Hysteresis and Stick-Slip Motion of Phase Boundaries in ...annav/papers/JNS.pdf · Hysteresis and Stick-Slip Motion of Phase Boundaries in Dynamic Models of Phase Transitions A. Vainchtein1

J. Nonlinear Sci. Vol. 9: pp. 697–719 (1999)

© 1999 Springer-Verlag New York Inc.

Hysteresis and Stick-Slip Motion of Phase Boundariesin Dynamic Models of Phase Transitions

A. Vainchtein1 and P. Rosakis21 Division of Mechanics and Computation, Department of Mechanical Engineering, Stanford

University, Stanford, CA 94305, USA2 Department of Theoretical and Applied Mechanics, Cornell University, Ithaca, NY, 14853,

USA

Received August 4, 1998; revised December 11, 1998Communicated by Robert Kohn

Summary. We investigate hysteretic behavior in two dynamic models for solid-solidphase transitions. An elastic bar with a nonconvex double-well elastic energy densityis subjected to time-dependent displacement boundary conditions. Both models includeinertia and a viscous stress term that provides energy dissipation. The first model involvesa strain-gradient term that models interfacial energy. In the second model this termis omitted. Numerical simulations combined with analytical results predict hystereticbehavior in the overall end-load versus end-displacement diagram for both models. Thehysteresis is largely due to metastability and nucleation; it persists even for very slowloading when viscous dissipation is quite small. In the model with interfacial energy,phase interfaces move smoothly. When this term is omitted, hysteresis is much morepronounced. In addition, phase boundaries move in an irregular, stick-slip fashion. Thecorresponding load-elongation curve exhibits serrations, in qualitative agreement withcertain experimental observations in shape-memory alloys.

Key words. Phase transition, dynamics, hysteresis, metastability, stick-slip

MSC numbers. 73C50, 73G25, 73H10, 73B30, 73F15

PC numbers. 64.60.My, 64.60.Qb, 64.70.Kb

1. Introduction

Materials undergoing stress-induced martensitic phase transformations often form avariety of finely layered microstructures and exhibit hysteretic behavior, e.g., [9], [31],[32], [33].

698 A. Vainchtein and P. Rosakis

In the last twenty years, a number of researchers have attempted to describe formationof microstructure and hysteresis in crystalline solids within the framework of elasticitytheory [1], [2], [6], [7], [14], [21], [22], [28], [29]. A common approach involves mini-mization of a nonconvex elastic energy for the material, following the pioneering analysisof Ericksen [10]. Although the absolute minimization of the total energy captures basicfeatures of the microstructure [17], it cannot account for hysteresis, which arises whenthe material gets locked in metastable states, as suggested by calculations in [12], [24];see also [1], [5], [7], [20], [28].

Recently, various studies of dynamic models have recognized the importance ofmetastable equilibria, namely local minimizers of the potential energy, for example, [6],[13], [34], [35]. The first two of these consider the dynamics of a viscoelastic bar with anonconvex elastic energy density, placed on an elastic foundation and subjected to zerodisplacement boundary conditions. In statics the elastic foundation makes finer and finerphase mixtures energetically more favorable [6]. However, numerical results in [35] showthat the dynamic solutions typically tend to weak local energy minimizers with finitelymany phase boundaries. This was confirmed analytically in [13] for the case of low initialenergy. The finite scale of phase layering also agrees with experimental results [32], [33].

Static models of phase transitions are important because they identify multiple meta-stable equilibrium states compatible with given boundary conditions. The complicatedstructure of branches of metastable equilibria under parametric loading was examined in[37], [38], [39], [40], [41]. On the other hand, in studying hysteresis under time-dependentloading, an additional mechanism is needed for switching from one equilibrium branchto another. Such a mechanism is essential for the study of hysteresis, since it determinesat what value of the loading the switching occurs and which of the equilibrium branchesis chosen.

In this paper, we consider a dynamic model that provides such a mechanism. A vis-coelastic bar, with a nonconvex double-well elastic energy density and viscous stresses,is subjected to time-dependent displacement boundary conditions. Each well of the en-ergy density represents a material phase. Inertia is taken into account. In the first part ofthe paper we include interfacial energy modeled by a strain-gradient term. In the secondpart, this term is omitted. Viscosity provides energy dissipation, while time-dependentdisplacement-controlled loading at the ends of the bar supplies energy into the system.

Similar dynamical models have been investigated before [3], [4], [6], [13], [15], [19],[27]; they all considertime-independentloads. This forces the energy to decrease withtime. This fact is crucial in proofs of global existence and facilitates the analysis of theasymptotic behavior of solutions.

The case of time-dependent displacement boundary conditions considered here isclearly relevant to the study of hysteresis, but is less tractable analytically. While localexistence, uniqueness, and regularity of strong solutions can still be established, we havenot succeeded in proving the global existence of solutions.

In the zero end-displacement case [6], [19], [35], one has to pick various nonzeroinitial conditions in order to obtain a nontrivial solution. These initial conditions oftencorrespond to unstable states, and it is far from clear whether the material can ever be inthese states in the first place. In this study, we start at a stress-free stable equilibrium withzero initial velocity and then load the bar at the ends. This is a physically appropriateway to model experiments such as the tension test, allows control of the loading rate,and facilitates a study of overall hysteretic behavior during a loading-unloading cycle.

Hysteresis and Stick-Slip Phase Boundary Motion 699

We present hysteresis loops in load-elongation curves obtained from numerical sim-ulations of the model when the system is subjected to cyclic loading. We find that forsufficiently slow loading, the system closely follows a branch of metastable equilibriathat emerges from the stress-free stable state, until the strain enters thespinodal region, orthe region where the elastic energy density is concave. At this point, the branch becomesunstable, and the instability causes the dissipation rate to exceed the loading power. Thedynamic solution switches to another metastable branch with lower potential energy. Theinitial switching event is from a single-phase to a two-phase branch and corresponds tonucleation.

The kinetic energy and the viscous dissipation are small, with inertia playing a minorrole, except during branch switching, which is a highly dynamical process. The majorsources of hysteresis are metastability and nucleation, with dissipation playing a minorrole. There is substantial hysteresis even for arbitrarily slow loading when viscosityeffects are minor.

We compare the hysteresis loops resulting from the models with and without interfa-cial energy. The results turn out to be very different.

In the case with interfacial energy there are only two stable branches of equilibria:a single-phase branch (without interfaces) and a two-phase branch with one smoothinterface (transition layer) and lower potential energy. After the system switches fromthe one-phase to the two-phase branch, it follows the latter, while the interface is moving.

