Marcello Ponsiglione Sapienza Università di Roma
Description of a dislocation line A deformed crystal C can be described by The displacement function u : C R 3. The strain function β = u. A dislocation in the crystal can be described by a line L in the crystal and by a strain β such that the circulation of β around L is a fixed vector b called Burgers vector. If b is parallel to L, the dislocation is called screw dislocation. If b is orthogonal to L, the dislocation is called edge dislocation. Locally in C \ L, β is the gradient of a multi-valued function u.
Dislocation
The anti-planar setting The reference configuration is an open set Ω R 2 which represents a horizontal section of an infinite cylindrical crystal. Screw dislocations can be represented by a measure on Ω which is a finite sum of Dirac masses of the type µ := i z i b δ xi. The class of admissible strains associated with a dislocation µ is given by the fields β : Ω R 2 whose circulation around the dislocations x i are equal to z i b (curl β = µ).
The core radius aproach Because of the singularities we have β 2 dx =! Ω To set up a variational formulation we introduce an internal scale ε referred to as core radius, proportional to the atomic distance. We remove B ε (x i ) around each singularity x i, and compute the energy outside the core region. E ε (µ, β) := β 2 dx, 2 where Ω ε (µ) := Ω \ i B ε (x i ). Ω ε(µ)
Ginzburg Landau energy functionals Ginzburg Landau Energy: GL ε : H (Ω; R 2 ) R defined by ( GL ε (v) := 2 v 2 + ) 4ε 2 W (v), () where W (v) := ( v 2 ) 2. Ω Bethuel F., Brezis H., Hélein F.: Ginzburg-Landau vortices. Progress in Nonlinear Differential Equations and their Applications. vol. 3. Birkhäuser Boston, Boston, 994....Of course, many other penalties can be devised. They all seem to lead to the same class of generalized solutions. For example, one other natural penalty consists of drilling a few little holes B(x i, ε i ) in Ω and considering the domain Ω ε := Ω \ i B ε (x i ). In this case there is no topological obstruction... min v 2. v Hg (Ω ε,s ) 2 Ω ε
The change of variables: Dislocations Vortices We fix the singularities {x,... x N }. Ginzburg Landau vortices. v : Ω ε S whose degree around the vortices x i are equal to z i Z. Ginzburg Landau energy. 2 Ω ε v 2. Screw dislocations. β : Ω ε R 2 whose circulation around the dislocations x i are equal to z i Z. Elastic energy. 2 Ω ε β 2. Change of variables: the displacement u = 2π θ, where θ is the lifting of v v = e iθ β = 2π θ e iθ 2 = θ 2 = 2π 2 β 2. 2 Ω ε 2 Ω ε Ω ε
The Γ convergence result by Jerrard and Soner Jerrard R., Soner H. M. The Jacobian and the Ginzburg-Landau energy. Calc. Var. Partial Differential Equations 4 (2002), no. 2, 5 9. Theorem (Jerrard and Soner) Let 0 < α be fixed. i) Compactness. Let log ε GL ε(v ε ) C. Then, (up to a subsequence) J(v ε ) µ with respect to dual norm of Cc 0,α (Ω), where µ := N i= z iδ xi for some x i Q, z i Z. ii) Γ-liminf. Let (v ε ) W,2 (Ω) be such that J(v ε ) µ := N i= z iδ xi. Then lim inf ε log ε GL ε(v ε ) π µ (Ω). iii) Γ-limsup. For every µ := N i= z iδ xi, there exists (v ε ) s.t. J(v ε ) µ and log ε GL ε(v ε ) π µ (Ω).
Γ-convergence of the energy functionals F ε. The class of admissible strains AS ε (µ) associated with µ: AS ε (µ) := {β L 2 (Ω ε (µ); R 2 ) : curl β = 0 in Ω ε (µ), β(s) τ(s) ds = µ(x i ) Z for every x i supp µ}. B ε(x i ) The (rescaled) elastic energy associated with µ is given by ( log ε F ε(µ) := ) min β(x) 2 dx + µ (Ω). log ε β AS ε(µ) 2 Ω ε(µ) Let X be the space of measures of the type µ := N i= z iδ xi. The Γ-limit of F ε is the functional F : X R defined by F(µ) := µ (Ω) for every µ X. 4π P.: Elastic energy stored in a crystal induced by screw dislocations, from discrete to continuous. SIAM J. Math. Anal. (2007).
