Now, we consider a free boundary problem which consists in determining the temperatureu=u(x, t) and the free boundaryx=s(t) with a control functionF which depends on time and the evolution of the heat flux at the boundaryx= 0, satisfying the following conditions
ρcut−kuxx=−γF(ux(0, t), t), 0< x < s(t), t >0, (3.1) u(0, t) =f = Const. >0, t >0, (3.2) u(s(t), t) = 0, kux(s(t), t) =−ρls. (t), t >0, (3.3)
s(0) = 0, (3.4)
Stefan Problem for Non-classical Heat Equation 121 where the thermal coefficientsk, ρ, c, l, γ >0 and the control functionF is given by the expression
F(V, t) = λ0
√t V (λ0>0). (3.5)
In order to obtain an explicit solution of a similarity type, we define Φ(η) =u(x, t), η= x
2a√
t (3.6)
wherea2=k/ρcis the diffusion coefficient.
After some elementary computations we obtain Φ(η) =f
1− E(η) E(η0)
, 0< η < η0, (3.7) where
E(x) =erf(x) + 4λ
√π x
0
f1(r)dr, λ=γλ0
ρca >0, [λ] = 1 (3.8) and
f1(x) = exp(−x2) x
0
exp(r2)dr (3.9)
is Dawson’s integral [1] andη0is an unknown positive parameter to be determined which characterizes the free boundary given by
s(t) = 2aη0√
t. (3.10)
We remark that Dawson’s integral also appears in the explicit solution for the su- percooled one-phase Stefan problem with a constant temperature boundary con- dition on the fixed face [16].
Taking into account the Stefan condition we have thatη0=η0(λ, Ste) must be the solution of the following equation
Ste√
π[exp(−x2) + 2λf1(x)] =x[erf(x) + 4λ
√π x
0
f1(z)dz] , x >0 (3.11) whereSte= f c
l >0 is the Stefan number and erf(x) = 2
√π x
0
exp(−z2)dz. (3.12)
The equation (3.11) is equivalent to the equation
W1(x) = 2λW2(x), x >0 (3.13) where functionsW1andW2 are defined by
W1(x) =Ste exp
−x2
−√
πerf(x)x (3.14)
W2(x) = 2x x
0
f1(r)dr−Ste f1(x). (3.15)
Remark 3.1. If λ= 0 (that is γ = 0) in the free boundary problem (3.1)–(3.4) we obtain the classical Lam´e-Clapeyron [11] solution and there exists a unique solutionη00of the equation (3.11) which is given now by
F0(x) = Ste
√π , x >0 (3.16)
where
F0(x) =erf(x) exp(x2)x . (3.17) Theorem 3.2. For eachλ >0there exists a unique solutionη0 of Eq.(3.13). This solutionη0=η0(λ)has the following properties
(i) η0(0+) =η00>0
(ii) η0(+∞) =x4<+∞ (3.18)
(iii) η0=η0(λ) is an increasing function onλ
where η00 is the unique solution of Equation (3.16) and x4 > 0 is the unique positive zero ofW2.
Theorem 3.3. For each λ >0 the free boundary problem(3.1)–(3.4) has a unique similarity solution of the type
u(x, t, λ) =f
1− E(n, λ) E(η0(λ), λ)
, 0< η= x 2a√
t < η0(λ) (3.19) s(t, λ) = 2a η0(λ)√
t (3.20)
where
E(η, λ) =erf(η) + 4λ
√π η
0
f1(r)dr (3.21)
andη0=η0(λ) is the unique solution of Eq.(3.13), withη00< η0(λ)< x4. Theorem 3.4. The explicit solution (3.19), (3.20) of the problem (3.1)–(3.4) has the following properties:
(i) ux(0, t, λ) = −f aE(η0(λ), λ)
√1
πt <0, ∀t >0 (ii) u(x, t, λ)≥u0(x, t), ∀0≤x≤s0(t), t >0 (iii) s(t, λ)≥s0(t), ∀t >0
where u0(x, t) =f
1− erf(η) erf(η00)
, 0< η= x 2a√
t < η00, t >0 s0(t) =s(t,0) = 2aη00√
t (iv) 1≤ u(x, t, λ)
u0(x, t) ≤ 1 1−η(x, t)
η00
1− 2 Ste
η0(λ) (1 + 2λf1∞)
exp (−η20(λ)) + 2λf1(η0(λ))η(x, t)
(v) lim
t→+∞
u(x, t, λ)
u0(x, t) = 1uniformly∀x∈compact sets ⊂[0, s0(t)).