In contrast, when interfacial energy is absent, there are infinitely many local mini-mizers of the potential energy with discontinuous strain. Moving interfaces are smooth,while strain discontinuities represent static interfaces that cannot move. This resultsin rather striking behavior, namelystick-slipmotion of the interfaces: A sharp phaseboundary, once formed, does not move (stickregime), until the time-dependent loadingcauses the strain to enter the spinodal region in part of the bar. The spinodal instabilitysmoothens the strain profile and moves the interface to its next location (slip regime).The interface alternates between stick and slip. This leads toserrations, or “teeth,” onthe overall load-elongation curve. Stick-slip interface motion and serrations of the load-elongation diagram have been observed experimentally in shape-memory alloys [21],[22], [25]. The overall shape of the hysteresis loop is in qualitative agreement with theseexperimental results, much more so than in the model with interfacial energy.

When interfacial energy is present, only one phase boundary forms. Without this term,there are multiple interfaces; their number is higher for lower values of the viscositycoefficient. Although local minima of the potential energy may have arbitrarily manyphase boundaries [10], dynamics with viscosity serves as a mechanism that selects afinite number of them, in agreement with an observation in [13].

Another phase transition model describing hysteresis is the sharp-interface theorywithout viscosity or higher gradients [2], but with a kinetic relation between the interfacedriving force and speed. When the kinetic relation is nonmonotone, stick-slip interfacemotion and serrated hysteresis loops are observed in both quasistatic and dynamic settings[29], [30].

It was pointed out to us by a referee that stick-slip boundary motion and serrationshave also been observed in [11], in numerical studies of a model with Maxwell viscosityin place of the Kelvin-Voight viscosity of the present model. That model involves anadditional term corresponding to the total stress rate in the constitutive law. This yieldsa hyperbolicsystem of equations that permits propagating as well as stationary strain


discontinuities, in contrast to the parabolic equation considered here. As noted in [11],the exponential growth of certain types of discontinuities that occurs in the Maxwell-type model makes it very sensitive to perturbations such as numerical rounding errors,in contrast to the model studied here.

The structure of the paper is as follows. In Section 2 we formulate the initial-boundary-value problem. Local existence and uniqueness theorems may be found in the Appendix.Results of numerical simulations for the case with viscosity and interfacial energy aredescribed in Section 3, and comparisons with analytical results regarding equilibriumbranches are made. This provides an explanation of the observed hysteresis in terms ofmetastability. In Section 4 we exhibit results for the case with viscosity only, in whichstick-slip motion of the phase boundaries is observed, resulting in serrated hysteresisloops. Analytical predictions aimed at understanding the mechanism responsible for thestick-slip phenomenon are presented in Section 5.

2. Formulation

Consider a bar of unit undeformed length, with constant reference densityρ > 0. Thedisplacement field is denoted byu(x, t), wherex ∈ [0,1] is the reference coordinate ofa point in the bar andt is time. The strain field isux(x, t). The elastic energy of the barequals

∫ 10 f (ux)dx, where

f (ux) = (u2x − 1)2/4 (1)

is a double-well energy density with minima atux = ±1. The two wells represent twodifferent material phases. The set of strainsux where f is locally concave, orf ′′ < 0, iscalled thespinodal region. For the energy density in (1), it is the interval [−1/

√3,1/√

3].In what follows, the− and+ phasesare the sets of strainsux > 1/

√3 andux < −1/

√3,

respectively, wheref is convex.Ericksen [10] considered the static problem of minimization of the potential energy

functional

E =∫ 1

0f (ux)dx, (2)

subject to the boundary conditionsu(0) = 0 andu(1) = d. For d in [−1,1], apartfrom the homogeneous equilibrium solutionu(x) = dx, which is unstable ford in[−1/√

3,1/√

3], there are infinitely many inhomogeneous solutions (weak local minima).In these solutionsu is continuous but the strain is piecewise constant, alternating betweentwo valuesux = e1 in the+ phase andux = e2 in the− phase, wheref ′(e1) = f ′(e2),so that the stressσ(ux) = f ′(ux) is constant in the bar. The number and location ofstrain discontinuities (phase boundaries) is arbitrary. Whene1 ande2 minimize f , i.e.,e1,2 = ±1, these solutions are global minimizers of (2) ford in the interval [−1,1]; theportion of the bar in each phase is determined byd.

In the first part of this paper we also account for interfacial energy, modeled by thestrain-gradient term

∫ 10 αu2

xx dx, with constantα > 0. In this case, the potential energy


is given by

E =∫ 1

0[ f (ux)+ αu2

xx] dx. (3)

The introduction of the strain-gradient term has been widely used to analyze spinodalregion decomposition, phase transitions, and other phenomena. See, e.g., [8], [23], [26].[36], where the problem of minimizing (3) subject to displacement boundary conditionswas considered. The strain-gradient term penalizes the formation of phase boundaries.In extremals of (3) strain discontinuities are replaced by smooth transition layers, ofwhich there can be a finite maximum number [41]. Moreover, only solutions with onephase boundary minimize the potential energy [8].

Our dynamic model includes a viscous stressγuxt, linearly proportional to the strainrate, with viscosity coefficientγ > 0. This term introduces energy dissipation. The totalstress6(x, t) is given by

6(x, t) = σ(ux)− 2αuxxx+ γuxt, u = u(x, t), (4)

whereσ(ux) = f ′(ux) is the elastic contribution to the stress; the term−2αuxxx is dueto strain gradients. Balance of linear momentum takes the formρutt = 6x.

We consider the following initial-boundary-value problem for the displacementu(x, t):

ρutt = [σ(ux)− 2αuxxx+ γuxt]x,

u(0, t) = d0(t),

u(1, t) = d1(t),

uxx(0, t) = uxx(1, t) = 0,

u(x,0) = u0(x),

ut (x,0) = 0.

(5)

Initially, the bar is in a stable equilibrium stateu0(x), with zero initial velocity. It is subjectto time-dependent displacement boundary conditions, chosen to be either symmetric,

d0(t) = −d(t)/2, d1(t) = d(t)/2, (6)

or nonsymmetric,

d0(t) = 0, d1(t) = d(t). (7)

The natural boundary conditions (5)4 are due to the strain-gradient term. The total (kineticplus potential) energy is give by

E =∫ 1

0

[1

2ρu2

t + f (ux)+ αu2xx

]dx. (8)

One can show that the rate of change of the total energy is given by

E′(t) = S(t)d′(t)− γ∫ 1

0u2

xt dx. (9)


In the case of nonsymmetric boundary conditions (7),S(t) equals the end load6(1, t)(recall (4)). In the case of symmetric boundary conditions (6), it is given by

S(t) = 1

2[6(0, t)+6(1, t)] . (10)

The termS(t)d′(t) in (9) is theloading power; it supplies energy into the system. Thesecond termγ

∫ 10 u2

xt dx in (9) is the energy dissipation rate due to the presence of viscousstresses.