The discrete model We consider a square lattice of size ε with N-N interactions Ariza M. P., Ortiz M.: Discrete crystal elasticity and discrete dislocations in crystals. Arch. Rat. Mech. Anal. 78 (2006). To introduce the dislocations we adopt the point of view of the additive decomposition of the strain: d u = β p + β e. β p is the plastic part of the strain and is valued in Z b = Z. The elastic energy corresponding to the decomposition d u = β p + β e is given by E el ε (u) := 2 (i,j) Ω ε β e i,j 2 = 2 (i,j) Ω ε d u i,j β p i,j 2. Minimizing Eε el with respect to the plastic strain we obtain the optimal βu p : project du on Z! (according with Pierls-Nabarro).
Discrete dislocations β p u is the projection of du on Z, while β e u = dist((u i u j ), Z). E el ε (u) := i,j ( dist((u i u j ), Z) ) 2 The discrete dislocation function α u : Ω 2 ε {, 0, } is defined by α u (Q i ) := curl β e u(q i ) := β e u( Q i ). It will be convenient also to represent α u as a sum of Dirac masses with weights in {, 0, } and supported on the centers of the squares where α 0, setting µ u := α u (Q i )δ i. (2) i Ω 2 ε
The asymptotic behaviour of the discrete energy functionals The discrete elastic energy associated with µ is given by log ε F ε(µ) := log ε min E µ ε el (u). u=µ P.: Elastic energy stored in a crystal induced by screw dislocations, from discrete to continuous. SIAM J. Math. Anal. (2007). Theorem 2 The functionals log ε F ε are equi-coercive and Γ-converge to F as ε 0 with respect to the flat norm.
Another discrete model: XY Spin systems Alicandro R., Cicalese M.: Variational analysis of the asymptotics of the XY model. Arch. Rational Mech. Anal. 92 (2009), 50 536. The space is a lattice εz 2 ; The variable is a function v : εz 2 S ; The energy is the discrete elastic energy (for instance N-N interactions). XY ε (v) := ( v i v j ) 2. 2 i j = Γ-convergence for GL vortices = Γ-convergence for XY spin systems The key estimate: let w be obtained by interpolation of v, then 4ε 2 W (w) C XY ε (v)
The log ε h energy regime of GL, SD and XY Alicandro R., Cicalese M., P.: Variational equivalence between Ginzburg-Landau, XY spin systems and screw dislocations energies. Indiana (To appear) Consider the following energy functionals, defined on the topological singularities (minimizing with respect to the order parameter). We consider the energetic regimes of order log ε h with h. { GL ε (µ) := GL ε (w), w H (Ω; R 2 ) : X Y ε (µ) := SD ε (µ) := log ε h inf log ε h inf 4π2 log ε h inf { XY ε (v), v AX Y ε : { SD ε (u), u ASD ε : Do these functionals share the same Γ-limit? J(w) π log ε h = µ µ v log ε h = µ µ u log ε h = µ }. }, }.
Identification between singularities: discrete dislocations discrete vortices Spin variable. v : εz 2 S. Energy for spin systems. XY ε (v) := 2 ( i,j v i v j ) 2. Screw dislocations. u : εz 2 R Elastic energy. SD ε (u) := ( 2 i,j dist((u i u j ), Z) ) 2. Identification of the order parameters: v = e iθ u = 2π θ If (θ i θ j ) is small, then (v i v j ) 2 (θ i θ j ) 2.
The variational equivalence argument If θ ε is a recovery sequence, then θ i θ j is small for most of the pairs.then log ε XY ε(v ε ) 4π2 log ε SD ε(u ε ). To conclude that X Y ε and SD ε have the same Γ-limit we need some estimate for sequences with bounded energy. There are some technicalities: If log ε XY ε C, we could have log ε small dipoles... log ε errors of order!
