Sharp-interface and phase-field theories of recrystallization in the plane

Physica D 130 (1999) 133–154

Sharp-interface and phase-field theories of recrystallization in the plane Morton E. Gurtin a , Mark T. Lusk b,∗ a

Department of Mathematics, Carnegie Mellon University, Pittsburgh, PA 15213, USA b Division of Engineering, Colorado School of Mines, Golden, CO 80401, USA

Received 5 December 1997; received in revised form 16 December 1998; accepted 18 December 1998 Communicated by C.K.R.T. Jones

Abstract A continuum framework is presented for recrystallization. The driving force is the energy stored in dislocation substructures, here characterized with the aid of a scalar measure, the dislocation content. Grain boundary kinetics are derived from a configurational force balance, a mechanical version of the second law, and suitable constitutive assumptions. A relation is obtained characterizing the efficiency with which dislocation substructure is eliminated by moving grain boundaries. Using a system of microforce balances, the sharp interface theory is shown to have a phase-field regularization that obviates the need to track individual grain boundaries. The sharp interface theory is recovered, via formal asymptotics, as a limiting case. ©1999 Elsevier Science B.V. All rights reserved.

1. Introduction In materials capable of sustaining inelastic strains, deformation beyond the elastic limit can result in the formation of complex networks of dislocations. The density of these dislocation substructures can be lowered by recrystallization, a process characterized, in part, 1 by the motion of grain boundaries: such boundaries sweep through damaged material rearranging the lattice into configurations of lower energy and hence fewer dislocations [1]. Fig. 1 shows a photograph of a grain boundary that has moved through a region of dense dislocation structure (dark upper area) leaving behind an expanse of nearly virgin crystal (lighter middle region) [2]. Note that for this recrystallization process, an existing grain boundary has begun to move, leaving in its wake a discontinuity in the dislocation structure. Our goal is a continuum–mechanical framework for the dynamics of recrystallization based on a balance law for configurational forces, a mechanical version of the second law, and suitable constitutive equations. In continuum mechanics standard (Newtonian) forces act in response to deformation; that additional configurational forces may ∗

Corresponding author. We do not discuss the nucleation of new grains, nor do we discuss recovery. These topics and the development of dislocation networks due to inelastic deformation will be the subject of future work. 1

0167-2789/99/$ – see front matter ©1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 2 7 8 9 ( 9 8 ) 0 0 3 2 3 - 6

134

M.E. Gurtin, M.T. Lusk / Physica D 130 (1999) 133–154

Fig. 1. Migration of an existing grain boundary. The boundary between grains, in the middle of the photo, has moved upwards leaving in its wake a region relatively free of dislocation substructure. Note that the original position of the grain boundary is observable as a discontinuity in the dislocation substructure.

be needed to describe phenomena associated with the material-structure is clear from the beautiful work of Eshelby [3], who studied the equilibria of defect-structures within a variational framework. A balance law for configurational forces, essential when the theory is dynamical and dissipative, was introduced in [4] (cf. [5,6]). At the lattice level, recrystallization is characterized by the short-range transport of atoms between adjacent mismatched lattices, a process that involves the breaking and forming of atomic bonds. These short-range forces and rearrangements imply an expenditure of work per unit time. At continuum length-scales the action of such short-range forces and motions manifest themselves via dissipative configurational forces that perform work over the macroscopic motion of grain boundaries. To avoid geometric complications that obscure the underlying physics, we restrict attention to planar processes. A material body is treated as a lattice together with dislocations that move, microscopically, relative to the lattice.

M.E. Gurtin, M.T. Lusk / Physica D 130 (1999) 133–154

135

Attention is focused not on individual dislocations but on substructures of dislocations as characterized by the dislocation content α(xx , t) ≥ 0. If the sole macroscopic manifestation of atomistic kinematics is this microparameter, then it seems reasonable that interatomic forces relevant to the motion of dislocations be characterized macroscopically by microforce fields that perform work when α changes. 2 We begin by discussing the motion of a single boundary between two grains. The framework is then extended to allow for the interaction of a finite number of grains in a manner that accounts explicitly for the relative mismatches in lattice orientations. The final system of equations consists of: (i) a constitutive equation W = W (α) for the bulk free energy; (ii) an evolution equation βij V = σij K + [W ]

(1)

relating the jump [W ] in W across the boundary Sij between adjacent grains i and j to the normal velocity V and the curvature K of Sij , with βij and σij , both constant, the kinetic modulus and surface tension for Sij ; (iii) an equation α − = Rij (α + )

(2)

giving the dislocation content α − just behind Sij as a function of its value α + just ahead of Sij ; (iv) a bulk equation α. = 0;

(3)

and (v) a force balance σij T¯ ij + σj k T¯ j k + σki T¯ ki = 0,

(4)

to be satisfied at triple junctions between grains i, j and k, where, e.g. T¯ ij is a unit tangent to the ij -interface directed away from the junction. The constitutive function Rij , assumed to satisfy Rij (α) ≤ α for all α, represents the efficiency with which the motion of Sij lowers the dislocation content; the relation (3) expresses the requirement that dislocation content at a point change only when that point is traversed by a grain boundary. (It is not difficult to extend the theory to allow for the bulk diffusion of dislocations – i.e. recovery.) In conjunction with the sharp-interface theory described above, we develop a phase-field theory that represents grain boundaries as smooth transition layers. A simple choice of constitutive relations then yields a system of Ginzburg–Landau equations having an additional driving term associated with dislocation substructure. This phasefield equation is coupled to an evolution equation for the dislocation content. A formal asymptotic analysis is performed to show that, granted a suitable scaling, the sharp-interface theory is recovered in the limit as the thickness of the transition layers tends to zero. The phase-field theory might be useful for computations, as it does not require the tracking of individual grain boundaries. Current numerical studies of recrystallization typically involve Monte Carlo simulations based on inherently discrete kinetics [9,10]. In that approach an initial orientation is randomly assigned (from a finite number of possibilities) to regions of discrete points. Dislocation energy is also assigned in a random fashion to each point. Individual points then change orientation to that of neighboring points and release their dislocation energy with a probability dependent on the concomitant reduction in total system energy. The continuum framework that we present is intended to be synergistic with such discrete kinetic models. 2

Cf. [7] and Footnotes 5 and 9 of [8].

