The origin of nucleation peak in

transformational plasticity

Lev Truskinovsky

Laboratoire de Mechanique des Solides, CNRS-UMR 7649, Ecole Polytechnique,

91128, Palaiseau, France

Anna Vainchtein ∗

Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA

Abstract

A typical stress-strain relation for martensitic materials exhibits a mismatch be-tween the nucleation and propagation thresholds leading to the formation of thenucleation peak. We develop an analytical model of this phenomenon and obtainspecific relations between the macroscopic parameters of the peak and the micro-scopic characteristics of the material. Although the nucleation peak appears in themodel as an interplay between discreteness and nonlocality, it does not disappearin the continuum limit. We verify the quantitative predictions of the model by com-parison with experimental data for cubic to monoclinic phase transformation inNiTi.

Key words: martensitic phase transitions, lattice models, nonlocal interactions,Peierls-Nabarro landscape, nucleation

1 Introduction

In displacement-controlled experiments shape memory alloys and other marten-sitic materials display a nucleation peak: prior to nucleation of a new phasethe load reaches a maximum but then drops to a distinctly lower value. Thesubsequent plateau is associated with phase boundaries propagating along the

∗ Corresponding authorEmail addresses: trusk@lms.polytechnique.fr (Lev Truskinovsky),

aav4@pitt.edu (Anna Vainchtein).

Preprint submitted to Elsevier Science 17 December 2003

specimen at essentially constant stress (Lexcellent and Tobushi, 1995; Shawand Kyriakides, 1995, 1997b; Sun and Zhong, 2000). The nucleation peak isnot unique to transformational plasticity and is also observed during initia-tion of a conventional plastic deformation in mild steels (Butler, 1962; Hall,1970; Kyriakides and Miller, 2000; Froli and Royer-Carfagni, 2000), where ithas been attributed to the fact that the stress required to release the trappeddislocations is higher than the stress needed to sustain their motion (Cot-trell and Bilby, 1949; Johnston and Gilman, 1959). For martensitic materialsthe nucleation-induced load drop has been observed in 3D numerical simu-lations based on various plasticity-like phenomenological models (Shaw andKyriakides, 1997a; Kyriakides and Miller, 2000; Sun and Zhong, 2000) but dueto the complexity of these models, the physical parameters responsible for thesize of the peak have not been identified. At a qualitative level, the nucleationpeak in these materials has been associated with the presence of sufficientlyfine grains and heuristically linked to the strong locking of phase boundariesand the relative ease of their glide upon release (Shaw and Kyriakides, 1995,1997b).

In this paper we develop an analytical model of the nucleation peak phe-nomenon in martensites: our model supports the intuition developed in plas-ticity theory and adapts it to the case when the principal carriers of inelas-tic deformation are phase boundaries. Specifically, we consider a prototypicalmass-spring system consisting of rigid elements (crystal planes) connectedby bi-stable elastic springs representing transforming shear layers. To mimicthe three-dimensional nature of the actual problem, we complement the up-down-up interactions between the nearest neighbors (NN) by a harmonic in-teraction of the next-to-nearest neighbors (NNN). The bi-stable discrete mod-els without NNN interactions (e.g. Muller and Villaggio, 1977; Fedelich andZanzotto, 1992; Puglisi and Truskinovsky, 2000, 2002a,b) capture many im-portant features of transformational plasticity but fail to predict the peakphenomenon. Recent numerical studies of the models incorporating NNN in-teractions showed that the nucleation peak can be recovered (e.g. Ye et al.,1991; Triantafyllidis and Bardenhagen, 1993; Rogers and Truskinovsky, 1997;Froli and Royer-Carfagni, 2000; Pagano and Paroni, 2003); none of these mod-els, however, have been developed analytically to the extent that they couldexplain the necessity of the peak phenomenon and identify the microparame-ters controlling the size of the stress drop.

We begin by finding the limits of instability of a homogeneous state and de-termine an analytical expression for the nucleation threshold in the mostgeneral case. In order to obtain an analytical characterization of the prop-agation threshold we use a piecewise linear approximation for the NN in-teractions. This simplification allows us to reconstruct the non-equilibriumPeierls-Nabarro landscape for the propagating phase boundaries and com-pute the martensitic analog of the Peierls stress. We show that the presence

2

of nonlocal interactions in the discrete model makes nucleation and propa-gation thresholds different and then prove that the nucleation peak does notdisappear in the continuum limit. An important question is whether the quasi-continuum strain-gradient approximation of the discrete model (Mindlin, 1965;Triantafyllidis and Bardenhagen, 1993) is capable of capturing the peak phe-nomenon. We show that the nucleation threshold is approximated well onlyin the case of long-wave instability (see also Triantafyllidis and Bardenhagen,1996) and that although the peak is captured, the stress drop is grossly exag-gerated. Finally, we verify the quantitative prediction of the model by usingthe experimental data for cubic to monoclinic phase transformation in NiTiwires. The comparison with experiment leads to the bounds for the measureof nonlocality which are in a good agreement with the independent estimatesbased on a realistic interatomic potential.

The proposed model stays in the same prototypical relation to transforma-tional plasticity as the well known Frenkel-Kontorova model to classical plas-ticity. A nontrivial formal correspondence between the two models in thecase of infinite systems has been established by Truskinovsky and Vainchtein(2003).

The structure of the paper is as follows. In Section 2 we formulate the discreteproblem for a finite system and specify the boundary conditions. In Section 3we analyze stability of the single-phase equilibrium in the general setting dis-covering the possibility of both macro and microinstabilities. In Section 4 weintroduce a piecewise linear model for the local interactions and obtain anexplicit representation of a generic metastable equilibrium state. In Section 5we study the energy barriers between the neighboring metastable states andshow that propagation always reduces to a succession of transformations insideindividual elements, while nucleation can involve transformation of several el-ements at once. Section 6 contains the derivation of the explicit formulae forthe size of the nucleation peak in both discrete and continuum problems. InSection 7 we establish a correspondence between the microparameters of thelattice and the experimental measurements of both the stress drop and the sizeof the nucleation band. Finally, Section 8 contains comparison of the discreteand strain-gradient models showing that the agreement is at most qualitative.The conclusions are summarized in Section 9.

2 The model

Consider a system of N + 1 particles linked to their nearest and next-to-nearest neighbors by elastic springs (see Fig. 1). Let uk, 0 ≤ k ≤ N , bethe displacements of the particles with respect to a load-free homogeneousreference configuration with spacing ε. Denote the strain in the kth NN spring

3

ii−1 i+10 1 N−1 N

ε

2ε

NNN springNN spring

Fig. 1. A finite chain of particles with nearest-neighbor (NN) andnext-to-nearest-neighbor (NNN) interactions.

by wk = (uk − uk−1)/ε. Then the total energy of the system can be written as

Ψ = εN∑

k=1

φ1(wk) + 2εN−1∑

k=1

φ2

(

wk+1 + wk

2

)

+ ΨB(w1, wN), (1)

where φ1(w) and φ2(w) are the energy densities of the NN and NNN interac-tions, respectively. The term ΨB corresponds to the energy of the boundaryelements. We assume that the chain is placed in a hard device with the totaldisplacement d:

uN − u0 = εN∑

k=1

wk = d. (2)

Due to the nonlocality of the model, the boundary condition (2) must be com-plemented by other constraints. For instance, to mimic an “extra hard device”one can additionally impose conditions w1 = 0 and wN = 0, making the termΨB irrelevant. Alternatively, one may consider a “zero-moment device” byassuming that ΨB = 0.