Change of variable in the scale ε δ Let GL ε (w ε ) C log ε. For ε 0 we have w ε in measure. If δ(ε) >> ε is a suitable mesoscale we have w ε uniformly on a suitable translation s ε of δ(ε)z 2. Set v δ(ε) := w ε (δ(ε)z 2 + s ε ) w ε If log δ(ε) log ε we can choose s ε so that log δ(ε) XY δ(ε)(v ε ) log ε GL ε(w ε ) Jw ε discrete vorticity of v ε
The notion of variational equivalence Let (F ε ) and (G ε ) be two families of functionals from X to R { } depending on the parameter ε R +. Definition 3 We set (G ε ) (F ε ) if there exists a continuous increasing function ε δ ε, with δ 0 = 0, such that the following holds. For every ε n 0, and (p n ) X such that F εn (p n ) C, there exists a family (q n ) X such that i) lim sup n (G δεn (q n ) F εn (p n )) 0; ii) Either (p n ) and (q n ) are unbounded or d(p n, q n ) 0 as n +. We set (F ε ) (G ε ), and we say that (F ε ) and (G ε ) are variationally equivalent (for ε 0) if (F ε ) (G ε ) and (G ε ) (F ε ).
Consequences of the variational equivalence Theorem 4 Assume that the functionals F ε are equi-coercive, and that Γ-converge to some functional H. Then (F ε ) is variationally equivalent to the constant sequence given by its Γ-limit (H ε H). Theorem 5 Let (F ε ) and (G ε ) be variationally equivalent. Then the following properties hold. ) F ε Γ-converge to some functional H if and only if G ε Γ-converge to H; 2) F ε are equi-coercive if and only if G ε are equi-coercive. Braides A., Truskinowsky L.: Asymptotic expansions by Γ-convergence. Contin. Mech. Thermodyn. 20 (2008), no., 2 62.
A first example of equivalent families For all positive s > 0 the following Ginzburg Landau functionals are variationally equivalent. GL s ε(w) := w 2 + s W (w). (3) log ε Q ε2 Let s < s 2. Since GL s ε GL s 2 ε, we have GL s ε GL s 2 ε. To prove GL s 2 ε GL s ε, set (ε n ) S ( ( ) ) s2 2 ε n. s Let p n = q n := w n X, GL s ε n (w n ) C, set (δ n ) = S(ε n ). = lim sup( ( ) n log ε s2 2 n s lim sup(gl s 2 δ n (q n ) GL s ε n (p n )) n log ε n )( w n 2 + s Q ε 2 W (w n )) = 0. n
Variational equivalence between GL, SD and XY Theorem 6 (Alicandro R., Cicalese M., P. (20)) For all energetic regimes of order log ε h with h, the following functionals are variationally equivalent GL ε (µ) := X Y ε (µ) := SD ε (µ) := { log ε h inf GL ε (w), w H (Ω; R 2 ) : { log ε h inf XY ε (v), v AX Y ε : 4π2 log ε h inf { SD ε (u), u ASD ε : } J(w) π log ε h = µ µ v log ε h = µ µ u log ε h = µ } }
A new result for homogenizing dislocations Jerrard R. L., Soner H. M.: Limiting behavior of the Ginzburg-Landau functional. J. Funct. Anal. 92 (2002), no. 2, 524 56. Theorem 7 (Alicandro, Cicalese, P. (20)) The functionals F ε : X [0, + ], defined by { F ε (µ) := SD ε (u), u ASD ε : log ε 2 inf } µ u log ε = µ, (4) are equi-coercive and Γ-converge as ε 0 to the functional F : X [0, + ] defined by F(µ) := 4π µ (Ω) + { } 2 Ω inf β 2, β L 2 (Ω; R 2 ), curl β = µ if µ is a measure in H (Ω) and infinity elsewhere.