136

M.E. Gurtin, M.T. Lusk / Physica D 130 (1999) 133–154

2. Sharp-interface theory 2.1. Motion of an interface between two grains 2.1.1. Preliminary definitions. Basic fields We consider a body B consisting of two grains separated by a sharp interfacial curve S = S(t) oriented by a choice of unit tangent T and unit normal N ; we let arc length s increase in the direction of T ; we denote by K and V , the curvature and (scalar) normal velocity of S, respectively; and we write [8] for the jump in a function 8 across S and 8S for the pair of limits of 8 at S: N , t) − 8(xx − 0N N , t), [8](xx , t) = 8(xx + 0N

(5)

N , t), 8(xx − 0N N , t)) . 8S (xx , t) = (8(xx + 0N

(6)

Given an arbitrary connected subcurve C of S, we write ϕinit and ϕterm for the values of ϕ at the initial and terminal points of C; in particular, vint and vterm designate the tangential velocities of the endpoints of C (i.e. the tangential components of the vector velocities of these endpoints). We consider interfacial motion in the presence of a dislocation field, and base our theory on five fields: σ , surface tension; f , internal configuration force; f ext , external configuration force; α, dislocation content; W , bulk-free energy. σ , assumed constant, represents the surface tension (free energy) of the interface. The field f , defined on S, represents internal forces that maintain the integrity of the interface when it is stationary and act in response to the rearangement of atoms during motion; the field f ext , also defined on S, represents forces exerted on S by sources external to the body. The fields α and W are defined in bulk; α≥0 measures the density of dislocations, while W is the bulk-free energy. 2.1.2. Dislocation kinetics Basic to our theory is the assumption that changes in the dislocation content are brought about by the motion of the interface as it sweeps through the bulk material. (Recovery is not addressed in the present form of the theory.) With this in mind, we assume that α. = 0

in bulk

(7)

and hence that α can change at a point x only at the instant the interface crosses x . Further, while the value of α behind the interface may, at that instant, change, the value just ahead cannot. Let x ∈ S(t) and assume that V (xx , t) 6= 0. Then the values of α ahead (α + ) and behind (α − ) the interface are defined as follows: N , t) α + (xx , t) = α(xx + 0N

if V (xx , t) > 0,

N , t) α + (xx , t) = α(xx − 0N

if V (xx , t) < 0,

while N , t) α − (xx , t) = α(xx − 0N

if V (xx , t) > 0,

N , t) α − (xx , t) = α(xx + 0N

if V (xx , t) < 0,

M.E. Gurtin, M.T. Lusk / Physica D 130 (1999) 133–154

137

As a consequence of these definitions [α] = α + − α −

if V > 0,

(8)

[α] = α − − α +

if V < 0.

(9)

2.1.3. Basic laws We assume that the motion of the interface is governed by the configurational balance Z T )init + (f f + f ext ) ds = 0, T )term − (σT (σT C

(10)

where C is an arbitrary subcurve of S. Basic to our theory is a mechanical version of the second law asserting that, for each part P (subregion of B), the total free energy of P increase at a rate not greater than the rate at which work is performed on P ; precisely, letting SP = S ∩ P denote the portion of the interface in P , . Z Z Z σ ds + W da ≤ f ext · (V N ) ds + (σ v)term − (σ v)init . (11) SP

SP

P

The force f does not enter Eq. (11), as it is internal; moreover, since the tangential component of f ext performs no work, we assume that f ext is normal: f ext · T = 0.

(12)

T /∂s = KN N , Eq. (10) is equivalent to the local balance Since ∂T N + f + f ext = 0. σ KN

(13)

Thus, by Eq. (12), the internal force f is necessarily normal to the interface: f = fN.

(14)

Next, the identities 3 · Z Z Z Z W da = W · da + W · da − Z

P

SP

σ ds

·

P1

Z

=−

SP

P2

SP

[W ]V ds,

σ KV ds + (σ v)term − (σ v)init ,

(15) (16)

where P1 and P2 are the portions of P in the individual grains, yield W· ≤ 0

(17)

in bulk and σ KV + [W ]V + f ext · (V N ) ≥ 0

(18)

on the interface. Thus, appealing to Eqs. (13) and (14), we are led to an internal dissipation inequality for the interface: (f − [W ])V ≤ 0. 3

The identity (15) is standard; for (16) cf. e.g. [11], Eq. (2.24).

(19)

138

M.E. Gurtin, M.T. Lusk / Physica D 130 (1999) 133–154

2.1.4. Constitutive equations We characterize the bulk material by a constitutive equation giving the free energy when the dislocation content is known: W = W (α).

(20)

(The same constitutive equation is used for both grains, as we presume that the undamaged grains differ only by a symmetry transformation of their lattices.) In view of Eq. (7), W (α)· ≡ 0 in bulk, and the inequality (17) is satisfied trivially. Since the surface tension σ has already been prescribed as constant, we need only consider two constitutive equations for the interface: an equation for the (normal) internal configurational force f and an equation for the dislocation content α − behind the interface. Generally one would expect f to depend on V , N , and the pair αS of interfacial values of α. Here, for convenience, we neglect a dependence on N and write f = f (V , αS ). A necessary and sufficient condition that a constitutive equation of this form be consistent with the dissipation inequality (19) is that it have the specific form f = [W ] − βV ,

β = β(V , αS ) ≥ 0;

(21)

in fact, we will assume that the kinetic modulus β satisfies the stronger hypothesis β > 0.

(22)

Our final constitutive equation, which characterizes the manner in which the interface S removes dislocations as it sweeps past material points, specifies the value of α just behind S in terms of its value just ahead: α − = R(α + )

for V 6= 0,

(23)

with R, the recrystallization function, assumed to satisfy R(α) < α

for α > 0,

(24)

an inequality ensuring that interfacial motion lower dislocation content. Although the dislocation content behind a recrystallization grain boundary is typically low, existing point defects can induce the generation of dislocations in newly recrystallized material [12]. Moreover, both simulations and experiments studying strain-induced boundarymigration indicate the dragging of dislocations behind a migrating grain-boundary [1,12]. The constitutive equation (24), which allows for the incomplete annihilation of dislocations, is a result of these observations. A simple type of recrystallization function is R(α) = δα

(0 < δ < 1)

(25)

with δ a constant, yielding α − = δα + .

(26)

Therefore, δ characterizes the efficiency with which the recrystallization front removes dislocation substructure. A special case of this is the clean sweeper R(α) ≡ 0, which renders the material just behind the interface free of dislocations: α − = 0.

(27)

M.E. Gurtin, M.T. Lusk / Physica D 130 (1999) 133–154

139

2.1.5. Evolution equations Dropping the external force f ext , the constitutive equations and the configurational force balance combine to form the basic evolution equations for the system: W = W (α),

(28)

α· = 0

(29)

in bulk and βV = σ K + [W ],

(30)

α − = R(α + )

(31)

on the interface. Note that, by Eqs. (28) and (31), we can rewrite Eq. (30) in the form βV = σ K + (sgn V ){W (α + ) − W (R(α + ))},

(32)

so that once the interface has begun to move, only its curvature and the value of α ahead affect its motion. (An interface initially at rest will begin to run in a direction consistent with Eq. (30); this initial direction would therefore depend on the values of α on both sides of the interface.) A further justification of the inequality (24) is as follows. If Eq. (32) is to be satisfied for t > 0 for any initial prescription of the interface, then, by considering an initially flat interace, Eq. (32) must be satisfied for arbitrarily small curvatures, and this implies that W (α + ) ≥ W (R(α + )); thus, if W is an increasing function of α (recall that α ≥ 0), then α + ≥ R(α + ). Interestingly, granted Eq. (24), the same argument renders W an increasing function of α. The left side of Eq. (19) represents the left side of Eq. (10) minus the right; thus, by Eq. (21), for f ext = 0 we can write Eq. (10) in the form · Z Z Z σ ds + W da = (σ v)term − (σ v)init − βV 2 ds, (33) SP

P

SP

so that βV 2 , the dissipation per unit interfacial length, represents the sole source of dissipation: there is no dissipation associated with changes in dislocation content. 2.2. Motion of a system of grain boundaries We consider a system of N grains, labelled i = 1, 2, . . . , N, with the lattice of adjacent grains differing only in orientation. The individual orientations are, in fact, irrelevant to the theory; what is important is the mismatch angle ϑij between adjacent grains i and j as well as the orientation of the grain boundary separating the lattices. (The large degree of incoherency of grain boundaries with large mismatch is well illustrated by the bubbleraft photograph of Fig. 2 [13].) The constitutive dependence of quantities such as surface energy on mismatch and orientation is known to be rather complex [14]. As discussed in Section 2.1, we neglect constitutive dependences on the orientation of grain boundaries. The characterization of ϑij depends on the symmetry of the underlying lattice: ϑij is the smallest rotation that, applied to one grain, causes it to be indistinguishable from the adjacent grain; e.g. ϑij must be less than 45◦ if the lattices are cubic and less than 90◦ if the lattices are tetragonal.