Let fi(w) = φ′

i(w), i = 1, 2, denote the forces in NN and NNN springs, respec-tively. The equations governing the equilibrium of the interior particles with2 ≤ k ≤ N − 1 have the form

f1(wk) + f2

(

wk+1 + wk

2

)

+ f2

(

wk + wk−1

2

)

= F, (3)

where F is the total force in the system. The natural boundary conditionsread

f1(w1) + f2

(

w2 + w1

2

)

+1

ε

∂ΨB

∂w1

= F,

f1(wN) + f2

(

wN + wN−1

2

)

+1

ε

∂ΨB

∂wN

= F,

(4)

An additional assumption adopted in what follows,

ΨB = εφ2(w1) + εφ2(wN) (5)

4

means that the boundary NNN springs are cut in half and reconnected parallelto the NN springs (see the dashed springs in Fig. 1). The formal advantageof this choice of ΨB is that the boundary equations (4) can be included intothe bulk equations (3) if we additionally assume the existence of fictitious 0thand (N + 1)th springs satisfying

w0 = w1, wN+1 = wN . (6)

The real advantage of (4) and (5), however, is that the corresponding boundaryconditions ensure the existence of a trivial solution with the uniform straindistribution (Triantafyllidis and Bardenhagen, 1993; Charlotte and Truski-novsky, 2002). While in the rest of the paper we will be using mostly (5), theeffect of switching to ΨB = 0 is briefly discussed in Section 6.

3 Nucleation threshold

It is easy to see that for arbitrary spring potentials the trivial solution ofthe problem (3), (4) and (5) is given by wk = d/L, where L = Nε. Thehomogeneous response of the system is then characterized by the formulaeF = f1(d/L) + 2f2(d/L) and Ψ = L(φ1(d/L) + 2φ2(d/L)). To analyze thestability of this solution we introduce the tangential moduli of NN and NNNsprings: K(w) = φ′′

1(w) and γ(w) = φ′′

2(w)/2. The homogeneous configurationis stable if and only if the quadratic form Bv · v with

B =

K + 3γ γ 0 . . . 0

γ K + 2γ γ. . . 0

0. . .

. . .. . . 0

... 0 γ K + 2γ γ

0 . . . 0 γ K + 3γ.

(7)

is positive definite for all v 6= 0 such that

N∑

k=1

vk = 0. (8)

Finding the boundaries of stability in the space of parameters is important forthe subject of the paper because they coincide with the limits of a barrierlessnucleation.

Before addressing the stability problem systematically, we observe that K +4γ > 0 and γ ≤ 0 are sufficient for stability. Indeed, in this case all terms in

5

the quadratic form

Bv · v = (K + 4γ)N∑

k=1

v2k − γ

N−1∑

k=1

(vk+1 − vk)2. (9)

are nonnegative. Similarly, by writing the quadratic form as

Bv · v = KN∑

k=1

v2k + γ

N−1∑

k=1

(vk+1 + vk)2 + 2γ(v2

1 + v2N).

we obtain that conditions K > 0 and γ ≥ 0 are also sufficient for stability. Interms of the main nondimensional parameter of the problem,

µ =K

4γ, (10)

these stability intervals can be written as −∞ < µ < −1 and 0 < µ < ∞ andhence the instability limits are located in the interval

−1 ≤ µ ≤ 0. (11)

To find the exact locations of the stability boundaries, consider the (zero)eigenvalue problem Bv = 0. In the bulk of the chain (1 ≤ k ≤ N − 1) thismeans

(K + 2γ)vk + γvk−1 + γvk+1 = 0. (12)

On the boundaries we obtain

(K + 3γ)v1 + γv2 = 0, (13)

(K + 3γ)vN + γvN−1 = 0. (14)

The eigenvector v must be nonzero and satisfy (8). We seek solution in theform vk = ρk and obtain the characteristic polynomial

ρ2 + (4µ + 2)ρ + 1 = 0. (15)

This means that ρ1,2 = −1−2µ±2√

µ(µ + 1), and according to (11), we haveto consider the following cases:

Special case µ = −1. Here ρ1 = ρ2 = 1 and vk = A1k + A2. The boundaryequations (13) and (14) reduce to v1 = v2 and vN = vN−1, implying thatA1 = 0 and thus vk = A2. The constraint (8) implies that A2 = 0. Since thereare no nontrivial solutions, there is also no stability change.

Special case µ = 0. Here ρ1 = ρ2 = −1 and vk = (A1k + A2)(−1)k. Applying(13) and (14), we obtain A1 = A2 = 0 which again means that there is nochange of stability.

6

Generic case −1 < µ < 0. We first observe that in this interval ρ1,2 can bewritten as

ρ1,2 = e±iω,

where ω is defined by

µ = − cos2 ω

2. (16)

Applying the boundary equation (13), we obtain (up to a multiple)

vk = cos[(k − 1/2)ω].

The second boundary equation (14) implies 2 sin(ω/2) sin(Nω) = 0, and since0 < ω < π, we obtain sin(Nω) = 0 and

ω =πn

N, (17)

for 1 ≤ n ≤ N −1. All these solutions are nontrivial and satisfy (8). Thereforethe instability of the trivial solution can take place at any of the bifurcationpoints

K + 4γ cos2 πn

2N= 0.

The corresponding unstable modes are

vk = cos[(k −1

2)πn

N]. (18)

To locate the stability boundary, we begin with the case γ < 0. Then

K + 4γ cos2 πn

2N≥ K + 4γ cos2 π

2N

and henceK + 4γ cos2 π

2N> 0 (19)

is both necessary and sufficient for stability. The instability develops throughthe growth of the long-wave mode

vk = cos[(k −1

2)π

N].

As N tends to infinity, the unstable wave length also becomes infinite and(19) reduces to K + 4γ > 0. At finite N we have K + 4γ cos2 π

2N> K + 4γ,

meaning that the discrete homogeneous configuration may still be stable whenthe macroscopic modulus E = K + 4γ is already negative. One can say thatin the discrete problem the macroscopic instability is delayed due to the finitesize of the system (see also Triantafyllidis and Bardenhagen, 1993).

The situation is different when γ > 0. In this case

K + 4γ cos2 πn

2N≥ K + 4γ cos2 π(N − 1)

2N= K + 4γ sin2 π

2N,

7

and hence the necessary and sufficient condition for stability is

K + 4γ sin2 π

2N> 0. (20)

The instability develops through the growth of the short-wave mode

vk = (−1)k sin[(k −1

2)π

N].

