In particular, in Ref. [19], the stability of steady state in the Lyapunov sense was shown. More precisely, the stationary state of Eq. (2) is expressed as a nonlocal nonlinear elliptic problem. If there exists a linearized stable critical point for the functional corresponding to the elliptic problem, then for each neighbourhood U of the equilibrium, we can find a neighbourhood W of the equilibrium such that the solution of Eq. (2) with the initial data in W stays in U for any time. The proof can be shown by the dual variation principle which appears in most of the models in non-equilibrium statistical thermodynamics (see [20]).
The existence of several non-trivial steady states for low-temperature phase and low-energy case is proved in Ref. [21]. The numerical simulation given in Ref. [17] exactly indicates the properties mentioned above. The dual variation structure appears also in a multi-dimensional case [22]; however, well-posedness of the multi-dimensional model corresponding to Eq. (2) is still open in large initial data case due to the propagation of singularity. That is one of our motivations of this problem.
We mention mathematical studies on the motion of curves governed by geometric evolution equations. One of the typical objects is curve-shortening flow derived as an L 2 gradient flow for the length functional of curve γ: where s denotes the arc length parameter of γ. By Gage [23], Gage and Hamilton [24] and Grayson [25], it is well known that the curve-shortening flow shrinks simple closed curves to a point in a finite time. Since the curve-shortening flow can be regarded as a one-dimensional case of mean curvature flow for surfaces, the flow is applicable to various mathematical analysis. For example, the curve-shortening flow plays an important role in studies on phase transition. We also mention the curve-straightening flow which has been attracted a great interest and studied actively in mathematical literature. The flow is derived as an L 2 gradient flow for the elastic energy where κ denotes the scalar curvature of γ. It is well known that the flow is applicable to studies on elastic curve inspired by Bernoulli and Euler. Indeed, the curve-straightening flow under the length constraint L(γ) C converges to a classical elastic curve so-called elastica. There is also an interest in the study on motion of curves governed by the L 2 -gradient flow for E under the inextensible condition. Under the condition, the length constraint L(γ) C is also satisfied for the condition means that the curve does not stretch. As we will state in Section 2.2, the constraint is imposed on each point of curves. Thus, a standard Lagrange multiplier theory does not work. Therefore, we have to make use of geometric properties of curves governed by the flow.
The purpose of this chapter is to derive a mathematical model describing thermoelastic deformation of shape memory wire in R 3 . In particular, we regard the wire as a closed space curve satisfying the inextensible condition. From the physical point of view, it may be unnatural that the wire does not stretch. However, the contribution of this chapter is to adopt a geometric analysis into a classical thermoelastic theory with phase transition inspired by Falk.
Throughout this chapter, we denote by L the length of Γ 0 , and hence, the length of Γ is also L from the non-stretching assumption. To apply the idea by Falk, we need to determine the form of strain and free energy (Eq. (1)) suitable for this setting.

Definition of strain
We first consider the strain. Let γ 0 (ξ) be a space closed curve, where ξ is a parameter (not necessary to be the arc length parameter). For γ 0 (ξ), we define the displacement vector by u(ξ), and we denote γðξÞ ¼ γ 0 ðξÞ þ uðξÞ. Since the relation "strain ≈ line element" holds, let us first pursue line element between γ 0 (ξ) and γ (ξ). From the direct calculation, we have Here, if we assume a smallness of deformation, then we may assume From now on, we regard the strain as γ 0 0 ðξÞ Á u 0 ðξÞ:

Definition of energy functional
Let γ 0 (ξ) be an initial closed curve and γ(ξ, t) denote a family of closed curves starting from γ 0 (ξ). Recalling that the arc length parameter s(ξ, t)of γ(ξ, t) is given by we can write kinetic energy M(γ) of γ as In a similar manner, thermal energy is defined by and the curvature energy is expressed as Observe that the scalar curvature κ is written as Lastly, the nonlinear elastic energy density is given by and then the nonlinear elastic energy is written as where u = γ À γ 0 . Thus, we obtain the Helmholtz energy for our setting as From now on, let s ∈ R=LZ ¼: S 1 L be the arc length parameter of the initial closed curve γ 0 = γ 0 (s). It follows from the property of arc length parameter that |γ 0 0 (s)| 1. In a similar fashion to the above equation, γ(s, t) means the closed curve deformed along evolution from γ 0 (s). Moreover, in what follows, we assume that γ(s, t) satisfies j∂ s γðs, tÞj 1 ð3Þ which means "s is arc length parameter of γ not only the initial time but also every time t". From the assumption, we can rewrite M, F 0 and F shortly as Therefore, the Helmholtz free energy density is denoted by We will explain that for the free energy under some assumptions, the following system of nonlinear partial differential equations is derived: where