140

M.E. Gurtin, M.T. Lusk / Physica D 130 (1999) 133–154

Fig. 2. Bubble raft picture depicting a triple-point between grain boundaries and the incoherency associated with interfaces between grains with a significant mismatch in orientation.

Guided by the theory developed in Section 2.1, we assume that the material within each grain is governed by the equations W = W (α), ·

α =0

(34) (35)

with the constitutive equation W = W (α) for the bulk-free energy independent of the grain in question. We assume further that any two adjacent grains i and j are separated by a smooth interface Sij whose motion is governed by the evolution equations βij V = σij K + [W ],

(36)

α − = Rij (α + ).

(37)

It is important to note that the kinetic modulus βij , the surface tension σij , and the recrystallization function Rij are allowed to depend on the adjacent grains i and j in question. In particular, one would expect Rij to depend on the mismatch angle ϑij , as grains with small mismatch do not generally lower the dislocation content as much as grains whose mismatch is large. We assume that all junctions (points at which interfaces meet) are triple junctions, and we supplement the equations described above by the requirement that configurational forces at each junction be balanced: if grains i, j , and k meet at a junction, then σij T¯ ij + σj k T¯ j k + σki T¯ ki = 0, where T¯ ij is a unit tangent to the ij -interface directed away from the junction, and similarly for T¯ j k and T¯ ki .

(38)

M.E. Gurtin, M.T. Lusk / Physica D 130 (1999) 133–154

141

3. Phase-field theory 3.1. Motion of an interfacial layer between two grains 3.1.1. Basic fields. Force balances. Second law 4 We base the theory on eight fields defined on the body for all time: ψ, free energy; α, dislocation content; γ and γext , internal and external microforces associated with dislocations; ϕ, phase-field; ξ , microstress; π and πext , internal and external microforce associated with the phase-field. In a strategy common to many phase-field theories, the underlying idea is to replace sharp grain-boundaries by interfacial layers associated with rapid changes in the phase-field ϕ, with the individual grains defined approximately by regions in which ϕ is roughly constant. The equilibrium values of ϕ represent individual grains, while values of ϕ between such equilibria represent regions associated with atoms as they detach from one lattice and re-attach to a second lattice. In association with this additional degree of freedom, we allow for forces that perform work in conjunction with changes in ϕ; these forces are characterized by the microstress ξ and by the internal and external microforces π and πext , and are presumed consistent with the microforce balance Z Z ξ · n ds + (π + πext ) da = 0 (39) ∂P

P

for each part P . Here n is the outward unit normal to ∂P . The balance (39) is, in essence, a counterpart of the configurational balance in the sharp-interface theory. Within the sharp theory, α· = 0 away from the interface S. Here, the interface is ‘smeared out’ and the individual grains characterized only approximately; for that reason the dislocation content α will generally vary with time throughout the body. As with the phase-field ϕ, we allow for forces that perform work when α changes; these forces are characterized by the internal and external microforces γ and γext , and are assumed consistent with the microforce balance Z (γ + γext ) da = 0 (40) P

for all P . Here, guided by the sharp theory for which, in bulk, spatial changes in α do not result in corresponding temporal changes, we assume that there is no interaction between neighboring dislocation structures; for that reason we have not introduced a microstress associated with the field of dislocations. We supplement the microforce balances by a mechanical version of the second law: · Z Z Z Z ψ da ≤ (ξξ · n )ϕ· ds + πext ϕ· da + γext α· da (41) P

∂P

P

P

for each P ; the microforces π and γ do not contribute, as they are internal. The balances (39) and (40) have the local forms div ξ + π + πext = 0,

(42)

γ + γext = 0,

(43)

and using Eqs. (42) and (43) we can reduce Eq. (41) to a local dissipation inequality ψ · + π ϕ· − ξ · ∇ ϕ· + γ α· ≤ 0. 4

This framework, based on a microforce balance and a mechanical version of the second law, originates in [7] (cf. [8]).

(44)

142

M.E. Gurtin, M.T. Lusk / Physica D 130 (1999) 133–154

3.1.2. General constitutive theory. Thermodynamic restrictions As constitutive equations we assume that ψ, π, ξ and γ are each a function of(ϕ, ∇ ϕ, ϕ· , α, α· ).

(45)

Conditions that are necessary and sufficient that all ‘processes’ consistent with the constitutive Eq. (45) be consistent with the second law, in the form of the dissipation inequality Eq. (44), are that 1. the free energy ψ and microstress ξ be independent of ϕ · and α· and satisfy ξ (ϕ, ∇ ϕ, α) = ∂ψ(ϕ, ∇ ϕ, α)/∂gg ,

g = ∇ ϕ;

(46)

2. writing (. . . ) for the partial list (ϕ, ∇ ϕ, α), the constitutive equations for the internal microforces satisfy π(. . . , ϕ· , α· ) = −

∂ψ(. . . ) + πnon (. . . , ϕ· , α· ), ∂ϕ

∂ψ(. . . ) + γnon (. . . , ϕ· , α· ), ∂α consistent with the inequality

γ (. . . , ϕ· , α· ) = − with πnon and γnon

πnon (. . . , ϕ· , α· )ϕ· + γnon (. . . , ϕ· , α· )α· ≤ 0.

(47) (48)

(49)

The functions πnon and γnon represent nonequilibrium contributions to the internal forces π and γ . The derivation of the restrictions (46)–(49) is based on the Coleman–Noll procedure [7,8,15]. We substitute the constitutive equations into Eq. (44). Since the resulting inequality is linear in ϕ.. , α.. , and ∇ ϕ· , it can be satisfied for all choices of the field ϕ if and only if the coefficients associated with these terms vanish. This renders the free energy and microstress independent of ϕ· and α· , and related through Eq. (46), and yields the requirement that π and γ be consistent with Eqs. (47)–(49); and conversely. 3.1.3. Application of the general theory to recrystallization We now introduce additional constitutive assumptions to capture the qualitative nature of recrystallization. Since the present discussion is limited to two grains, i and j , say, the phase-field might be viewed as the volume fraction of grain i; the value ϕ = 1 therefore corresponds to grain i, the value ϕ = 0 to grain j . For the free energy we consider the standard Ginzburg–Landau energy modified to include an energy W (α) associated with the dislocation field: ∇ ϕ|2 + W (α). ψ = f (ϕ) + 21 E|∇

(50)

We assume that: (GL1) f (ϕ), the exchange energy, is a double-well potential with well-bottoms at ϕ = 0 and ϕ = 1 (so that f (0) = f (1) = 0, f (ϕ) > 0 for ϕ 6= 0, 1); (GL2) E is a strictly positive constant; (GL3) W (α) is strictly convex in α ≥ 0 with W 0 (0) = 0. Then by Eq. (45), ξ = E∇ ∇ ϕ.

