Dynamics for Systems of Screw Dislocations

DYNAMICS FOR SYSTEMS OF SCREW DISLOCATIONS TIMOTHY BLASS, IRENE FONSECA, GIOVANNI LEONI, MARCO MORANDOTTI Abstract. The goal of this paper is the analytical validation of a model of Cermelli and Gurtin [12] for an evolution law for systems of screw dislocations under the assumption of antiplane shear. The motion of the dislocations is restricted to a discrete set of glide directions, which are properties of the material. The evolution law is given by a “maximal dissipation criterion”, leading to a system of differential inclusions. Short time existence, uniqueness, cross-slip, and fine cross-slip of solutions are proved. 1. Introduction Dislocations are one-dimensional defects in crystalline materials [27]. Their modeling is of great interest in materials science since important material properties, such as rigidity and conductivity, can be strongly affected by the presence of dislocations. For example, large collections of dislocations can result in plastic deformations in solids under applied loads. In this paper we study the motion of screw dislocations in cylindrical crystalline materials using a continuum model introduced by Cermelli and Gurtin [12]. One of our main contributions is the analytical validation to this model by proving local existence and uniqueness of solutions to the equations of motions for a system of dislocations. In particular, we prove rigorously the phenomena of cross-slip and fine cross-slip. We refer to the work of Armano and Cermelli [4, 11] for the case of a single dislocation. Following the work of Cermelli and Gurtin [12], we consider an elastic body B := Ω × R, where Ω ⊂ R2 is a bounded simply connected open set with C 2,α boundary. The body B undergoes antiplane shear deformations Φ : B → B of the form Φ(x1 , x2 , x3 ) := (x1 , x2 , x3 + u(x1 , x2 )), with u : Ω → R. The deformation gradient F is given by     1 0 0 ∇u 1 0  = I + e3 ⊗ . F := ∇Φ =  0 0 ∂u ∂u 1 ∂x1 ∂x2 (1.1) The assumption of antiplane shear allows us to reduce the three-dimensional problem to a two-dimensional problem. We will consider strain fields h that are defined on the cross-section Ω, taking values in R2 . In the absence of dislocations, the strain h is the gradient of a function, h = ∇u. If dislocations are present, then the strain field is singular at the sites of the dislocations, and in the case of screw dislocations this will be a line singularity. In the antiplane shear setting, this line is parallel to the x3 axis and the screw dislocation is represented as a point singularity on the cross-section Ω. 1 2 TIMOTHY BLASS, IRENE FONSECA, GIOVANNI LEONI, MARCO MORANDOTTI A screw dislocation is characterized by a position z ∈ Ω and a vector b ∈ R3 , called the Burgers vector. The position z ∈ Ω is a point where the strain field fails to be the gradient of a smooth function and the Burgers vector measures the severity of this failure. To be precise, a strain field associated with a system of N screw dislocations at positions Z := {z1 , . . . , zN } ⊂ Ω with corresponding Burgers vectors satisfies the relation B := {b1 e3 , . . . , bN e3 } curl h = N X bi δ zi in Ω (1.2) i=1 ∂h1 2 in the sense of distributions. Here curl h is the scalar curl ∂h ∂x1 − ∂x2 , δx is the Dirac mass at the point x, and the scalar bi is called the Burgers modulus for the dislocation at zi , and in view of (1.2) it is given by ˆ bi = h · t ds, ℓi where ℓi is any counterclockwise loop surrounding the dislocation point zi and no other dislocation points, t is the tangent to ℓi , and ds is the line element. When dislocations are present, (1.1) is replaced with   h F = I + e3 ⊗ . 0 To derive a motion law for the system of dislocations we need to introduce the free energy associated to the system. We work in the context of linear elasticity. The energy density W is given by 1 W (h) := h · Lh 2 where the elasticity tensor L is a symmetric, positive-definite matrix, which, in suitable coordinates, can be written in terms of the Lamé moduli λ, µ of the material as   µ 0 . L := 0 µλ2 We require µ > 0, and the energy is isotropic if and only if λ2 = 1. The energy of a strain field h is given by ˆ W (h(x)) dx, (1.3) J(h) := Ω and the equilibrium equation is div Lh = 0 in Ω. (1.4) Equations (1.2) and (1.4) provide a characterization of strain fields describing screw dislocation systems in linearly elastic materials. To be precise, we say that a strain field h ∈ L2 (Ω; R2 ) corresponds to a system of dislocations at the positions Z with Burgers vectors B if h satisfies  PN curl h = i=1 bi δzi in Ω, (1.5) div Lh = 0 SCREW DISLOCATION DYNAMICS 3 in the sense of distributions. In analogy to the theory of Ginzburg-Landau vortices [6], no variational principle can be associated with (1.5) because the elastic energy of a system of screw dislocations is not finite (see, e.g., [13, 12, 27]), therefore the study of (1.5) cannot be undertaken in terms of energy minimization. Indeed, the simultaneous requirements of finite energy and (1.2) are incompatible, since if curl h = δz0 , z0 ∈ Ω, and if Bε (z0 ) ⊂⊂ Ω, then ˆ |h|2 dx = O(| log ε|). Ω\Bε (z0 ) In the engineering literature (see, e.g., [12, 27]), this problem is usually overcome by regularizing the energy, namely, by replacing the energy J in (1.3) with a new energy Jε obtained by removing small cores of size ε > 0 centered at the dislocations points zi . This allows to obtain finite-energy strains hε as minimizers of Jε . It was shown in [7] that Jε (hε ) = C| log ε| + U (z1 , . . . , zN ) + O(ε), (1.6) where U is the renormalized energy associated with the limiting strain h0 = limε→0 hε , satisfying (1.5). This type of asymptotic expansion was first proved by Bethuel, Brezis, and Hélein in [5] for Ginzburg-Landau vortices. The case of edge dislocations was studied in [13]. Asymptotic expansions of the type (1.6) can also be derived using Γ-convergence techniques (see, e.g., [3, 30] and the references therein for GinzburgLandau vortices, [15, 24, 21] for edge dislocations, and [1, 9, 14, 20, 22, 23, 31] for other dislocations models). Finally, it is important to mention that we ignore here the core energy, that is, the energy contribution proportional to | log ε| in (1.6), which comes from the small cores that were removed to obtain Jε . We refer to [27, 33, 35] for a more detailed discussion of the core energy. The force on a dislocation at zi due to the elastic strain is called the Peach-Köhler force, and is denoted by ji (see [12], [28]). The renormalized energy U is a function only of the positions {z1 , . . . , zN } (and of the Burgers moduli), and it is shown in [7] that its gradient with respect to zi gives the negative of the Peach-Köhler force on zi . Specifically, ˆ {W (h0 )I − h0 ⊗(Lh0 )} n ds, (1.7) ji = −∇zi U = ℓi where ℓi is a suitably chosen loop around zi and n is the outer unit normal to the set bounded by ℓi and containing zi . The quantity W (h0 )I − h0 ⊗(Lh0 ) is the Eshelby stress tensor, see [17, 25]. To study the motion of dislocations it is more convenient to rewrite ji in the form i hX kj (zi ; zj ) + ∇u0 (zi ; z1 , . . . , zN ) (1.8) ji (zi ) = bi JL j6=i (see [7] for a proof of this derivation). Here kj (·; zj ) is the fundamental singular strain generated by the dislocation zj , where kj (x; y) := bj λJT (x − y) , 2π |Λ(x − y)|2 (x, y) ∈ R2 ×R2 , x 6= y, (1.9) 4 TIMOTHY BLASS, IRENE FONSECA, GIOVANNI LEONI, MARCO MORANDOTTI with J :=  0 1 −1 0  , Λ :=  λ 0 0 1  . (1.10) Straightforward calculations show that, for (x, y) ∈ R2 ×R2 , x 6= y, we have divy (L∇y kj (x; y)) = 0, (1.11a) divx (Lkj (x; y)) = 0, (1.11b) and, for (x, y) ∈ R2 ×R2 , curlx kj (x; y) = bj δy (x). (1.11c) Also, for fixed z1 , . . . , zN ∈ Ω, the function u0 (·; z1 , . . . , zN ) is a solution of the Neumann problem  divx (L∇x u0 (x; z1 , . . . , zN )) = 0, x ∈ Ω,
PN (1.12) L ∇x u0 (x; z1 , . . . , zN ) + i=1 ki (x; zi ) · n(x) = 0, x ∈ ∂Ω. The expression of (1.8) contains two contributions accounting for the two different kinds of forces acting on a dislocation when other dislocations are present: the interactions with the other dislocations and the interactions with ∂Ω. The latter balances the tractions of the forces generated by all the dislocations. Indeed, the function ∇u0 (x; z1 , . . . , zN ) represents the elastic strain at the point x ∈ Ω due to the presence of ∂Ω and the dislocations at zi with Burgers moduli bi . For this reason, we refer to ∇u0 (x; z1 , . . . , zN ) as the boundary-response strain at x due to Z. Following [12], we will assume the dislocations will move in the glide direction that maximally dissipates the (renormalized) energy. The set of glide directions, G := {g1 , . . . , gM }, is crystallographically determined and is discrete. When many dislocations are present, the dynamics is non-trivial. Dislocations whose Burgers moduli have the same sign will repel each other, while attraction occurs if the Burgers moduli have opposite signs. This can be seen by investigating (1.8) in the case of two dislocations, and extended to an arbitrary number of dislocations by superposition, since the system (1.5) is linear. In addition, because G a discrete set, the motion need not be continuous with respect to the direction. Cross-slip and fine cross-slip may occur whenever it is more convenient for the system to switch direction, in the former case, or to bounce at a faster and faster time scale between two glide directions, in the latter. In this last situation, macroscopically, a dislocation is able to move along a direction which is not in G, but belongs to the convex hull of two glide directions. We discuss this in more detail in Section 2.5. Since the direction of the motion of dislocations can change discontinuously and may not be uniquely determined, we cannot use the standard theory of ordinary differential equations to study the dynamics. Instead we will use differential inclusions (see [19]). We refer to [2, 8, 29, 34, 36] and the references contained therein for other results on the dynamics of dislocations. In particular, it is important to point out that, due to the discrete set of glide directions and the maximal dissipation criterion introduced in [25], our analysis significantly departs from that of Ginzburg-Landau vortices, where the motion of vortices can be derived from a gradient flow (see the review paper of Serfaty [32], see also [2]). SCREW DISLOCATION DYNAMICS 5 In forthcoming work and in collaboration with Thomas Hudson, we plan to study the behavior of dislocations as they approach the boundary and at collisions. In particular, preliminary results show that dislocations are attracted to the boundary. The structure of the paper is as follows. Section 2 addresses the dynamics for a system of dislocations: a brief introduction on differential inclusion is presented in Subsection 2.1, and the framework for the dynamics is presented in Subsection 2.2. Local existence of the solutions to the dynamics problem is addressed in Subsection 2.3, while Subsection 2.4 deals with local uniqueness of the solution. A description of cross-slip and fine cross-slip is presented in Subsection 2.5, where we give analytic proofs of the scenarios presented in [12]. In Section 3 we discuss the case of multiple dislocations simultaneously exhibiting fine cross-slip and provide numerical simulations of the dynamics. Some special cases are discussed in Section 4, namely the unit disk (Subsection 4.1), the half-plane and the plane (Subsections 4.2, 4.3), and finally the notion of mirror dislocations is introduced in Subsection 4.1. We collect some technical proofs in the appendix. 