In the limit of infinite N , the wave length approaches the interatomic dis-tance. Also (20) becomes K > 0, and since now K + 4γ sin2 π

2N< K + 4γ,

this microscopic instability develops before the macroscopic (or homogenized)system becomes unstable (see also Triantafyllidis and Bardenhagen, 1996). Inthe context of martensitic phase transitions, this effect may be linked to theobservation of the pre-martensitic tweed microstructures (Kartha et al., 1995).

-4 -2 0 2 4

-0.5

0

0.5

�

K

(a) (b)

0.1 0.2 0.3 0.4

-1

-0.5

0

�

N ��

�����

����� � ��

��

microinstability

macroinstability macroinstability

microinstabilityK

�

K �� � �

Fig. 2. (a) Domain of stability for the trivial solution in K-γ plane: solid lines -discrete model, N = 3; dashed lines - continuum limit, N → ∞. (b) The dependenceof the stability limits on the size of the system. In both figures stability domain isin gray.

We now summarize the necessary and sufficient conditions for stability of thetrivial solution. In terms of elastic moduli K and γ we obtain the followingstability intervals

K + 4γ cos2 π

2N> 0, K + 4γ sin2 π

2N> 0. (21)

In the limit of infinite N the inequalities (21) reduce to the known conditionsK > 0, K + 4γ > 0 (e.g. Mindlin, 1965). In terms of the nondimensionalparameter µ the stability intervals take the form

µ < − cos2 π

2N(γ < 0), µ > − sin2 π

2N(γ > 0), (22)

which in the limit of infinite N gives µ < −1 (γ < 0) and µ > 0 (γ > 0). Thestructure of the stability domain in the plane K-γ is illustrated in Fig. 2a for

8

f1

wwc

K

K

phase I

phase II

Fig. 3. Up-down-up force-strain relation in an individual NN spring (solid line) andits bilinear approximation (dashed lines).

both finite and infinite N ; in Fig. 2b, showing the plane µ-N−1, we illustratethe dependence of the stability boundaries on the size of the system. For theNNN system with the “moment free” boundary conditions (ΨB = 0) similaranalysis of the stability for trivial equilibria (which now have boundary layers)can be found in Charlotte and Truskinovsky (2002).

4 Nontrivial solutions and metastability

To model martensitic phase transitions, we assume that each NN spring hasa double-well energy generating a non-monotone “up-down-up” force-strainrelation depicted in Fig. 3. If the negative slope of the force-strain relation forthe NN spring is sufficiently steep, it is easy to show that the total strain canreach the threshold where the homogeneous phase is absolutely unstable. Tostudy the resulting nucleation and the subsequent growth of the new phase,we need to describe the postcritical behavior of the system. More precisely, inthe domain of loadings where trivial solution is unstable, we need to find thenontrivial solutions of the equilibrium equations (3) corresponding to eitherlocal or global minima of the energy.

Although the instability limits for the trivial solution can be found in thegeneral case, we succeeded in obtaining a complete analytical description ofthe postcritical behavior only in the case when the functions f1(w) and f2(w)are either linear or piecewise linear. Specifically, we considered

f1(w) =

Kw for w < wc

K(w − a) for w ≥ wc

(23)

andf2(w) = 2γw. (24)

9

The parameters of the NN potential are the critical strain wc, transformationstrain a and elastic modulus K (see Fig. 3). The linear NNN spring is char-acterized by the elastic modulus γ which will also be used as the measure ofnonlocality. We assume that K > 0 and γ < 0, which is suggested by thelinearization of the Lennard-Jones potential (see Charlotte and Truskinovsky,2002); to ensure that independently of N the homogeneous states are stablein their domain of definition, we also assume that K + 4γ > 0. In termsof the nondimensional parameter µ, the assumptions on the moduli can besummarized as

−∞ < µ < −1. (25)

It is not hard to see that under these assumptions the microinstability isexcluded while the macroinstability always takes place at w = wc.

For the piecewise linear model (23), (24) the total energy (1) reduces to

Ψ = ε

{

1

2Bw · w − q · (w − wc)

}

, (26)

where the vector qk = Kaθ(wk − wck) prescribes distribution of phases, θ(x)is a unit step function and wck = wc, k = 1, ..., N . The equilibrium equations(3) together with the boundary conditions (6) can then be rewritten as

Bw = F + q, (27)

where Fk = F , k = 1, ..., N . To eliminate the redundant parameters we rescalethe variables. By selecting L = Nε as the length scale and K as the scale offorce, we define

uk =uk

Nε, d =

d

Nε, F =

F

K, Ψ =

Ψ

KNε, B =

1

KB. (28)

Unless specially mentioned, in what follows, we will be using only rescaledvariables with the bars dropped. The dimensionless problem depends on thetwo main parameters: µ and N .

To find the metastable states we can simply prescribe the distribution of phasesq and solve the linear problem (27). The resulting equilibrium configurationsare automatically local minimizers of the energy (26), because the stiffnessmatrix B is positive definite. To obtain the global minima of the energy, weneed additionally to minimize Ψ with respect to the phase geometry q.

The solution of the linear system (27) can be written as

w = B−1F + B−1q = w0 + w1. (29)

The first term in (29) corresponds to the uniform configuration with all springs

10

in the first phase. Without explicitly computing B−1 one can see that

w0k =

Fµ

1 + µ. (30)

It is convenient to express w1 not in terms of the variables q, but in terms oftheir discrete derivatives p defined by

pi =qi+1 − qi

a. (31)

Notice that these relations can always be explicitly inverted, yielding

qk = q1 + aN∑

i=1

piθ(k − i − 1). (32)

The physical meaning of the variables pi is clear from the representation

pi =

1 if wi < wc and wi+1 > wc (I to II phase switch)

0 if sign(wi − wc) =sign(wi+1 − wc) (no phase switch)

−1 if wi > wc and wi+1 < wc (II to I phase switch).

(33)

To simplify the subsequent formulae we also set pN = 0.

In the interior nodes (2 ≤ k ≤ N − 1) w1k must satisfy the difference equation

(1 +1

2µ)w1

k +1

4µ(w1

k+1 + w1k−1) = q1 + a

∑

i

piθ(k − i − 1). (34)

We seek the general solution of (34) in the form

w1k = wh

k + wink , (35)

where whk satisfies the homogeneous equation and win

k is a particular solution.The general solution of the homogeneous problem can be represented (Mick-ens, 1990) as a linear combination of ρk

1 and ρk2, where ρ1,2 are the roots of the

characteristic polynomial (15). By writing ρ1,2 = e±λ, where

λ = 2arccosh√

|µ|, (36)

we obtainwh

k = C1eλk + C2e

−λk. (37)

The constants C1 and C2 are to be found from the boundary conditions (6).