(51) , ϕ · , α· )

represents internal forces associated with dislocations, forces that perform work The function γnon (. . . in association with changes in the dislocation content. Guided by continuum theories of plasticity, we assume that this force is rate-independent: γnon (. . . , ϕ· , α· ) = γnon (. . . , λϕ· , λα· )

for all λ > 0.

(52)

M.E. Gurtin, M.T. Lusk / Physica D 130 (1999) 133–154

143

The particular choice λ = |ϕ· |−1 yields γnon (. . . , ϕ· , α· ) = γnon (. . . , |ϕ· |−1 ϕ· , |ϕ· |−1 α· )

(53)

for ϕ· 6= 0, an inequality that will be tacit throughout what follows. To simplify the theory, we restrict attention to a dependence of γnon on |ϕ· |−1 α· , in fact, to a dependence of the simple form γnon (. . . , ϕ· , α· ) = −

Dα· , |ϕ· |

(54)

with D constant. When external forces are neglected, Eq. (43) yields γ = 0, so that, by Eqs. (48), (50), (54), Dα· = −|ϕ· |W 0 (α),

(55)

which represents the evolution equation for dislocation substructure. Roughly speaking, the individual grains are characterized by regions in which ϕ is approximately constant; granted this, α· ≈ 0, which should be compared to the assumption α· = 0 that forms a basis of the sharp-interface theory. Further, as W 0 (α) > 0 for α > 0, sgn α· = −sgn D; since dislocation content generally decreases with time during recrystallization, we assume that D > 0. By Eq. (48), πnon (. . . , ϕ· , α· ) represents internal forces that are work-conjugate to ϕ· and hence to the evolution of the transition layer; bearing this in mind, we consider πnon as the sum of a force πco associated with a coherent transition and a force πnco associated with incoherency: πnon = πco + πnco .

(56)

We assume that πco is independent of changes in dislocation content; in fact that πco is dependent only on ϕ· , and that this dependence is linear: πco (. . . , ϕ· , α· ) = −Bϕ· ,

(57)

with B constant and strictly positive (cf. Eq. (49)). The force πnco represents coupling between the layer and the field of dislocations. In the sharp-interface theory there is no dissipation associated with changes in dislocation content (cf. Eq. (33)). Based on this observation, we assume that there is no net working of the dissipative internal forces associated with changes in dislocation content: πnco ϕ· + γnon α· = 0.

(58)

(One might roughly view γnon as representing forces exerted on the dislocation field by the transition layer, and πnco as representing forces exerted on the transition layer by the dislocation field. In mechanics such forces are equal and opposite, but here that cannot be true, as γnon and πnco perform work on objects of a completely different nature.) In view of Eq. (54), this yields πnco =

D|α· |2 ϕ· , |ϕ· |3

(59)

which is also rate-independent. Further, when external forces are neglected, we may use Eq. (55) to conclude that πnco = −

W 0 (α)α· ϕ· W (α)· ϕ· = − , |ϕ· |2 |ϕ· |2

(60)

and hence, by Eq. (58), that W (α)· = γnon α· ;

(61)

144

M.E. Gurtin, M.T. Lusk / Physica D 130 (1999) 133–154

the working of the nonequilibrium internal-force γnon associated with the dislocation content α is balanced by the change in free energy associated with changes in α. This also follows directly from Eqs. (43), (48) and (50) with external forces neglected. 3.1.4. Evolution equations for recrystallization Neglecting external forces, the balance laws and constitutive equations yield the basic PDEs of the theory, which are Bϕ· = E1ϕ − f 0 (ϕ) −

W (α)· ϕ· |ϕ· |2

(1 = Laplacian),

Dα· = −|ϕ· |W 0 (α),

(62) (63)

with B, D, E > 0. Note that, as a consequence of Eq. (63), W (α)· ≤ 0

(64)

and the dislocation content at each point decreases monotonically with time. Note further that by Eqs. (50) and (51), ∇ ϕ · n )ϕ· = 0 on ∂B, then if (∇ Z ∇ ϕ|2 + W (α)} da {f (ϕ) + 21 E|∇ (65) B

decreases with time. In writing Eq. (62), it is tacit that ϕ· not vanish. (Note that by Eq. (63), the term G = −W (α)· ϕ· /|ϕ· |2 may be written in the form G = D −1 W 0 (α)2 sgn ϕ· ; by (GL3), Eq. (64) renders α and hence W 0 (α)2 decreasing in t; G is therefore bounded.) 3.1.5. Relation to the sharp-interface theory Consider the phase-field equations (62) and (63) rescaled with B, E, and f replaced by εB, εE, and ε−1 f , where ε > 0 is a small parameter. Let ϕε and αε denote a solution of the rescaled equations: εBϕε· = εE1ϕε − ε−1 f 0 (ϕε ) −

W (αε )· ϕε· , |ϕε· |2

Dαε· = −|ϕε· |W 0 (αε ).

(66) (67)

As we will show, in the limit ε → 0 these equations are formally asymptotic to the equations of the sharp-interface theory discussed in Section 2.1. Precisely, the limiting solution partitions the body into regions {ϕ ≡ 1} and {ϕ ≡ 0} within which α· = 0. These regions, the two grains, are separated by a sharp interface which evolves according to βV = σ K + [W ] and α − = R(α + ) (in the notation of Sections 2.1.1 and 2.1.2). Here β and σ are the constants 1/2 Z 1 Z +∞ dη(r) 2 2 dη, (68) β = kB, σ = kE, k= dr dr = Ef (η) −∞ 0 with η(r) the unique solution of E

d2 η = f 0 (η), dr 2

η(−∞) = 0,

η(+∞) = 1;

while R = R(α) is the function defined uniquely by the equation Z α D = 1. 0 (λ)dλ W R

(69)

(70)

M.E. Gurtin, M.T. Lusk / Physica D 130 (1999) 133–154

For the special case in which W (α) = (1/2)Gα 2 , Eq. (67) takes the form   G . α − = δα + , δ = exp − D

145

(71)

(The specific form of the recrystallizaiton function R(α) defined via Eq. (70) depends on W (α); this stands in contrast to the sharp-interface theory, where R(α) is independent of W (α). This limitation is a consequence of the assumption that D be constant. Letting D depend on α leaves the results unaltered, but allows for a large class of functions R(α), a class dependent on W (α) only through the requirement that W 0 (α)−1 D(α) have a nonintegrable singularity at α = 0; cf. the sentence containing Eq. (A.14).) The analysis leading to these results is sketched in Appendix A. 3.1.6. Configurational force balance 5 Our formal verification of the asymptotic results described above makes use of the configurational force balance div C + k = 0

(72)

where C , a tensor field, and k , a vector field, are defined by  ·   α 0 · ∇ ϕ ⊗ ∇ ϕ, C = ψ11 − E∇ k = Bϕ ∇ ϕ + W (α) ∇ϕ − ∇α . ϕ·

(73)