2. Dislocation Dynamics We now turn our attention to the dynamics of the system Z. As explained in the introduction, the direction of the motion of dislocations can change discontinuously and this motivates its study using differential inclusions. We begin this section with some preliminaries on the theory developed by Filippov [19]. We introduce the setting for dislocation dynamics in Subsections 2.1 and 2.2, and prove local existence and uniqueness in Subsections 2.3 and 2.4, respectively. 2.1. Preliminaries on Differential Inclusions. The theory developed by Filippov [19] provides a notion of solution to an ordinary differential inclusion. Given an interval I and a set-valued function H : D → P(Rd ), where D ⊂ Rd+1 and P(Rd ) is the power set of Rd , a solution on I of the differential inclusion ẋ ∈ H(t, x) (2.1) d is an absolutely continuous function x : I → R such that (t, x(t)) ∈ D and ẋ(t) ∈ H(t, x(t)) for almost every t ∈ I. In order to state a local existence theorem for (2.1), we need to introduce the definition of continuity for a set valued map (see [19]). Given two nonempty sets A, B ⊆ Rd , we recall that the Hausdorff distance between A and B is given by n o dH (A, B) := max sup dist(a, B), sup dist(b, A) . a∈A b∈B Remark 2.1. In the special case in which the sets A and B are cartesian products, that is, A = A1 ×A2 ⊆ Rd1 × Rd2 and B = B1 ×B2 ⊆ Rd1 × Rd2 , we have that dH (A, B) 6 dH (A1 , B1 ) + dH (A2 , B2 ). (2.2) To see this, let a = (a1 , a2 ) ∈ A and fix ε > 0. Then there exist bε1 ∈ B1 and bε2 ∈ B2 such that ||ai − bεi || 6 dist(ai , Bi ) + ε Since bε := (bε1 , bε2 ) ∈ B, we have that for i = 1, 2. dist(a, B) 6 ||a − bε || 6 ||a1 − bε1 || + ||a2 − bε2 || 6 dist(a1 , B1 ) + dist(a2 , B2 ) + 2ε. 6 TIMOTHY BLASS, IRENE FONSECA, GIOVANNI LEONI, MARCO MORANDOTTI Letting ε → 0 and taking the supremum over all a ∈ A, it follows that sup dist(a, B) 6 sup dist(a1 , B1 ) + sup dist(a2 , B2 ) a2 ∈A2 a1 ∈A1 a∈A 6 dH (A1 , B1 ) + dH (A2 , B2 ). By exchanging the roles of A and B, we obtain (2.2). Definition 2.2 (Continuity and Upper Semicontinuity). Given D ⊂ Rd+1 and a set-valued function H : D → P(Rd ), we say that H is continuous if dH (H(yn ), H(y)) → 0 for every y, yn ∈ D such that yn → y. We say that H is upper semicontinuous if sup a∈H(yn ) dist(a, H(y)) → 0 for every y, yn ∈ D such that yn → y. It follows from the definition that any continuous set-valued function is upper semicontinuous. The proof of the following theorem can be found in [19, pg. 77]. Theorem 2.3 (Local Existence). Let D ⊂ Rd+1 be open and let H : D → P(Rd ) be upper semicontinuous, and such that H(t, x) is nonempty, closed, bounded, and convex for every (t, x) ∈ D. Then for every (t0 , x0 ) ∈ D there exist h > 0 and a solution x : [t0 − h, t0 + h] → Rd of the problem ẋ(t) ∈ H(t, x(t)), x(t0 ) = x0 . (2.3) Moreover, if D contains a cylinder C := [t0 − T, t0 + T ] × Br (x0 ), for some r, T > 0, then h ≥ min{T, r/m}, where m := sup(t,x)∈C |H(t, x)|. Next we address uniqueness of solutions to (2.3). We say that right uniqueness holds for (2.3) at a point (t0 , x0 ) if there exists t1 > t0 such that any two solutions to the Cauchy problem (2.3) coincide on the subset of [t0 , t1 ] on which they are both defined. Similarly, we say that left uniqueness holds for (2.3) at a point (t0 , x0 ) if there exists t1 < t0 such that any two solutions to the Cauchy problem (2.3) coincide on the subset of [t1 , t0 ] on which they are both defined. We we say that uniqueness holds for (2.3) at a point (t0 , x0 ) if both left and right uniqueness hold for (2.3) at (t0 , x0 ). Unlike the case of ordinary differential equations, for differential inclusions the question of uniqueness is significantly more delicate We will consider here a very special case. Suppose that V ⊂ Rd is an open set and is separated into open domains V ± by a (d − 1)-dimensional C 2 surface S. Let f : (a, b) × (V \ S) → Rd , and define f ± : (a, b) × V ± → Rd as f ± (t, x) := f (t, x) for x ∈ V ± . Assume that f ± can both be extended in a C 1 way to (a, b) × V , and denote these extensions by b f ± . Define  {f (t, x)} for x ∈ / S, (2.4) H(t, x) := co{b f − (t, x), b f + (t, x)} for x ∈ S, and consider the differential inclusion (2.3). Here for a set E ⊂ Rd we denote by coE the convex hull of E, that is, the smallest convex set that contains E. It can be shown that the function H defined in (2.4) satisfies the conditions of Theorem 2.3, and local existence follows. In the following theorems, we denote by n(x0 ) the unit normal to S at x0 ∈ S directed from V − to V + . The following theorem can be found in [19, pg. 110]. SCREW DISLOCATION DYNAMICS 7 Theorem 2.4 (Local Uniqueness). Let H : (a, b)×V → P(Rd ) be given as in (2.4), where f , V , and S are as above. If (t0 , x0 ) ∈ (a, b) × S is such that b f − (t0 , x0 ) · n(x0 ) > 0 or b f + (t0 , x0 ) · n(x0 ) < 0, then right uniqueness holds for (2.3) at the point (t0 , x0 ). Similarly, if b f − (t0 , x0 ) · n(x0 ) < 0 or b f + (t0 , x0 ) · n(x0 ) > 0, then left uniqueness holds for (2.3) at the point (t0 , x0 ). Next we discuss cross-slip and fine cross-slip. Theorem 2.5 (Cross-Slip; [19] Corollary 1, p.107). Let (t0 , x0 ) ∈ (a, b) × S be such that b f − (t0 , x0 ) · n(x0 ) > 0 and b f + (t0 , x0 ) · n(x0 ) > 0. Then uniqueness holds for (2.3) at the point (t0 , x0 ). Moreover, the unique solution x to (2.3) passes from V − to V + , that is, there exist t1 < t0 < t2 such that x(t) belongs to V − for t ∈ [t1 , t0 ) and to V − for t ∈ (t0 , t1 ]. Similarly, if b f − (t0 , x0 ) · n(x0 ) < 0 and + b f (t0 , x0 ) · n(x0 ) < 0, then uniqueness holds for (2.3) at the point (t0 , x0 ) and the unique solution passes from V + to V − . Theorem 2.6 ([19] Corollary 2, p.108). Let (t0 , x0 ) ∈ (a, b) × S be such that b f − (t0 , x0 ) · n(x0 ) > 0 and b f + (t0 , x0 ) · n(x0 ) < 0. (2.5) Then there exists a ≤ t1 < t0 such that the problem (2.1) admits exactly one solution curve x− with x− (t) ∈ V − for t ∈ (t1 , t0 ) and x− (t0 ) = x0 , and exactly one solution curve x+ with x+ (t) ∈ V + for t ∈ (t1 , t0 ) and x+ (t0 ) = x0 . Lemma 2.7. Assume that the conditions (2.5) hold for (t0 , x0 ) ∈ (a, b) × S. Let x(t) be a solution to ẋ = b f + (t, x) on an interval [t0 , T ] with x(t0 ) = x0 ∈ S. Then there exists δ > 0 such that x(t) ∈ V − ∩ U for t ∈ (t0 , t0 + δ). Similarly, if ẋ = b f − (t, x) on an interval [t0 , T ] with x(t0 ) = x0 ∈ S, then there exists δ > 0 such that x(t) ∈ V + ∩ U for t ∈ (t0 , t0 + δ). Proof. Let h := min{−b f + (t0 , x0 ) · n(x0 ), b f − (t0 x0 ) · n(x0 )}. Then h > 0 by hypoth± esis, and therefore, by continuity of b f and n, there exist neighborhoods I0 and U0 of t0 and x0 , respectively, such that b f + (t, x) · n(x̃) < − 12 h and b f − (t, x) · n(x̃) > 21 h for (t, x) ∈ I0 × U0 and x̃ ∈ U0 ∩ S. We can write S locally as the graph of a function. Denoting points x = (ξ, y) ∈ Rd−1 ×R, there is r > 0 such that we can write (without loss of generality) S ∩ Br (x0 ) = {(ξ, y) ∈ Br (x0 ) : y = Φ(ξ)} for some Φ of class C 2 . The sets V ± are locally defined as V + ∩ Br (x0 ) = {(ξ, y) ∈ Br (Z0 ) : y > Φ(ξ)} and V − ∩ Br (x0 ) = {(ξ, y) ∈ Br (Z0 ) : y < Φ(ξ)}. By rotating the coordinate axes, if necessary, we can assume that the tangent hyperplane to S at x0 is {(ξ, y) : y = 0}, so that ∇Φ(ξ 0 ) = 0, where x0 = (ξ 0 , y0 ). Then the unit normal to S at x0 is n(x0 ) = n(ξ 0 , Φ(ξ 0 )) = (0, 1). Consider the solution to ẋ = b f + (t, x) with x(t0 ) = x0 . Since x is continuous, there is δ1 > 0 such that x(t) ∈ U0 for t ∈ (t0 , t0 +δ1 ), and in this interval it satisfies ´t + x(t) = x0 + t0 b f (s, x(s))ds. Hence, ˆ t h b f + (s, x(s)) · n(x0 ) ds < y0 − (t − t0 ). (2.6) y(t) = x(t) · n(x0 ) = x0 · n(x0 ) + 2 t0 Writing x(t) = (ξ(t), y(t)), we have x(t) · n(x0 ) = y(t). Additionally, Φ(ξ(t)) = Φ(ξ(t0 )) + ∇Φ(ξ(t0 )) · (ξ(t) − ξ(t0 )) + o(t − t0 ) = y0 + o(t − t0 ). Therefore, (2.6) 8 TIMOTHY BLASS, IRENE FONSECA, GIOVANNI LEONI, MARCO MORANDOTTI implies there is δ < δ1 such that y(t) < Φ(ξ(t)) − h (t − t0 ) + o(t − t0 ) < Φ(ξ(t)) 2 for t ∈ (t0 , t0 + δ). Thus, x(t) = (ξ(t), y(t)) ∈ V − ∩ Br (x0 ) for t ∈ (t0 , t0 + δ). The proof of the result for solutions to ẋ = b f − (t, x) is similar.  Corollary 2.8 (Fine Cross-Slip). Assume that the conditions (2.5) hold for (t0 , x0 ) ∈ (a, b) × S. Then there exist δ > 0 and a unique solution x defined on [t0 , t0 + δ) to the initial value problem (2.3) that is confined to S. Proof. Existence and uniqueness are consequences of Theorems 2.3 and 2.4. Let T be the maximal existence time provided by Theorem 2.3. As in the proof of Lemma 2.7, there are neighborhoods I0 and U0 of t0 and f − (t, x) · n(x̃) > 21 h for x0 , respectively, such that b f + (t, x) · n(x̃) < − 21 h and b + b (t, x) ∈ I0 × U0 and x̃ ∈ U0 ∩ S, with h = min{−f (t0 , x0 ) · n(x0 ), b f − (t0 x0 ) · n(x0 )}. By continuity of x(t), there exists a δ > 0 such that x(t) ∈ U0 for t ∈ (t0 , t0 + δ). Suppose there is t1 ∈ (t0 , t0 + δ) such that x(t1 ) ∈ / S. Without loss of generality, we can assume x(t1 ) ∈ V + , and we define s1 := sup{s ∈ [t0 , t1 ) : x(s) ∈ / V + }, i.e., s1 is the last time x(t) belongs to S before entering V + and remaining in V + for t ∈ (s1 , t1 ]. It follows that x(t) solves ẋ = b f + (t, x) on [s1 , t1 ] with x(s1 ) ∈ S. Since the hypotheses of Lemma 2.7 are satisfied, there is a unique solution to ẋ = b f + (t, x) − on [s1 , s1 + δ̂] for some δ̂ > 0 , where x(t) ∈ V for t ∈ (s1 , s1 + δ̂). This contradicts the fact that x(t) ∈ V + on [s1 , t1 ]. We conclude that x(t) ∈ S for t ∈ [t0 , t0 +δ).  