To obtain a particular solution of (34) with q1 = 0 and pj = 0 for j 6= i,we observe that in this case the right hand side of (34) is zero for k ≤ i and

11

constant for k ≥ i + 1. Therefore we can write

wpi

k =

wp−k = A1e

λk + A2e−λk for k ≤ i

wp+k = A3e

λk + A4e−λk +

aµpi

1 + µfor k ≥ i,

(38)

and then “glue” the two sides together by requiring that

wp−i = wp+

i , wp−i+1 = wp+

i+1. (39)

Since we are looking for a particular solution, we may always let A1 = A2 = 0,and thus consider wp−

k = 0. Solving (39) for A3 and A4, we obtain

wpi

k = ∆

{

θ(k − i − 1/2)

[

1 −cosh[(k − i − 1/2)λ]

cosh (λ/2)

]}

,

where

∆ =aµ

µ + 1(40)

is the macroscopic transformation strain. Finally, by superposition, we obtainthe particular solution in the form

wink =

q1µ

1 + µ+ ∆

N∑

i=1

piθ(k − i − 1/2)

[

1 −cosh[(k − i − 1/2)λ]

cosh(λ/2)

]

. (41)

The general solution of (34) is now given by (35), (37) and (41). Applying theboundary conditions (6), we obtain

w1k =

q1µ

1 + µ+ ∆

N∑

i=1

pi

{

sinh[(N − i)λ] cosh[(k − 1/2)λ]

cosh(λ/2) sinh(Nλ)

+θ(k − i − 1/2)

(

1 −cosh[(k − i − 1/2)λ]

cosh(λ/2)

)}

.

(42)

Before combining (30) and (42), we notice that the relation between the forceF and the total displacement d can be written in the form:

F =1 + µ

µ(d − ∆

l

N), (43)

where l is the number of springs in phase II related to q1 and p via

l =Nq1

a+

N∑

i=1

(N − i)pi. (44)

Finally, substituting (43), (44) into (29), (30) and (42), we obtain the repre-

12

sentation for a generic metastable configuration

wk =d + ∆N∑

i=1

pi

{

sinh[(N − i)λ] cosh[(k − 1/2)λ]

cosh(λ/2) sinh(Nλ)+

i

N− 1

+θ(k − i − 1/2)

(

1 −cosh[(k − i − 1/2)λ]

cosh(λ/2)

)}

.

(45)

To compute the energy we introduce a new variable w with components wk =wk − d representing the p-dependent part of the strain field w. Then (26) canbe rewritten as

Ψ =1 + µ

2µd2 − a(d − wc)

l

N+ Ψ, (46)

where

Ψ =1

N

(

1

2Bw − q

)

· w (47)

is the contribution due to phase changes. Observing that in equilibrium

Ψ = −1

2Nq · w

and using (32) to eliminate q, we obtain

Ψ =1 + µ

µ

{

1

2d2 − ∆(d − wc)

l

N+

∆2

2NJp · p

}

, (48)

where the kernel J is given by

Jki =sinh[(N − i)λ] sinh[λk]

sinh(Nλ) sinh λ+k

(

i

N−1

)

+θ(k−i−1)

[

k−i−sinh[(k − i)λ]

sinh λ

]

.

(49)

For a given distribution of phases the loading parameter d cannot take arbi-trary values since the strains must satisfy the constraint

pi(wi − wc) < 0, pi(wi+1 − wc) > 0 for i: pi 6= 0. (50)

Conditions (50) generate bounds on d(i1, i2, ..., in), where i1, i2, ..., in are loca-tions of the phase boundaries, parameterizing a particular metastable branch.For example, if n = 1 we obtain the limits d−(i) < d(i) < d+(i), where

d±(i) = wc + ∆

{

sinh[(N − i)λ] cosh[(i ± 1/2)λ]

coshλ

2sinh(Nλ)

+i

N− 1

}

. (51)

Formulae (43), (48) and (51) describing single-interface solutions are illus-trated in Fig. 4. An equilibrium branch parametrized by i begins at d = d−(i),

13

0.5 1 1.5 2

0.05

0.1

0.15

0.2

0.5 1 1.5 2

-0.2

0.2

Ψ

d

F

d

i=0i=1

i=10

(a) (b)

i=1

i=0

i=9

i=10

Fig. 4. The overall energy-strain and force-strain relations along the trivial (i = 0,N) and the single-interface (1 < i < N) metastable solutions. Here N = 10, µ = −2,wc = a = 1.

where wi = wc, and terminates at d = d+(i), where wi+1 = wc. In the intervald−(i) < d < d+(i) the force is linear, while the energy is quadratic. Due tothe symmetry of the problem, the ith branch is indistinguishable from the(N − i)th branch.

Equilibria with two interfaces are illustrated in Fig. 5 for a chain with N =6. Notice that unlike the case without NNN interactions (e.g. Puglisi and

0 0.5 1 1.5 2 d

0.05

0.1

0.15

0.2

1.2 1.25 1.3 1.35 1.4 1.45

0.057

0.058

0.059

d

(a) (b)

b

(-1 1 0 0 0 0) (0 -1 1 0 0 0) (0 0 -1 1 0 0) (0 0 0 -1 1 0)

(-1 0 1 0 0 0) (0 -1 0 1 0 0) (0 0 -1 0 1 0) (1 0 0 0 -1 0)

(-1 0 0 1 0 0) (0 -1 0 0 1 0) (1 0 0 -1 0 0) (0 1 0 0 -1 0)

(-1 0 0 0 1 0) (1 0 -1 0 0 0) (0 1 0 -1 0 0) (0 0 1 0 -1 0)

(1 -1 0 0 0 0) (0 1 -1 0 0 0) (0 0 1 -1 0 0) (0 0 0 1 -1 0)

(0 -1 0 1 0 0)

(-1 0 1 0 0 0)

p =Ψ Ψ

Fig. 5. (a) Energy-strain relation for the homogeneous solution (thin lines) and thetwo-interface solutions with various locations of the interfaces (thick lines). (b) Theblow-up of (a) around configurations with p = (0 −1 0 1 0 0) and p = (−1 0 1 0 0 0).Parameters: µ = −2, wc = a = 1, N = 6.

Truskinovsky, 2002b), here not only the volume fractions but also the actuallocations of the interfaces distinguish the branches. For example, the blow-upin Fig. 5b shows that the energy of the branch p = (0 −1 0 1 0 0) is higherthan the energy of a branch p = (−1 0 1 0 0 0) although both branches havethe same fraction of phase II.

To find the global minimum of the energy we can first minimize amongmetastable solutions with a given number of interfaces n. The resulting lowerenvelopes Ψ(n, d) are shown in Fig. 6 for N = 6 and n ≤ 4. One can see

14

that the single-interface solutions have the lowest energy, which is expectedsince our choice of nonlocal interactions leads to the penalization of interfaces.Similar observation has been made by Rogers and Truskinovsky (1997) for adiscrete model with long-range forces.

0.5 1 1.5 2

0.02

0.04

0.06

0.08

0.1

0.12

n = 1

n = 2

n = 3

n = 4

d0

Ψ

Fig. 6. Energy-strain relation for the energy minimizers in the families of solutionswith up to four interfaces. Here N = 6, µ = −2, wc = a = 1.

5 Peierls-Nabarro landscape

To understand the degree of relative stability of various metastable states weneed to evaluate the energy barriers. Since configurations with at most oneinterface are expected to have the lowest energy, it is instructive to start theanalysis with the corresponding section of the energy landscape. By focusingon the configurations with a single phase boundary we obtain an analog of thePeierls-Nabarro landscape known in the theory of dislocations; it is also consis-tent with experimental observations for martensitic materials (e.g. Krishnan,1985).