This identity is a consequence of the evolution equation (62). ∇ ϕ|, and K¯ = −div N¯ represent the unit normal, normal velocity, and ∇ ϕ/|∇ ∇ ϕ|, V¯ = −ϕ· /|∇ The fields N¯ = −∇ ◦ · ∇ α) · V¯ N¯ represents the time-derivative of α following the normal curvature for level sets of ϕ, while α = α + (∇ trajectories of such level sets. If we take the inner product of Eq. (72) with N¯ we arrive at the normal configurational balance ∇ ϕ|2 }K¯ + N¯ · ∇ W (α) + J, ∇ ϕ|2 V¯ = {f (ϕ) + 21 E|∇ B|∇

(74)

∇ ϕ|2 )N¯ ⊗ N¯ }. J = −V¯ −1 W 0 (α)α ◦ + N¯ · div{(f (ϕ) − 21 E|∇

(75)

Note the similarity between Eq. (75) and the evolution equation (30) for a sharp interface. We exploit this similarity in our discussion of the formal asymptotics; as shown in Appendix A, the term J becomes insignificant as ε → 0. ∇ ϕ|2 and P = 1 − N¯ ⊗ N¯ , note that N¯ · divP P = To verify Eqs. (74) and (75), we let F ± = f (ϕ) ± (1/2)E|∇ ¯ ¯ −divN = K, and use the identity ∇ ϕ|2 + F + )(N¯ ⊗ N¯ ) + W (α)1} N¯ · div C = N¯ · div{F +P − (E|∇ = F + K¯ + N¯ · div{F − (N¯ ⊗ N¯ )} + N¯ · ∇ W (α). 3.2. Phase-field theory for a system of grain boundaries 3.2.1. The constraint. Basic laws We consider a system of N grains, labelled i = 1, 2, . . . , N, with phase-field ϕ1 ≥ 0 the volume fraction of grain i, so that X i 5

ϕi = 1,

X i

=

N X . i=1

Cf. [16,17]. See also [6] for a general discussion of configurational forces.

(76)

146

M.E. Gurtin, M.T. Lusk / Physica D 130 (1999) 133–154

Further, we let 8 denote the list of ϕ’s and, for i 6= j , we write 8ij for the list 8 with all entries except ϕi and ϕj set equal to zero: 8ij = (0, . . . , 0, ϕi , 0, . . . , 0, ϕj , 0 . . . , 0). We let ϑij denote the mismatch angle between grains i and j ; we let ϕij denote the volume fraction of the ij -interaction, ϕij = ϕi + ϕj ;

(77)

and we write 2=

N X

ϕij ϑij .

(78)

i,j =1 i
Then 2 = ϑij

(79)

is a pure ij -interaction in the sense that ϕij = 1. The simpler theory developed in Section 3.1 was based on two grains i and j , with ϕ the volume fraction of grain i and 1 − ϕ the volume fraction of grain j ; the constraint (76) was tacitly used to eliminate ϕj . Here, because we allow for an arbitrary number of grains, it seems preferable to consider the complete list 8 of volume fractions supplemented by the constraint (76). Even so, the theory is not much different than that developed in Section 3.1, and for that reason we shall simply sketch the underlying discussion. We now have a system analogous to Eqs. (39)–(41) consisting of N microforce balances associated with the phase-fields, a balance associated with dislocation substructure, and a version of the second law; locally these have the form div ξ i + πi + πext(i) = 0,

i = 1, 2, . . . , N,

γ + γext = 0, X (πi ϕi· − ξ i · ∇ ϕi· ) + γ α· ≤ 0. ψ· +

(80) (81) (82)

i

P P P By Eq. (76), i ϕi· = 0; hence i ξ i and i πi do not contribute to Eq. (82); therefore, without loss in generality, we normalize these fields with the requirement X X X ξ i = 0, πi = πext(i) = 0, (83) i

i

i

where the normalization for the external force balance follows from (80). (Note that Eqs. (80)–(82) are invariant P under the transformation ξ j → (ξξ j − N −1 i ξ i ) (for each j ) in conjunction with similar transformations of πj and πext(j ) .) 3.2.2. Constitutive theory Guided by Eq. (45), we consider constitutive equations in which ϕ, π, ξ and γ are each functions of (8, ∇ 8, 8· , α, α· ). When restricted by the dissipation inequality (82), these relations have the reduced form ξ i (8, ∇ 8, α) =

∂ψ(8, ∇ 8, α) , ∂gg i

g i = ∇ ϕi ,

M.E. Gurtin, M.T. Lusk / Physica D 130 (1999) 133–154

πi (. . . , 8· , α· ) = − γ (. . . , 8· , α· ) = −

147

∂ψ(8, ∇ 8, α) + πnon(i) (. . . , 8· , α· ), ∂ϕi

∂ψ(8, ∇ 8, α) + γnon (. . . , 8· , α· ), ∂α

with γnon (. . . , 8· , α· )α· +

X {πnon(i) (. . . , 8· , α· )ϕi· } ≤ 0. i

Here ∂ψ/∂gg i and ∂ψ/∂ϕi represent derivatives on the hyperplanes We restrict attention to a simple free energy

(84) P

ig i

= 0 and Eq. (76), respectively.

∇ 8 |2 + W (α) ψ = F (8) + 21 E(2)|∇

(85)

consistent with the assumptions: (GLN1) The exchange energy F (8) is an N-well potential with F (8) = 0 at the well-bottoms (1, 0, . . . , 0, 0), (0, 1, . . . , 0, 0), . . . , (0, 0, . . . , 0, 1), with F (8) > 0 otherwise, and with the wells distributed symmetrically (i.e., F (8) = F (perm8) for all 8 and all permutations, perm 8, of the list 8); (GLN2) E(2) is strictly positive. We continue to assume that W (α) satisfies (GL3). Note that in contrast to the theory of Section 3.1, where E was constant, we here allow E to depend on the phase-field 8 through a dependence on 2; by Eq. (79), E(2) reduces to the constant E(ϑij ) in a pure ij -interaction, rendering this assumption consistent with the discussion of Section 3.1, and more generally providing a dependence on the orientational mismatch between grains [14]. Arguing as in Eqs. (54)–(57), we assume that γnon = −D(2)α· /|8· |, πnon(i) = πco(i) + πnco(i) , πco(i) = −B(2)ϕi· , πnco(i) = D(2)|α· |2 ϕi· /|8· |3 with B(2) and D(2) strictly positive. Therefore, when external forces are dropped, D(2)α· = −|8· |W 0 (α),

πnco(i) = −

W (α)· ϕi· . |8· |2

(86)

3.2.3. Evolution equations Neglecting external forces, we are led to the PDEs B(2)ϕi· = E(2)1ϕi −

∂F (8) W (α)· ϕi· − , ∂ϕi |8· |2

i = 1, 2, . . . , N,

D(2)α· = −|8· |W 0 (α), P P ∇ ϕi · n )ϕi· = 0 on ∂B, subject to the constraint i ϕi = 1. As before, W (α)· ≤ 0, and, if i (∇ Z ∇ 8|2 + W (α)}da {F (8) + 21 E(2)|∇ B

(87) (88)

(89)

decreases with time. Another consequence of Eqs. (87) and (88) is the following generalization of the configurational force balance (72): div C + k = 0, where C = ψ11 −

X i

∇ ϕi ⊗ ∇ ϕi }, {E∇

(90)

(91)

148

M.E. Gurtin, M.T. Lusk / Physica D 130 (1999) 133–154

(

· −2

k = {B + |8 |

X W (α) } ϕi· ∇ ϕi

)

·

∇ α. − W 0 (α)∇

(92)

i

3.2.4. Pure ij-interactions In preparation for a discussion of the asymptotic behavior of the phase-field theory, consider a pure ij -interaction for which 8 = 8ij = (0, . . . , 0, ϕi , 0, . . . , 0, ϕj , 0, . . . , 0),

ϕij = ϕi + ϕj = 1,

2 = ϑij .