Remark 2.9. In view of Corollary 2.8, the velocity field ẋ is tangent to S, therefore it must be orthogonal to n(x), for x ∈ S. Moreover, by (2.4), ẋ belongs to co{b f − (t, x), b f + (t, x)}, and so, ẋ = f 0 (t, x) ∈ H(t, x), where and α = α(t, x) ∈ (0, 1) is given by α= since f 0 (t, x) · n(x) = 0. f 0 (t, x) := αb f + (t, x) + (1 − α)b f − (t, x) b f − (t, x) · n(x) , b f − (t, x) · n(x) − b f + (t, x) · n(x) 2.2. Setting for the Dynamics. We now turn our attention to the dynamics of the system Z. We will neglect inertia and any external body forces, and consider only the Peach-Köhler force ji as given in (1.8). Recall that a screw dislocation is a line in a three-dimensional cylindrical body B, and is represented by a point in the cross-section Ω. The motion of dislocations (often called dislocation glide) in crystalline materials is restricted to a discrete set of crystallographic planes called glide planes, which are spanned by e3 and vectors g called glide directions, determined by the lattice structure of that material. We will consider the glide directions as a fixed finite collection of unit vectors in R2 , denoted by G := {g1 , . . . , gM } ⊂ S 1 , SCREW DISLOCATION DYNAMICS 9 with the requirement that if g ∈ G then −g ∈ G. The dislocation glide is restricted to the directions in G, so the equation of motion for zi has the form żi = Vi gi , gi ∈ G and Vi is a scalar velocity. In [12] motion laws are proposed, where a variable mobility M (g) and Peierls force F (g) are incorporated to obtain equations of the form żi = M (gi )[max{ji · gi − P (gi ), 0}]p gi , (2.7) with the exponent p > 0 allowing for various “power-law kinetics”. The mobility function M favors some directions of dislocation glide. The Peierls force, P > 0, is a threshold force, acting as a static friction. If the Peach-Köhler force along gi is below the threshold, then the dislocation will not move. Glide initiates when ji · gi > P (gi ). In this paper we will assume the simplest form of linear kinetics (p = 1) with vanishing Peierls force (P ≡ 0) and isotropic mobility (M ≡ 1). Thus (2.7) takes the form żi = (ji (zi ) · gi )gi for gi ∈ G, (2.8) where we recall that ji (zi ) = bi JL hX j6=i i kj (zi ; zj ) + ∇u0 (zi ; z1 , . . . , zN ) , (2.9) with kj and u0 given in (1.9) and (1.12), respectively. Remark 2.10. The formula (2.9) gives the force on the dislocation at zi , and it shows that, as a function of zi , the force ji is smooth in the interior of Ω \ {z1 , . . . , zi−1 , zi+1 , . . . , zN }. That is, provided zi is not colliding with another dislocation or with ∂Ω, then the force is given by a smooth function. Of course, ji depends on the positions of all the dislocations, and the same reasoning applies to ji as a function of any zj . Following the model presented in [12], the choice of glide direction in (2.8) is determined by a maximal dissipation inequality for dislocation glide. This means that the direction of motion of zi is the glide direction that is most closely aligned with ji . Thus, since ji is determined by all the dislocations z1 , . . . , zN , and since G is discrete, the selection of the glide direction gi ∈ G depends in a discontinuous fashion on the dislocations positions. To stress this fact, we will often write gi = gi (z1 , . . . , zN ), i ∈ {1, . . . , N }. We note that, at any point where zi (t) is differentiable and where (2.8) is satisfied, we have żi = −(∇zi U · gi )gi (see (1.7)), and the energy dissipation inequality N N X X d (∇zi U · gi )2 6 0 ∇zi U · żi = − U (z1 , . . . , zN ) = dt i=1 i=1 (2.10) holds. The dissipation in (2.10) is maximal when gi maximizes {ji · g | g ∈ G}. Note, however, that when there is more than one glide direction g that maximizes ji · g, then (2.8) becomes ill-defined . This leads us to consider differential inclusions in place of differential equations. The problem consists in solving the system of differential inclusions  żℓ ∈ Fℓ (Z), zℓ (0) = zℓ,0 , 10 TIMOTHY BLASS, IRENE FONSECA, GIOVANNI LEONI, MARCO MORANDOTTI where Z := (z1 , . . . , zN ) belong to Ω N ⊂R and Z0 := (z1,0 , . . . , zN,0 ) 2N and, for ℓ = 1, . . . , N , n o ′ Fℓ (Z) := (jℓ (Z) · g) g : g ∈ arg max {j (Z) · g } . ℓ ′ (2.11) Gℓ (Z) := arg max {jℓ (Z) · g′ }, ′ (2.12) g ∈G Setting g ∈G the vectors g ∈ Gℓ (Z) represent the glide directions closest to jℓ (Z) (see [12]), that is, jℓ (Z) · g > jℓ (Z) · g′ , for all g′ ∈ G. (2.13) We are interested in the physically realistic case where the span of the glide directions is all of R2 , otherwise dislocations are restricted to one-dimensional motion and cannot abruptly change direction. Therefore, we assume that span(G) = R2 . (2.14) When jℓ (Z) 6= 0, the set Fℓ can either contain a single element, which we will call gℓ (Z), or two distinct elements, denoted by gℓ− (Z) and gℓ+ (Z), and in this case jℓ (zℓ ) is the bisector of the angle formed by gℓ− and gℓ+ . Remark 2.11. Notice that if jℓ (Z) = 0, then any glide direction g ∈ G satisfies (2.13) and therefore Gℓ (Z) = G. In view of the comments above, we   {0} Fℓ (Z) = {(jℓ (Z) · gℓ (Z)) gℓ (Z)}   {(jℓ (Z) · gℓ± (Z)) gℓ± (Z)} and the problem becomes ( have if jℓ (Z) = 0, if jℓ (Z) 6= 0 and Gℓ (Z) = {gℓ (Z)}, if jℓ (Z) = 6 0 and Gℓ (Z) = {gℓ± (Z)}, (2.15) Ż ∈ F (Z), Z(0) = Z0 , (2.16) F (Z) := F1 (Z)× · · · ×FN (Z) ⊂ R2N . (2.17) where The domain of the set-valued function F must be chosen in such a way that the forces jℓ (Z) are well-defined, and so collisions must be avoided. We denote by Πjk := {Z ∈ ΩN : zj = zk , j 6= k} (2.18) the set where dislocations zj and zk collide, and we define the domain of F to be [ D(F ) := ΩN \ Πjk . (2.19) j<k Recall that the force ji is not defined for zℓ ∈ ∂Ω. Since Ω is open, boundary collisions are also excluded from D(F ). SCREW DISLOCATION DYNAMICS 11 2.3. Local Existence. Following Section 2.2, and in view of (2.16) and (2.17), we consider the differential inclusion  Ż ∈ co F (Z), (2.20) Z(0) = Z0 . The following lemma, whose proof is given in Section 5.1, shows that the convex hull of F (Z) is given by F̂ (Z) := (co F1 (Z))× · · · ×(co FN (Z)), (2.21) where, by (2.15),   if jℓ (Z) = 0, {0} co Fℓ (Z) = {(jℓ (Z) · gℓ (Z)) gℓ (Z)} if jℓ (Z) 6= 0 and Gℓ (Z) = {gℓ (Z)}, (2.22)   Σℓ (Z) if jℓ (Z) 6= 0 and Gℓ (Z) = {gℓ± (Z)}, with Σℓ (Z) the segment of endpoints (jℓ (Z)·gℓ− (Z)) gℓ− (Z) and (jℓ (Z)·gℓ+ (Z)) gℓ+ (Z). Lemma 2.12. Let Fℓ (Z) be defined as in (2.11) for ℓ = 1, . . . , N , and let F (Z) be as in (2.17).Then co F (Z) = F̂ (Z), where F̂ (Z) is defined in (2.21). Lemma 2.12 is useful for understanding the dynamics in Ω rather than in ΩN . Each zi moves in some direction gi ∈ G, unless the arg max in (2.12) is multivalued, in which case zi moves in a direction belonging to the convex hull of gi+ and gi− . Lemma 2.12 makes this precise and validates the use of (2.20) as our model for dislocation motion. Lemma 2.13. Let D(F ) be defined in (2.19). Then the set-valued map F : D(F ) → P(R2N ) defined in (2.17) is continuous (according to Definition 2.2). Proof. Let Z, Zn ∈ D(F ) be such that Zn → Z as n → ∞. In view of Remark 2.1, it suffices to show that for every ℓ ∈ {1, . . . , N }, dH (Fℓ (Zn ), Fℓ (Z)) → 0 as n → ∞. Fix ℓ ∈ {1, . . . , N }. We consider the two cases jℓ (Z) = 0 and jℓ (Z) 6= 0. If jℓ (Z) = 0, then by (2.15) Fℓ (Z) = {0}. In turn, again by (2.15) the continuity of jℓ (cf. Remark 2.10 and (2.19)), dH (Fℓ (Zn ), 0) 6 ||jℓ (Zn )|| → 0 as n → ∞. If jℓ (Z) 6= 0, then, again by continuity of jℓ , jℓ (Zn ) 6= 0 for all n > n̄, for some n̄ ∈ N. Taking n̄ larger, if necessary, we claim that gℓ− (Zn ), gℓ+ (Zn ) ∈ {gℓ− (Z), gℓ+ (Z)} for n > n̄. Arguing by contradiction, if the claim fails, since G is finite, there exists e ∈ G \ {gℓ± (Z)} such that gℓ− (Zn ) = e or gℓ+ (Zn ) = e for infinitely many n. By (2.13) and (2.12), jℓ (Zn ) · e > jℓ (Zn ) · g for all g ∈ G and for infinitely many n. Letting n → ∞ and using the continuity of jℓ , it follows that jℓ (Z) · e > jℓ (Z) · g for all g ∈ G, which implies that e ∈ Gℓ (Z), which is a contradiction. Thus the claim holds. In particular, we have shown that Fℓ (Zn ) = {(jℓ (Zn ) · gℓ± (Z))gℓ± (Z)} for n > n̄, hence dH (Fℓ (Zn ), Fℓ (Z)) 6 ||jℓ (Zn ) − jℓ (Z)|| → 0 as n → ∞. This concludes the proof.  Corollary 2.14. Let F : D(F ) → P(R2N ) be defined by (2.17) and (2.19), and consider the set valued map co F (Z), Z ∈ D(G). Then co F (Z) is nonempty, closed, convex for every Z ∈ D(F ), and co F is continuous. 12 TIMOTHY BLASS, IRENE FONSECA, GIOVANNI LEONI, MARCO MORANDOTTI Proof. For all Z ∈ D(F ), the set co F (Z) is nonempty because F (Z) is nonempty. By definition of convexification, co F (Z) is closed and convex. By Lemma 2.13, the set valued map F is continuous, and therefore so is co F (see Lemma 16, page 66 in [19]). This corollary is proved.  Note that co F is not bounded on D(F ) because |zi − zj | and dist(zi , ∂Ω) can become arbitrarily small, and thus ji can become unbounded (see (1.8) and (1.9)). Theorem 2.15 (Local existence). Let Ω ⊂ R2 be a connected open set. Let F : D(F ) → P(R2N ) be defined as in (2.17) and (2.19) with each Fℓ as in (2.15), and let Z0 ∈ D(F ) be a given initial configuration of dislocations. Then there exists a solution Z : [−T, T ] → D(F ) to (2.20), with T ≥ r0 /m0 , where !1/2 N X 2 |jℓ (Z)| . (2.23) 0 < r0 < dist(Z0 , ∂D(F )) and m0 := max Z∈B(Z0 ,r0 ) ℓ=1 Proof. The function F is bounded on the ball B(Z0 , r0 ) ⊂ D(F ). Hence, by Corollary 2.14, the set valued map co F satisfies the conditions of Theorem 2.3 in B(Z0 , r0 ), and thus local existence holds.  Remark 2.16. In view of (2.19) and (2.23), solutions to the problem (2.20) exist as long as dislocations stay away from ∂Ω and do not collide. 