Open access peer-reviewed chapter

Shape Memory Wires in R3

Written By

Shinya Okabe, Takashi Suzuki and Shuji Yoshikawa

Submitted: November 23rd, 2016 Reviewed: April 12th, 2017 Published: September 20th, 2017

DOI: 10.5772/intechopen.69175

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Abstract

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.

Keywords

  • shape memory alloys
  • elastic curve
  • thermoelastic system
  • nonlinear partial differential equations
  • Ginzburg-Landau theory
  • phase transition
  • stability
  • dual variation principle

1. 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 ϵas an order parameter (see e.g., [1]). That is, the Helmholz free energy density proposed by Falk is given by

f˜(ϵ,θ,x2u):=12|x2u|2+f(ϵ,θ)+f0(θ)=12|x2u|2+(16|ϵ|614|ϵ|4+12(θθc)|ϵ|2)+f0(θ)E1

where uand θare displacement and absolute temperature, respectively, and a positive constant θcdenotes the critical temperature of the phase transition. We call the first term |x2 u|22curvature energy density, the second term f(ϵ, θ) nonlinear elastic energy densityand the third term f0(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 uin that direction to depend only on a coordinate xperpendicular to the habit plane, that is, the variable xruns 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 ϵ= ∂xu. Moreover, we take f0 as the following typical form:

f0(θ) :=θθlogθ.

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:

{t2u +x4u=x{(xu)5(xu)3+(θθc)xu},tθx2θ=θxu txu.E2

Here, unknowns are displacement uand absolute temperature θ, and ∂xand ∂trepresent partial differential operator with respect to xand 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, [1113] 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 Uof the equilibrium, we can find a neighbourhood Wof the equilibrium such that the solution of Eq. (2) with the initial data in Wstays in Ufor any time. The proof can be shown by the dual variation principlewhich 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 L2 gradient flow for the length functional of curve γ:

L(γ):=γds

where sdenotes 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 L2 gradient flow for the elastic energy

K(γ):=12γκ2 ds

where κdenotes 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) ≡ Cconverges to a classical elastic curve so-called elastica. There is also an interest in the study on motion of curves governed by the L2-gradient flow for Eunder the inextensible condition. Under the condition, the length constraint L(γ) ≡ Cis 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 R3. 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.

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2. Setting and derivation of equations

We denote the closed curve representing shape of wire by Γ= {γ(ξ): ξ∈ Ξ}, where the variable ξis an arbitrary parameter not necessarily the arc length parameter. Let us define a displacement vector from a point ξin an original shape Γ0 = {γ0(ξ): ξ∈ Ξ} by u(ξ) (see Figure 1); namely, it holds that

Figure 1.

Original shape Γ0 and deformed curve Γ.

γ(ξ)=γ0(ξ)+u(ξ)

for

  • γ(ξ) := (γ1(ξ), γ2(ξ), γ3(ξ)): vector representing the shape,

  • γ0(ξ):=(γ10(ξ),γ20(ξ),γ30(ξ)): vector representing the original shape,

  • u(ξ):=(u1(ξ),u2(ξ),u3(ξ)): displacement vector.

Throughout this chapter, we denote by Lthe length of Γ0, and hence, the length of Γis also Lfrom 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.

2.1. Definition of strain

We first consider the strain. Let γ0(ξ) be a space closed curve, where ξis a parameter (not necessary to be the arc length parameter). For γ0 (ξ), we define the displacement vector by u(ξ), and we denote γ(ξ)=γ0(ξ)+u(ξ). Since the relation “strain ≈ line element” holds, let us first pursue line element between γ0 (ξ) and γ(ξ). From the direct calculation, we have

|γ(ξ)|2|γ0(ξ)|2={γ0(ξ)+u(ξ)}{γ0(ξ)+u(ξ)}γ0(ξ)γ0(ξ)=2γ0(ξ)u(ξ)+|u(ξ)|2.

Here, if we assume a smallness of deformation, then we may assume

|γ0(ξ)u(ξ)||u(ξ)|2.

From now on, we regard the strain as

γ0(ξ)u(ξ).