Equation of motion
By using the Hamilton principle, we derive an equation of the motion of γ. Namely, we will derive the Euler-Lagrange equation for the functional: Let us denote the variation of γ by γðs, t; θÞ :¼ γðs, tÞ þ ε ϕðs, tÞ where ε is a sufficiently small positive parameter and ϕ is sufficiently smooth and satisfies ϕ(s, t 1 ) = ϕ(s, t 2 ) = 0. Moreover, from the assumption Eq. (3), it is also necessary to hold that ϕ has to satisfy ∂ s γðs, tÞ Á ∂ s ϕðs, tÞ ¼ 0 for any s ∈ S 1 L and t > 0. Calculating the first variation of the energy functional, we have From the integral by parts, the right-hand side is rewritten as follows: Then the integral Eq. (6) is equal to 0 for any ϕ satisfying ϕðs, For the purpose, we define The orthogonal complement V ⊥ of the space V with respect to L 2 ðS 1 L Þ inner product is given by Here, we remark that in the case where γ is a curve embedded in three-dimensional space (not a planar curve), ∂ 2 s γ 6 ¼ 0 has to be satisfied for every ðs, tÞ ∈ S 1 L Â R þ . In the end of this section, we will show the reason why V ⊥ is given as above. Consequently, if for the direction ∂ 2 t γ þ ∂ 4 s γ À ∂ s f , ∂sγ ð∂ s γ, θ; ∂ s γ 0 Þ, there exists a scalar function w = w(s, t) such that then Eq. (6) is equal to 0.
The assumption ∂ 2 s γ 6 ¼ 0 in Lemma 1 means that the curvature is always non-zero. If ∂ 2 s γ ¼ 0 at some point, we cannot determine the tangential vector ∂ s γ at the point uniquely. Therefore, we need the assumption in order to give a coordinate system at every point of Γ. On the other hand, in the case of a planar curve, we do not need the assumption. Indeed, by rotating the tangential vector, we can construct a coordinate system. Remark 2. We mention the elastic flow with the inextensible condition (Eq. (3)), more precisely, L 2 gradient flow for K(γ) under the constraint (Eq. (3)). To the best of our knowledge, the problem was first considered by N. Koiso [26] for planar closed curves. With the aid of smoothing effect of the elastic energy E, the Cauchy problem on the elastic flow has a unique classical solution and the solution converges to an equilibrium state as t ! ∞ in the C ∞topology. The result can be extended to the following case: (i) L 2 gradient flow for E under the area-preserving condition and (C) [27] and (ii) L 2 gradient flow for Tadjbakhsh-Odeh energy functional under the constraint (C) [28]. Moreover, the result [26] was also extended to the case of space curves [29].