Consider a generic metastable configuration with a single phase boundaryat k = i. To evaluate the barrier between this and the neighboring localminimum we need to choose a path connecting the configuration wk(i) to theconfiguration wk(i + 1) with one extra spring in phase II. Since the (i + 1)thspring must change phase, it is natural to choose the strain wi+1 as a parameterand minimize the total energy with respect to all wk with k ≤ i and k ≥ i+2.We obtain the system of equations

(1 +1

2µ)wk +

1

4µ(wk+1 + wk−1) =

F + a for k ≤ i

F for k ≥ i + 2,(52)

which must be supplemented by the boundary conditions (2) and (6).

15

The configurations satisfying (52), (2) and (6) have the lowest energy amongall states with a given wi+1. This follows from the positive definiteness ofthe matrix B(i+1) obtained from (7) by deleting the (i + 1)th row and the(i+1)th column. It is convenient to write the explicit representation for theseconfigurations by using instead of wi+1 another parameter ν:

wk(ν) =

d − ∆{ ν

N− 1 + C1(sinh(λk) − sinh(λ(k − 1)))

}

, k ≤ [ν] + 1

d − ∆{ ν

N+ C2(e

λk − eλ(k−1) − eλ(2N−k) + eλ(2N−k+1))}

, k ≥ [ν] + 1,

(53)where [ν] denotes the integer part of ν. It can be directly checked that (53)solves (52) for k ≤ [ν] and k ≥ [ν] + 2, with [ν] = i and the total force Fgiven by

F =1 + µ

µ

(

d −ν

N∆)

. (54)

By matching the strains at k = i + 1 = [ν] + 1 and imposing (2), we computethe values of the constants

C1 =2{e(2N−[ν])λ[ν − [ν] − eλ(ν − [ν] − 1)] − eλ([ν]+1)[ν − [ν] − 1 − eλ(ν − [ν])]}

(e2λ − 1)(e2Nλ − 1)

and

C2 =2{(ν − [ν]) sinh[([ν] + 1)λ] − (ν − [ν] − 1) sinh[[ν]λ]}

(e2λ − 1)(e2Nλ − 1).

In particular, we obtain an explicit relation between the parameters wi+1 andν:

w[ν]+1(ν) = d − ∆( ν

N− 1

)

+ C1(sinh[([ν] + 1)λ] − sinh[[ν]λ]). (55)

One can see that parameter wi+1 oscillates as the function of i with period 1,while ν increases monotonically. Since the integer values of ν correspond ex-actly to metastable configurations, the function Ψ(ν) obtained by substituting(53) in (46) is exactly the Peierls-Nabarro (PN) potential of our system. Inorder to move from one valley of this potential at ν = i to the neighboringone at ν = i + 1 the system must overcome the Peierls barrier. It is located atν = νi defined by

wi+1(νi) = wc. (56)

The height of the Peierls barrier δΨi→i+1 = Ψ(νi) − Ψ(i) can be explicitlycomputed from (46) and (47).

The typical structure of the PN landscape is illustrated in Fig. 7 for a chainwith N = 10. One can see that at d = 0.5 the metastable configurationscorrespond to the integer values ν = 2 (or 3), ν = 1 (or 4), ν = 0, ν = 5 andν = 6. The corresponding strain profiles are shown in Fig. 8 for the metastablestates ν = 0, ν = 1 and ν = 2 and for the saddle point configurations withν = ν0 and ν = ν1.

16

1 2 3 4 5 6

0.05

0.1

0.15

0.5

0.05

0.1

d

i=1

i=0

i=2 i=3 i=4 i=5 i=6

50

1,4

2,3

ν

ν=0 ν=ν0

ν=1ν=2

ν=ν1

ν=6

ν=5

(a) (b)

ΨΨ

Fig. 7. (a) Energies of the single- and zero-interface branches of equilibria availableat d = 0.5. (b) Peierls-Nabarro energy landscape along the path wk(ν) connectingmetastable equilibria at d = 0.5. Parameters: N = 10, µ = −2, wc = a = 1.

(a) (b)

w2=wc

ν=ν1ν=2

ν=1

ν=1

2 4 6 8

0.5

1.5

2 4 6 8

0.5

1.5

1.0 1.0

w1=wc

ν=0

ν=ν0

wk

k

wk

k

Fig. 8. Strain profiles associated with nucleation and incremental growth of a newphase. Configurations ν = 0, ν = 1 and ν = 2 are local minima; ν = ν0 and ν = ν1

are saddle points. Parameters are the same as in Fig. 7.

6 Propagation threshold and nucleation peak

It is realistic to assume that the system driven by external loading d remainsin a local minimum configuration until the minimal energy barrier aroundthis state reaches a critical threshold H determined by the level of fluctu-ations (imperfections). In particular, the maximal delay strategy, associatedwith the gradient-flow dynamics, requires that the system stays on a givenmetastable branch until it becomes absolutely unstable (H = 0) and thenevolves towards the nearest local minimum along the path of steepest de-scent. Various resulting force-strain paths for the NN system are discussed inPuglisi and Truskinovsky (2002a,b). For the NNN system the two characteris-tic paths with and without a threshold are shown in Fig. 9. One can see that asthe elongation increases the system initially stays in the trivial configurationbut eventually reaches the state where the smallest energy barrier becomesequal to H. Then nucleation takes place and the system escapes from the

17

0.5 1 1.5 2

-0.2

0.2

F

d

H = 0

H = 0.003

Fig. 9. Branch-switching sequence for the path of maximal delay, H = 0, and for thepath with a finite critical energy barrier H = 0.003. Parameters: µ = −2, N = 10,wc = a = 1.

local minimum through the first saddle point with a subcritical height. Af-ter the nucleation event the phase boundary propagates along the chain ina stick-slip fashion, with the system getting temporarily trapped in each ofthe single interface metastable equilibria (parametrized by i). The resultinggraphs of the force F (d) exhibit characteristic serrations. A visible nucleationpeak originates from the fact that the nucleation event involves two springstransforming at once while the propagation involves only one spring changingphase at a time.

To understand why the first two springs change phase simultaneously, whichnever happens in the NN system (Puglisi and Truskinovsky, 2002a,b), weneed to compare the height of the energy barrier for the transition 0 → 1 withthe height of the barriers for several subsequent transitions. The height of thebarriers is shown in Fig. 10 for the chain with N = 10 and µ = −2. Notice thatthe barrier for the transition 1 → 2 is lower than the barrier for the transition0 → 1, moreover, the barrier 1 → 2 vanishes at d+(1) and beyond this pointthe branch i = 1 does not exist any more (transition 0 → 1 deteriorates into0 → 2). A similar calculation for N = 20 and µ = −2 shows that the barrier forthe transition 0 → 1 is higher than two subsequent barriers for the transitions1 → 2 and 2 → 3; as the last two barriers vanish at sufficiently large d, thetransition 0 → 1 first deteriorates into 0 → 2 and then into 0 → 3. One cansee that in this case nucleation event involves simultaneous transformation ofthree springs.