Stefan Problem for Non-classical Heat Equation 123
References
[1] M. Abramowitz, I.E. Stegun (Eds.),Handbook of Mathematical Functions with For- mulas, Graphs, and Mathematical Tables,National Bureau of Standards, Washington (1972).
[2] V. Alexiades, A.D. Solomon, Mathematical modeling of melting and freezing pro- cesses,Hemisphere – Taylor & Francis, Washington (1983).
[3] L.R. Berrone, D.A. Tarzia and L.T. Villa, Asymptotic behavior of a non-classical heat conduction problem for a semi-infinite material, Math. Meth. Appl. Sci., 23 (2000), 1161–1177.
[4] J.R. Cannon, The one-dimensional heat equation, Addison-Wesley, Menlo Park (1984).
[5] J.R. Cannon, H.M. Yin, A class of non-linear non-classical parabolic equations, J.
Diff. Eq., 79 (1989), 266–288.
[6] J. Crank,Free and moving boundary problems,Clarendon Press Oxford (1984).
[7] A. Friedman,Free boundary problems for parabolic equations I. Melting of solids, J.
Math. Mech., 8 (1959), 499–517.
[8] G. Gripenberg, S.O. Londen and O. Staffans,Volterra integral and functional equa- tions, Cambridge Univ. Press, Cambridge (1990).
[9] S.C. Gupta, The classical Stefan problem. Basic concepts, modelling and analysis, Elsevier, Amsterdam (2003).
[10] N. Kenmochi, M. Primicerio,One-dimensional heat conduction with a class of auto- matic heat source controls, IMA J. Appl. Math. 40 (1988), 205–216.
[11] G. Lam´e, B.P. Clapeyron, M´emoire sur la solidification par refroidissement d’un globe liquide, Annales chimie Physique 47 (1831), 250–256.
[12] V.J. Lunardini,Heat transfer with freezing and thawing ,Elsevier, Amsterdam (1991).
[13] W.R. Mann, F. Wolf,Heat transfer between solid and gases under nonlinear boundary conditions, Quart. Appl. Math., 9 (1951), 163–184.
[14] R.K. Miller, Nonlinear Volterra integral equations, W. A. Benjamin, Menlo Park (1971).
[15] W.E. Olmstead, R.A. Handelsman,Diffusion in a semi-infinite region with nonlinear surface dissipation, SIAM Rev., 18 (1976), 275–291.
[16] A. Petrova, D.A. Tarzia and C.V. Turner,The one-phase supercooled Stefan problem with temperature boundary condition, Adv. Math. Sci. and Appl., 4 (1994), 35–50.
[17] J.H. Roberts, W.R. Mann,A certain nonlinear integral equation of the Volterra type, Pacific J. Math., 1 (1951), 431–445.
[18] L.I. Rubinstein,The Stefan problem, Trans. Math. Monographs # 27, Amer. Math.
Soc., Providence (1971).
[19] B. Sherman,A free boundary problem for the heat equation with prescribed flux at both fixed face and melting interface, Quart. Appl. Math., 25 (1967), 53–63.
[20] D.A. Tarzia,A bibliography on moving-free boundary problems for the heat-diffusion equation. The Stefan and related problems, MAT-Serie A, Rosario, # 2 (2000), (with 5869 titles on the subject, 300 pages).
See www.austral.edu.ar/MAT-SerieA/2(2000)/
[21] D.A. Tarzia, A Stefan problem for a non-classical heat equation, MAT-Serie A, Rosario, # 3 (2001), 21–26.
[22] D.A. Tarzia, L.T. Villa,Some nonlinear heat conduction problems for a semi-infinite strip with a non-uniform heat source, Rev. Un. Mat. Argentina, 41 (2000), 99–114.
[23] L.T. Villa,Problemas de control para una ecuaci´on unidimensional del calor, Rev.
Un. Mat. Argentina, 32 (1986), 163–169.