Derivation of heat equation
In this subsection, we study the energy law. We confirm thermal energy conservation law (the first law of thermodynamics) and the increasing law of entropy (the second law of thermodynamics). To begin with, we consider the first law of thermodynamics. According to Ref. [30], thermal energy conservation law for thermoelastic system is given by where S, q and h are entropy, thermal velocity and external heat, respectively. In our setting, thermal transfer occurs only on wire, and the wire does not expand. Then, we may regard ∇ Á q as ∂ s q as the same as one-dimensional case, where s is necessary to be arc length parameter. By the same reason, the Fourier law q = ∇θ is replaced by The Helmholtz free energy density and the entropy S are connected with the relation Then, the conservation law, Eq. (14), is rewritten as follows: We note that the Clausius-Duhem inequality holds automatically: Indeed, we observe from Eqs. (14) and (15) that The Clausius-Duhem inequality corresponds to the second law of thermodynamics. For more precise information of the inequality, we refer to, for example, 1.11 of chapter 4 in Ref. [2].
Here, we assume external heat source h = 0 and adopt the well-known form: Then since f 0 0 0 ðθÞ ¼ À1=θ, Eq. (16) is reduced to We thus obtain the system of equation as Eq. (4).
We assume that the system has sufficiently smooth solution (γ, θ) satisfying θ > 0. Then initial data also has to satisfy we have the relation f ¼ θf 1 þ f 2 . Multiplying Eq. (17) by ∂ t γ and integrating it with respect to s, we obtain d dt Integrating Eq. (19), we find d dt Then for the quantity Eðγ, ∂ t γ, θÞ :¼ Finally, we confirm an important structure of the system (17)- (20). We denote the stationary state of θ by θ > 0, and the corresponding equilibrium by γ satisfy the following constraint: b Eðγ 0 , γ 1 , θ 0 Þ ¼ Eðγ, 0, θÞ, Eliminating θ by the relation the stationary state of this problem satisfies the following nonlinear nonlocal problem: Eq. (21) is derived as the Euler-Lagrange equation of the functional where y ¼ ∂ s γ ∈ H 2 ðS 1 L , S 2 Þ and S 2 :¼ fω ∈ R 3 jjωj ¼ 1g: We remark that the following relation between J b and W holds true: Wð∂ s γ, θÞ ≥ LJ b ð∂ s γÞ: The relation is called semi-unfolding minimality. Thus, if (γ, θ) is a non-stationary state, b ¼ Eðγ 0 , γ 1 , θ 0 Þ, y ¼ ∂ s γ is a linearized stable critical point of J b ¼ J b ðyÞ, y ∈ H 1 ðS 1 L , S 2 Þ and θ > 0 is a constant satisfying Eðγ, 0, θÞ ¼ b, then it holds that for y ¼ ∂ s γ J b ðyÞ À J b ðyÞ ≤ Wð∂ s γ 0 , θ 0 Þ À Wðy, θÞ: By this structure, we can infer that any infinitesimally stable stationary state is dynamically stable, that is, stable in the Lyapunov sense. A critical point y ¼ ∂ s γ of J b for γ ∈ H 2 ðS 1 L , S 2 Þ is infinitesimally stable if there exists ε 0 > 0 such that any ε 1 ∈ 0, ε0 2 À Ã admits δ 0 > 0 such that if jj∂ s ðγ À γÞjj H 1 < ε 0 and J b ð∂ s γÞ À J b ð∂ s γÞ < δ 0 then jj∂ s ðγ À γÞjj H 1 < ε 1 : The definition of infinitesimally stable is obviously weaker than the one of well-known linearized stable which means that for a critical point y ¼ ∂ s γ of J b , the quadratic form is a positive definite for any w ∈ H 2 ðS 1 L , S 2 Þ.
From the semi-unfolding minimality (Eqs. (24) and (25)) and the linearized stability of J b , we observe that where δ is small enough such that δ < ε / C. This completes the proof. Q.E.D.
Remark 3. Both the existence of solution for evolution equations (Eq. (4)) and non-trivial solutions for stationary problem (Eq. (21)) are open problems. In the straight material case (i.e. the problem (2)), smooth solution for Eq. (2) is assured in [11] (we also refer to chapter 5 in [2]). The existence results of non-trivial solution for stationary problem (Eq. (21)) in lowtemperature and low-energy cases can be found in [21,31].

Concluding remarks
In this chapter, we propose the new mathematical model describing the movement of wire made of shape memory alloys. The derived system of nonlinear partial differential equations is a thermoelastic system with phase transition and non-stretching constraint. The Falk model (Eq. (2)) represents the dynamics for crystal as a stack of layers, whose displacement is restricted to move only on one direction. On the other hand, our model describes the dynamics of wire. We emphasize that our model allows the displacement of each direction. Thus, our model may describe a more realistic motion of wire made of shape memory alloys. Moreover, it is also interesting to regard our model as a mathematical problem on elastic curve with heat conduction. To the best of our knowledge, there is no result considering such a mathematical problem.
We mention the mathematical contribution of the present chapter. We prove the dynamical stability of an infinitesimally stable stationary state by finding the dual variation structure in our model. This property shall be applicable, for example, to assure the strength of not only a wire in an original shape but also a deformed wire. Indeed, in the straight material case, namely in the Falk model (Eq. (2)), numerical simulation shows the stability in this sense (see [17]).