(93) (94)

By (GLN1), we can define a function f (ϕ), independent of i and j , through f (ϕ) = F (8ij )

(95)

ϕ = ϕi (so that ϕj = 1 − ϕ).

(96)

with

Then f (ϕ) is consistent with (GL1). By Eqs. (93), (95) and (96), for 8 = 8ij , |8· |2 = 2|ϕ· |2 ,

f 0 (ϕ) =

2∂F (8) ; ∂ϕi

(97)

the definitions Bij = 2B(ϑij ),

Eij = 2E(ϑij ),

Dij = 2−1/2 D(ϑij )

(98)

therefore, reduce the general equations (87) to the Eqs. (62) and (63) of the two-grain theory; viz. Bij ϕ· = Eij 1ϕ − f 0 (ϕ) −

W (α)· ϕ· , |ϕ· |2

(99)

Dij α· = −|ϕ· |W 0 (α)

(100)

3.2.5. Relation to the sharp-interface theory As in Section 3.1.5, we consider the rescaled equations εB(2)ϕi· = εE(2)1ϕi − ε−1 D(2)α· = −|8· |W (α),

∂F (8) − W (α)· ϕi· /|8· |2 , ∂ϕi

i = 1, 2, . . . , N,

(101) (102)

for 8 = 8ε and α = αε . In the limit ε → 0, these equations are formally asymptotic to the equations of the sharpinterface theory discussed in Section 2.2. Precisely, the limiting solution partitions the body into (possibly empty) regions {ϕ1 ≡ 1}, {ϕ2 ≡ 1}, . . . , {ϕN ≡ 1}, which represent the individual grains. We assume that each grain is a connected region, that the interface between any two grains is a smooth curve, and that the junctions between interfaces are triple junctions. (Although our analysis is valid almost without change for junctions of higher degree, which we conjecture to be dynamically unstable.) The limiting solution then satisfies α· = 0

(103)

M.E. Gurtin, M.T. Lusk / Physica D 130 (1999) 133–154

149

within each grain and, for any two grains i and j , the corresponding interface evolves according to βij V = σij K + [W ],

(104)

α − = Rij (α + ),

(105)

with the constants βij and σij and the function Rij (α) defined by Eqs. (68)–(70) with B, E, and D replaced by Bij , Eij , and Dij , where the latter are defined in Eq. (98) with f (ϕ) given by Eq. (95). For the special case in which W is quadratic Eq. (105) takes the form   G − + (106) α = δij α , δij = exp − Dij (cf. (4.33)). Finally, the triple-junction condition σij T¯ ij + σj k T¯ j k + σki T¯ ki = 0

(107)

holds at each junction, where the T¯ ’s are unit tangents directed away from the junction. In fact, this result can be immediately extended to junctions involving N grains which suggests that this phase-field model can be used to study the stability of such junctions as a function of the number of intersecting grains. The actual analysis is sketched in Appendix B. The formal verification of Eqs. (104) and (105) is based on the observation that, in the stretched coordinates, the interfacial layer between the grains i and j is, to lowest order in ε, a pure ij -interaction; the verification of the triple-junction condition utilizes the configurational balance Eq. (90).

Acknowledgements We are grateful to R. James, M. Soner, G. Stiehl, and, especially, H.-J. Jou, for numerous helpful discussions. This work was supported by the National Science Foundation, the Army Research Office, and the Department of Energy.

Appendix A. Formal asymptotics 6 A.1. Asymptotics of the interfacial layer Let ϕε and αε denote a solution of the rescaled Eqs. (65) and (66). We assume that, as ε → 0, ϕε → ϕ

and

αε → α

on B for all time.

We will refer to ϕ and α as far fields. In view of Eq. (65), the far field ϕ satisfies f 0 (ϕ) ≡ 0, thus by (GL1), B consists of time-dependent regions {ϕ ≡ 0} and {ϕ ≡ 1}, which we assume to represent the grains. By Eq. (66), α· = 0 within the grains. 6 The formal asymptotics of the scalar Ginzburg–Landau equation is discussed in [18]; rigorous treatments are given in [19–21]. The vector Ginzburg–Landau equation is studied in [22], where a formal derivation of the triple-junction condition is given. See also [23–25]. Our discussion, which is formal, differs from the above by focusing on configurational forces. Specifically, the approach used makes clear the link between each component of the sharp-interface configurational force balance of Eqs. (29) and (30) and its phase field counterpart in Eqs. (61) and (62) (cf. [16,17]).

150

M.E. Gurtin, M.T. Lusk / Physica D 130 (1999) 133–154

We assume that the interface between grains is a smoothly evolving curve S(t). We write d(xx , t) for the signed distance to S(t) with d < 0 in the region {φ ≡ 0},

d > 0 in the region {ϕ ≡ 1}.

(A.1)

For any (xx , t) such that x has a unique closest point x ∗ on S(t), we let ζ (xx , t) denote the arc-length value on S(t) that corresponds to x ∗ . On the set of such (xx , t)’s – to which we henceforth restrict attention – (d(xx , t), ζ (xx , t)) provides a coordinate field; moreover, the fields defined by V = −d · ,

N = ∇ d,

K = −1d

(A.2)

depend only on (ζ, t) and represent the unit normal, normal velocity, and curvature of S. We assume that for small ε > 0, S(t) is ‘replaced by’ a thin transition layer. Within this layer we stretch the coordinate normal to S by letting d(xx , t) ε and, for uε = ϕε or uε = αε , we write r(xx , t) =

(A.3)

uε (xx , t) = u¯ ε (r, ζ, t).

(A.4)

Then, for D = ∂/∂r, u·ε = −ε−1 V Du¯ ε + O(1), −1

(A.5)

N + O(1), (Du¯ ε )N

(A.6)

1uε = ε−2 D2 u¯ ε + O(ε−1 ).

(A.7)

∇ uε = ε

Further, letting N¯ ε (r, ζ, t), V¯ε (r, ζ, t), and K¯ ε (r, ζ, t) denote the unit normal, normal velocity, and curvature of level sets of ϕε as defined in the text following Eq. (73), we have the additional estimates N¯ ε = N + O(ε),

V¯ε = V + O(ε),

K¯ ε = K + O(ε),

(A.8)

and these with Eqs. (A.5)–(A.7) and the definition of α ◦ , appearing after Eq. (73), yield (αε)◦ = O(1).

(A.9)

The estimates for N¯ ε and V¯ε follow directly from their definitiions and Eqs. (A.5) and (A.6); the verification of the estimate for K¯ ε is not so straightforward (cf., e.g., Fried, Gurtin [17], Eq. (7.38)). Passing to the limit in Eqs. (66) and (67), we find, using Eq. (A.4) and the estimates (A.5) and (A.6), ¯ ED2 ϕ¯ = f 0 (ϕ),

(A.10)

¯ DDα¯ = (sgn V )|Dϕ|W ¯ 0 (α),

(A.11)

where ϕ¯ = ϕ(r, ¯ ζ, t) and α¯ = α(r, ¯ ζ, t), the stretched fields, denote the limits, as ε → 0, of ϕ¯ε and α¯ ε . By Eq. (A.1), to match conditions in the far field requires a solution of Eq. (A.10) satisfying ϕ¯ = 0 at r = −∞ and ϕ¯ = 1 at r = +∞. We assume that such a solution ϕ¯ exists and is unique; then, necessarily, ϕ¯ depends only on r. Further, multiplying Eq. (A.10) by Dϕ¯ and integrating the resulting equation from r = −∞ to an arbitrary r yields the conclusion ¯ 2, f (ϕ) ¯ = 21 E|Dϕ| so that Dϕ¯ > 0.