2.2. Definition of energy functional

Let γ0(ξ) be an initial closed curve and γ(ξ, t) denote a family of closed curves starting from γ0(ξ). Recalling that the arc length parameter s(ξ, t)of γ(ξ, t) is given by

ds=|γξ|dξ,

we can write kinetic energy M(γ) of γas

M(γ):=|γt|2|γξ|dξ.

In a similar manner, thermal energy is defined by

F0(γ):=f0(ξ)|γξ|dξ,

and the curvature energy is expressed as

K(γ):=κ2|γξ|dξ.

Observe that the scalar curvature κis written as

κ={|γξ| 2γξ2(γξ2γξ2)γξ }|γξ|3.

Lastly, the nonlinear elastic energy density is given by

f(ξ γ, θ;ξ γ0) :=16(γ0ξuξ)614(γ0ξuξ)4+12(θθc)(γ0ξuξ)2

and then the nonlinear elastic energy is written as

F(ξγ,θ;ξγ0):=f(ξγ, θ;ξ γ0)|γξ|dξ

where u= γγ0. Thus, we obtain the Helmholtz energy for our setting as

H(γ,θ,γ0):=M(γ)+K(γ)+F(ξγ,θ;ξγ0)+F0(θ).

From now on, let sR/LZ=: SL1be the arc length parameter of the initial closed curve γ0 = γ0(s). It follows from the property of arc length parameter that |γ0′(s)| ≡ 1. In a similar fashion to the above equation, γ(s, t) means the closed curve deformed along evolution from γ0(s). Moreover, in what follows, we assume that γ(s, t) satisfies

|sγ(s,t)|1E3

which means “sis arc length parameter of γnot only the initial time but also every time t”. From the assumption, we can rewrite M, F0 and Fshortly as

M(γ)=0L| tγ |2 ds,F0(θ)=0Lf0(θ)ds,F(s γ,θ;s γ0)=0Lf(sγ,θ;sγ0) ds.

Moreover, since sγs2γ= 0, from Eq. (3), it holds that

K(γ)=0L| s2γ|2ds.

Therefore, the Helmholtz free energy density is denoted by

H(t):=12||tγ(,t)||L2(SL1)2+12||s2γ(,t)||L2(SL1)2+F(sγ(,t), θ(,t);sγ0())+F0(θ(,t)).

We will explain that for the free energy under some assumptions, the following system of nonlinear partial differential equations is derived:

{t2γ+s4γ+sf,sγ(sγ, θ;sγ0)s {(v2 |s2γ|2) sγ}=0, s2v + |s2γ|2v=2 |s2γ|4|s3γ|2+|stγ|2+s2 f,sγ(sγ, θ;sγ0)sγ,tθ s2θ=θ(sγ0s(γ γ0))(tsγsγ0)E4

where

f,sγ(sγ, θ;s γ0)=(fsγ1, fsγ2, fsγ3)  ={( sγ0su )5( sγ0su )3+(θθc)sγ0su }sγ0.

2.3. 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:

H˜(γ, θ;γ0)=t2t1{12||tγ(,t)||L2(SL1)2+12||s2γ(,t)||L2(SL1)2F(sγ(,t), θ(,t);sγ0())F0(θ(,t))}dt.E5

Let us denote the variation of γby

γ(s,t;θ):=γ(s,t)+ε ϕ(s,t)

where εis a sufficiently small positive parameter and ϕis sufficiently smooth and satisfies ϕ(s, t1) = ϕ(s, t2) = 0. Moreover, from the assumption Eq. (3), it is also necessary to hold that

ddε|sγ(s,t;ε)||ε=0=0.

Since

ddε|sγ(s,t;θ)||ε=0=sγ(s,t)sϕ(s,t)

ϕhas to satisfy

sγ(s,t)sϕ(s,t)=0

for any sSL1 and t>0. Calculating the first variation of the energy functional, we have

ddεH˜(γ, θ;γ0)|ε=0=t1t2{tγ,tϕs2γ,s2ϕf,sγ(sγ,θ;sγ0),sϕ}dt.