2.4. Local Uniqueness. The set where dislocations can move in either of two different glide directions is called ambiguity set and denoted by A. To be precise, we define A := N [ ℓ=1 Aℓ , where Aℓ := {Z ∈ D(F ) : card(Gℓ (Z)) = 2} , (2.24) and Gℓ (Z) is defined in (2.12). On Aℓ the direction of the Peach-Köhler force jℓ bisects two different glide directions that are closest to it. Note that jℓ (Z) 6= 0 for Z ∈ Aℓ , because card(G) > 4 by assumption (2.14) and since g ∈ G implies −g ∈ G. The uniqueness results in Subsection 2.1 can only be applied at points Z0 ∈ A in which the ambiguity set A is locally a (2N − 1)-dimensional smooth surface separating D(F ) into two open sets in a neighborhood of Z0 . In this subsection we show that A is a (2N − 1)-dimensional smooth surface outside of a “singular set” and we estimate the Hausdorff dimension of this set. Lemma 2.17. For all ℓ ∈ {1, . . . , N } the functions jℓ (z1 , . . . , zN ) are analytic on any compact subset of D(F ). Proof. Observe that if a smooth function v satisfies the partial differential equation div (L∇v) = 0 in Ω, then the function w(x1 , x2 ) := v(λx1 , x2 ) satisfies the partial differential equation ∆w = 0 in an open set U . Hence, without loss of generality, we may assume that λ = 1 (i.e. L = µI), so that (1.11a) and (1.12) reduce to ∆y kj (x; y) = 0, (x, y) ∈ R2 ×R2 , x 6= y, and, for fixed z1 , . . . , zN ∈ Ω,  ∆x u0 (x; z1 , . . . , zN ) = 0, PN ∇x u0 (x; z1 , . . . , zN ) · n(x) = − i=1 ki (x; zi ) · n(x), x ∈ Ω, x ∈ ∂Ω. (2.25) (2.26) SCREW DISLOCATION DYNAMICS 13 A solution to (2.26) is given by u0 (x; z1 , . . . , zN ) = ˆ ∂Ω G(x, y) N X i=1 ki (y; zi ) · n(y) ds(y), (2.27) where G is the Green’s function for the Neumann problem. Consider u0 as a function in ΩN +1 ⊂ R2N +2 . Fix Ki ⊂⊂ Ω for i = 0, . . . , N . If (x, Z) ∈ K := K0 × K1 × · · · × KN , then the integrand in (2.27) is uniformly bounded, and we can find the derivatives of u0 with respect to each zi,m by differentiating under the integral sign in (2.27). Using (2.25), (2.26), and (2.27) we have ∆(x,Z) u0 = ∆x u0 + ∆z1 u0 + · · · + ∆zN u0 N ˆ X =0+ G(x, y)∆zi (ki (y; zi ) · n(y)) ds(y) = 0. i=1 ∂Ω Observe that in a small ball around (x, Z) ∈ K, u0 is a C 2 function in each variable because the formula (2.27) has singularities only on the boundary. Since a harmonic C 2 function on an open set is analytic in that set (cf. [18, Chapter 2]), we deduce that u0 is analytic in the interior of ΩN +1 , and thus u0 (zi ; Z) is also analytic (though, possibly no longer harmonic). By (2.9) we have that jℓ is analytic away from the boundary and away from collisions, because in this case each ki (zℓ ; zi ) is harmonic in both zℓ and zi .  Fix Z∗ ∈ Aℓ . There are two maximizing glide directions for zℓ , denoted by and gℓ− (Z∗ ) (i.e. Gℓ (Z∗ ) = {gℓ+ (Z∗ ), gℓ− (Z∗ )}, as defined in (2.12)). For simplicity we will write gℓ± := gℓ± (Z∗ ). Let Bh (Z∗ ) be a ball around Z∗ with radius h > 0 small enough so that Bh (Z∗ ) ⊂ D(F ), and for any Z ∈ Bh (Z∗ ) one of the following three possibilities holds: Gℓ (Z) = {gℓ+ }, Gℓ (Z) = {gℓ− }, or Gℓ (Z) = {gℓ+ , gℓ− }. Such h exists because of the continuity of jℓ and the fact that jℓ (Z∗ ) 6= 0 (cf. the discussion following (2.24)). We denote by g0 ∈ R2 the vector gℓ+ (Z∗ ) g0 := gℓ+ − gℓ− , (2.28) ∗ which is a well-defined constant vector for Z ∈ Bh (Z ) (see the proof of Lemma (2.13)). Note that if ∂ β jℓ (Z∗ ) · g0 6= 0 for some multi-index β = (β1 , . . . , βN ) ∈ NN 0 with |β| = 1, then Aℓ is locally a smooth manifold. With g0 as in (2.28), we define the singular sets Sℓ := {Z ∈ Aℓ : jℓ (Z) · g0 = 0, ∇Z (jℓ (Z) · g0 ) = 0}, ℓ = 1, . . . , N. (2.29) Each Sℓ contains the points where Aℓ could fail to be a manifold, and is an obstruction to uniqueness of solutions to (2.20). We now estimate the Hausdorff dimension of the singular sets. We adapt an argument from [26], which follows [10]; recall that Sℓ ⊂ R2N , ℓ = 1, . . . , N . Lemma 2.18. Let Sℓ be defined as in (2.29). Then dim(Sℓ ) 6 2N − 2. Proof. Fix ℓ ∈ {1, . . . , N } and Z∗ ∈ Aℓ . As in the discussion above, set g0 := gℓ+ − gℓ ∈ R2 \ {0}, where gℓ± are uniquely defined in Bh (Z∗ ) for h > 0 small enough. We will be considering derivatives in all the zi directions except for i = ℓ. For this 2 purpose, we introduce the notations ∆Z b ℓ , ∇Z b ℓ , and DZ b to denote the Laplacian, the ℓ 14 TIMOTHY BLASS, IRENE FONSECA, GIOVANNI LEONI, MARCO MORANDOTTI gradient, and the Hessian with respect to z1 , . . . , zℓ−1 , zℓ+1 , . . . , zN , respectively. We also write Nℓ for the set of multi-indices α such that ∂ α does not contain any derivatives in the zℓ directions, that is, Nℓ := {α ∈ N2N : α = (α1 , . . . , αℓ−1 , 0, αℓ+1 , . . . , αN )}. 0 (2.30) For m > 2 we define fℓm := {Z : jℓ (Z) · g0 = 0, ∂ α (jℓ (Z) · g0 ) = 0 for all α ∈ Nℓ such that |α| < m, M and ∂ α (jℓ (Z) · g0 ) 6= 0 for some α ∈ Nℓ , with |α| = m}, and also fℓ∞ := {Z : jℓ (Z) · g0 = 0, ∂ α (jℓ (Z) · g0 ) = 0 for all α ∈ Nℓ }. M Therefore f∞ Sℓ ⊂ {Z : jℓ (Z) · g0 = 0, ∇Z b ℓ (jℓ (Z) · g0 ) = 0} = Mℓ ∪ f∞ M ℓ [ m>2 (2.31) ! m f Mℓ . By Lemma 5.3 in the appendix, we have that = ∅. m f Let m > 2 and let Z0 ∈ Mℓ . Then there exists β ∈ Nℓ such that |β| = m − 2, and 2 β DZ b (∂ jℓ (Z0 ) · g0 ) 6= 0. ℓ 2 Thus, if we define v(Z) := ∂ jℓ (Z) · g0 , then DZ b ℓ v(Z0 ) is a symmetric matrix that is not identically zero, so it must have at least one non-zero eigenvalue, say λi . β 2 Observe that Trace(DZ b ℓ (∂ jℓ (Z) · g0 ) = 0 because ∆Z b ℓ (jℓ (Z) · g0 ) = b ℓ v(Z)) = ∆Z P 2N −2 2 0. But Trace(DZ k=1 λk , where λk are the eigenvalues, and λi 6= 0, b ℓ v(Z0 )) = and so there is another non-zero eigenvalue, say λj . Define w(Y) := v(RY), where R is a rotation matrix such that   λ1 · · · 0   .. .. D2b w(Y0 ) =  ... , . . β Yℓ 0 −1 ··· λ2N −2 where Y0 := R Z0 . Since λi and λj are different from zero, there are two distinct multi-indices α1 , α2 ∈ Nℓ with |αk | = 1 such that αk ∇Y w(Y0 ) 6= 0, bℓ∂ k = 1, 2. Hence, applying the Implicit Function Theorem to ∂ α1 w and ∂ α2 w, we conclude that M = {Y : ∂ α1 w(Y) = 0, ∂ α2 w(Y) = 0} is a (2N − 2)-dimensional manifold fm ⊂ M, we have that Sℓ is contained in a in a neighborhood of Y0 . Since M ℓ countable union of manifolds with dimension at most 2N − 2.  We proved that the collection of singular points Esing := N [ ℓ=1 Sℓ , with Sℓ defined in (2.29), has dimension at most 2N − 2. Further, each Aℓ is a (2N − 1)-dimensional smooth manifold away from points on Sℓ but, in general, the set A defined in (2.24) will not be a manifold at points Z ∈ Aℓ ∩ Aj for ℓ 6= j. For this reason we need to exclude the set Eint := {Z ∈ R2N : Z ∈ Aℓ ∩ Aj for some ℓ, j ∈ {1, . . . , N }, ℓ 6= j}. (2.32) SCREW DISLOCATION DYNAMICS 15 Uniqueness at points in Eint is significantly more delicate and will be discussed in Section 3. If Z ∈ Aℓ , then jℓ (Z) 6= 0, but it could be that ji (Z) = 0 for some i 6= ℓ. This would mean that the glide direction for zi would not be well-defined at Z, and could cause an obstruction to uniqueness. In view of this, we set Ezero := {Z ∈ D(F ) : jk (Z) = 0 for some k ∈ {1, . . . , N }} . Reasoning as in Lemma 2.18, dim(Ezero ∩{∇jk has rank 0}) 6 2N −2. On the other hand, dim(Ezero ∩ {∇jk has rank 2}) = 2N − 2, by the Implicit Function Theorem. The set Ezero ∩ {∇jk has rank 1} could have dimension at most 2N − 1. For each ℓ ∈ {1, . . . , N } define Iℓ := Aℓ \ (Sℓ ∪ Eint ∪ Ezero ). (2.33) Let Ẑ ∈ Iℓ . Since Ẑ ∈ / Sℓ (see (2.29)), there is an r > 0 so that Br (Ẑ) ∩ Aℓ is a (2N −1)-dimensional smooth manifold, and Aℓ divides Br (Ẑ) into two disjoint, open sets V ± . Since the functions jk are continuous by Lemma 2.17 for all k ∈ {1, . . . , N }, and Ẑ ∈ / Ezero , by taking r smaller, if necessary, we can assume that jk (Z) 6= 0 for all Z ∈ Br (Ẑ) and for all k ∈ {1, . . . , N }. In turn, since Ẑ ∈ / Eint , again by continuity and by taking r even smaller, gk (Z) ≡ gk (Ẑ) for all Z ∈ Br (Ẑ) and for all k 6= ℓ, and gℓ (Z) ≡ gℓ± (Ẑ) for Z ∈ V ± . Let now f : Br (Ẑ) \ Aℓ → R2N , f = (f1 , . . . , fN ), be the function defined by fk (Z) := (jk (Z) · gk (Ẑ))gk (Ẑ) fℓ (Z) := (jℓ (Z) · gℓ± (Ẑ))gℓ± (Ẑ) if k 6= ℓ, if Z ∈ V ± . (2.34) We define f ± as the restrictions of f to V ± , and we extend them smoothly to the fℓ± (Z) := (jℓ (Z) · gℓ± (Ẑ))gℓ± (Ẑ). ball Br (Ẑ) by setting b fk± (Z) := fk (Z) if k 6= ℓ and b Let n(Ẑ) denote the unit normal vector to Aℓ at Ẑ directed from V − to V + . Motions starting in V + will move towards or away from Aℓ according to whether b f + (Ẑ) · n(Ẑ) < 0 or b f + (Ẑ) · n(Ẑ) > 0. Similarly, motions starting in V − will move towards or away from Aℓ according to whether b f − (Ẑ)·n(Ẑ) > 0 or b f − (Ẑ)·n(Ẑ) < 0. We define the set of source points Esrc := {Z ∈ ΩN : Z ∈ Iℓ for some ℓ ∈ {1, . . . , N }, b f + (Z)·n(Z) > 0 and b f − (Z)·n(Z) < 0}. If Ẑ ∈ Esrc there are two solution curves originating at Ẑ, one that moves into V + and one that moves into V − . Thus there is no uniqueness at source points. Theorem 2.19 (Local Uniqueness). Let T > 0 and let Z : [−T, T ] → R2N be a solution to (2.20). Assume that there exist t1 ∈ [−T, T ) and Z1 ∈ Iℓ , for some ℓ ∈ {1, . . . , N }, such that Z(t1 ) = Z1 and b± b f − (Z1 ) · n(Z1 ) > 0 or b f + (Z1 ) · n(Z1 ) < 0, (2.35) ± where f are the extensions of the functions f defined in terms of the function f given in (2.34) with Ẑ = Z1 . Then right uniqueness holds for (2.20) at the point (t1 , Z1 ). Proof. By (2.35), Z0 ∈ / Esrc , therefore, by the previous discussion, the result follows from Theorem 2.4.  16 TIMOTHY BLASS, IRENE FONSECA, GIOVANNI LEONI, MARCO MORANDOTTI Remark 2.20. Existence time is limited by the possibility of collisions between dislocations, that is, |zi − zj | → 0, or between a dislocation and ∂Ω, that is, dist(zi , ∂Ω) → 0. Additionally, uniqueness is limited by possible intersections of Z(t) with Sℓ ∪ Eint ∪ Ezero ∪ Esrc . The ambiguity set A is smooth except possibly on the singular sets Sℓ , which are at most (2N − 2)-dimensional by Lemma 2.18, or points in Eint . 