(A.12)

M.E. Gurtin, M.T. Lusk / Physica D 130 (1999) 133–154

151

We use the notation of Section 2.1.1 and 2.1.2 for the interface S, oriented by N (= ∇ d), and for the limits at S and the jump across S of the far fields ϕ and α. Matching the stretched field ϕ¯ and the far field α then requires that [α] = {α¯ at r = +∞} − {α¯ at r = −∞}. By Eqs. (A.10) and (A.11), [α](sgn V ) = α + − α − ; Eqs. (A.10) and (A.11) therefore yield Z α+ D dλ = 1. (A.13) − W 0 (λ) α Choose α + . Then the left side of Eqs. (A.10) and (A.11) increases strictly from zero as α − decreases from α + to zero. Moreover, granted sufficient smoothness, (GL1) implies that W 0 (α)−1 has a nonintergrable singularity at α = 0. The integral in Eq. (A.13), therefore, tends to +∞ as α − tends to zero; thus there is a solution of Eq. (A.13): α − = Rij (α + ).

(A.14)

To complete our verification of the formal convergence of the phase-field equations to the sharp-interface equations we have only to show that βV = σ K + [W ]. Thus consider the normal configurational balance (73) and (74), multiplied by ε, as applied (with B, E, and f replaced by εB, εE, and ε−1 f ) to the fields ϕε and αε n o ∇ ϕε |2 V¯ε = f (ϕε ) + 21 ε2 E|∇ ∇ ϕε |2 K¯ ε + εN¯ ε · ∇ W (αε ) + Jε , ε2 B|∇ (A.15) ∇ ϕε |2 )N¯ ε ⊗ N¯ ε }. Jε = −ε(V¯ε )−1 W 0 (αε )αε◦ + N¯ ε · div{(f (ϕε ) − 21 Eε2 |∇

(A.16)

∇ ϕε |2 V¯ε → B|Dϕ| ∇ ϕε |2 }K¯ ε → E|Dϕ| ¯ 2 V , {f (ϕε )+ 21 ε2 E|∇ ¯ 2 K, εN¯ ε · Then Eqs. (A.5)–(A.9) and (A.12) yield ε 2 B|∇ 1 −1 0 ◦ 2 2 ¯ ∇ ϕε | → f (ϕ) ¯ and ε(Vε ) W (αε )αε → 0. Further, since f (ϕε ) − 2 Eε |∇ ¯ − (1/2)E|Dϕ| ¯2= ∇ W (αε ) → DW (α), ¯ 0, one might expect that the remaining term in (Eq. (A.16), N ε · div{. . . .}, also tends to zero. This is indeed the case, but its verification is more difficult (cf., e.g., Fried and Gurtin [17], p.178). Passing to the limit in Eqs. (A.15) and (A.16), therefore, yields ¯ 2 K + DW (α), ¯ B|Dϕ| ¯ 2 V = E|Dϕ|

(A.17)

and, since Dϕ¯ = dϕ/dr, ¯ while V and K are independent of r, integrating Eqs. (A.15) and (A.16) from r = −∞ to r = +∞ yields βV = σ K + [W ] (supplemented by Eqs. (67) and (68) with η = ϕ). ¯

Appendix B. Asymptotics for multiple grains Let 8ε and αε be a solution of Eqs. (100) and (101), and assume that, as ε → 0, 8ε → 8 and αε → α on B for all time. By Eq. (100), ∂F (8)/∂ϕi ≡ 0 for all i. Thus, in view of (GLN1), B consists of time-dependent regions {ϕ1 ≡ 1}, {ϕ2 ≡ 1}, . . . , {ϕN ≡ 1}, which represent the grains; by Eq. (101), α· = 0 within each grain. We assume that the interface between any two adjacent grains i and j is a smoothly evolving curve Sij (t). If the behavior in the stretched coordinates between the grains i and j were a pure ij -interaction, then we could use the results of Appendix A to conclude that the desired interface conditions (103) and (104) are satisfied. Unfortunately, we cannot assert a priori that the volume fractions of grains other than i and j vanish within the ij -layer. Restricting attention to the ij -interface S = Sij , we let d denote the signed distance to S with d < 0 in grain j and d > 0 in grain i, and we continue to use the notation introduced in Eqs. (A.1)–(A.4), so that (analogs of) the estimates (A.5)–(A.7) are valid. Then passing to the limit in Eqs. (100) and (101), we find that, within the ij -layer, ED2 ϕ¯k =

¯ ∂F (8) , ∂ ϕ¯k

k = 1, 2, . . . , N.

(B.1)

152

M.E. Gurtin, M.T. Lusk / Physica D 130 (1999) 133–154

Fig. 3. A triple-junction showing the notation used in Appendix B.

This construction yields the matching conditions ϕ¯j = 1 at r = −∞ and ϕ¯i = 1 at r = +∞, and since P k ϕ¯ k=1 , ϕ¯ k ≥ 0, all ϕ¯ other than ϕ¯ i and ϕ¯ j vanish at r = ±∞. Thus, assuming that the system described has a unique solution, the only nonzero ϕ’s ¯ within the layer are those corresponding to grains i and j . The ij -layer must therefore correspond to a pure ij -interaction. Hence, Eqs. (103) and (104) are satisfied. We have only to establish the asymptotic validity of the triple-junction condition (106). Let z = z (t) be such a junction and assume, without loss in generality, that subscripts 1, 2 and 3 label the three grains involved. Let T¯ ij , i < j, denote the unit tangent to the ij -interface directed away from z , and let N¯ ij be the unit vector that is perpendicular to T¯ ij and directed into grain j . We will refer to a unit vector T as pointing strictly into grain i if T ‘lies between’ T¯ ij and T¯ ik but is equal to neither. To discuss the immediate neighborhood of the junction, we let y = ε−1 [xx − z (t)]; y denotes the position vector, measured from the junction, when the material around the junction has been stretched by the factor ε−1 . For uε = ϕiε or uε = αε , let u¯ ε (yy , t) = uε (xx , t), so that u·ε = u¯ ε /∂t − ε−1z· · ∇ u¯ ε , ∇ uε = ε−1∇ u¯ ε , and 1uε = ε−2 1u¯ ε . (Here and in what follows, differential operations on u¯ ε are associated with the variables (yy , t), while operations on uε are associated with (xx , t).) Thus, passing to the limit in Eq. (100) yields ¯ ϕ¯i = E(2)1

¯ ∂F (8) , ∂ ϕ¯i

i = 1, 2, . . . , N,

(B.2)