From the integral by parts, the right-hand side is rewritten as follows:

t1t2t2γ+s4γsf,sγ(sγ,θ;sγ0),ϕdt.E6

Then the integral Eq. (6) is equal to 0 for any ϕsatisfying ϕ(s,t1)=ϕ(s,t2)=0and sγsϕ0. For the purpose, we define

V:={ϕ | sγsϕ0}.

The orthogonal complement V of the space V with respect to L2(SL1)inner product is given by

V={s(wsγ)w=w(s,t) is a scalar function}.E7

Here, we remark that in the case where γis a curve embedded in three-dimensional space (not a planar curve), s2γ0has to be satisfied for every (s,t)SL1×R+. In the end of this section, we will show the reason why V is given as above. Consequently, if for the direction t2γ+s4γsf,sγ(sγ,θ;sγ0), there exists a scalar function w= w(s, t) such that

t2γ+s4γsf,sγ(sγ,θ;sγ0)=s(wsγ)E8

then Eq. (6) is equal to 0.

Next, we derive the equation for w. From the assumption Eq. (3), we see that

0=t2|sγ|2=2st2γsγ+2|stγ|2

It follows from Eq. (8) that

{s5γ+s2f,sγ(sγ,θ;sγ0)+s2(wsγ)}sγ=|stγ|2. E9

Differentiating Eq. (3), we obtain

sγs2γ=0,sγs3γ=|s2γ|2,sγs4γ=32s(|s2γ|2),sγs5γ= 2s2(|s2γ|2)+ |s3γ|2.

By the relations, we will rewrite Eq. (9). It follows from the direct calculation that

s2 (wsγ)sγ=s2w  w|s2γ|2.

Since from the definition

f,sγ(sγ, θ;sγ0)={( sγ0s(γγ0) )5( sγ0s(γγ0) )3+(θθc)(sγ0s(γγ0))}sγ0

we also obtain

s2f,sγ(sγ, θ;sγ0)sγ=s2[{(sγ0s(γγ0))5(sγ0s(γγ0))3+(θθc)(sγ0s(γγ0))}sγ0]sγ.

Therefore, substituting these into Eq. (9), we find

0=2s2(|s2γ|2)|s3γ|2+s2ww|s2γ|2+|stγ|2+s2[{(sγ0s(γγ0))5( sγ0s (γγ0))3+(θθc)(s γ0s (γγ0))}sγ0]sγ.

Here setting the new unknown vby v:=w+2|s2γ|2,we can rewrite the equation as follows:

s2v+|s2γ|2v=2 |s2γ|4|s3γ|2+|stγ|2+s2[{(sγ0s(γγ0))5(sγ0s(γγ0))3+(θθc)(sγ0s(γγ0))}sγ0]sγ.

Consequently, under given temperature θ,the equation of motion is given by

{t2γ+s4γsf,sγ(sγ,θ;sγ0)s{(v2|s2γ|2)sγ}=0, s2v+|s2γ|2v= 2|s2γ|4|s3γ|2+|stγ|2s2f,sγ(sγ, θ;sγ0)sγ.

At the rest of this section, we prove that the orthogonal complement of Vis given by Eq. (7).

Lemma 1. Letγ(s,t)be a smooth curve inR3 andsSL1be an arc length parameter of γ for any t. Suppose thats2γ0holds for all(s,t)SL1×R+then the orthogonal complement V of the spaceV={ϕ|sγsϕ0}with respect toL2(SL1)inner product is represented by

V={s(wsγ)w=w(s,t) is a scalar function}.

Proof. Observe that sγ,s2γ and the outer product sγ×s2γare orthogonal each other. Under the assumption s2γ0,a coordinate system defined on γconsists of the vectors. Then arbitrary vector η= η(s, t) can be represented as

η(s,t)=η1(s,t)sγ(s,t)+η2(s,t) s2γ(s,t)+η3(s,t) sγ(s,t)×s2γ(s,t).

