moving phase boundaries: a lattice model and its continuum ... › ckm2011 › talks ›...

1

Moving phase boundaries: a lattice model and itscontinuum implications

Johannes Zimmer

Work with Hartmut Schwetlick, Daniel Sutton, Christine Venney (Bath)and Michael Herrmann (Saarbrucken)

Motivation 1

Background: Martensitic phase transitionsWeak phase transitions

One class of martensiticmaterials exhibits theshape-memory effect:

cool

deform

heat

Motivation (1D case only)As in Landau theory, phase transitions are here modelled by a nonconvexpotential V . Then the stress σ = V ′ is non-monotone; the classicequations of elasticity are then of mixed (elliptic-hyperbolic) type,

utt = (σ (ux))x . (1)

Eq. (1) is ill-posed (there are generically infinitely many solutions).Mathematical approach in the 80s and 90s: introduce regularisation, e.g.,by viscosity or by interfacial energy.Problem: there is no universally accepted regularisation.

Motivation 2

Mathematical challenge: PDE models for martensitesMicroscopic picture

Figure: Needles and wedges inNiTi (weak transition)

Figure: Martensite in 0.8%carbon steel (reconstructivetransition)

Applied mechanics viewpointThe missing bit of information which renders (1) ill-posed is the velocityof the interface(s) as a function of forces; there is no physical lawprescribing such a relation.=⇒ One approach is to postulate a kinetic relation, which relates theconfigurational force f and the velocity c : f = f (c).Question of this talk: Can we derive a kinetic relation from “firstprinciples”, that is, an atomistic model?

Subsonic waves on the real line: Existence of solutions 3

The problem setting

Model

Discrete model of elasticity: chain of particles coupled by elastic springs.The equations of motion are

uk(t) = V ′(uk+1(t)− uk(t))− V ′(uk(t)− uk−1(t)), k ∈ Z

(uk(t) is longitudinal motion of atom k).We consider only nearest neighbour interaction: the argument of elasticpotential V : R→ R is uk+1(t)− uk(t).Fermi-Pasta-Ulam 1953: (non-)equipartitioning of energy.

Travelling wave ansatzWe seek travelling waves uj(t) = u(j − ct) for j ∈ Z and t ∈ R:

c2u(x) = V ′(u(x + 1)− u(x))− V ′(u(x)− u(x − 1)). (2)


Supersonic speeds: existence of soliton solutions for FPU

Existence of soliton solutions, convex V and supersonic c

I Constrained minimisation: Friesecke, Wattis, Comm. Math. Phys.,161 (1994), 391–418:12

∫R u(t)2 dt

!= min with prescribed K =

∫R V (u(t + 1)− u(t))dt.

⇒ c given by Lagrange multiplier associated with constraint.

Main assumption:(

V (ε)ε2

)′> 0.

I Centre manifold analysis: Iooss, Nonlinearity, 13 (2000), 849–866.

I Mountain pass methods: Smets, Willem, J. Funct. Anal., 149(1997), 266–275; Arioli, Gazzola, Nonlinear Anal., 26 (1996),1103–1114.Mountain pass argument, using convexity of energy. Monotonewaves are found as saddle points of the action functional.

I Nonlinear eigenvalue problem: Filip, Venakides, Comm. Pure Appl.Math., 52 (1999), 693–735

Heteroclinic supersonic solution, minimal action: Rademacher &Herrmann (2009); Herrmann (2010)


Subsonic waves: existence of waves?Literature

I Balk, Cherkaev, Slepyan, J. Mech. Phys. Solids, 49 (2001), 131–148

I Slepyan, Cherkaev, Cherkaev, J. Mech. Phys. Solids, 53 (2005),407–436

I Truskinovsky, Vainchtein, SIAM J. Appl. Math., 66 (2005), 533–553

Use piecewise bi-quadratic energy, equal depth wells and elastic modulus.

The problem formulationExpress equation for discrete strain ε(x) = u(x)− u(x − 1):

c2ε′′(x) = ∆1V ′(ε(x)) (∆1g(x) := g(x +1)−2g(x)+g(x−1)). (3)

Special potential V (ε) = 12 min{(ε+ 1)2, (ε− 1)2}.