Gradient flow for discrete screw dislocations The discrete equation: u i = j N N u j dist 2 (u i u j, Z) Notice that dist 2 (u i u j, Z) is not smooth (with singularities for u i u j Z). Regularize the equation Pass from discrete to continuum.
The Γ-convergence approach by Sandier-Serfaty Sandier E., Serfaty S.: Gamma-convergence of gradient flows with applications to Ginzburg-Landau (2003) We follow the approach by Sandier and Serfaty for our discrete dislocation energies (work in progress with R. Alicandro). Theorem 8 Fix a boundary datum ū on Ω. Let N N deg(e 2πiū ) be fixed. The functionals SDūε(µ) N log ε Γ-convergence to the renormalized energy W GL (g, µ) + C discr. (if µ = N, otherwise to ± ) Here C discr. is a constant depending on the discrete lattice, and does not affect the dynamics.
The asymptotic dynamics of dislocations Theorem 9 Let u ε be a family of solutions to log ε u ε = j N N u j dist 2 Reg (u i u j, Z) + boundary conditions and well prepared initial data. Then µ uε(t) µ(t) := 2π i d iδ ai (t) where d i = and ȧ i = i W (g, µ(t)) for t T
Case of planar elasticity The reference configuration is an open set Ω R 2 which represents an horizontal section of the cylindrical crystal. The Burgers vectors are a finite set S R 2 S := {b,..., b s }. The dislocations are represented by a measure on Ω, which is a finite sum of Dirac masses, µ := i z ib i δ xi. The class of admissible strains corresponding to µ is given by the functions β : Ω ε M 2 2 whose circulation around each x i is equal to z i b i. The elastic energy corresponding to a pair (µ ε, β ε ) is given by F ε (µ ε, β ε ) = W (β ε )dx = Ω ε Cβ ε : β ε dx c βε sym 2 2! Ω ε
Planar elasticity The log ε energy regime: L. de Luca (work in progress) Non linear energy of order log ε : L. Scardia, C. Zeppieri (Preprint) The log ε 2 regime: Garroni A., Leoni G., P.: Gradient theory for plasticity via homogenization of discrete dislocations. JEMS 200 The Γ-limit: ( ) dµ F(µ, β) := < Cβ : β > dx + ϕ d µ. d µ Ω µ is the dislocation density measure, β is the strain (Curl β = µ) and ϕ is the density of the plastic energy.) Ω Concerning vortices... E. Sandier and S. Serfaty, Vortices in the Magnetic Ginzburg-Landau Model, Progress in Nonlinear Differential Equations and their Applications, 70. Birkhuser Boston, Inc., Boston, MA, 2007. Jerrard R. L., Soner H. M.: Limiting behavior of the Ginzburg-Landau functional. J. Funct. Anal. 92 (2002), no. 2, 524 56.
Three dimensional dislocations In the context of GL functionals: Alberti G., Baldo S., Orlandi G.: Variational convergence for functionals of Ginzburg-Landau type. Indiana Univ. Math. J. 54 (2005), no. 5, 4 472. Baldo S., Jerrard R., Orlandi G., Soner H. M.: Convergence of Ginzburg-Landau functionals in 3-d superconductivity. Preprint Consider the three dimensional torus T := R 3 /(εz 3 ), and consider the fields w : Ω T R. We denote by u the first three components of w, representing the displacement function, and by r the fourth one, representing the modulus of the order parameter, so that ( r) represents the distance by the torus. Therefore, we define the Ginzburg-Landau energy functionals by GL ε (w) := r 2 < C u : u > + r 2 + ε 2 r 2. (5) Ω
A Ginzburg-Landau energy for 3 D dislocations GL ε (w) := Ω r 2 < C u : u > + r 2 + ε 2 r 2. A variant suggested to me very recently by Adriano Pisante and Jean Van Schaftingen: Let Φ : Ω C 3 GL ε (Φ) := < CΦ Φ : Φ Φ > + Φ 2 + ε 2 W ( Φ ). Ω Does GL ε provide a good model for dislocations? (simulations, variational equivalence with a discrete model...); Asymptotic behaviour of GL ε.