If we assume additionally ηV, then we obtain

0=sηsγ={sη1sγ+η1s2γ+sη2s2γ+η2s3γ+sη3sγ×s2γ+η3 (s2γ×s2γ+sγ×s3γ)}sγ=sη1+η2sγs3γ=sη1η2|s2γ|2

that is

sη1=|s2γ|2η2. E10

In other words, an element of Vconsists of η1, η2 satisfying Eq. (10) and arbitrary η3. However, we remark that we cannot take η2 freely. Indeed, in order to verify Lperiodicity of η1, η2, we have to show the following condition:

0L|s2γ|2η2ds=0.E11

If ζ(s,t)=ζ1(s,t)sγ(s,t)+ζ2(s,t)s2γ(s,t)+ζ3(s,t)sγ(s,t)×s2γ(s,t)satisfies ⟨ζ, η⟩ = 0, then we see that η1 and η2 satisfy Eqs. (10) and (11) and any η3 satisfies

0L{ζ1η1+ζ2η2|s2γ|2+ζ3η3(sγ×s2γ)2}ds=0. E12

In particular, if we assume η2 ≡ 0 and η3 ≡ 0, then we infer from Eq. (10) that η1Cholds true. Then, we deduce from Eq. (12) that

0Lζ1ds=0.

Now we define

ϕ(s,t)=ζ1(0,t)+0sζ1(s,t)ds.

Then, ϕhas the period Land satisfies sϕ=ζ1. Substituting it into Eq. (12) and using Eq. (10), we have

0=0L{sϕ η1+ζ2η2|s2γ|2+ζ3η3(sγ×s2γ)2}ds=0L{ϕsη1+ζ2η2|s2γ|2+ζ3η3(sγ×s2γ)2}ds=0L{ϕ |s2γ|2η2+ζ2η2|s2γ|2+ζ3η3(sγ×s2γ)2}ds=0L{(ϕ+ζ2)η2|s2γ|2+ζ3η3(sγ×s2γ)2}ds.

Recalling Eq. (11), we see that the vector-valued function (η2, η3) is orthogonal with (|s2γ|2,0)in the sense of L2 inner product. Therefore, there exists some function μ= μ(t) depending only on tsuch that

( {ϕ+ζ2} |s2γ|2, (sγ×s2γ)2ζ3)=μ(|s2γ|2,0)

that is,

ϕ+ζ2=μ, ζ30.E13

Setting

μ+ϕ(s,t)=w(s,t),

the function w(s, t) is the Lperiodic function and satisfies sw=ζ1. It follows from Eq. (13) that

ζ2(s,t)=w(s,t).

Then ζ(s) is orthogonal with elements of V with respect to L2 inner product. Thus, we get

ζ(s,t)=sw(s,t)sγ(s,t)+w(s,t)s2γ(s,t)=s(w(s,t)sγ(s))

which completes the proof.

Q.E.D.

Remark 1. The assumption s2γ0in Lemma 1 means that the curvature is always non-zero. If s2γ=0at some point, we cannot determine the tangential vector ∂sγat 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, L2 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) L2 gradient flow for E under the area-preserving condition and (C) [27] and (ii) L2gradient 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].

2.4. 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

θtS+q=hE14

where S, qand hare 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 ∇ ⋅ qas ∂sqas the same as one-dimensional case, where sis necessary to be arc length parameter. By the same reason, the Fourier law q= ∇θis replaced by

q=sθ.E15

The Helmholtz free energy density

f˜=f˜(s2γ, sγ, θ ;γ0)=12|s2γ |2+f(sγ, θ ;γ0)+f0(θ)

and the entropy Sare connected with the relation

S=f˜ θ.

Then, the conservation law, Eq. (14), is rewritten as follows:

 θ f0''(θ) tθ s2θ=θ(sγ0s(γγ0))(tsγsγ0)+h.E16

We note that the Clausius-Duhem inequality holds automatically:

tS+s(qθ)hθ .

Indeed, we observe from Eqs. (14) and (15) that

tS+s(qθ)=hsqθ+s(qθ)=hθqsθθ2=hθ+|sθθ|2hθ .

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:

f0(θ)=θθlogθ.