2.5. Cross-Slip and Fine Cross-Slip. We expect to see two kinds of motion at points where the force is not single-valued. If a dislocation point zℓ is moving in the direction gℓ− and the configuration Z = (z1 , . . . , zN ) arrives at a point on Aℓ where gℓ± are two glide directions that are equally favorable to zℓ , then zℓ could abruptly transition from motion along gℓ− to motion along gℓ+ . Such a motion is called cross-slip (see Figure 1). Heuristically, cross-slip occurs when, on one side of Aℓ , the vector field F (see (2.20)) is pointing toward Aℓ , while the other side F is pointing away from Aℓ . If the configuration Z is in the region where F points towards Aℓ , then Z approaches Aℓ and arrives at it in a finite time. The configuration then leaves Aℓ , moving into the region where F points away from Aℓ . z3 z2 V+ G z1 (a) Ω V− A1 Z R2N (b) Figure 1. Cross-slip. The glide directions are G = {±e1 , ±e2 }, where ei is the i-th basis vector. In (a), dislocation z1 ∈ Ω is undergoing cross-slip, switching direction from g1− = e2 to g1+ = e1 , while dislocations z2 and z3 glide normally along directions g2 = e1 and g3 = −e2 , respectively. In (b) the same motion is represented in R2N : the motion of Z changes direction while crossing the surface A1 , where the velocity field is multivalued. (Here, N = 3.) Another possibility is that the vector field F points towards Aℓ on both sides of Aℓ . In this case, at a point on Aℓ , a motion by zℓ in the gℓ+ direction will drive the configuration Z to a region where jℓ is most closely aligned with gℓ− , but then motion by zℓ along gℓ− immediately forces Z to intersect the surface Aℓ again. Motion by zℓ along gℓ− then pushes Z into a region where jℓ is most closely aligned with gℓ+ , which forces Z back to Aℓ . A motion such as this one on a finer and finer scale will appear as motion along the surface Aℓ . Following [12], such a motion is called fine cross-slip. See Figure 2, where the dislocation z1 is undergoing fine cross-slip. In part (a) it is shown how it follows a curve l rather than one of the glide directions g ∈ G. In part (b) the same phenomenon is shown in R2N (N = 3), where the point Z hits A1 and starts moving along it. SCREW DISLOCATION DYNAMICS 17 z3 z2 V l G z1 Ω A1 + V− R2N Z (a) (b) Figure 2. Fine cross-slip. Let G be the same as in Figure 1. In (a), dislocation z1 ∈ Ω is undergoing fine cross-slip, switching direction from g1− = e2 to a curved one which is not in G, while dislocations z2 and z3 glide normally along directions g2 = e1 and e3 = −e2 , respectively. In (b) the same motion is represented in R2N : the motion of Z, after hitting the surface A1 continues on the surface following the tangent direction. (Here, N = 3.) The following theorems formalize the behaviors described above and provide an analytical validation of the notions of cross-slip and fine cross-slip introduced in [12]. We refer to the discussion preceding Theorem 2.19 for the definitions of n(Z) and V ± for Z ∈ Iℓ . Theorem 2.21 (Cross-Slip). Let T > 0 and let Z : [−T, T ] → R2N be a solution to (2.20). Assume that there exist t1 ∈ (−T, T ) and Z1 ∈ Iℓ , for some ℓ ∈ {1, . . . , N }, such that Z(t1 ) = Z1 , b f − (Z1 ) · n(Z1 ) > 0, and b f − (Z1 ) · n(Z1 ) < 0 and b f + (Z1 ) · n(Z1 ) > 0, (2.36) b f + (Z1 ) · n(Z1 ) < 0, (2.37) where f is the function defined in (2.34). Then uniqueness holds for (2.20) at the point (t1 , Z1 ) and the solution passes from V − to V + . Similarly, if then uniqueness holds for (2.20) at the point (t1 , Z1 ) and the solution passes from V + to V − . Proof. Since b f ± are C 1 extensions of f ± := f 2.5. V± , the result follows from Theorem  Theorem 2.22 (Fine Cross-Slip). Let T > 0 and let Z : [−T, T ] → R2N be a solution to (2.20). Assume that there exist t1 ∈ (−T, T ) and Z1 ∈ Iℓ , for some ℓ ∈ {1, . . . , N }, such that Z(t1 ) = Z1 , b f − (Z1 ) · n(Z1 ) > 0 and b f + (Z1 ) · n(Z1 ) < 0, where f is the function defined in (2.34). Then right uniqueness holds for (2.20) at the point (t1 , Z1 ) and there exists δ > 0 such that Z belongs to Aℓ and solves the ordinary differential equation for all t ∈ [t1 , t1 + δ], Ż = f 0 (Z) ∈ co F (Z), where f 0 (Z) := α(Z)b f + (Z) + (1 − α(Z))b f − (Z) 18 TIMOTHY BLASS, IRENE FONSECA, GIOVANNI LEONI, MARCO MORANDOTTI and α(Z) ∈ (0, 1) is defined by α(Z) := b f − (Z) · n(Z) . b f − (Z) · n(Z) − b f + (Z) · n(Z) Proof. The result follows from Corollary 2.8.  Note that the cross-slip and fine cross-slip trajectories that we have described in Theorems 2.21 and 2.22 satisfy the conditions for right uniqueness in Theorem 2.19. Specifically, if (2.36) or (2.37) holds, then (2.35) holds (i.e., Z1 ∈ / Esrc ). 3. More on Fine Cross-Slip In Subsection 2.1 we have discussed uniqueness only in the special case in which f is discontinuous across a (d − 1)-dimensional hypersurface. The case when two or more such (d − 1)-dimensional hypersurfaces meet is significantly more involved and can lead to non-uniqueness of solutions for Filippov systems (see, e.g., [16]). In our setting, this situation arises at points in the set Eint defined in (2.32). Indeed, in Theorem 2.22 we assumed that Z1 does not belong to the intersection of two hypersurfaces (see (2.32) and (2.33)). In this section we study fine cross-slip in the case in which Z1 belongs to Eint . For simplicity, we consider only the case in which only two hypersurfaces intersect at a point. See Figure 3. A3 z2 A1 z3 G z1 (a) Ω R2N Z (b) Figure 3. Simultaneous fine cross-slip. Let G be the same as in Figure 1. In (a), dislocations z1 , z3 ∈ Ω are undergoing fine cross-slip, switching directions from g1− = e2 and g3− = −e1 , respectively, to curved ones l1 , l3 which are not in G, while dislocation z2 glides normally along direction g2 = e1 . In (b) the same motion is represented in R2N : the motion of Z, after hitting A1 ∩ A3 , continues on the intersection of the two surfaces. Assume that there exists Z1 ∈ Aℓ ∩ Ak for k 6= ℓ, with Z1 ∈ / Ai for i 6= k, ℓ and Z1 ∈ / Ezero ∪ Sℓ ∪ Sk . Consider the case of fine cross-slip conditions along both Aℓ and Ak . Specifically, at Z1 , the vectors jℓ (Z1 ) and jk (Z1 ) are well-defined and bisect two maximally dissipative glide directions gℓ± and gk± , respectively. By assumption, the other ji (Z1 ) have uniquely defined maximally dissipative glide directions. By Lemma 2.12, the set-valued vector field has the form co F (Z1 ) = (co F1 (Z1 ), . . . , co FN (Z1 )), where (see (2.22)), co Fi (Z1 ) = Fi (Z1 ) = {(ji (Z1 ) · gi (Z1 ))gi (Z1 )} SCREW DISLOCATION DYNAMICS 19 for i 6= k, ℓ and co Fi (Z1 ) = {si (ji (Z1 )·gi+ (Z1 ))gi+ (Z1 )+(1−si )(ji (Z1 )·gi− (Z1 ))gi− (Z1 ), si ∈ [0, 1]}, for i = k, ℓ. Additionally, there is a ball Bh (Z1 ) ⊂ D(F ) that is separated into two open sets Vℓ± by Aℓ , such that, for Z ∈ Vℓ+ , Fℓ (Z) = {(jℓ (Z1 ) · gℓ+ , (Z1 ))gℓ+ , (Z1 )} and for Z ∈ Vℓ− , Fℓ (Z) = {(jℓ (Z1 ) · gℓ− , (Z1 ))gℓ− , (Z1 )}. Similarly, Bh (Z1 ) is separated into two open sets Vk± by Ak where the corresponding equalities hold. Since we are avoiding singular points, let nℓ (Z1 ) and nk (Z1 ) denote the normals to Aℓ and Ak at Z1 , where ni (Z1 ) points from Vi− to Vi+ for i = k, ℓ. Now, at the intersection of two surfaces, Bh is divided into four regions, so there will be four vector fields that will need to satisfy some projection conditions in order for fine cross-slip to occur. For i 6= k, ℓ, set fi (Z) := (ji (Z) · gi (Z))gi (Z) for Z ∈ Bh (Z1 ). Set fk± (Z) := (jk (Z) · gk± (Z))gk± (Z) for Z ∈ Vk± and fℓ± (Z) := (jℓ (Z) · gℓ± (Z))gℓ± (Z) for Z ∈ Vℓ± . By assumption, fk± and fℓ± can be extended in a C 1 way to Bh (Z1 ), we denote these extensions by b fk± and b fℓ± . Define the extended vector fields in Bh (Z1 ), f (+,+) (Z) = (f1 (Z), . . . , b fk+ (Z), . . . , b fℓ+ (Z), . . . , fN (Z)), f (+,−) (Z) = fℓ− (Z), . . . , fN (Z)), (f1 (Z), . . . , b fk+ (Z), . . . , b f (−,+) (Z) = (f1 (Z), . . . , b fk− (Z), . . . , b fℓ+ (Z), . . . , fN (Z)), f (−,−) (Z) = (f1 (Z), . . . , b fk− (Z), . . . , b fℓ− (Z), . . . , fN (Z)). (3.1a) (3.1b) (3.1c) (3.1d) The fine cross-slip conditions are that the surfaces Ak and Aℓ are attracting at Z1 , so that f (+,+) (Z1 ) · nk (Z1 ) < 0, f (+,−) (Z1 ) · nk (Z1 ) < 0, f (−,+) f (−,−) f (+,+) (Z1 ) · nℓ (Z1 ) < 0, (3.2a) f (+,−) (Z1 ) · nℓ (Z1 ) > 0, (3.2b) (Z1 ) · nℓ (Z1 ) < 0, (3.2c) (Z1 ) · nℓ (Z1 ) > 0. (3.2d) (Z1 ) · nk (Z1 ) > 0, f (−,+) (Z1 ) · nk (Z1 ) > 0, f (−,−) By taking h smaller, if necessary, we can assume that Z ∈ / Ai for i 6= k, ℓ, that Z∈ / Ezero ∪ Sℓ ∪ Sk , and that (3.2a)-(3.2d) continue to hold for all Z ∈ Bh (Z1 ). We now show that the only possible motion is along the intersection Ak ∩ Aℓ . Theorem 3.1. Let T > 0 and let Z : [−T, T ] → R2N be a solution to (2.20). Assume that there exist t1 ∈ [−T, T ) and Z1 as above such that Z(t1 ) = Z1 . Then there exists δ > 0 such that Z is unique in [t1 , t1 + δ] and Z(t) belongs to Ak ∩ Aℓ for all t ∈ [t1 , t1 + δ]. Proof. Step 1. Since Z(t1 ) = Z1 , by continuity we can find t2 > t1 such that Z(t) ∈ Bh (Z1 ) for all t ∈ [t1 , t2 ]. We claim that Z(t) belongs to Ak ∩ Aℓ for all t ∈ [t1 , t2 ]. Indeed, suppose by contradiction that there exists t3 ∈ [t1 , t2 ] such that Z leaves Ak , that is, Z(t3 ) ∈ Vk+ (the case of Vk− is similar, as well as the case of leaving Aℓ and going into Vℓ± ). Define τ1 := sup{s ∈ [t1 , t3 ) : Z(s) ∈ / Vk+ }, which is the last time Z was in Ak before entering and remaining in Vk+ . Case 1. Suppose that Z(τ1 ) ∈ / Aℓ . Then Z(τ1 ) belongs to either Vℓ+ or Vℓ− . Without loss of generality, we assume that Z(τ1 ) ∈ Vℓ+ . Since Z(τ1 ) ∈ Ak by 20 TIMOTHY BLASS, IRENE FONSECA, GIOVANNI LEONI, MARCO MORANDOTTI definition, and it does not belong to any other Ai , only the k-th component of the force is double-valued at Z(τ1 ). Thus, Z(τ1 ) is a point satisfying the hypotheses of Theorem 2.22 because f (+,±) (Z(τ1 )) · nk (Z(τ1 )) < 0. Therefore there is δ > 0 such that Z(t) ∈ Ak for t ∈ [τ1 , τ1 + δ], which contradicts the definition of τ1 . Case 2. By Case 1, Z(τ1 ) ∈ Aℓ . We claim that Z(t) ∈ Aℓ for all t ∈ [τ1 , t3 ]. (3.3) If (3.3) fails, then there is t4 ∈ (τ1 , t3 ] such that Z(t4 ) ∈ / Aℓ , and so Z(t4 ) is in Vℓ+ ∪ Vℓ− . Without loss of generality, assume Z(t4 ) ∈ Vℓ+ , and define τ2 := sup{s ∈ [τ1 , t4 ] : Z(s) ∈ / Vℓ+ }. which is the last time Z was in Aℓ . If τ2 > τ1 , then Z(τ2 ) ∈ Aℓ and Z(τ2 ) ∈ / Ak because Z(t) ∈ Vk+ on [τ1 , t4 ]. Hence Z(τ2 ) is a point that satisfies the hypotheses of the fine cross-slip theorem because f (+,±) (Z(τ2 )) · n(Z(τ2 )) < 0, and so there is δ > 0 such that Z(t) ∈ Aℓ for t ∈ [τ2 , τ2 + δ]. This contradicts the definition of τ2 . Therefore τ2 = τ1 , Z(τ2 ) ∈ Ak ∩ Aℓ , and Z(t) ∈ Vk+ ∩ Vℓ+ for t ∈ (τ2 , t1 ]. We deduce that Z satisfies Ż = f (+,+) (Z) on (τ2 , t3 ], thus ˆ t Z(t) = Z(τ2 ) + f (+,+) (Z(s)) ds, t ∈ [τ2 , t4 ]. (3.4) τ2 Applying the argument from the proof of Corollary 2.8, we can reach a contradiction as follows. Locally Ak is given by the graph of a function, so without loss of generality we can write Ak ∩ Br (Z(τ2 )) = {Z = (ξ, y) ∈ Br (Z(τ2 )) : y = Φ(ξ)} for a function Φ of class C 2 . Denote Z(τ2 ) as (ξ 0 , y0 ) = Z(τ2 ). Without loss of generality, we can assume that ∇Φ(ξ 0 ) = 0 so nk (Z(τ2 )) = (0, 1) and Vk+ ∩ Br (Z(τ2 )) = {(ξ, y) ∈ Br (Z(τ2 )) : y > Φ(ξ)}, Vk− ∩ Br (Z(τ2 )) = {(ξ, y) ∈ Br (Z(τ2 )) : y < Φ(ξ)}. From (3.2a), which holds in Bh (Ẑ), we have the same condition as (3.2a) at the point Z(τ2 ) ∈ Bh (Ẑ). Set h := −f (+,+) (Z(τ2 )) · n(Z(τ2 )) > 0, and find a neighborhood V e > 1 h for Z ∈ V and Z e ∈ V ∩ Ak . From (3.4) of Z(τ2 ) such that −f (+,+) (Z) · n(Z) 2 we have ˆ t f (+,+) (Z(s)) · n(Z(τ2 )) ds Z(t) · n(Z(τ2 )) = Z(τ2 ) · n(Z(τ2 )) + τ2 