In Section 4 we focus on a special case of the model, with higher gradient termsomitted (α = 0) but with viscosity and inertia still present (viscoelastic bar). In this casethe stress is6 = σ(ux)+ γuxt and the initial-boundary value problem reduces to

ρutt = [σ(ux)+ γuxt]x,

u(0, t) = −d(t)/2,u(1, t) = d(t)/2,u(x,0) = u0(x),ut (x,0) = 0,

(11)

whereu0(x) is a stable equilibrium.

3. The Case with Interfacial Energy

We now investigate the behavior of the dynamic model including interfacial energy. Fromthe analytical point of view, local existence and uniqueness of solutions to problem (5)is shown in Theorem 1 in the Appendix. To study the mechanical behavior of the modelunder cyclic loading, problem (5) was solved numerically. An implicit finite differencecode was adapted from [18], [19].

The time-dependent displacement boundary conditions (5)2,3 are specified by thefollowing choice of the loading functiond(t) in (6), (7), given by

d(t) =

(1T −1) t2

ti tT+1 for 0≤ t ≤ ti ,

(1T −1)[− (t−ti )2

tT (tT−ti )+ 2 t

tT− ti

tT

]+1 for ti ≤ t ≤ 2tT − ti ,

(1T −1)(t−2tT )2

ti tT+1 for 2tT − ti ≤ t ≤ 2tT .

(12)

Here1,1T , ti , tT are constants. Graphs of the loading functiond(t) and the loading rated′(t) are shown in Figure 1. The initial end displacement isd(0) = 1. Loading occurswith d increasing up to a maximumd(tT ) = 1T at t = tT , followed by unloading to theinitial valued(2tT ) = 1. The parametertT is inversely proportional to the loading rateamplitude. Parameterti controls the initial accelerationd′′(0). The graphs of the loadingfunctiond(t) and the loading rated′(t) are shown in Figure 1.

We turn momentarily to the description of branches of equilibrium states with the enddisplacementd viewed as a parameter. Here we quote results form [41]. Equilibriau(x)satisfy the Euler-Lagrange equation,

f ′′(ux)uxx − 2αuxxxx= 0, (13)


Fig. 1. (a) Loading functiond(t), and (b) loading rated′(t) used in numerical simulations.

natural boundary conditionsuxx(0) = uxx(1) = 0 and boundary conditionsu(0) =0, u(1) = d. Figure 2 displays equilibrium branches relevant here. Each branch isrepresented by plotting the potential energyE in (3) and the end loadSas functions ofd. Branches are labeled by the numbern of phase boundaries, which are in the form ofsmooth transition layers. For anyd, there is a constant strain solutionu(x) = dx, withno phase boundaries. The main branchn = 0 consists of these solutions. It is unstablebetween the pointsA andD and stable elsewhere. Here and below, bystablesolutions wemean local minima of the potential energy. From this branch a finite number of nontrivialsolution branches bifurcate, with solutions satisfyingun(x) = dx+ ε sin(nπx)+ o(ε)in smallε-neighborhoods of the bifurcation points; for a more complete description see[41]. Among these branches, only then = 1 branch contains stable solutions on the partBC [8]. These solutions have one phase boundary.

Fig. 2. (a) Diagrams of potential energyE, and (b) stressS versus end displacementd for then = 0 (dotted line) andn = 1 (solid line) equilibrium branches (bar with interfacial energy):α = 0.0005,β = 0. PartAD of then = 0 branch is unstable, the rest is stable. PartsAB andC Dof then = 1 branch are unstable;BC is stable.


Fig. 3. (a) Potential energyE, and (b) end loadSversusd from numerical solutions to dynamicsproblem (5) (thick solid line) shown over the staticsn = 0 (dotted line) andn = 1 (solid line)equilibrium branches:α = 0.0005,β = 0, γ = 0.1, ρ = 0.05, tT = 100.

Next we consider numerical dynamic solutions of system (5) (nonzero inertia andviscosity terms), with the bar initially at the equilibriumu0(x) = −x with d(0) = −1(stress-free global minimum of the+phase), and boundary displacements given by (5)2,3,(7), (12). For sufficiently low loading rates, the numerically computed dynamic solutioninitially follows the staticn = 0 branch rather closely. It goes past the bifurcation pointA in Figure 2, thus entering the statically unstable part of the branch, with strains inthe spinodal region. Then it suddenly drops onto the stablen = 1 branch, as shown inFigure 3. At this point, suddennucleationof a finite interval of the− phase occurs at oneend of the bar, resulting in a solution with one phase boundary, of the form(−,0,+).This notation means that the right and left ends of the bar are in the+ and− phase,respectively, and the phases are separated by a “thick” phase boundary (transition layer)within which strain is in the spinodal region, denoted by 0. The solution then follows then = 1 branch, with the phase boundary movingsmoothlyto the right, until the branchceases to exist for the current value ofd (at the turning pointC), and drops onto then = 0 branch again. At this point the entire bar has transformed into the− phase. Thereverse behavior is observed during unloading.

Nucleation and branch switching takes place after the current branch becomes un-stable. To see this, consider the linearization of (5)1 about the constant strain solutionu0(x) = dx and setu(x, t) = u0(x)+ v(x, t), to obtain

ρvt t = −θvxx − 2αvxxxx+ γ vxxt. (14)

Hereθ = − f ′′(d) andv is subject tov(0, t) = v(1, t) = vxx(0, t) = vxx(1, t) = 0.Seek solutions in the formv(x, t) = exp(ωt) sin(kx). The boundary conditions dictatethatk = πn, whileω must equal one of

ω1,2 = γ k2

2ρ

(−1±

√1+ 4θρ

γ 2k2− 8αρ

γ 2

), (15)

provided the expression under the square root is nonnegative. One easily sees that when


Fig. 4. (a) EnergyE, and (b) end loadS from numerical solutions to dynamics problem (5) (thicksolid line) shown over the staticsn = 0 (dotted line) andn = 1 (solid line) equilibrium branches:α = 0.0005,β = 0, γ = 0.1, ρ = 0.05, tT = 1000.