Then since f0(θ)=1/θ,Eq. (16) is reduced to

tθs2θ=θ(sγ0s(γγ0))(tsγsγ0).

We thus obtain the system of equation as Eq. (4).

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3. Dual variation structure

Let us rewrite Eq. (4):

t2γ+s4γ +sf,sγ(sγ, θ;sγ0)s{(v2 |s2γ|2)sγ}=0,E17
 s2v+|s2γ|2v=2|s2γ|4|s3γ|2+|stγ|2+s2 f,sγ(sγ, θ;sγ0)sγ,E18
tθs2θ=θ(sγ0s(γγ0))(tsγsγ0), (t,s)(0,T)×SL1,E19
γ(0,s)=γ0(s), tγ(0,s)=γ1, θ(0,s)=θ0, sSL1.E20

In this section, we show that the problem in Eqs. (17)–(20) has also dual variation structure as well as the problem (Eq. (2)). The structure plays an important role to prove the dynamical stability of infinitesimally stable stationary state.

We assume that the system has sufficiently smooth solution (γ, θ) satisfying θ> 0. Then initial data also has to satisfy

|sγ0|=1, sγ0sγ1=0.

Setting

f1(sγ) :=12(γ0s(γγ0)s)2,f2(sγ) :=16(γ0s(γγ0)s)614(γ0s(γγ0)s)4θc2(γ0s(γγ0)s)2,

we have the relation f=θf1+f2. Multiplying Eq. (17) by ∂tγand integrating it with respect to s, we obtain

ddt(12||tγ||L22+12||s2γ||L22)=f,sγ, stγ=ddt0Lf2(sγ)ds0Lθtf1(sγ)ds.

Integrating Eq. (19), we find

ddt0Lθds=0Lθtf1(sγ)ds.

Then for the quantity

E(γ,tγ,θ):=12||tγ||L22+12||s2γ||L22+0Lθds+0Lf2(sγ)ds,

it holds that

ddtE(γ,tγ,θ)=0.

Moreover, for the quantity

W(sγ,θ):=0L{f1(sγ)logθ}ds

we deduce from Eq. (19) that

ddtW(sγ,θ)=0Ls2θθds=0L|sθθ|2ds0.

Finally, we confirm an important structure of the system (17)–(20). We denote the stationary state of θby θ¯>0, and the corresponding equilibrium by γsatisfy the following constraint:

bE(γ0, γ1, θ0)=E(γ, 0, θ¯),s4γ=s{θ¯f1,sγ+f2,sγ}.

Eliminating θ¯by the relation

b=Lθ¯+12||s2γ||L22+0Lf2(sγ) ds

the stationary state of this problem satisfies the following nonlinear nonlocal problem:

s4γ=s{1L(b12||s2γ||L220Lf2(sγ)ds)f1,sγ(sγ)+f2,sγ(sγ)}. E21

Eq. (21) is derived as the Euler-Lagrange equation of the functional

Jb(y):=1L0Lf1(y)dslog(b12||sy||L220Lf2(y)ds)+logL

where y= sγH2(SL1,S2)and

S2:={ωR3||ω|=1}.

We remark that the following relation between Jband Wholds true:

W(sγ,θ)LJb(sγ).

The relation is called semi-unfolding minimality. Thus, if (γ, θ) is a non-stationary state, b=E(γ0,γ1,θ0),y¯=sγ¯is a linearized stable critical point of Jb=Jb(y),y H1(SL1, S2) and θ¯> 0is a constant satisfying E(γ¯,0,θ¯) = b,then it holds that for y=sγ

Jb(y)Jb(y¯)W(sγ0, θ0)W(y¯,θ¯).

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¯=sγ¯of Jbfor γ¯H2(SL1,S2)is infinitesimally stableif there exists ε0 > 0 such that any ε1(0,ε02]admits δ0 > 0 such that if ||s(γγ¯)||H1<ε0 and Jb(sγ) Jb(sγ¯)<δ0then

||s(γγ¯)||H1<ε1.