t − τ2 < Z(τ2 ) · n(Z(τ2 )) − h. 2 Using n(Z(τ2 )) = (0, 1) and writing Z(t) = (ξ(t), y(t)), we obtain t − τ2 h. (3.5) 2 But Φ(ξ(t)) = Φ(ξ(τ2 )) + 0 + o(t − τ2 ) = Φ(ξ 0 ) + o(t − τ2 ) = y0 + o(t − τ2 ). So (3.5) becomes y(t) < y0 − y(t) < y0 − t − τ2 t − τ2 h = Φ(ξ(t)) − h + o(t − τ2 ) < Φ(ξ(t)) 2 2 for 0 < t − τ2 < δ for some δ > 0. This implies that Z(t) ∈ Vk− for t ∈ (τ2 , τ2 + δ], which contradicts the fact that Z(t) ∈ Vk+ , for t ∈ (τ2 , t4 ]. SCREW DISLOCATION DYNAMICS 21 Thus, we have shown that (3.3) holds. Since Z(t3 ) ∈ Vk+ by the definition of τ1 , Z(t) ∈ Vk+ for all t ∈ (τ1 , t3 ]. This, together with (3.3) and Theorem 2.22, implies that Ż(t) = f (+,0) (Z(t)) = α(Z(t))f (+,+) (Z(t)) + (1 − α(Z(t)))f (+,−) (Z(t)) for t ∈ (τ1 , t3 ], where α(Z(t)) = f (+,−) (Z(t)) · nℓ (Z(t)) . f (+,−) (Z(t)) · nℓ (Z(t)) − f (+,+) (Z(t)) · nℓ (Z(t)) (3.6) Using the same argument with Φ as above (starting from (3.4)) and the fact that f (+,0) (Z(t)) · nk (Z(t)) < 0, we conclude that Z(t) ∈ Vk− , yielding a contradiction. This shows that t3 cannot exist, and, in turn, that Z(t) ∈ Ak ∩ Aℓ for all t ∈ [t1 , t2 ]. Step 2. In view of the previous step, we have that Z(t) ∈ Ak ∩Aℓ for all t ∈ [t1 , t2 ]. In turn, Ż(t) · nk (Z(t)) = 0 and Ż(t) · nℓ (Z(t)) = 0 for L1 -a.e. t ∈ [t1 , t2 ]. Moreover, Ż(t) ∈ coF (Z(t)) for L1 -a.e. t ∈ [t1 , t2 ]. Finally, since Z(t) ∈ Bh (Z1 ) for all t ∈ [t1 , t2 ], we have that (3.2a)-(3.2d) hold with Z(t) in place of Z1 for all t ∈ [t1 , t2 ] and Z(t) ∈ / Ai for i 6= k, ℓ and Z(t) ∈ / Ezero ∪ Sℓ ∪ Sk for all t ∈ [t1 , t2 ]. Hence, we can apply Lemma 5.4 in the appendix with Z(t) in place of Z1 to conclude that Ż(t) is uniquely determined for L1 -a.e. t ∈ [t1 , t2 ]. This concludes the proof.  Remark 3.2. The argument in Step 1 does not rely on the fact that only two surfaces are intersecting. Any number of surfaces would be treated the same way, but with more subcases for showing the motion does not leave the intersection. However, establishing uniqueness would require a different argument from the one in Lemma 5.4. 3.1. Identification of Aℓ with a curve in Ω. Each dislocation point zℓ moves in Ω ⊂ R2 according to żℓ = (jℓ (Z) · gℓ (Z))gℓ (Z), but the dynamics is understood in the larger space ΩN ⊂ R2N . If zℓ is exhibiting fine cross-slip, then zℓ moves along a curve that is not a straight line parallel to a glide direction. In this section, we describe the fine cross-slip motion of zℓ in Ω in terms of the dynamics of the system in ΩN . That is, we will examine fine cross-slip for zℓ , which occurs when the solution curve Z(t) ∈ ΩN lies inside the set Aℓ , via a projection into Ω. The projection zℓ (t) of Z(t) onto its ℓ-th components is the fine cross-slip curve in Ω, with zℓ (t) = (zℓ,1 (t), zℓ,2 (t)) for t ∈ [t0 , t1 ]. Recall that Aℓ is locally given by the zero-level set of the function jℓ · g0 . Specifically, if Z0 = (z0,1 , . . . , z0,N ) ∈ Aℓ , then there exists r > 0 such that Aℓ ∩ Br (Z0 ) = {Z ∈ ΩN : jℓ (Z) · gℓ0 (Z) = 0}, gℓ0 (Z) gℓ+ gℓ− where = − given (up to a sign) by (3.7) is constant in Br (Z0 ). Additionally, the normal to Aℓ is
∇ jℓ (Z) · gℓ0 (Z) ∈ R2N , n := |∇ (jℓ (Z) · gℓ0 (Z))| (3.8) which is assumed to be non-zero in Aℓ ∩ Br (Z0 ). We write n = (n1 , . . . , nN ), with ni ∈ R2 , for i = 1, . . . , N . 22 TIMOTHY BLASS, IRENE FONSECA, GIOVANNI LEONI, MARCO MORANDOTTI Assuming that no other dislocations exhibit fine cross-slip, the fine cross-slip conditions at Z0 ∈ Aℓ are (with the appropriate sign for n)   n · (j1 (Z0 ) · g1 )g1 , . . . , (jℓ (Z0 ) · gℓ+ )gℓ+ , . . . , (jN (Z0 ) · gN )gN < 0,   n · (j1 (Z0 ) · g1 )g1 , . . . , (jℓ (Z0 ) · gℓ− )gℓ− , . . . , (jN (Z0 ) · gN )gN > 0. Note that we dropped the explicit dependence of each gi on Z because they are constant in Br (Z0 ). Thus, since (jℓ (Z0 ) · gℓ+ ) = (jℓ (Z0 ) · gℓ− ),   0 > n · 0, . . . , (jℓ (Z0 ) · gℓ± )gℓ0 , . . . , 0 = (jℓ (Z0 ) · gℓ± )nℓ · gℓ0 . This implies nℓ 6= 0 ∈ R2 , i.e., by (3.8), we have
∂ jℓ (z1 , . . . , zN ) · gℓ0 (z1 , . . . , zN ) 6= 0. ∂zℓ,1 (3.9) Let us write Ž for points in R2N −1 of the form Ž := (z1 , . . . , zℓ−1 , zℓ,2 , zℓ+1 , . . . , zN ), where the zℓ,1 component is omitted. From (3.7) and (3.9), the Implicit Function Theorem yields r1 > 0, r2 ∈ (0, r), and a function ϕ : Br1 (Ž0 ) ⊂ R2N −1 → R, where Ž0 := (z0,1 , . . . , z0,ℓ,2 , . . . , z0,N ), such that ϕ(Ž0 ) = z0,ℓ,1 and Aℓ ∩ Br2 (Z0 ) = {Z ∈ ΩN : zℓ,1 = ϕ(Ž)}. That is, locally, Aℓ is the graph of ϕ. If Z(t) is a solution curve lying in Aℓ ∩Br2 (Z0 ) for t ∈ [t0 , t1 ] with Z(t0 ) = Z0 , then Z(t) = (z1 (t), . . . , ϕ(Ž(t)), zℓ,2 (t), . . . , zN (t)) ∈ Aℓ for t ∈ [t0 , t1 ]. In particular, the projection of Z(t) onto its ℓ-th components gives the fine cross-slip curve zℓ (t) = (zℓ,1 (t), zℓ,2 (t)) = (ϕ(Ž(t)), zℓ,2 (t)), t ∈ [t0 , t1 ]. (3.10) Note that nℓ (Z(t)) is not directly related to the fine cross-slip curve given by (3.10) because nℓ (Z(t)) is not orthogonal to żℓ (t), in general. We have 0 = n(Z(t)) · Ż(t) = so nℓ (Z(t)) · żℓ (t) = − N X i=1 X i6=ℓ ni (Z(t)) · żi (t), ni (Z(t)) · żi (t), and the sum on the right-hand side need not be zero. 3.2. Numerical Simulations. The simulation of (2.20) may be undertaken using standard numerical ODE integrators, provided sufficient care is taken in resolving the evolution near the “ambiguity surfaces” Aℓ . A discrete time step leads to a numerical integration that oscillates back and forth across an attracting ambiguity surface in case of fine cross-slip. On the macro-scale, this appears as fine crossslip since the small oscillations across the surface average out and what remains is motion approximately tangent to Aℓ . To compute the vector field, one must solve the Neumann problem (1.12) at each time step, so a fast elliptic PDE solver is needed in practice. SCREW DISLOCATION DYNAMICS 23 An example is shown in Figures 4 and 5, where we have simulated a system of N = 12 screw dislocations with each Burgers modulus bi = 1 for i = 1, . . . , 12, and where the domain is the unit disk. The integration is done in Ω12 ⊂ R24 , but the graphics depict the path each zi takes in Ω ⊂ R2 . All but one dislocation exhibit normal glide motions, while the dislocation at the center exhibits fine cross-slip, as is visible in Figure 5. In this case, the solution to the Neumann problem is explicit (cf. (4.3)), so it is not difficult to simulate systems with more dislocations and observe more complicate behavior, such as multiple dislocations simultaneously exhibiting fine cross-slip, corresponding to motion along the intersection of multiple ambiguity surfaces in the full space ΩN . The simulation depicted in Figures 4 and 5 was run until a dislocation collided with the boundary. Since all dislocations have positive Burgers moduli, they repel each other, and no collision between dislocations occurs, and the dynamics can be continued until a boundary collision. 0.8 z12 0.6 z11 0.4 z2 0.2 z3 0 z10 z9 z1 ց z8 −0.2 −0.4 z5 z4 −0.6 z6 G z7 −0.8 −1 −0.5 0 0.5 1 Figure 4. The forces are repulsive and the dislocations move mostly along the glide directions G = {±e1 , ±e2 , ± √12 (e1 + e2 )}. All but one (the one at the center) move along a glide direction until one of them hits the boundary. The dislocation in the middle moves along −e1 but then exhibits fine cross-slip. 4. Special Cases In this section we consider some special domains Ω for which the Peach-Köhler force can be explicitly determined (i.e. the solution to the Neumann problem (1.12) is known), specifically the unit disk B1 , the half-plane R2+ , and the plane R2 . The last two cases do not technically fit in our previous discussion, because Ω is unbounded. However, the Neumann problem is well-defined for these settings and we are able to discuss the dislocation dynamics. In what follows we will use the fact that the boundary-response strains generated from each dislocation are “decoupled” in the following sense. Define ui0 as ˆ i G(x, y)Lki (y; zi ) · n(y) ds(y), u0 (x; zi ) := ∂Ω 24 TIMOTHY BLASS, IRENE FONSECA, GIOVANNI LEONI, MARCO MORANDOTTI 0.02 0.05 z10 (0) 0 −0.02 0 z1 (T ) z1 (0) −0.04 z3 (0) z1 (T ) −0.05 z9 (0) z1 (0) −0.06 G −0.08 −0.1 z5 (0) z4 (0) z6 (0) G −0.15 −0.12 −0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 −0.15 −0.1 −0.05 0 0.05 0.1 Figure 5. These plots are magnified views of the motion of z1 . The motion begins at the dot on the right and ends at the square on the left. The motion abruptly begins to fine cross-slip and eventually moves back to a gliding motion as the fine cross-slip motion becomes aligned with −e1 . where G is the Green’s function for the Neumann problem. Then ui0 (·; zi ) solves (1.12) with only one dislocation, i.e.,
 divx L∇x ui0 (x; zi ) = 0,
x ∈ Ω, L ∇x ui0 (x; zi ) + ki (x; zi ) · n(x) = 0, x ∈ ∂Ω. Thus the boundary-response strain at x due to a dislocation at zi with Burgers modulus bi is given by ∇x ui0 (x; zi ), and the total boundary-response strain at x PN due to the system Z is ∇x u0 (x; z1 , . . . , zN ) = i=1 ∇x ui0 (x; zi ). If we consider two dislocations z1 and z2 with Burgers moduli b1 and b2 , respectively, that collide in Ω, then by (1.9) the boundary data in (1.12) satisfies   b2 L(k1 (x; z1 ) + k2 (x; z2 )) · n(x) → L k1 (x; z1 ) + k1 (x; z1 ) · n(x), as z2 → z1 . b1 Notice that k1 (·; z1 ) + (b2 /b1 )k1 (·; z1 ) is the singular strain generated by a single dislocation located at z1 with Burgers modulus b1 + b2 . The same argument applies to an arbitrary number N of dislocation by linearity of (1.12). Thus, unlike the singular strain which becomes infinite if any two dislocations collide in Ω (see (1.9)), the boundary-response strain is oblivious to collisions between dislocations. Although the boundary-response strain is well-defined when dislocations collide with each other, it is not well-defined if a dislocation collides with ∂Ω. 4.1. The Unit Disk. Consider the case Ω = B1 = {x ∈ R2 : |x| < 1} and λ = µ = 1, so that L = I. For z ∈ B1 we define z ∈ B1c to be the reflection of z across the unit circle ∂B1 ,  z  if z ∈ B1 \ {0}, z := |z|2 ∞ if z = 0. For fixed zi ∈ B1 , it can be seen that the function    x2 − z i,2 − bi arctan if z 6= 0, π x1 − z i,1 + |x − zi | ui0 (x; zi ) :=  0 if z = 0 (4.1) SCREW DISLOCATION DYNAMICS satisfies and  ∆x ui0 (x; zi ) = 0, ∇x ui0 (x; zi ) · n(x) = −ki (x; zi ) · n(x), 25 x ∈ B1 , x ∈ ∂B1 , (4.2) ∇x ui0 (x; zi ) = −ki (x; zi ) for all x ∈ B1 . i Note that ∇u0 is singular only at the point x = zi ∈ / B1 . As discussed at the beginning of Section 4, for a system of dislocations given by Z and B, the solution to the Neumann problem (1.12) is given by u0 (x; z1 , . . . , zN ) = N X ui0 (x; zi ) i=1 with ui0 as in (4.1). Thus, combining (2.9) and (4.2), we have jℓ (z1 , . . . , zN ) = bℓ J X i6=ℓ ki (zℓ ; zi ) − N X ! ki (zℓ ; zi ) . i=1 (4.3) Formula (4.3) greatly simplifies numerical simulations of the dislocation dynamics. Without an explicit formula, one must solve the Neumann problem at each timestep. From (4.3), we can see that the boundary of B1 attracts dislocations. If N = 1 and z1 ∈ B1 \ {0}, then j1 (z1 ) = −b1 Jk1 (z1 ; z1 ) = − b21 z1 − z1 b21 z1 = 2π |z1 − z1 |2 2π (1 − |z1 |2 ) since z − z = z(1 − |z|−2 ). Thus, the force is directed radially outward (toward the nearest boundary point to z1 ) and diverges as z1 → ∂B1 . If z1 = 0 then j1 = 0 and z1 will not move. Otherwise, a single dislocation in B1 will be pulled to ∂B1 , and will collide with ∂B1 in a finite time (assuming the glide directions span R2 ). If N > 1, then the other dislocations produce boundary forces that will pull on zℓ in the directions −bℓ bi (zℓ − zi ) for each i. The sets Aℓ as given in (2.24) are smooth, because they are locally given by jℓ · g0 = 0 for a fixed vector g0 (cf. equation (2.28)), and by (4.3), jℓ · g0 is a rational function with singularities only at collision points. 