Then (3) becomes

c2ε′′(x) = ∆1ε(x)− 2∆1H(ε(x)), (4)

with H denoting the Heaviside function.

-2 -1 1 2

0.1

0.2

0.3

0.4

0.5

Figure: Special choiceof interaction potential


(Non-)existence resultsBounded travelling waves with one phase boundaryOne can show:

I Existence for almost sonic waves (SIAM Math Anal. ’09).

I Nonexistence of travelling waves for some slow velocities(Schwetlick, Sutton, Z., ’10).

(Foster, Scheil, Z. Metallkd., 32 (1940), 165–201: distinction betweentwo kinds of martensitic transformations: fast (umklapp): move with avelocity close to the speed of sound; slow (schiebung): observable underan optical microscope)

Relevant part of the (technical) proofsRewrite equations for discrete strain ε(x) := u(x)− u(x − 1); then withdistance ≥ 1 from interface spatially discrete wave equation. Assumethere is only one interface at 0; split solution ε := εpr − εcor with explicitfunction εpr so that corrector εcor is solution of

(c2∂2 −∆1)[εcor](x) = Φ(x) ∈ L2(R) (5)

Fourier estimates to show that single interface assumption is (not) met.


Nonexistence of subsonic waves on the real line

Existence vs. nonexistenceConjecture for related models: Decisive difference: existence for fastsubsonic waves, nonexistence for slow waves. Nonexistence of slow wavesin dislocation models: Peyrard, Kruskal, Phys. D, 14 (1984), 88–102

Theorem (Schwetlick, Sutton & Z., ’10)There is a range of subsonic wave speeds c such that there is no”heteroclinic” travelling waves satisfying

ε > 0 on R+ and ε < 0 on R−.

Proof similar to existence result:

I Decomposition in εpr and εcor, first for given εpr.

I Dispersion relation dictates “wrong” sign of εpr.

I Corrector not large enough to compensate

I Independence of original choice of εpr: distribution theory.


Macrosopic non-uniquenessSolution family and selection criteriaNontrivial kernel of corrector equation on previous slide,

(c2∂2 −∆1)[εcor](x) = Φ(x) ∈ L2(R)

=⇒ three-parameter family of solutions (Schwetlick & Z.’10).In Mechanics / Physics literature: extend Sommerfeld’s selection criteriafor Helmholtz equation with source at the origin. These are:

Sources have to be sources, not sinks of the energy. (SOM1)

The energy radiated from the sources has to scatter to infinity,energy must not be radiating from infinity into the prescribedsingularities of the field.

(SOM2)

(Both implied by a third condition which is specific to the linear problem.)Used in mechanics literature: causality principle, motivation:

The energy has to radiate away from the interface.

Our scepticism: passage to travelling wave coordinates is not Galilei

transformation, so flux in these coordinates has no direct physical meaning.

Thermodynamic limit 9

Thermodynamic framework to extend Sommerfeld’s criteriaMicroscopic conservation lawsNew notation: rj := uj+1 − uj discrete strain, Φ potential. Rewritegoverning equations (velocity vj := uj) as first order equations

rj = vj+1 − vj , vj = Φ′(rj)− Φ′(rj−1). (6)

Hyperbolic scaling: define macroscopic time τ and the macroscopicparticle index ξ (distances and velocities unscaled) by τ = ht, ξ = hj .

Macroscopic conservation laws for mass, momentum & energyThermodynamic limit (Herrmann, Schwetlick, Z. ’10) h→ 0 yields easily

∂τR − ∂ξV = 0, ∂τV + ∂ξP = 0, ∂τE + ∂ξF = 0, (7)

with macroscopic strain R = 〈r〉, macroscopic velocity V = 〈v〉, pressureP = −

⟨Φ′(r)

⟩, energy density E =

⟨12v2 + Φ(r)

⟩, and energy flux

F = −⟨vΦ′(r)

⟩. Separate Galilean invariant part from E and F : with

internal energy density U and heat flux Q,

E = 12V 2 + U, F = VP + Q.