−θ = f ′′(d) > −2απ2, bothω1 andω2 are negative, so thatu0(x) = dx is stable.This is the case whend is in the+ phase and a small portion of the spinodal region.The bifurcation pointA, where−θ = f ′′(d) = −2απ2, lies in the spinodal region andcorresponds to the margin of stability of the constant strain equilibrium branch. Whenθ

exceeds 2απ2, ω2 < 0 butω1 > 0, thus resulting in exponential growth and instability.Hence, when the dynamic solution passes pointA, the instability causesuxt to grow.This increases the dissipation rate in (9). The kinetic energy also grows. When the energydissipation rate exceeds the loading powerS(t)d′′(t), the total energy, and therefore thepotential energy, start to decrease; the system thereby switches to then = 1 branch thathas lower energy. How far into the spinodal region the strain gets before the nucleationoccurs depends on the loading rate. For higher values of the loading rated′(t), the systemis carried further into the spinodal region before the instability has time to develop. Forexample, in Figure 4, where the loading is slower than in Figure 3 (tT is ten times larger),nucleation occurs earlier than in the faster case (closer to the bifurcation point).

In the simulations described above, the kinetic energy is close to zero and inertia playsa minor role except during the branch-switching process.

Notice that the hysteresis in the middle range|d| < 0.4 observed in Figure 3b,which was due to viscous effects, is almost absent at the lower loading rate in Figure 4.However, the hysteresis due to metastability of then = 0 branch is still present, andwill remain in the quasistatic limit (very slow loading). Observe also that the end loaddrops during nucleation. This is a common characteristic of experimentally observednucleation events [32]. However, the load drop is quite severe and the phase boundary,once formed, proceeds at almost zero load. Hysteresis is largely confined to the beginningand the end of the loop, and is small in the middle range of strains. This does not resembletypical experimental behavior [21], [22], [25], where one observes a smaller load dropand a thicker hysteresis loop. The present model fails to capture important qualitativeaspects of hysteresis.

Note that in the present modelinstabilityprovides a mechanism for switching betweenbranches. The solution must get on an unstable part of the branch before the dissipation


can decrease the potential energy. This is different form anad hocbranch-switchingmechanism based on barrier estimation that was postulated in [41]. In that model, thesystem could switch from one branch of local minima to anotherbeforethe branch itwas on became unstable, provided the energy barrier that had to be overcome was lessthan some critical value. The barrier calculation was based on the energy level of anunstable branch connecting the two stable ones. For lower loading rates in model (5), thesolution gets less far on the unstable branch before switching to a stable one, since theinstability has more time to develop before the system is carried away by the loading.In the quasistatic limit, the branch switching will occur at the end of the stable part of abranch. This corresponds to a zero critical barrier.

If, instead of (7), symmetric boundary conditions (6) are imposed, the results of thesimulations are similar, except that instead of solutions with one phase boundary, solu-tions with two interfaces (symmetric with respect to the center of the bar) are observed.These are of type(+,0,−,0,+). This happens because symmetric boundary conditions(6) introduce the symmetryu(x) = −u(1−x) into the problem. Since the initial conditionu0(x) = −x possesses that symmetry, so does the solution of the initial-boundary-valueproblem, since it is unique by Theorem 1; see the Appendix.

4. The Case without Interfacial Energy: Stick-Slip Phenomenon

In this section we turn to the case without interfacial energy (α = 0), but with inertia andviscosity terms maintained. For results on local existence, uniqueness, and regularityof solutions to the initial-boundary-value problem (11), see Theorems 2 and 3 in theAppendix. We solve problem (11) numerically, using an adapted version of an implicitfinite-difference code from [34], [35]. We employ the loading given by (12) as in theprevious section. The simulations start atd = −1 with the bar in the stable stress-freeequilibrium stateu0(x) = −x (the global energy minimum in the+ phase).

A typical load-displacement diagram for a loading-unloading cycle is shown in Fig-ure 5. Observe the overall substantial width of the hysteresis loop, which resemblesresults of tensile tests in certain shape memory alloys [21], [22]. The most striking andunexpected characteristic is the presence of oscillations in the end load (serrations).Serrated load-elongation curves have been observed experimentally in tensile tests ofshape-memory alloys [25].

We proceed with a description of some results of the computations. As we start loadingthe bar, the strainux at eachd is close to constant; nonuniformities due to inertia effectsare kept small by viscosity. When the strain enters the spinodal region (past the firstlocal maximum in Figure 5), the uniform state becomes unstable, as can be shown bya linearized analysis similar to the one presented for the previous model. As a result,the strain gradient increases and phase boundaries start to form. Eventually, two sharpphase boundaries form in the middle of the bar. The formation of the boundaries isaccompanied by a drop in end load. Now the bar is occupied by the− phase in themiddle and the+ phases at the ends:(+,−,+). Across each phase boundary, the strainis close to discontinuous. Once the boundaries have been formed, they do not move (stickregime). Continued loading results in changing the strains in the regions separated by theboundaries. Thus, the strain in the+ phase at the ends of the bar increases, eventually


Fig. 5. End loadSversusd during a loading-unloading cycle froma numerical solution of problem (11)γ = 0.1,ρ = 0.05,tT = 100.

entering the spinodal region. Instability increases the strain gradient once again. Thiscauses smoothening of existing discontinuities, which turn into mobile transition layersand move further toward the ends of the bar, so that a larger portion of the bar is nowoccupied by the− phase. After that, the phase boundaries rapidly sharpen and becomestuck again. One can say that the boundaries have slipped to new positions. During theslip, the end load drops once again. The scenario described above repeats several times.The hysteresis loop thus contains several serrations, or “teeth.” Each tooth is an increaseof the end load followed by a load drop.

This process is exhibited more clearly in Figure 6, which shows how the strain profileevolves along a tooth. At the time instant labeled 1, the strain profile is close to discon-tinuous, with a sharp phase boundary atx = 0.08 (the size of the mesh inx used in thenumerical computations was 0.005). As we continue loading, the strains on each sideof the interface adjust to the loading almost uniformly, while potential energy and theend load increase (times 1–3). At time 3, the strain to the left of the phase boundary hasincreased and is already in the spinodal region. The resulting instability increases thestrain gradient there (times 4, 5), thus smoothening the strain profile (time 6). Observethat both the end load and the potential energy drop. By time 7, a sharp phase boundaryhas formed at a new positionx = 0.045. During this time, it remains fixed, while theend load and potential energy increase again.