4.2. The Half-Plane. Although the theory developed in this paper only applies to bounded domains, the equation for the Peach-Köhler force (1.8) is still welldefined, provided there is a weak solution to the Neumann problem (1.12). For the special cases of the half-plane and the plane we present an explicit expression for the Peach-Köhler force without resorting to the renormalized energy. Let Ω = R2+ := {x ∈ R2 : x2 > 0} and let λ = µ = 1. The solution to (1.12) is given in terms of the inverse tangent, using a reflected point across ∂R2+ = {x2 = 0}. For all z = (z1 , z2 ) ∈ R2 define z̃ := (z1 , −z2 ). Then for zi ∈ R2+ ,   x2 − z̃i,2 bi (4.4) ui0 (x; zi ) := − arctan π x1 − z̃i,1 + |x − z̃i | satisfies and  ∆x ui0 (x; zi ) = 0, ∇x ui0 (x; zi ) · n(x) = −ki (x; zi ) · n(x), ∇x ui0 (x; zi ) = −ki (x; z̃i ) x ∈ R2+ , x ∈ ∂R2+ , for all x ∈ R2+ . 26 TIMOTHY BLASS, IRENE FONSECA, GIOVANNI LEONI, MARCO MORANDOTTI Again, we have u0 (x; z1 , . . . , zN ) = Peach-Köhler force is jℓ (z1 , . . . , zN ) = bℓ J PN i=1 X i6=ℓ ui0 (x; zi ) with ui0 as in (4.4), and the ki (zℓ ; zi ) − N X ! ki (zℓ ; z̃i ) . i=1 (4.5) From (4.5) it is again not difficult to see that a single dislocation z1 in R2+ with Burgers modulus b1 is attracted to ∂R2+ . As in the case of the disk, the ambiguity set A is smooth except at the intersections of the Aℓ . 4.3. The Plane. The case Ω = R2 and λ = µ = 1 is the simplest case. There is no boundary so u0 ≡ 0 and, by (1.8), the Peach-Köhler force is then X jℓ (z1 , . . . , zN ) = bℓ J ki (zℓ ; zi ). (4.6) i6=ℓ Even though the renormalized energy has not been defined for unbounded domains, in the case of the plane we can formally write jℓ = −∇zℓ U , where, up to an additive constant, N N −1 X X b i bj U (z1 , . . . , zN ) = − log |Λ(zi − zj )|, 2π i=1 j=i+1 with Λ defined in (1.10). In general, it can be difficult to exhibit an example that shows analytically fine cross-slip (though it is regularly observed in numerical simulations). However, in the case Ω = R2 , this can be done with two dislocations as follows. Suppose we have a system of two dislocations Z = (z, w) ∈ R4 with Burgers moduli b1 = −b2 =: b > 0, respectively. Under these assumptions, (4.6) reduces to j1 (z, w) = − b2 z − w = −j2 (z, w). 2π |z − w|2 Assume that the glide directions are along the lines x2 = ±x1 ,     1 1 1 1 G = {±g1 , ±g2 } , g1 := √ , g2 := √ . 1 −1 2 2 (4.7) (4.8) There are two cases of initial conditions Z0 = (z0 , w0 ) with z0 = (z0,1 , z0,2 ), w0 = (w0,1 , w0,2 ) to consider: either z0 and w0 are aligned along a vertical or horizontal line, or they are not. That is, either z0,1 = w0,1 or z0,2 = w0,2 (but not both), or z0,i 6= w0,i for i = 1, 2. We begin by considering the case z0,2 = w0,2 . Let y := z0,2 = w0,2 , and without loss of generality take w0,1 > z0,1 . From (4.7) we have   1 b2 1 = −j2 (Z0 ). (4.9) j1 (Z0 ) = j1 (z0,1 , y, w0,1 , y) = 0 2π w0,1 − z0,1 Since w0,1 − z0,1 > 0, we see that j1 (Z0 ) is aligned with (1, 0) and j2 (Z0 ) is aligned with (−1, 0). Thus, the maximally dissipative glide directions for z are g1 and g2 (see (4.8)) and the maximally √ dissipative glide directions for w are −g1 and −g2 . Define g10 := g1 − g2 = (0, 2) and g20 := −g1 + g2 = −g10 , so that locally, near Z0 , the ambiguity surfaces are A1 ∩ Br (Z0 ) = {Z : j1 (Z) · g10 = 0}, A2 ∩ Br (Z0 ) = SCREW DISLOCATION DYNAMICS 27 {Z : j2 (Z) · g20 = 0} for some small r > 0. From (4.7) we see that j1 (Z) · g10 = 0 if and only if z2 = w2 , and the same holds for j2 (Z) · g20 = 0, so that A1 ∩ Br (Z0 ) = A2 ∩ Br (Z0 ) = {Z = (z, w) ∈ Br (Z0 ) : z2 = w2 }. This is a degenerate situation, since the ambiguity surfaces A1 and A2 coincide locally, and instead of having four vector fields near the intersection, we have two vector fields. That is, the fields f (+,+) and f (−,−) (see (3.1)) are defined on either side of the surface A1 , but since A1 = A2 , there are no regions where the fields f (−,+) or f (+,−) are defined. We choose a sign for the normal to A1 and A2 at Z0 and set 1 (4.10) n := √ (0, 1, 0, −1). 2 Recall the convention that A1 (and A2 ) divides Br (Z0 ) into two regions, V ± − + + and √ n points from√V to V . A point in V is of the form Z0 + εn = (z0,1 , y + ε/ 2, w0,1 , y − ε/ 2), and from (4.7)   b2 1 w0,1√ − z0,1 = −j2 (Z0 + εn), j1 (Z0 + εn) = − 2ε 2π (z0,1 − w0,1 )2 + 2ε2 so g2 is the maximally dissipative glide direction for z, and −g2 is the maximally + dissipative glide direction√for w if Z ∈ V√ . Similarly, a point in V − is of the form Z0 − εn = (z0,1 , y − ε/ 2, w0,1 , y + ε/ 2), and the maximally dissipative glide directions for z and w in this case are g1 and −g1 , respectively. Thus, we have for Z ∈ Br (Z0 ), f (+,+) (Z) := ((j1 (Z) · g2 )g2 , (j2 (Z) · (−g2 ))(−g2 )), f (−,−) (Z) := ((j1 (Z) · g1 )g1 , (j2 (Z) · (−g1 ))(−g1 )). Since j1 (Z) = −j2 (Z) we have f (+,+) (Z) := (j1 (Z) · g2 )(g2 , −g2 ), f (−,−) (Z) := (j1 (Z) · g1 )(g1 , −g1 ). (4.11) 2 From (4.8) and (4.9) we have j1 (Z0 ) · g1 = j1 (Z0 ) · g2 = 2√b 2π (w0,1 − z0,1 )−1 > 0, and from (4.8) and (4.10) we have n · (g2 , −g2 ) = −1 and n · (g1 , −g1 ) = 1. Thus, b2 < 0, n·f (+,+) (Z0 ) = − √ 2 2π(w0,1 − z0,1 ) b2 n·f (−,−) (Z0 ) = √ > 0, 2 2π(w0,1 − z0,1 ) so the fine cross-slip conditions (3.2) are satisfied (there are no conditions for f (+,−) or f (−,+) since locally A1 = A2 ). By (3.6), Ż must be a convex combination of f (+,+) and f (−,−) , Ż = αf (+,+) (Z) + (1 − α)f (−,−) (Z), and the trajectory Z(t) ∈ A1 = A2 for some time interval [0, T ]. Therefore, Z(t) = (z(t), w(t)) = (z1 (t), z2 (t), w1 (t), w2 (t)) and z2 (t) = w2 (t) for t ∈ [0, T ]. From (4.11) and the fact that j1 (Z) · g1 = j1 (Z) · g2 whenever z2 = w2 , we have Ż = α(j1 (Z) · g2 )(g2 , −g2 ) + (1 − α)(j1 (Z) · g1 )(g1 , −g1 ) = b2 (1, 1 − 2α, −1, 2α − 1) . 4π(w1 − z1 ) The condition n · Ż = 0 yields α = 21 , so the equations of motion (3.6) are  b2 1  (+,+) f (Z) + f (−,−) (Z) = (1, 0, −1, 0). Ż = (ż1 , ż2 , ẇ1 , ẇ2 ) = 2 4π(w1 − z1 ) 28 TIMOTHY BLASS, IRENE FONSECA, GIOVANNI LEONI, MARCO MORANDOTTI In particular, ż2 = 0, ẇ2 = 0, and z2 (0) = y = w2 (0), so z2 (t) = y = w2 (t) for t ∈ [0, T ]. The equations for z1 and w1 are easily solved with  1 b2 2 1 1 2 (w0,1 − z0,1 ) − t z1 (t) = − + (z0,1 + w0,1 ) 2 π 2   12 2 b 1 1 2 w1 (t) = + (z0,1 + w0,1 ). (w0,1 − z0,1 ) − t 2 π 2 This implies that the trajectory Z(t) moves on A1 = A2 up to the maximal time T = bπ2 (w0,1 − z0,1 )2 , and z1 (t) increases from z0,1 while w1 (t) decreases from w0,1 , with the two meeting at z1 (T ) = w1 (T ) = 21 (z0,1 + w0,1 ). At this collision, the dynamics are no longer well-defined. If the initial condition has z0 and w0 vertically aligned, then the same analysis applies, but the situation is rotated. If z0 and w0 are not aligned vertically or horizontally, then a regular glide motion occurs until either z1 = w1 or z2 = w2 , and then the above analysis applies. To see this, consider z0 = (z0,1 , z0,2 ) and w0 = (w0,1 , w0,2 ), and without loss of generality, assume that w0,1 > z0,1 and w0,2 > z0,2 (the other cases are similar). In this case b2 j1 (Z0 ) = 2π|z0 − w0 |2  w0,1 − z0,1 w0,2 − z0,2  = −j2 (Z0 ). Since w0,1 − z0,1 > 0 and w0,2 − z0,2 > 0, the maximally dissipative glide directions for j1 and j2 are g1 and −g1 , respectively. Thus, z glides in the g1 direction, so that z1 and z2 increase from z0,1 and z0,2 , while w glides in the −g1 direction, so w1 and w2 decrease from w0,1 and w0,2 . At some time t1 we must obtain either z1 (t1 ) = w1 (t1 ) or z2 (t1 ) = w2 (t1 ). If only one of these equalities holds, we are in the situations described above and fine cross-slip occurs. If both of these equalities hold, then z and w have collided and the dynamics is no longer defined. Remark 4.1 (Mirror Dislocations). A direct inspection of equations (4.3) and (4.5) shows that the force on zℓ in Ω = B1 and Ω = R2+ is the same as the force on zℓ in R2 if one adds N dislocations with opposite Burgers moduli at the points z̄i in the case Ω = B1 , and at z̃i in the case Ω = R2+ , for i = 1, . . . , N . 5. Appendix We collect some technical results that are needed in the proofs from Section 2. 5.1. Proof of Lemma 2.12. Proof of Lemma 2.12. Let Z ∈ R2N be fixed. For simplicity, in this proof we drop the explicit dependence on Z. By (2.15) we can write Fℓ = {pℓ , qℓ }, with SCREW DISLOCATION DYNAMICS 29 pℓ , qℓ ∈ R2 , for all ℓ = 1, . . . , N . By definition, we have    s1 p1 + (1 − s1 )q1       . .. F̂ =  and  , s1 , . . . , sN ∈ [0, 1]     sN pN + (1 − sN )qN        q1 q1 p1     p2   q2    p2        co F = V ∈ R2N : V = α1  .  + α2  .  + . . . + α2N  .  ,   ..   ..   ..     