“Heat” in the chainOscillatory energySplit the energy density E since weak limit (local mean values) andnonlinearities do not commute for oscillations. We write

E = Enon + Eosc, Enon = 12V 2 + Φ(R),

thus Eosc = 12

⟨(v − 〈v〉

)2⟩+⟨Φ(r)− Φ

(〈r〉)⟩

.

Corresponding energy balancesPartial energies are balanced by

∂τEosc + ∂ξQ = Ξ, ∂τEnon + ∂ξ(PV ) = −Ξ; (8)

the production Ξ = −(P + Φ′(R))∂ξV =(⟨

Φ′(r)⟩− Φ′

(〈r〉))∂ξ〈v〉

describes transfer of non-oscillatory energy into oscillatory energy. Phasetransition waves are driven by a constant transfer between the oscillatoryand the non-oscillatory energy (Q, PV below exchange with exterior):

d

dτ

∫ b

a

Eoscdξ + Q|ξ=bξ=a =

∫ b

a

Ξdξ = − d

dτ

∫ b

a

Enondξ − (PV )|ξ=bξ=a .


Insight I: Selection criteria, piecewise harmonic latticesReformulation of (SOM1)

The interface has to be a source rather than a sink of oscillatory energy.The production Ξ therefore has to be non-negative. So with group speedcgr and phase speed cph, Ξ = (cgr − cph) [[Eosc]] ≥ 0. This inequality isequivalent to the usual entropy condition for phase transition waves,

cphΥ ≥ 0, with Υ := [[Φ(R)]]− {Φ′(R)} [[R]] . (9)

� � � � � � � � � � ��

� � � � � � ��

r j

j

Type-I wave with 0 < cgr < cph

radiation flux Q−∞

radiation flux Q+∞

wave speed cph � � � � � � ��

� � � � � � � � � � ��

r j

j

wave speed cph

radiation flux Q+∞

radiation flux Q−∞

Type-II wave with cgr < 0 < cph

Validity of (SOM2) for moving inhomogeneity, used in physics

Provably violated for every (bounded single-interface) wave travellingwith almost sonic speed satisfying the entropy inequality. We suggest toreject this criterion.


Insight II: Kinetic relation “heat”-dependentKinetic relation as a function of heat(with Schwetlick; key suggestion by Kaushik Dayal).In the thermodynamic framework established above: kinetic relationdepends on “heat”: Rewrite a continuum mechanics expression for thekinetic relation as

f = [[Φ(〈r〉)]]− {Φ′(〈r〉)} [[〈r〉]] , (10)

and a short calculation shows that (α, κ0 constants > 0)

f = −2κ20α

[[〈Φ(ε)〉 − Φ (〈ε〉)]] . (11)

Interpretation:I Microscopic oscillations “lost” in continuum limit lead to

macroscopic dissipation R = f · c > 0I Thermodynamic limit suggests that there is a meaningful

thermochemical macroscopic framework for mechanical microscopicmodel: f depends (on speed c and) on Eosc (“heat”).

I For travelling waves: no temperature-like decay of oscillations overtime, so at most “non-equilibrium temperature”.


Summary phase transition waves

Microscopic theory

I Existence and nonexistence for the same model (special interactionpotential), different subsonic velocities.

I Extension to wider class of potentials (hard); stability open.

I Fast subsonic velocities: for fixed c , two-parameter family;asymmetry of solution defines macroscopic dissipation (kineticrelation).

Scale-bridging

I Split of energy in oscillatory and non-oscillatory part; oscillatoryenergy proportional to squared amplitude.

I Thermodynamic framework with radiation heat flux leads tomacroscopic thermodynamic model.

I Kinetic relation alone does not make macroscopic model well-posed(since kinetic relation depends on parameter). Oscillatoryenergy/heat could be one selection criterion, vanishing viscosityanother one.

NNN interaction 14

Toward dynamics with a Lennard-Jones potential

NNN interactionWith next to nearest neighbour (NNN) interaction, the equations ofmotion are

uj(t) = V ′(uj+1(t)− uj(t))− V ′(uj(t)− uj−1(t))

+ g0 (uj+2(t)− 2uj(t) + uj−2(t)) (12)

(uk(t) is longitudinal motion of atom k); g0 ∈ R is constant.