Figure 7 shows plots of the end load, potential and kinetic energy during the loading.The stick and slip regimes are shown respectively by the dashed and solid lines. Potentialenergy grows in the stick regimes and drops suddenly when the boundaries slip. Recallthat the rate of total energy is given by (9). In the stick regime, the loading power exceedsthe dissipation rate. This results in an increase of total energy. In particular, potentialenergy grows while kinetic energy remains small. When part of the bar has strain inthe spinodal region, the instability causes the kinetic energy and the dissipation rate to


Fig. 6. Potential energyEp versusd, end loadSversusd, and strain profileux versusx at differenttime instants along a tooth.

increase. Eventually, the dissipation rate exceeds the loading power, and the total energydrops. Observe the spikes of kinetic energy during the slip regimes, which show that slipis a highly dynamic process.

Kinetic energy is almost zero in the stick regimes, suggesting that the process is essen-tially quasistatic (close to equilibrium) during those times. This motivates the followingway to explain the shape of the teeth. Consider an equilibrium state with a part of thebar, of total lengths, occupied by strainw+, while the rest of the bar, of length 1− s,has strainw−. Then equilibrium dictates

σ(w+) = σ(w−) = σ . (16)

On the other hand, since the bar has unit length, the average strain must be equal to the


Fig. 7. End loadS, potential energyEp, and kinetic energyEk versusd during loading,γ = 0.1, ρ = 0.05, tT = 100. Stick regimes—dashed lines, slip regimes—solid lines.

end displacement:

sw+ + (1− s)w− = d. (17)

For a fixed value ofs, one can show that there is a one-parameter family of equilibriumstates, the parameter beingd. For each fixeds, one may expressσ as a function ofd.The resulting curve represents a static load-displacement diagram for fixed positionsof the interfaces. Figure 8 shows several such constant-s curves for different valuesof s. On each curve, the stressσ decreases, then increases and decreases again. Thepart of the curve whereσ increases withd contains states withw± in the± phases,respectively. On the rest of the curve, whereσ decreases, one ofw± is in the spinodalregion. Figure 9 compares the end load curve from the dynamical simulations with thequasistatic constant-s curves. During each stick regime (stationary sharp interfaces),the end load follows one of thes-curves quite closely. When part of the bar has strainsufficiently well within the spinodal region (past thes-curve maximum) for the instabilityto move the boundary, the dynamic solution switches to a differents-curve, and the endload drops. Observe that the teeth get thinner ass decreases. The values ofs used inFigure 9 were extracted from measurement of boundary positions during stick in theresults of the simulations. One can see that dynamic and quasistatic curves are close,except during slip, which is highly dynamic in nature. The dynamic end load is slightlylarger than the quasistatic one due to the viscous stress termγuxt.

As in the model with nonzero interfacial energy, here instability provides a mechanismfor switching between branches of equilibria. However, unlike the case withα > 0, wherethere is a finite number of separate equilibria, the viscoelastic model has an uncountableinfinity of equilibrium states. Pego [27] has shown that an equilibrium solutionu(x)


Fig. 8. Constant-s curves of stress versusd for equilibria withpiecewise constant strain and fixed interfaces.

Fig. 9. Comparison of dynamic (thick solid line,γ = 0.1, ρ =0.05, tT = 100) and quasistatic solutions (thin solid and dottedlines): end loadS versusd. The volume fractions of the+ phaseis constant on each quasistatic curve; its values, from left to right,ares= 1,0.74,0.58,0.45,0.38,0.25,0.17,0.1.

(σ(ux) = const. andu piecewise linear) is linearly stable (that is, perturbations small in(W1,2, L∞) decay exponentially) as long asσ ′(ux) > 0.

We remark that while the numerical results indicate the presence of discontinuitiesin strain during the stick regime, we may infer from Theorem 3 in the Appendix thatsolutions must preserve their initial smoothness, at least until some timeT . In his analysisof this model with constant loading, Pego [27] has shown that while the global solutionsapproach equilibria with discontinuous strain very fast, they become discontinuous onlyin the limit of infinite time. So we suspect that in our solutions the strain profile is smoothbut very close to discontinuous.


5. Some Analytical Predictions and Discussion

To obtain some analytical understanding of the stick-slip phenomenon observed in nu-merical computations and described above, we now consider a simplification of theproblem. To dispense with the troublesome nonlinearity due to theσ(ux) term in (11),we replace the elastic stressσ(ux) = f ′(ux) by a piecewise-linear function, correspond-ing to a so-called trilinear material, employed, e.g., in [2],

σ(ux) =σ+(ux) = 2(ux + 1) for ux ≤ −τ,σ0(ux) = −θux for − τ < ux < τ,

σ−(ux) = 2(ux − 1) for ux ≥ τ.(18)

Here−θ = 2(τ −1)/τ < 0 is the negative slope of the stress-strain curve in the spinodalrange. In effect this replacesf in (1) by a piecewise quadratic function, concave in thespinodal region|ux| < τ , but convex elsewhere (phases). The advantage of this model isthat the governing equation becomes linear in each part of the bar, and a standard Fourieranalysis can be applied.

As was observed in the previous section, in the dynamic simulations, the strain initiallyincreases but remains largely uniform until after it has entered the spinodal region, atwhich point its gradient grows and phase boundaries develop. To see how the phaseboundaries form, suppose that the entire bar initially has constant strain in the spinodalregion. This leads to the following boundary-initial-value problem:

ρutt = −θuxx + γuxxt,

u(0, t) = −d(t)/2,u(1, t) = d(t)/2,u(x,0) = d(0)(x − 1/2),ut (x,0) = d′(0)(x − 1/2) = 0,

(19)

with d′(0) = 0 and|d(0)| < τ . This is valid as long as|ux| < τ , i.e., the strain is in thespinodal region. Introducing the change of variables

u(x, t) = d(t)(x − 1/2)+ v(x, t), (20)

we obtain ρvt t = −θvxx + γ vxxt + f (x, t),v(0, t) = v(1, t) = 0,v(x,0) = 0,vt (x,0) = 0,

(21)

where f (x, t) = −ρd′′(t)(x − 1/2). Using the Fourier series method, one shows that

v(x, t) =∞∑

n=1

vn(t) sinπnx, (22)

with vn(t) given by

vn(t) = 1

πn(ωn2 − ωn

1)

∫ t

0

[eω

n2(t−ξ) − eω

n1(t−ξ)

]d′′(ξ)dξ (23)


Fig. 10.Graphs ofωn1(n) andAn = |vn(t0)| for t0 fixed, at different values ofγ .

for evenn, andvn(t) ≡ 0 for oddn. Hereωn1,2 are given by

ωn1,2 =

(πn)2γ

2ρ

(−1±

√1+ 4θρ

(πnγ )2

), (24)

with ωn1 > 0 andωn

2 < 0.The graphs ofωn

1 and|vn(t0)| versusn for differentγ and fixedt0 (with d(t) as in (12))are shown in Figure 10. Asγ tends to zero,ωn

1(n)→ πn√θ /ρ; hence atγ = 0 and fixed

t , vn(t) grows withn. In other words, the higher the mode is, the more it is amplifiedin time. Whenγ is nonzero,vn tends to zero for highn; hence one expects the modeswith finite n to grow the fastest. This results in afinite number of phase boundaries.The number of the interfaces increases asγ becomes smaller. This was also observedin numerical simulations. For example, atγ = 0.1 we observed two phase boundaries,whereas atγ = 0.01, with other parameters kept the same, six phase boundaries wereformed.