Our computations show that “massive” nucleation and the associated nucle-ation peak phenomenon occur only when NNN interactions are sufficientlystrong (large |µ−1|). Physically, it is the consequence of the nonlocal characterof the model. The nonlocality does not have to be of the NNN type: a differentnonlocal model with long-range interactions also exhibits the nucleation peak(Rogers and Truskinovsky, 1997).

To understand what is going on, we observe that before the nucleation allsprings are stretched uniformly, whereas after the nucleation, due to the pres-ence of the internal boundary layers, the springs that are closer to the phase

18

0.4 0.5 0.6 0.7 0.8 0.9 1

0.01

0.02

0.03

0.04

δΨ

d

0 1

2 3

3 4

1 2

0.85 0.9 0.95 1

0.001

0.002

1 2

0 1

0 2

2 3

Fig. 10. Peierls barriers for several transitions i → i + 1 parametrized by d. Thedashed lines indicate the intervals where the transition is energetically unfavorable.The insert is the blow-up of the selected region. Parameters: N = 10, µ = −2,wc = a = 1.

boundary have higher strain and hence are closer to the critical threshold thanthe springs far away. This facilitates the subsequent switching events and re-sults in the smaller force required for the propagation of a phase boundarycomparing to nucleation. In the system without nonlocality (µ = −∞), allsprings outside the interface are stretched uniformly, and therefore the prop-agation of the interface does not take place until the critical strain is reachedin all springs simultaneously. In this case the nucleation peak is absent: phasepropagation effectively reduces to successive nucleation events in the shorterand shorter chains which requires the same critical force.

To obtain the upper bound for the size of the nucleation peak it is enoughto consider the maximum delay strategy. In our model barrierless nucleationtakes place when the force reaches the spinodal limit Fmax = (1 + µ−1)wc. Onthe other hand, the advance of the interface from k = i to k = i + 1 takesplace at F (i, N) = FM + FP(i, N), where

FM =1 + µ

µwc −

a

2(57)

is the Maxwell force and FP(i, N) is the Peierls force given by

FP(i, N) = F (d+(i), i) − FM = a

(cosh[(1

2+ i)

λ]

sinh[λ(N − i)]

coshλ

2sinh(Nλ)

−1

2

)

. (58)

One can show that the function FP(i, N) depends on i weakly away from thenarrow boundary layers near i = 1 and i = N − 1. In the limit of infiniteN the Peierls force (58) approaches the constant value (Truskinovsky andVainchtein, 2003)

limN→∞

FP(i, N) = FP =a

2

√

1 + µ

µ. (59)

19

As the NNN interactions get weaker (µ → −∞), the Peierls force tends to thespinodal limit Fmax − FM and the nucleation peak disappears.

The configuration of the resulting hysteresis loop in the continuum limit isshown in Fig. 11. Although the serrations disappeared, the nucleation peakremains finite with the force dropping by the amount

τ =1 + µ

µwc − FP =

1

2a

(

1 −

√

1 + µ

µ

)

. (60)

This quantity is positive as long as −∞ < µ < −1 and is always less than a/2- the difference between the spinodal and Maxwell forces. The half-height of

F

d

∆

τ

FM

Fmax

wc

FP

Fig. 11. The maximum hysteresis loop in the continuum limit for the nonlocal (NNN)model with −∞ < µ < −1.

the narrow part of the hysteresis loop is given in the continuum limit by (59).

To estimate the number of springs involved in the nucleation event, we recallthat the nucleation always takes place at d = wc. Therefore we must find themetastable branch with the smallest nonzero i which is defined at this valueof d. Setting d± from (51) equal to wc, we obtain the equation for the numberof participating springs i

sinh[(N − i)λ] cosh[(i ± 1/2)λ]

coshλ

2sinh(Nλ)

= 1 −i

N. (61)

For sufficiently large N and generic µ the solution of this equation can beapproximated by

inuc =N

1 + eλ. (62)

In particular, (62) implies that in the continuum limit the number of springsinvolved in the nucleation event is infinite. The dimensionless size of the trans-

20

0.6 0.7 0.8 0.9 1 1.1

0.3

0.4

0.5A

B

C

2 4 6 8 10 12 14

0.5

1

1.5

2

A

B

C

d

F wk

k

(a) (b)

{}{7,8} {13} {12} {6,9} {11}

Fig. 12. Nucleation behavior of the system with external boundary layers: (a) sin-gle-interface (solid lines) and two-interface (dashed lines) metastable states; (b) thecorresponding strain profiles. The numbers in brackets indicate locations of phasethe boundaries for configurations with phase I on the left end. Parameters: N = 15,µ = −2, wc = a = 1.

formed portion of the chain (martensite band) remains finite:

l0 =1

1 + eλ. (63)

When nonlocal interactions are absent (µ = −∞), the nucleus contains onlyone spring and l0 = 0.

To illustrate the effect of the outside boundary layers on the nucleation phe-nomenon we replace the special boundary conditions (6), which suppressboundary layers, by the “zero-moment” conditions (4) with ΨB = 0. In thiscase the zero-interface solution is no longer trivial because the strain decreasesexponentially near the boundaries (see Fig. 12b).

Suppose that the chain is originally in phase I. When the critical value d =d+(0) is reached, the strains in the middle of the chain pass the thresholdw = wc and the one-phase solution becomes unstable (point A in Fig. 12).This leads to the formation of either two symmetric interfaces in the center(point C in Fig. 12) or of a single interface near one of the the boundaries (pointB in Fig. 12). Our computations show that the single-interface configurationB has a lower energy than the two-interface configuration C. However, duringtransition from A to C only two springs (7th and 9th) transform into phase II,while transition from A to B requires transformation of at least three springswhich are also initially farther below the threshold. As a result, transitionfrom A to B may encounter a higher barrier than transition from A to C,and then the nucleation will take place in the interior of the chain. Whilethis possibility can be investigated rigorously, for the subject of this paperit is enough to mention that the nucleation peak survives in both cases (seeFig. 12a).

21

7 Verification of the model

The explicit formulae from the previous section can be used to obtain boundson the value of the nonlocality measure γ. The expression for γ from (10) and(60) has the form

γ =E

4

[

1 −

(

1 −2τ

E∆

)−2]

, (64)

where E = K + 4γ is the macroscopic elastic modulus of the homogeneouschain and ∆ is the macroscopic transformation strain. The magnitude of thestress drop at the peak τ and the transformation strain ∆ are available fromthe experimental data of Shaw and Kyriakides (1995) on NiTi wires. For in-stance, in the experiment conducted at 70◦ and the loading rate 4× 10−5 s−1,the measurements gave τ = 0.039 GPa, and ∆ = 3.97%. The Young’s moduliof the two participating phases are different, E = 56.7 GPa and E = 27.5GPa. By using two separate values we estimate γ to be between −1.1 and −1GPa, which implies that µ is in the range −14.7 < µ < −7.3. Similar com-parison of experiment and theory for CuAlNi yields µ = −33.4 (Truskinovskyand Vainchtein (2003)) .