......

Figure: Next to nearest-neighbour (NNN) interaction

Motivation for g0 < 0: Lennard-Jones potential, has attractive andrepulsive regimes.

NNN interaction 15

NNN: what types of solutions can we expect?Travelling wave ansatzSeek travelling waves, uj(t) = u(j − ct) (j ∈ Z, t ∈ R), write g0 = gc2

0 :

c2u(x) = V ′(u(x + 1)− u(x))− V ′(u(x)− u(x − 1))

+gc20 [u(x + 2)− 2u(x) + u(x − 2)]. (13)

Linearisation dispersion relation 1√µ

q2 = sin

(q2

)√1 + 4g cos2

(q2

),

where µ =c20c2 and c0 =

√V ′′(0) is the NN sound speed;

the NNN sound speed is c0√

1 + 4g .

Figure: Plots of 1√µ

q2

and sin(q2

)√1 + 4g cos2

(q2

), for g = 0 (left) and

g = − 316

(right)

The interesting case is when the NN springs act against the NNN springs;we consider the regime − 1

4 < g < − 116 .

NNN interaction 16

Existence of periodic supersonic solutions for −14< g < − 1

16

NN caseThere are subsonic periodic solutions. What about the NNN case?

Variational approach for the NNN caseLet V (r) = 1

2c20 r2 + U(r) and c0 is the sound speed for the NN case.

TheoremUnder assumptions on smoothness and growth of V , for the range− 1

4 < g < − 116 , there exists an ηg > 0, such that, if

c2 < c20 (1 + 4g) + ηg , there are periodic solutions to

c2u(t)) = V ′(u(t + 1)− u(t))− V ′(u(t)− u(t − 1))

+ gc20 (u(t + 2)− 2u(t) + u(t − 2)) . (14)

So in the NNN case there are supersonic periodic solutions. The proofrequires g ≤ − 1

16 , which excludes the NN interaction case.

NNN interaction 17

Existence of periodic supersonic solutions for −14< g < − 1

16

Variational approach for the NNN caseMotivation for choice of parameter:

I The NN springs should act against the NNN springs.

I In the regime − 14 < g < − 1

16 , there are supersonic periodic solutionswith mean zero for harmonic potentials.

The argument is different from the NN case: the spectrum of

Lu := −c2u′′ + c20 [u(t + 1)− 2u(t) + u(t − 1)]

+ gc20 [u(t + 2)− 2u(t) + u(t − 2)]

has eigenvalues of different signs due to g < 0, suggesting adecomposition of a suitable function space in positive and negativeeigenfunctions.The proof relies on a linking argument: it can be shown thatXk = {u ∈ H1

loc(R)∣∣ u′(t + 2k) = u′(t), u(0) = 0} (k denotes the

period; we consider k ≥ 2) can be decomposed so that the actionfunction corresponding to the Euler-Lagrange equation (14) possesses thelinking geometry.

NNN interaction 18

Existence of subsonic homoclinic solutions for −14< g < − 1

16

µ < 1 µ = 1 µ > 1

g = 0

µ <1

1 + 4gµ =

11 + 4g

µ >1

1 + 4g

!14

< g < ! 116

!

= single eigenvalue, = double eigenvalue, = quadruple eigenvalue, =sixfold eigenvalue.

µ

g

!43,! 1

16

"

!14

! 116

Figure: Eigenvalues at the bifurcation µ = 11+4g

in C (right) and bifurcation forthe NN case (g = 0).

The figure shows (recall µ =c20c2 ): Real eigenvalues appear

I for µ < 1 in the NN case ( supersonic homoclinic waves), and

I for µ > 11+4g for the NNN case ( subsonic homoclinic waves)

TheoremThere are subsonic solutions which are homoclinic to exponentially smallperiodic solutions (unlike the constant solution in the NN case).

Proof: Centre manifold analysis; Lombardi, Oscillatory integrals andphenomena beyond all algebraic orders, 2000

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