Figure 11 compares Fourier series solutions of (19) with numerical solutions of thenonlinear problem (11) for the trilinear material (18) and the same boundary and initialconditions. There is excellent agreement (e.g., att = 0.9, t = 1) while the strain isstill entirely in the spinodal region. When part of the bar has strain outside the spinodalregion, the linear problem (19) is no longer valid, and the stress nonlinearity becomesimportant. Hence numerical and analytical solutions diverge (e.g., att = 1.2). Note,however, that the linear problem does capture the beginning of interface formation. Theassociated drop in potential energy is also captured in Figure 11b.

At a later stage in the simulations of the previous section, it was observed that sharpinterfaces have formed and the strain is approximately piecewise constant. We are inter-ested in modeling the smoothening of interfaces observed in the sequel, using the trilinearmaterial. Thus we should consider the situation when there are two phase boundaries(discontinuities of strain), atx = 1 andx = 1 − l , the middle of the bar is in thespinodal region (−τ < ux < 0), while the ends are occupied by the− phase (ux > τ ).The bar is initially in equilibrium with piecewise constant strain. The interfaces are sta-tionary as long as the strain profile is discontinuous. This follows from our assumption


Fig. 11. Comparison of Fourier series solutions (solid lines) and numerical solutions (dashedlines) for trilinear material, withγ = 0.1, ρ = 0.05, τ = 0.75,1 = −0.4,1T = 1, tT = 100:(a) strain profiles at different time instants; (b) potential energy versusd.

that equation (11)1 holds everywhere in the bar. If the strain discontinuities were tomove, there would be nonzero jumps in the velocityut and the total stressσ(ux)+ γuxt

across the moving interface. This follows from considerations of the jump conditionsdescribing continuity ofu and momentum balance across a moving strain discontinuity.However, Theorem 2 in the Appendix implies that these quantities are continuous aslong as the solution exists. One can do a Fourier analysis in each part of the bar, andthen match the solutions together using the smoothness conditions. However, this resultsin an integro-differential equation which is difficult to solve. Instead, we employ thefollowing approximation. Based on the numerical observation that inertia effects are notsignificant at the beginning of interface smoothening, we neglect the inertia term in orderto find the displacement at the boundaryul (t) ≡ u(l , t). After it has been found, wesolve the full dynamic equations, with inertia, in each part of the bar, treating the partsas separate bars.

Neglecting inertia implies that the strain is piecewise constant, and its valueswp andws in the− phase and spinodal parts are, respectively,

wp(t) = (ul (t)+ d(t)/2)/l , ws(t) = −ul (t)/(1/2− l ), (25)

in view of the boundary conditions. The continuity of the total stress across the phaseboundaries requires that

σ(wp)+ γw′p(t) = σ(ws)+ γw′s(t). (26)

Use of trilinear stress-strain law (18) and (25) in (26) yields a linear ODE foru1(t):

u′l −2θ l + 4l − 2

γul + [γd′ + 2d − 4l ](1/2− l )

γ= 0. (27)

The initial valueu1(0) is found from the requirement that the bar is initially in equilib-rium:

ul (0) = l (1− 2l )(d(0)/2l − 1)

θ l + 2l − 1. (28)


Solving (27), we obtain

ul (t) = e−Atul (0)+ (1/2− l )[ 4l

γ A(1− e−At)− d(t)+ d(0)e−At

+(

A− 2

γ

)e−At

∫ t

0eAξ d(ξ)dξ

], (29)

wherea = −(2θ l + 4l − 2)/γ . Now thatu1(t) is known, we can solve the linear initial-boundary-value problem for the displacement in the middle of the bar (l < x < 1− l ):

ρutt = −θuxx + γuxxt,

u(l , t) = ul (t),u(1− l , t) = −u1(t),u(x,0) = ul (0)(1/2− x)/(1/2− l ),ut (x,0) = 0.

(30)

Note that the inertia term is now included. The solution is

u(x, t) =∞∑

n=1

vn(t) sin1/2− x

1/2− lπn+ ul (t)

1/2− x

1/2− l, (31)

where

vn(t) = 2(−1)n

πn(ωn2 − ωn

1)

∫ t

0

[eω

n2(t−ξ) − eω

n1(t−ξ)

]u′′l (ξ)dξ. (32)

Here

ωn1,2 =

k2nγ

2ρ

(−1±

√1+ 4θρ

k2nγ

2

), (33)

with kn = πn/(1/2− l ). This describes the evolution of strain in the spinodal region.The strain in the− phase region is obtained similarly.

Figure 12 compares the analytical and numerical solutions. Despite the fact that wehave neglected inertia to findul (t), there is excellent agreement of the two solutionsuntil the strain in the middle of the bar enters the+ phase and the nonlinear effectsbecome important. The analysis described above thus shows how the strain gradientstarts increasing near the phase boundary, due to instability in the spinodal region; thiseventually leads to smoothening of strain discontinuities.

Once a discontinuity is smoothed out, the resulting transition layer can propagate,resulting in interface slip. This can be explained by performing an analysis of travellingwaves in an infinite viscoelastic bar, e.g., [39]. These are solutions of (11)1 in which thestrain is a function of the variablex − V t, V being a constant propagation speed. Theyhave the form of travelling transition layers, connecting strains in two phases, or oneof the phases and the spinodal region. One can show that the speedV is related to thesharpness of the layer, or the maximum value of the strain gradient. The sharper a layeris, the slower it propagates; in the limit of infinite gradient, discontinuities are stationary[6], [27]. On the other hand, numerical studies of (11)1 indicate that travelling wavesrepresenting transition layers between a strain in the spinodal region and one in a phase


Fig. 12. Comparison of strain profiles from Fourier series solutions (solid lines) and numericalsolutions (dashed lines) for trilinear material at three different time instants:γ = 0.1, ρ = 0.05,d(t) = 0.07+ 0.0093t2, l = 0.2, τ = 0.75.

are dynamicallyunstable[39] (the amplitude of initial perturbations of these travellingwave solutions grows with time). As a result, strain gradients will increase; hence theinterface will slow down by the observation just made, and will approach a stationarystrain discontinuity. This argument provides a heuristic explanation of the observed stickphenomenon.