To obtain an independent estimate of γ we can assume that the interactionsbetween particles are governed by the Lennard-Jones potential. In this casewe have εφ1(

rε− 1) = 2εφ2(

r2ε

− 1) = U(r), where φ1(w) and φ2(w) are theenergies of the NN and NNN springs, respectively, and U(r) has the form

U(r) =Kε

72

[(

ε

r

)12

− 2

(

ε

r

)6]

. (65)

The coefficients in (65) are chosen to ensure that the elastic modulus at equi-librium r = ε equals K. Linearizing around the unstretched homogeneous statewith the spacings r = ε and r = 2ε, we obtain (Charlotte and Truskinovsky,2002)

µ =U ′′(ε)

4U ′′(2ε).

This yields µ = −56.5. Taking E = 56.7 GPa as the value of the Young’smodulus we obtain γ = −0.255 GPa which despite the rather rigid form ofthe potential (65) is within a reasonable range from the values obtained above.

Finally, we can use the above estimates of µ to predict the initial size of themartensite band L0. By taking −14.7 < µ < −7.3 we obtain 0.001 < L0/L <0.005, where L is the size of the specimen. While we could not find directexperimental measurements for L0 in the literature, this parameter is usuallyestimated to be of the order of the specimen’s diameter D (e.g. Sun and Zhong,2000). This would be in agreement with our estimate since for thin NiTi wiresused in experiment D/L ∼ 0.0025 (Leo et al., 1993)).

22

8 Strain-gradient approximation

Now we can compare the exact results obtained in the discrete model withthe predictions of the quasi-continuum strain gradient approximation whichneglects discreteness but retains the original internal length scale. To obtainthe approximate model we temporarily reintroduce dimensional variables andperform the Taylor expansion in the small parameter ε = L/N . By preservingthe first nonlocal term in the energy functional we obtain

Ψ =∫ L

0[φ(w) +

1

2Aε2(w′)2]dx, (66)

where we omitted nonessential null Lagrangian contributions and introduced

A = −1

12(K + 16γ). (67)

The functional (66) must be minimized subject to the constraint

∫ L

0w(x)dx = d (68)

and the clamping boundary conditions

w′(0) = w′(L) = 0 (69)

which represent the continuum analog of (6). The Euler-Lagrange equationscan be written in the form φ′(w)−Aε2w′′ = F (analog of (3)), where φ′(w) =Ew − Kaθ(w − wc). Recall that E = K + 4γ is the macroscopic modulus.

The trivial solution of the Euler-Lagrange equations w(x) = d/L is stablewhenever the second variation of energy

δ2Ψ =∫ L

0(Ev2 + ε2Av′2)dx (70)

is positive for all v(x) 6= 0 satisfying the constraint

∫ L

0vdx = 0. (71)

The zero eigenvalue problem determining the stability boundaries reduces tosolving linear equation Ev′ − ε2Av′′′ = 0. The general nontrivial solution ofthis equation compatible with (71) takes the form

v(x) = cosπnx

L,

23

where n ≥ 1 is an integer. The solution exists for

E +(πnε

L

)2A = 0. (72)

If A = 0, the stability condition obviously reduces to E > 0. When A > 0, weobtain E + (πnε/L)2A ≥ E + (πε/L)2A and the trivial state is stable if andonly if

E +(πε

L

)2A > 0. (73)

The instability takes place through the growth of a long-wave mode v =cos(πx/L). When A < 0, the trivial solution is always unstable because onecan always find large enough n at which E+(πnε/L)2A < 0; the correspondinginstability is of the short-wave type.

To compare the stability limits in the strain gradient model with the onesobtained in the discrete model, we shall first rewrite them in terms of dimen-sionless µ, and replace L/ε by N . In the case when A > 0, we obtain

−4 < µ < −1 − π2

3N2

1 − π2

12N2

(γ < 0). (74)

Recall that in the discrete case the corresponding interval is −∞ < µ <− cos2 π

2N(γ < 0). It is easy to see that for large N both models predict

the same upper boundary µ = −1 (E = 0), which means that the strain-gradient model captures the onset of macroscopic instability in the discretemodel rather well.

As we showed above, when A < 0 the stability range does not exist due to theshort-wave instability. However, in view of the underlying discrete structure,a short-wave instability is unphysical if its wave length is less than the lengthscale of the lattice ε. If we assume that the mode number cannot exceed n = N ,we obtain the following stability condition:

E + π2A > 0, (75)

which in terms of µ can be written as a combination of two intervals,

µ < −4 (γ < 0) (76)

and

−1 − π2

3

1 − π2

12

< µ < ∞ (γ > 0). (77)

While the former stability range obviously complements (74), the latter has itsown analog in the discrete model, − sin2 π

2N< µ < ∞ (γ > 0). Observe that

the lower limit in the strain-gradient model is significantly higher than in thediscrete model suggesting that even with the restriction on the minimum wave

24

length, the strain-gradient approximation grossly exaggerates the instabilitydomain on the side of the short-wave instability. The general trend, however,is predicted correctly: as in the discrete model, the short-wave instability inthe gradient model develops before the strain enters the macroscopic spinodalregion (see also Bardenhagen and Triantafyllidis (1994)).

Fig. 13 summarizes the stability analysis in both the K-γ and µ-N−1 planes.It should be compared with the analogous stability diagrams for the discretemodel (Figure 2).

-4 -2 0 2 4

-0.5

0

0.5

A > 0

A < 0

γ

K

(a) (b)

0.1 0.2 0.3 0.4

-5

5

10

-1

�

N ��

�����

�����

A < 0

A > 0

A < 0

���

���

microinstability

macroinstability

microinstability

macroinstability K �� ����

K ��

Fig. 13. (a) Domain of stability for the trivial solution in K-γ plane in thestrain-gradient model with L/ε = N = 3 and in the continuum limit (N → ∞,dashed lines). (b) The dependence of the stability boundaries on the size of thesystem. In both figures stability domain is in gray.

Now consider the nontrivial solutions of the variational problem (66), (68).As shown in Carr et al. (1984), the strain-gradient model allows for localminima with at most one interface. For the piecewise linear model all thesemetastable configurations can be obtained analytically (Truskinovsky andZanzotto, 1996). A representative picture is shown in Fig. 14, with the pa-rameters chosen to match those for the discrete model with N = 20. Weobserve that while the strain-gradient approximation captures the structureof the absolute minimizers of the discrete problem, it fails to reproduce therich structure of the metastable equilibria which in the quasi-continuum modelall collapse into a single branch. This is the reason why the strain-gradientapproximation does not generate a realistic hysteresis. Also, while all multiple-interface equilibria of the discrete model are metastable, configurations withmore than one interface are absolutely unstable in the quasi-continuum ap-proximation. This difference is due to the fact that in the discrete model onecannot vary the volume fraction of phases continuously, and thus certain per-turbations that make multiple-interface solutions unstable in the continuumcase are impossible in the discrete model. Finally, we remark that the estimate−14.7 < µ < −7.3 obtained in the previous section from the experimental dataimplies that A < 0, contrary to the typical assumption in the strain-gradientmodels that the coefficient A is positive.