The definition of infinitesimally stable is obviously weaker than the one of well-known linearized stable which means that for a critical point y¯=sγ¯of Jb, the quadratic form

Qy(w,w)=ddε2Jb(y¯+εw)|ε=0

is a positive definite for any wH2(SL1,S2).

Theorem 1. Assume thatθ¯>0is a constant and thatγ¯is an infinitesimally stable critical point of Jbwith constraintsγH2(SL1,S2). Then(γ¯,θ¯)is a dynamically stable in the sense that for any ε> 0, there exists δ> 0 such that if

E(γ0, γ1, θ0)= b,||s(γ0γ¯)||H1<δ, |1L0Llogθ0(s)dslogθ¯|<δE22

then

supt0||s(γ(t)γ¯)||H1<ε, |1L0Llogθ(t,s)dslogθ¯|<ε.

Proof. We first show the semi-unfolding minimality. From the energy conservation law, we see that

b =12||tγ||L22+12||s2γ||L22+0Lθds+0Lf2(sγ)ds.

It follows from the Jensen inequality that

1L0Llogθdslog(1L0Lθds).

Then we have

W(sγ, θ)0Lf1(sγ)dslog(1L0Lθds)0Lf1(sγ)dsLlog1L(b12||sy||L220Lf2(y)ds)=LJb(sγ).

We have thus completed to show the semi-unfolding minimality. Recall that γ¯H2(SL1,S2)is an infinitesimally stable critical point of Jb(sγ). Thus, we find ε0 > 0such that any ε1(0,ε02] admits δ0 > 0 such that if ||s(γγ¯)||H1<ε0 and Jb(sγ) Jb(sγ¯)<δ0then

||s(γγ¯)||H1<ε1.E23

From the above properties, it holds that

Jb(sγ(t))1LW(sγ(t), θ(t))1LW(sγ0, θ0).E24

Moreover, for the constant θ¯>0,we obtain

W(sγ¯, θ¯)=L[1L0Lf1(sγ)ds log1L(b12||sy¯||L220Lf2(y¯)ds)]=LJb(s γ¯),

namely,

Jb(sγ¯)=1LW(sγ¯,θ¯). E25

Given ε> 0, setting δ(0,ε02]and satisfying Eq. (22), we have

1L|W(sγ0, θ0)W(sγ¯,θ¯)|1L(f1, sγ(sγ0)L+f1, sγ(sγ¯)L)s(γ0γ¯)L1+|1L0Llogθ0dslogθ¯|<min(δ0,ε).E26

Therefore, it follows from Eq. (24) to Eq. (26) that

Jb(sγ(t))Jb(sγ¯)=1L(W(sγ0, θ0)W(sγ¯,θ¯))<δ0.

If ||s(γ(t)γ¯)||H1=δ(ε0/2 <ε0), then we apply Eq. (23) for ε1 = δ, and hence

s(γ(t)γ¯)H1<δ,

which is a contradiction. Thus, we have

s(γ(t)γ¯)H1δ.

Here, from γC([0,);H2) and s(γ0γ¯)H1<δ,it follows that

||s(γ(t)γ¯)||H1<δE27

for any t≥ 0.

From the semi-unfolding minimality (Eqs. (24) and (25)) and the linearized stability of Jb, we observe that

W(sγ,θ)LJb(sγ)LJb(sγ¯)=W(sγ¯,θ¯).

Then, combining Eq. (26) with Eq. (27), we have

|1L0Llogθ(t,s)dslogθ¯|1L(W(sγ,θ)W(sγ¯,θ¯))+1L|0L{f1(sγ)f1(sγ¯)}ds|1L(W(sγ0,θ0)W(sγ¯,θ¯))+C||s(γγ¯)||H1ε+Cδ2ε

where δis 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 low-temperature and low-energy cases can be found in [21, 31].

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4. 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]).

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Acknowledgments

This work was partially supported by JSPS KAKENHI Grant-in-Aid for Scientific Research (A) and (C), Grant Numbers 26247013 and 16K05234.

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Written By

Shinya Okabe, Takashi Suzuki and Shuji Yoshikawa

Submitted: November 23rd, 2016 Reviewed: April 12th, 2017 Published: September 20th, 2017