¯ ε . Eq. (B.2) is identical to Eq. (86) with W (α) ¯ and 2 ¯ the formal limits of 8 ¯ ε and 2 ¯ and with the stretched fields 8 ¯ B(2) omitted; thus, by Eqs. (90)–(92), Eq. (B.2) is associated with the configurational force balance X ¯ ∇ 8| ¯ 2 }11 − ¯ ∇ ϕ¯i ⊗ ∇ ϕ¯i }. ¯ + 1 E(2)|∇ {E(2)∇ (B.3) div C¯ = 0, C¯ = {F (8) 2 i

M.E. Gurtin, M.T. Lusk / Physica D 130 (1999) 133–154

Note that for a pure ij -interaction C¯ has the form o n ∇ ϕ| ¯ 2 1 − Eij ∇ ϕ¯ ⊗ ∇ ϕ, ¯ C¯ = f (ϕ) ¯ + 21 Eij |∇

ϕ¯ = ϕ¯j ,

153

(B.4)

where we have used the notation specified in Eqs. (94), (95) and (97). For convenience, we restrict i and j to 1, 2 and 3 but let k denote any integer between 1 and N ; and we suppress the argument t. Appropriate matching conditions for the stretched fields may then be stated as follows: 1. for T any unit vector that points strictly into grain i, T) → 1 ϕ¯ i (λT

as λ → ∞,

(B.5)

T ) → 0 as λ → ∞ for k 6= i; so that, by Eq. (B.4), ϕ¯k (λT 2. for i < j, k 6= i, j , and any r, ϕ¯j (λT¯ ij + r N¯ ij ) → η(r) ϕ¯k (λT¯ ij + r N¯ ij ) → 0

as λ → ∞,

as λ → ∞,

(B.6) (B.7)

with η defined by Eq. (68) (and hence equal to ϕ¯ for the layer solution). Eqs. (B.6) and (B.7) render the stretched fields near the triple junction consistent with those for the layer. A formal consequence of Eq. (B.6) is that ∇ ϕ¯j (λT¯ ij + r N¯ ij ) →

dη ¯ N ij dr

as λ → ∞.

(B.8)

We are now in a position to establish the triple-junction condition at z . Choose λ > 0 and let lij (λ), i < j , denote the straight line through the point λT¯ ij parallel to N¯ ij . Further, let 0(λ) denote the boundary of the triangular region bounded by segments of the lines lij (λ), and let 0ij (λ) denote the portion of 0(λ) that lies on lij (λ). By Eq. (B.3), Z C¯ n ds = 0 , (B.9) 0(λ)

where n is the outward unit normal to 0(λ). By Eqs. (B.6) and (B.7), lij (λ) is formally asymptotic, as λ → ∞, to an ij -layer which is a pure ij -interaction. Thus, since n = T¯ ij on 0ij (λ), we may use Eq. (B.3) to conclude that, as λ → ∞, the integral of C¯ n over 0ij (λ) is formally asymptotic to (Z  2 ! ) +∞ 1 dη f (η) + Eij dr T¯ ij . (B.10) 2 dr −∞ On the other hand, Eq. (A.12) holds with ϕ¯ replaced by η, thus, by Eq. (67) and the sentence containing Eq. (103), (B.10) reduces to σij T¯ ij . Thus Eq. (B.8) yields the triple-junction condition Eq. (106) as its formal asymptotic limit. References [1] [2] [3] [4]

P.G. Shewmon, Transformations in Metals, McGraw-Hill, New York, 1969. J.E. Bailey, P.B. Hirsch, Recrystallization processes in some polycrystalline metals, Proc. Roy. Soc. London 267 (1962) 11–30. J.D. Eshelby, The force on an elastic singularity, Phil. Trans. A 244 (1951) 87–112. M.E. Gurtin, Multiphase thermomechanics with interfacial structure. 1. Heat conduction and the capillary balance law, Arch. Rational Mech. Anal. 104 (1988) 185–221. [5] M.E. Gurtin, A. Struthers, Multiphase thermomechanics with interfacial structure. 3. Evolving phase boundaries in the presence of bulk deformation, Arch. Rational Mech. Anal. 112 (1990) 97–160.

154 [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]

M.E. Gurtin, M.T. Lusk / Physica D 130 (1999) 133–154 M.E. Gurtin, The nature of configurational forces, Arch. Rational Mech. Anal. 131 (1995) 67–100. E. Fried, M.E. Gurtin, Continuum theory of thermally induced phase transitions based on an order parameter, Physica D 68 (1993) 326–343. M.E. Gurtin, Generalized Ginzburg–Landau and Cahn–Hilliard equations based on a microforce balance, Physica D 92 (1996) 178–192. M.P. Anderson, D.J. Srolovitz, G.S. Grest, P.S. Sahni, Computer simulation of grain growth-I. kinetics, Acta Met. 32 (1984) 783–791. M.P. Anderson, D.J. Srolovitz, G.S. Grest, P.S. Sahni, Computer simulation of recrystallization-I. Homogeneous nucleation and growth, Acta Met. 34 (1986) 1833–1845. M.E. Gurtin, Thermomechanics of Evolving Phase Boundaries in the Plane, Oxford Press, Oxford, 1993. F.J. Humphreys, M. Hatherly, Recrystallization and Related Annealing Phenomena, Pergamon Press, New York, 1995. L. Bragg, J.F. Nye, A dynamical model of a crystal structure, Proc. Roy. Soc. London 190 (1947) 474–481. D.A. Porter, K.E. Easterling, Phase Transformations in Metals and Alloys, Chapman and Hall, London, 1993. B.D. Coleman, W. Noll, The thermodynamics of elastic materials with heat conduction and viscosity, Arch. Rational Mech. Anal. 13 (1963) 245–261. E. Fried, Correspondence between a phase-field theory and a sharp-interface theory for crystal growth, Cont. Mechs. Thermo., 1996, in press. E. Fried, M.E. Gurtin, A phase-field theory for solidification based on a general anisotropic sharp-interface theory with interfacial energy and entropy, Physica D 72 (1996) 287–308. J. Rubinstein, P. Sternberg, J. Keller, Fast reaction, slow diffusion, and curve shortening, SIAM J. Appl. Math. 49 (1989) 116–133. P. De Mottoni, M. Schatzman, Development of surfaces in RN , Proc. Roy. Soc. Edinberg A 116 (1990) 207–220. L.C. Evans, H.M. Soner, P. Souganidis, Phase transitions and generalized motion by mean curvature, Comm. Pure Appl. Math. 45 (1992) 1097–1123. L. Bronsard, R. Kohn, Motion by mean curvature as the singular limit of Ginzburg–Landau dynamics, J. Diff. Eqts. 90 (1991) 211–237. L. Bronsard, F. Reitich, On three-phase boundary motion and the singular limit of a vector-valued Ginzburg–Landau equation, Arch. Rational Mech. Anal. 124 (1993) 355–379. I. Steinbach, F. Pezolla, B. Nestler, M. Seebelberg, R. Prieler, G.J. Schmitz, J.L.L. Rezende, A phase-field concept for multi phase systems, Physica D (1996) 35–147. H. Garcke, B. Nestler, B. Stoth, A multi phase-field concept: numerical simulations of moving phase boundaries and multiple junctions, SIAM J. Appl. Math., 1998, submitted for publication. A.A. Wheeler, G.B. McFadden, On the notion of a ζ -vector and a stress tensor for a general class of anisotropic diffuse interface models, Proc. Roy Soc. London A453 (1997) 1611–1630.