25

0.5 1 1.5 2

0.05

0.1

0.15

0.2

0.5 1 1.5 2

-0.2

0.2

Ψ

dF

d1

0.5 1 1.5 2

0.05

0.1

0.15

0.2

d

0.5 1 1.5 2

-0.2

0.2

F

d

(a) (b)

trivial solution

single interface

Ψ

Fig. 14. The rescaled energy-strain and force-strain relations for the absolute energyminimizers (thick line) and the metastable single-interface solutions (thin lines) for(a) the discrete chain with N = 20 and (b) the strain-gradient approximation withε = 1/20. Other parameters: µ = −2, wc = a = 1. Unstable single-interface solutionsare shown by dashed lines.

9 Conclusions

Our study clarifies why the behavior of the local and nonlocal systems withbi-stable elements is different. In the case of nonlocal interactions the ele-ments around the phase-boundary are “pre-conditioned” due to the presenceof the boundary layers. This results in smaller energy barriers for the propa-gation of the phase boundary compared to the local model and gives rise tothe finite mismatch between the spinodal stress and the Peierls stress. An-other difference is that in the nonlocal models the nucleation event typicallyinvolves transformation of more than one element. This produces nucleationpeak which has been long known from experiment. We showed that the nucle-ation peak persists in the continuum limit where it manifests itself through aninstantaneous formation of a finite band scaled with the length of the sample.Although the nucleation peak is captured by the strain-gradient approxima-tion, the resulting shape of the hysteresis loop is unrealistic. This suggeststhat the nonlocal coupling in the actual physical system is weaker than whatis required for the gradient model to be applicable.

Acknowledgments. This work was supported by the NSF grants DMS-0102841 (L.T.) and DMS-0137634 (A.V.).

26

References

Bardenhagen, S., Triantafyllidis, N., 1994. Derivation of higher order gradi-ent continuum theories in 2,3-D non-linear elasticity from periodic latticemodels. Journal of the Mechanics and Physics of Solids 42 (1), 111–139.

Butler, J. F., 1962. Luders front propagation in low carbon steels. Journal ofthe Mechanics and Physics of Solids 10, 313–334.

Carr, J., Gurtin, M., Slemrod, M., 1984. Structured phase transitions on afinite interval. Archive for Rational and Applied Mechanics and Analysis86, 317–351.

Charlotte, M., Truskinovsky, L., 2002. Linear elastic chain with a hyper-pre-stress. Journal of the Mechanics and Physics of Solids 50, 217–251.

Cottrell, A. H., Bilby, B. A., 1949. Dislocation theory of yielding and strainageing of iron. Proceedings of Physical Society London 62/I-A, 49–62.

Fedelich, B., Zanzotto, G., 1992. Hysteresis in discrete systems of possiblyinteracting elements with a two well energy. Journal of Nonlinear Science 2,319–342.

Froli, M., Royer-Carfagni, G., 2000. A mechanical model for the elastic-plasticbehavior of metallic bars. International Journal of Solids and Structures 37,3901–3918.

Hall, E. O., 1970. Yield point phenomena in metals and alloys. Plenum Press,New York.

Johnston, W. G., Gilman, J. J., 1959. Dislocation velocities, dislocation densi-ties and plastic flow in lithium fluoride crystals. Journal of Applied Physics30, 129–144.

Kartha, S., Krumhansl, J., Sethna, J., Wickham, L., 1995. Disorder driven pre-transitional tweed pattern in martensitic transformations. Physical ReviewB: Condensed Matter 52 (2), 803.

Krishnan, R., 1985. Stress induced martensitic transformations. Material Sci-ence Forum 3, 387–398.

Kyriakides, S., Miller, J. E., 2000. On the propagation of luders bands in steelstrips. Journal of Applied Mechanics 67 (4), 645–654.

Leo, P. H., Shield, T. W., Bruno, O. P., 1993. Transient heat transfer effectson the pseudoelastic behavior of shape-memory wires. Acta Metallurgica etMaterialia 41, 2477–2485.

Lexcellent, C., Tobushi, H., 1995. Internal loops in pseudoelastic behaviour ofti-ni shape memory alloys: experiment and modelling. Meccanica 30, 459–466.

Mickens, R. E., 1990. Difference equations: theory and applications, 2nd Edi-tion. Van Nostrand Reinhold Co., New York.

Mindlin, R. D., 1965. Second gradient of strain and surface-tension in linearelasticity. International Journal of of Solids and Structures 1, 417–438.

Muller, I., Villaggio, P., 1977. A model for an elastic-plastic body. Archive forRational and Applied Mechanics and Analysis 65, 25–46.

Pagano, S., Paroni, S., 2003. A simple model for phase transformations: from

27

discrete to the continuum problem. Quarterly of Applied Mathematics 54,328–348.

Puglisi, G., Truskinovsky, L., 2000. Mechanics of a discrete chain with bi-stableelements. Journal of the Mechanics and Physics of Solids 48, 1–27.

Puglisi, G., Truskinovsky, L., 2002a. A mechanism of transformational plas-ticity. Continuum Mechanics and Thermodynamics 14, 437–457.

Puglisi, G., Truskinovsky, L., 2002b. Rate independent hysteresis in a bi-stablechain. Journal of the Mechanics and Physics of Solids 50, 165–187.

Rogers, R., Truskinovsky, L., 1997. Discretization and hysteresis. Physica B233, 370–375.

Shaw, J. A., Kyriakides, S., 1995. Thermomechanical aspects of NiTi. Journalof the Mechanics and Physics of Solids 43, 1243–1281.

Shaw, J. A., Kyriakides, S., 1997a. Initiation and propagation of localized de-formation in elasto-plastic strips under uniaxial tension. International jour-nal of plasticity 13 (10), 837–871.

Shaw, J. A., Kyriakides, S., 1997b. On the nucleation and propagation of phasetransformation fronts in a NiTi alloy. Acta Materialia 45, 683–700.

Sun, Q.-P., Zhong, Z., 2000. An inclusion theory for the propagation of marten-site band in NiTi shape memory alloy wires under tension. InternationalJournal of Plasticity 16, 1169–1187.

Triantafyllidis, N., Bardenhagen, S., 1993. On higher order gradient continuumtheories in 1-D nonlinear elasticity. Derivation from and comparison to thecorresponding discrete models. Journal of Elasticity 33 (3), 259–293.

Triantafyllidis, N., Bardenhagen, S., 1996. The influence of scale size on thestability of periodic solids and the role of associated higher order gradientcontinuum models. Journal of the Mechanics and Physics of Solids 44, 1891–1928.

Truskinovsky, L., Vainchtein, A., 2003. Peierls-Nabarro landscape for marten-sitic phase transitions. Physical Review B 67, 172103.

Truskinovsky, L., Zanzotto, G., 1996. Ericksen’s bar revisited: Energy wiggles.Journal of the Mechanics and Physics of Solids 44 (8), 1371–1408.

Ye, Y. Y., Chan, C. T., Ho, K. M., 1991. Effect of phonon anomalies on theshear response of martensitic crystals. Physical Review Letters 66, 2018–2021.

28