While the viscosity term provides the dissipation, the inertia is also important for theslip phenomenon. For example, if one considers (30) withρ = 0, u(x, t) = ul (t)(1/2−x)/(1/2− l ) is the solution. Hence if the initial state has piecewise constant strain, it willremain piecewise constant for all times. No strain gradient in the spinodal region willoccur, and therefore, the interfaces will not slip unless the initial condition has nonzerostrain gradient. This is also the case for the fully nonlinear stress-strain law. On the otherhand, in the presence of viscosity, inertia generates strain gradients automatically, evenif the initial strain is constant.

Another observation can be made regarding the effect of the loading rate on theserrated form of the hysteresis loop. For example, in Figure 13, where the loading is tentimes slower than in Figure 5, there are more serrations but their amplitude is smaller.Under slower loading , the strain gets less far into the spinodal region before slip occurs.


Fig. 13. End loadS versusd for a loading-unloading cycle, vis-coelastic bar:γ = 0.1, ρ = 0.05, tT = 1000.

As a result, the system settles at a nearer equilibrium state during the slip event, sincethe instability is less severe. This causes the serrations to be shallower and the interfaceslip distance to be smaller. In turn, this means that more slip events are required for theinterfaces to traverse the entire length of the bar. The number of serrations is hence largerand their depth smaller for slower loading. Although we were not able to prove it, wesuspect that in the quasistatic limit as the loading rate goes to zero, the serrations willdisappear and the hysteresis loop will become flat. The dependence of the amplitude ofthe serrations on the loading rate remains an open question.

Appendix: Local Existence and Uniqueness Results

In this section we state theorems of local existence and uniqueness for problems (5) and(11). The proofs, which are omitted, can be found in [39]. They are similar to the onesgiven in [6], [19], [27] for the case of time-independent loading and employ the resultsof [16] for an abstract parabolic initial-value problem.

In what follows,Ck,θ (S, X) denotes the space ofk times continuously differentiablefunctions formS, an open subset of a Banach space, to another Banach spaceX, wherethek-th derivative satisfies a H¨older condition with exponentθ . The norm is

‖ f ‖Ck,θ (S,X) =k∑

j=0

supS‖D j f ‖X + sup

x 6=y

‖Dk f (x)− Dk f (y)‖X

‖x − y‖θS.

The last term is omitted ifθ = 0, and we writeCk ≡ Ck,0 andC ≡ C0. If k = 0 andθ = 1, a function is called Lipschitz continuous.


Consider the initial-boundary value problem,

utt = (σ (ux)− 2αuxxx+ γuxt)x,

u(0, t) = 0,u(1, t) = d(t),uxx(0, t) = uxx(1, t) = 0,u(x,0) = u0(x),ut (x,0) = u1(x).

(A.1)

Note that (5) with the nonsymmetric boundary conditions (7) can be reduced to thissystem by rescaling time:t = t /

√ρ, γ = γ /

√ρ, u(x, t) = u(x,

√ρ t), d(t) = d

(√ρ t),

and then omitting the hats. The case of symmetric boundary conditions (6) is treatedsimilarly.

Theorem 1. Assume thatσ(w) is locally Lipschitz continuous and d(t) is locally C2,θ .Suppose u0 − d(0)x ∈ H2 ∩ H1

0 and U1 ∈ L2. Then there is T> 0, such that thereexists a unique strong solution of (A.1) on[0, T ] in the following sense:

u ∈ C([0, T ], H2) ∩ C1((0, T ],C2) ∩ C((0, T ], H3), (A.2)

ut ∈ C([0, T ], L2) ∩ C1((0, T ],C) ∩ C((0, T ], H1), (A.3)

utt ∈ C((0, T ],C), (A.4)

σ (ux)+ γuxt − 2αuxxx ∈ C((0, T ],C1), (A.5)

and equation (A.1)1 holds pointwise for0≤ t ≤ T and0< x < 1.

In the case with no interfacial energy, consider the initial-boundary value problemutt = (σ (ux)+ γuxt)x,

u(0, t) = 0,u(1, t) = d(t),u(x,0) = u0(x),ut (x,0) = u1(x).

(A.6)

Theorem 2. Assume thatσ(w) is locally Lipschitz continuous and d(t) is locally C2,θ .Suppose u0 ∈ W1,∞ and u1 ∈ L2. Then there is T> 0 such that there exists a uniquestrong solution of (A.6) on[0, T ] in the following sense:

u ∈ C([0, T ],W1,∞), (A.7)

uxt ∈ C((0, T ], L∞), (A.8)

utt ∈ C((0, T ],C), (A.9)

σ (ux)+ γuxt ∈ C((0, T ],C1), (A.10)

and equation (A.6)1 holds pointwise for0≤ t ≤ T and a.e. x.


As in [6], [27], solutions preserve their initial smoothness:

Theorem 3(Classical Solutions).Supposeσ(w) is C1, with σ ′(w) locally Lipschitzcontinuous. Let d(t) ∈ C2,θ and assume that u0 ∈ C2[0,1], u1 ∈ H1[0,1], withu0(0) = u1(0) = 0, u0(1) = d(0), and u1(0) = d′(0). Then for some T> 0, a classicalsolution u(x, t) of (A.6) exists for0≤ t ≤ T , with

u ∈ C([0, T ],C2),

utt ,uxtx continuous in(x, t) for t > 0.

References

[1] R. Abeyaratne, C. Chu, and R. D. James. Kinetics and hysteresis in martensitic single crystals.In Mechanics of Phase Transformations and Shape Memory Alloys. ASME, New York, 1994.

[2] R. Abeyaratne and J. K. Knowles. A continuum model of a thermoelastic solid capable ofundergoing phase transitions.J. Mech. Phys. Solids, 41:541–571, 1993.

[3] G. Andrews. On the existence of solutions to the equationutt = uxxt + σ(ux)x. J. Diff. Eq.,35:200–231, 1980.

[4] G. Andrews and J. M. Ball. Asymptotic behavior and changes in phase in one-dimensionalviscoelasticity.J. Diff. Eq., 44:306–341, 1982.

[5] J. M. Ball, C. Chu, and R. D. James, Hysteresis during stress-induced variant rearrangement.In ICOMAT 95, J. de Physique IV, 5(8), 1995.