pN pN qN  2N  X αi = 1 . where αi ∈ [0, 1] for all i = 1, . . . , 2N , and  i=1 To see that co F ⊆ F̂ , first note that F ⊆ F̂ because if X ∈ F then each component is either pi or qi , which is a point in F̂ with si = 1 or 0. Next we show that F̂ is convex. Let V, W ∈ F̂ . Then their i-th components are vi = si pi + (1 − si )qi , wi = ri pi + (1 − ri )qi , respectively. Let λ ∈ [0, 1], then the i-th component of λV + (1 − λ)W is λvi + (1 − λ)wi = λ(si pi + (1 − si )qi ) + (1 − λ)(ri pi + (1 − ri )qi ) = (λsi + (1 − λ)ri )pi + (λ(1 − si ) + (1 − λ)(1 − ri ))qi . Setting θi := λsi + (1 − λ)ri , then θi ∈ [0, 1] because si , ri ∈ [0, 1] and λ(1 − si ) + (1 − λ)(1 − ri ) = 1 − (λsi + (1 − λ)ri ) = 1 − θi , so λvi + (1 − λ)wi = θi pi + (1 − θi )qi , with θi ∈ [0, 1], for every i = 1, . . . , 2N. Hence, λV + (1 − λ)W ∈ F̂ , so F̂ is convex. We prove that F̂ (Z) ⊆ co F (Z) by induction on N . To highlight the dependence on the dimension, we write F (N ) (Z) ⊆ R2N and F̂ (N ) (Z) ⊆ R2N for the sets F (Z) and F̂ (Z) defined in (2.17) and (2.21). The case N = 1 is trivial since F (1) (Z) = {p1 , q1 } and any V(1) ∈ F̂ (1) (Z) is of the form V(1) = s1 p1 + (1 − s1 )q1 ∈ co F (1) (Z). Now assume that F̂ (N −1) (Z) ⊆ co F (N −1) (Z) for some N . Let V(N ) ∈ F̂ (N ) (Z), so   s1 p1 + (1 − s1 )q1   V(N −1)   .. = V(N ) =   . sN pN + (1 − sN )qN sN pN + (1 − sN )qN for V(N −1) ∈ F̂ (N −1) (Z). By the induction hypothesis, V(N −1) ∈ co F (N −1) (Z), so P2N −1 (N −1) 2N −1 and V̂i ∈ F (N −1) (Z) such that αi ∈ [0, 1], i=1 αi = 1 there exist {αi }i=1 and   N −1 s1 p1 + (1 − s1 )q1 2X   (N −1) . (N −1) .. . αi V̂i V = = (N ) sN −1 pN −1 + (1 − sN −1 )qN −1 ∈ F (N ) (Z) for i = 1, . . . , 2N as  (N −1)   (N −1)  (N ) V̂i V̂i , V̂i+2N −1 := := qN pN i=1 We define V̂i (N ) V̂i for i = 1, . . . , 2N −1 , 30 TIMOTHY BLASS, IRENE FONSECA, GIOVANNI LEONI, MARCO MORANDOTTI and we define the coefficients βi ∈ [0, 1] for i = 1, . . . , 2N as βi := sN αi , Hence, P 2N βi+2N −1 := (1 − sN )αi for i = 1, . . . , 2N −1 . βi = 1 and !   P2N −1 (N −1) V(N −1) V̂ α i i i=1 = = sN pN + (1 − sN )qN sN pN + (1 − sN )qN ! P2N −1 P2N −1 (N −1) (N −1) sN αi V̂i + i=1 (1 − sN )αi V̂i i=1 = P2N −1 P2N −1 i=1 αi sN pN + i=1 αi (1 − sN )qN N −1 N −1   (N −1)   2X 2X (N −1) V̂i V̂i + = (1 − sN )αi sN αi qN p N i=1 i=1 i=1 V(N ) = N −1 2X i=1 (N ) βi V̂i + N −1 2X N (N ) βi+2N −1 V̂i+2N −1 = 2 X i=1 i=1 (N ) βi V̂i ∈ co F (N ) (Z).  5.2. Lemmas on the Singular Set. Lemma 5.1. The set D(F ), as defined in (2.19), is open and connected. Proof. From (2.19) and (2.18), it is clear that D(F ) is open. We will now show that D(F ) is path connected. Let w, z1 , . . . , zN ∈ Ω be distinct points, and let b ∈ D(F ) be given by Z = (z1 , . . . , zN ) and Z b = (z1 , . . . , zℓ−1 , w, zℓ+1 , . . . , zN ). Z, Z b as We construct a continuous path γ : [0, 1] → D(F ) with γ(0) = Z and γ(1) = Z follows. Note that Ω \ {z1 , . . . , zℓ−1 , zℓ+1 , . . . , zN } is path connected. Thus there is a path γℓ : [0, 1] → Ω \ {z1 , . . . , zℓ−1 , zℓ+1 , . . . , zN } with γℓ (0) = zℓ and γℓ (1) = w. Then setting γ(t) = (z1 , . . . , zℓ−1 , γℓ (t), zℓ+1 , . . . , zN ) for each t ∈ [0, 1] gives a path b in D(F ) from Z to Z. We can now connect any Z = (z1 , . . . , zN ) ∈ D(F ) to any other W = (w1 , . . . , wN ) ∈ D(F ) by first moving z1 to w1 as above, then z2 to w2 , and so on, until all the zi are moved to wi , producing a path from Z to W.  To prove the following lemma we will use the fact that the renormalized energy (see (1.6)) diverges logarithmically with the relative distance between the dislocations, that is, U (z1 , . . . , zN ) = − N −1 X i=1 N X µλbi bj log |Λ(zi − zj )| + O(1) 4π j=i+1 (5.1) as |zi − zj | −→ 0. We refer to [7] for a proof. Lemma 5.2. Fix ℓ ∈ {1, . . . , N } and let e ∈ R2 \ {0} be fixed. Then the set V = {Z ∈ D(F ) : jℓ (Z) · e = 0} has empty interior. Proof. The set V is closed because jℓ is continuous. Suppose there is a ball B ⊂ V . From Lemma 2.17, we have that jℓ (Z) · e is analytic in B and is constant, therefore jℓ (Z) · e is constant in the largest connected component of D(F ) containing B. Hence, by Lemma 5.1, jℓ (Z) · e = 0 in D(F ). From (1.7), we have that ∇zℓ U (Z) · e = 0 in D(F ), (5.2) SCREW DISLOCATION DYNAMICS 31 so U is constant when zℓ varies along the direction e. Consider a fixed Z∗ = (z1 , z2 , . . . , zN ) ∈ D(F ). Let h > 0, and for δ ∈ (0, h] define zδℓ := zℓ + δe. We assume that h0 small enough so that zδℓ ∈ Ω \ {z1 , . . . , zN } for δ ∈ (0, h0 ]. Fix a k 6= ℓ and h ∈ (0, h0 ], and let Zh be the point in D(F ) obtained by replacing zk in Z∗ with zhℓ , i.e., Zh := {z1 , . . . , zℓ , . . . , zk−1 , zhℓ , zk+1 , . . . , zN }.
Letting δn = 1 − n1 h, we construct the sequence {Zn } ⊂ D(F ) given by  Zn := z1 , . . . , zℓ + δn e, . . . , zk−1 , zhℓ , zk+1 , . . . , zN . We have Z1 = Zh , and Zn → Z∞ := {z1 , . . . , zhℓ , . . . , zk−1 , zhℓ , zk+1 , . . . , zN } as n → ∞. Note that Z∞ ∈ / D(F ) because zℓ and zk are colliding as n → ∞. In particular, by (5.1), |U (Zn )| → ∞ as n → ∞. On the other hand, in the sequence {Zn }, only the ℓ-th dislocation is moving, and it is moving along the direction e, so from (5.2), U (Zn ) remains constant for all n. We have reached a contradiction and we conclude that V does not contain any ball.  f∞ , as defined in (2.31), is empty. Lemma 5.3. The set M ℓ Proof. Without loss of generality, let ℓ = 1. Recall that f∞ = {Z : j1 (Z) · g0 = 0, ∂ α (j1 (Z) · g0 ) = 0 for all α ∈ N1 }, M 1 f∞ . f∞ 6= ∅ and Z̃ = (z̃1 , . . . , z̃N ) ∈ M with N1 defined in (2.30). Suppose that M 1 1 ∞ f Since j1 · g0 is analytic and Z̃ ∈ M1 , we have that j1 (Z) · g0 = j1 (Z̃) · g0 for Z ∈ {z̃1 } × V , where V is open in R2N −2 . Take V to be the largest connected component of D(F ) with z1 = z̃1 , which, by the same argument as Lemma 5.1, can be written {z̃1 } × V = {Z ∈ D(F ) : z1 = z̃1 }. We cannot follow the energy approach of Lemma 5.2, because that would require moving z1 , which is fixed. Instead, let 0 < ε0 ≪ 1 and construct a sequence {Zn } ⊂ V0 , where   V0 := Z ∈ V : min dist(zi , ∂Ω) > ε0 , i∈{1,...,N } (ε0 is only required to assure we do not have boundary collisions). To be precise, choose z3 , . . . , zN ∈ Ω pairwise distinct and such that zk 6= z̃1 and dist(zk , ∂Ω) > ε0 for every k = 3, . . . , N . Therefore, for n > 1 and δ0 > 0 sufficiently small, Zn := (z̃1 , z̃1 + δn g0 , z3 , . . . , zN ) belongs to V0 , where where δn = δ0 /n. Then j1 (Zn ) · g0 = j1 (Z̃) · g0 by construction, but Z∞ ∈ / D(F ), where Z∞ = limn→∞ Zn , because the first and second dislocations have collided. For each n, all the components of Zn are a bounded distance from ∂Ω. Thus, by (1.9), (1.12), and standard elliptic estimates, there exists C > 0 such that |∇u(z̃1 ; Zn )| 6 C for all n. For each n the singular strains |ki (z̃1 ; zi )| are bounded for i > 3. However, |k2 (z̃1 ; z̃1 + δn g0 )| > c/δn → ∞ as n → ∞, for some c > 0. Thus, we see from (1.9) and (1.8) that for large n, the force j1 (Zn ) will be large in
magnitude and aligned closely with b1 b2 z̃1 −(z̃1 +δn g0 ) (i.e., j1 (Zn ) will be nearly parallel or anti-parallel to g0 ). Therefore, j1 (Z̃)·g0 = j1 (Zn )·g0 > c1 |j1 (Zn )|·|g0 | → ∞ as n → ∞, for some c1 > 0, which contradicts the fact that j1 (Zn )·g0 = j1 (Z̃)·g0 f∞ = ∅. We conclude that M  1 32 TIMOTHY BLASS, IRENE FONSECA, GIOVANNI LEONI, MARCO MORANDOTTI The following lemma was used in the proof of Theorem 3.1. Lemma 5.4. Let Z1 ∈ Aℓ ∩ Ak for k 6= ℓ, but Z1 ∈ / Ai for i 6= k, ℓ. Also, assume that Z1 ∈ / Ezero ∪ Sℓ ∪ Sk and that (3.2a)-(3.2d) hold. Then there is at most one Z ∈ co F (Z1 ) such that nk (Z1 ) · Z = 0 nℓ (Z1 ) · Z = 0, and (5.3) where nk and nℓ are normal vectors for Ak and Aℓ , respectively. Proof. Without loss of generality, assume that k = 1 and ℓ = 2. Then (3.1) become f (+,+) (Z1 ) = (b f1+ (Z1 ), b f2+ (Z1 ), f3 (Z1 ), . . . , fN (Z1 )), f (+,−) (Z1 ) = (b f1+ (Z1 ), b f2− (Z1 ), f3 (Z1 ), . . . , fN (Z1 )), f (−,+) (Z1 ) = (b f1− (Z1 ), b f2+ (Z1 ), f3 (Z1 ), . . . , fN (Z1 )), f (−,−) (Z1 ) = (b f1− (Z1 ), b f2− (Z1 ), f3 (Z1 ), . . . , fN (Z1 )). From now on, we will omit the dependence on Z1 and simply write f (+,+) , etc. The important feature of the form of these four fields is that (b f+ − b f − , 0, . . . , 0) = f (+,+) − f (−,+) = f (+,−) − f (−,−) , (5.4a) 1 1 (0, b f2+ − b f2− , . . . , 0) = f (+,+) − f (+,−) = f (−,+) − f (−,−) . (5.4b) Then (3.2a)-(3.2d) become n1 · f (+,+) < 0, n1 · f (+,−) < 0, n1 · f (−,+) > 0, n1 · f (−,−) > 0 (5.5a) n2 · f n2 · f n2 · f n2 · f (5.5b) (+,+) < 0, (+,−) > 0, (−,+) < 0, (−,−) > 0. Let Z ∈ co F (Z1 ) satisfy (5.3). From Lemma 2.12, we have that there exist s, t ∈ [0, 1] such that Z = (sb f1+ + (1 − s)b f1− , tb f2+ + (1 − t)b f2− , f3 , . . . , fN ) f2− , f3 , . . . , fN ) f1− , b f2− ), 0) + (b f2+ − b f1− ), t(b = (s(b f1+ − b (5.6) = s(f (+,+) − f (−,+) ) + t(f (+,+) − f (+,−) ) + f (−,−) , where we used (5.4). By (5.6), the conditions in (5.3) become n1 · (f (+,+) − f (−,+) )s + n1 · (f (+,+) − f (+,−) )t = −n1 · f (−,−) , n2 · (f (+,+) − f (−,+) )s + n2 · (f (+,+) − f (+,−) )t = −n2 · f (−,−) , or, equivalently, A where A := and  a11 a21 a12 a22  =   s t  = b, n1 · (f (+,+) − f (−,+) ) n2 · (f (+,+) − f (−,+) ) b :=  −n1 · f (−,−) −n2 · f (−,−) (5.7) n1 · (f (+,+) − f (+,−) ) n2 · (f (+,+) − f (+,−) )   (5.8) . To prove the lemma it is enough to show that there is a unique choice of s and t that satisfy (5.7). We prove this by using (5.5) to show that detA > 0. SCREW DISLOCATION DYNAMICS 33 From (5.5a) we have a11 = n1 · (f (+,+) − f (−,+) ) < 0, and n1 · (f (+,+) − f (−,−) ) < 0 (5.9) Thus, by (5.4b) and (5.8) 0 > n1 · (f (+,+) − f (−,−) ) = n1 · (f (+,+) − f (−,+) ) + n1 · (f (−,+) − f (−,−) ) (5.10) = a11 + a12 . Again using (5.5a), we have n1 · (f (+,−) − f (−,+) ) < 0. Thus 0 > n1 · (f (+,−) − f (−,+) ) =n1 · (f (+,−) − f (+,+) ) + n1 · (f (+,+) − f (−,+) ) = −a12 + a11 . (5.11) Combining equations (5.10) and (5.11) we have a11 < a12 < −a11 =⇒ |a12 | < −a11 = |a11 |. (5.12) Similarly, a22 =n2 · (f (+,+) − f (+,−) ) < 0, n2 · (f (−,+) −f (+,−) n2 · (f (+,+) − f (−,−) ) < 0, (5.13) ) < 0. Noting that, from (5.4) and (5.8), we have a21 = n2 ·(f (+,+) −f (−,+) ) = n2 ·(f (+,−) − f (−,−) ) so 0 > n2 · (f (+,+) − f (−,−) ) = n2 · (f (+,+) − f (+,−) ) + n2 · (f (+,−) − f (−,−) ) (5.14) = a22 + a21 , and 0 > n2 · (f (−,+) − f (+,−) ) = n2 · (f (−,+) − f (+,+) ) + n2 · (f (+,+) − f (+,−) ) (5.15) = −a21 + a22 . Combining equations (5.14) and (5.15) we have a22 < a21 < −a22 =⇒ |a21 | < −a22 = |a22 |. (5.16) From (5.12) and (5.16) we have 0 < |a11 ||a22 | − |a12 ||a21 | 6 |a11 ||a22 | − a12 a21 = a11 a22 − a12 a21 = detA, where we also used that a11 , a22 < 0 from (5.9) and (5.13).  Acknowledgments The authors warmly thank the Center for Nonlinear Analysis (NSF Grant No. DMS-0635983), where part of this research was carried out. The research of I. Fonseca was partially funded by the National Science Foundation under Grant No. DMS-0905778 and that of G. Leoni under Grant No. DMS-1007989. T. Blass, I. Fonseca, and G. 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