[6] J. M. Ball, P. J. Holmes, R. D. James, R. L. Pego, and P. J. Swart. On the dynamics of finestructure.J. Nonlin. Sci., 1:17–70, 1991.

[7] D. Brandon, T. Lin, and R. Rogers. Phase transitions and hysteresis in nonlocal and orderparameter models.Meccanica, 30:541–565, 1995.

[8] J. Carr, M. E. Gurtin, and M. Slemrod. Structured phase transitions on a finite interval.Arch.Ratl. Mech. Anal., 86:317–351, 1984.

[9] C. Chu.Hysteresis and Microstructures: A Study of Biaxial Loading on Compound Twins ofCopper-Aluminum-Nickel Single Crystals. PhD thesis, University of Minnesota, 1993.

[10] J. L. Ericksen. Equilibrium of bars.J. Elasticity, 5:191–202, 1975.[11] C. Faciu. Initiation and growth of strain bands in rate-type viscoelastic materials.Eur. J.

Mech. A/Solids, 15(6):969–1011, 1996.[12] B. Fedelich and G. Zanzotto. Hysteresis in discrete systems of possibly interacting elements

with a two well energy.J. Nonlin. Sci, 2:319–342, 1992.[13] G. Friesecke and J. B. McLeod. Dynamics as a mechanism preventing the formation of finer

and finer microstructure.Arch. Ratl. Mech. Anal., 133:199–247, 1996.[14] S. Fu, I. Muller, and H. Xu. The interior of the pseudoelastic hysteresis. In C. Y. Liu,

H. Kunsmann, K. Otsuka, and M. Wuttig, editors,Mat. Res. Soc. Symp. Proc., vol. 246,pages 39–42, Elsevier North-Holland, Amsterdam, 1992.

[15] H. Hattori and K. Mischaikow. A dynamical system approach to a phase transition problem.J. Diff. Eq., 94:340–378, 1991.

[16] D. Henry.Geometric Theory of Semilinear Parabolic Equations. Springer-Verlag, Berlin,1981.

[17] R. D. James. Deformation of shape-memory materials. In C. Y. Liu, H. Kunsmann, K. Otsuka,and M. Wuttig, editors,Mat. Res. Soc. Symp. Proc, vol. 246, pages 81–90, 1992.

[18] W. Kalies, 1994. Private communication.[19] W. Kalies. Regularized Models of Phase Transformation in One-Dimensional Nonlinear

Elasticity. PhD thesis, Cornell University, 1994.[20] D. Kinderlehrer and L. Ma. The hysteretic event in the computation of magnetization and

magnetostriction. In H. Bresis and J.-L. Lions, editors,Proc. Nonlinear Diff. Eqs. and TheirAppl., College de France Sem., Boston, 1994.


[21] R. V. Krishnan. Stress induced martensitic transformations.Mat. Sci. Forum, 3:387–398,1985.

[22] R. V. Krishnan and L. C. Brown. Pseudo-elasticity and the strain-memory effect in a Ag-45at.pct. Cd alloy.Metallurgical Trans., 4:423–429, 1973.

[23] A. E. Lifshitz and G. L. Rybnikov. Dissipative structures and couette flow in a nonnewtonianfluid. Sov. Phys. Dokl., 30(4):275–278, 1985.

[24] I. Muller and P. Villaggio. A model for an elastic-plastic body.Arch. Ratl. Mech. Anal.,65:25–46, 1977.

[25] N. Nakanishi. Lattice softening and the origin of SME. In J. Perkins, editor,Shape MemoryEffects in Alloys, pages 147–175. AIME, 1975.

[26] A. Novick-Cohen and L. A. Segel. Nonlinear aspects of the Cahn-Hilliard equation.PhysicaD, 10:277–298, 1984.

[27] R. Pego. Phase transitions in one-dimensional nonlinear viscoelasticity: Admissibility andstability.Arch. Ratl. Mech. Anal., 97:353–394, 1987.

[28] R. Rogers and L. Truskinovsky. Discretization and hysteresis.Physica B, 233:370–375, 1997.[29] P. Rosakis and J. K. Knowles. Unstable kinetic relations and the dynamics of solid-solid

phase transitions.J. Mech. Phys. Solids, 45:2055–2081, 1998.[30] P. Rosakis and J. K. Knowles. Continuum models for irregular phase boundary motion in

shape-memory tensile bars. ONR Technical Report No. 11, Division of Engineering andApplied Science, California Institute of Technology, Pasadena, 1997.

[31] D. Schryvers, Y. Ma, L. Toth, and L. Tanner. Nucleation and growth of bainitic Ni5Al3 in B2austenite. InPTM Solid-Solid Phase Transitions, 1994.

[32] J. A. Shaw and S. Kyriakides. On the thermomechanical behavior of NiTi. Technical Re-port 94/12, EMRL, Department of Aerospace Engineering and Engineering Mechanics, TheUniversity of Texas at Austin, 1994.

[33] T. W. Shield. Orientation dependence of the pseudoelastic behavior of single crystals ofCu-Al-Ni in tension.J. Mech. Phys. Solids, 43(6):869–895, 1995.

[34] P. Swart.The Dynamical Creation of Microstructure in Material Phase Transitions. PhDthesis, Cornell University, 1991.

[35] P. Swart and P. Holmes. Energy minimization and the formation of the fine structure indynamic antiplane shear.Arch. Ratl. Mech. Anal., 121:37–85, 1992.

[36] N. Triantafyllidis and S. Bardenhagen. On higher order gradient continuum theories in 1-Dnonlinear elasticity. Derivation from and comparison to the corresponding discrete models.J. Elasticity, 33(3):259–293, 1993.

[37] L. Truskinovsky and G. Zanzotto. Finite-scale microstructures and metastability in one-dimensional elasticity.Meccanica, 30:577–589, 1995.

[38] L. Truskinovsky and G. Zanzotto. Ericksen’s bar revisited: Energy wiggles.J. Mech. Phys.Solids, 44(8):1371–1408, 1996.

[39] A. Vainchtein.Static and Dynamic One-dimensional Continuum Models of Martensitic Phasetransitions. PhD thesis, Cornell University, 1998.

[40] A. Vainchtein, T. J. Healey and P. Rosakis. The use of bifurcation theory in coupled-bar mod-els of martensitic phase transitions. To appear inComputer Methods in Applied Mechanicsand Engineering, 1999 (in press).

[41] A. Vainchtein, T. Healey, P. Rosakis, and L. Truskinovsky. The role of the spinodal region inone-dimensional martensitic phase transitions.Physica D, 115:29–48, 1998.

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