Adriana C. Briozzo Depto. Matem´atica FCE, Univ. Austral
Paraguay 1950, S2000FZF Rosario, Argentina e-mail:Adriana.Briozzo@fce.austral.edu.ar Domingo A. Tarzia
Depto. Matem´atica – CONICET FCE, Univ. Austral
Paraguay 1950, S2000FZF Rosario, Argentina e-mail:Domingo.Tarzia@fce.austral.edu.ar
International Series of Numerical Mathematics, Vol. 154, 125–135 c 2006 Birkh¨auser Verlag Basel/Switzerland
Dislocation Dynamics:
a Non-local Moving Boundary
P. Cardaliaguet, F. Da Lio, N. Forcadel and R. Monneau
Abstract. In this article, we present briefly the mathematical study of the dynamics of line defects called dislocations, in crystals. The mathematical model is an eikonal equation describing the motion of the dislocation line with a velocity which is a non-local function of the whole shape of the dislocation.
We present some partial existence and uniqueness results. Finally we also show that the self-dynamics of a dislocation line at large scale is asymptotically described by an anisotropic mean curvature motion.
Mathematics Subject Classification (2000).Primary 35F25; Secondary 35Q99.
Keywords.Dislocations dynamics, non-local equations, viscosity solutions.
1. Introduction
1.1. What are dislocations?
The crystal defects called dislocations are lines whose typical length in metallic alloys is of the order of 10−6m, with thickness of the order of 10−9m (see Figure 1 for an example of observations of dislocations by electron microscopy).
In the face centered cubic structure, dislocations move at low temperature in well defined crystallographic planes (the slip planes), at velocities of the order of 10ms−1. We refer for instance to Hirth and Lothe [17] for a description at the atomic level of these dislocations.
The concept of dislocations has been introduced and developed in the XXth century, as the main microscopic explanation of the macroscopic plastic behavior of metallic crystals (see for instance the physical monographs Nabarro [20], Hirth and Lothe [17], or Lardner [19] for a mathematical presentation). Since the beginning of the 90’s, the research field of dislocations has enjoyed a new boom based on the increasing power of computers, allowing simulations with a large number of dislocations (see for instance Kubin et al. [18]). This simultaneously motivated new theoretical developments for the modelling of dislocations. Recently Rodney,
Figure 1. Dislocations in a Al-Mg alloy (from [23])
Le Bouar and Finel introduced in [21] a new model that we present and study mathematically in this paper. We also refer the reader to [6] and the references therein for a more detailed introduction to dislocations dynamics. This model has also been numerically studied by Alvarez, Carlini, Monneau and Rouy in [3] and [4]; see also Alvarez, Carlini, Hoch, Le Bouar and Monneau [2]
1.2. Mathematical modelling of dislocations dynamics
An idealization consists in assuming that the thickness of these lines is zero, and in the case of a single line, in assuming that this line is contained and moves in thex= (x1, x2) plane. The motion of the line Γt(where the subscript t denotes the time) is simply given by the normal velocityc(see Figure 2).
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1111111111111111111111111 1111111111111111111111111 1111111111111111111111111 1111111111111111111111111 1111111111111111111111111 1111111111111111111111111 1111111111111111111111111 1111111111111111111111111 1111111111111111111111111 1111111111111111111111111 1111111111111111111111111 1111111111111111111111111 1111111111111111111111111 1111111111111111111111111 1111111111111111111111111 1111111111111111111111111 1111111111111111111111111 1111111111111111111111111
c ∆ t nΓ
t
t
Γ t
Γ t+ ∆ t
Ω
Figure 2. Schematic evolution of a dislocation line Γt by normal ve- locitycbetween the times tandt+ ∆t with unit normalnΓt.
The velocityc is proportional to the shear stress in the material. This stress can be computed solving the equations of linearized elasticity where the shape of the dislocation line appears as a source term. This gives a coupled system where the dislocation line evolution is a function of the velocityc, and the velocitycis a function of the dislocation line Γtitself. In the case of a single dislocation line it
Dislocation Dynamics: a Non-local Moving Boundary 127 is possible to write the velocitycas a non-local quantity depending on the whole shape of the dislocation line (see Alvarez et al. [6]):
c(x, t) = (c0 ρ(ã, t)) (x) +c1(x, t)
whereρis the characteristic function of an open set Ωt⊂R2 whose the boundary is the dislocation line Γt=∂Ωt:
ρ(x, t) = 1Ωt := 1 if x∈Ωt
0 if x∈R2\Ωt,
andc0(x) is a given kernel depending on the material. Here the convolution is only done in space onR2.
It can be easily checked (at least formally), that the evolution on the time interval (0, T) of the dislocation line Γtis described by the equation of dislocations dynamics:
∂ρ
∂t = (c0 ρ+c1) |Dρ| on R2×(0, T)
ρ(ã,0) =ρ0(ã) := 1Ω0 on R2 (1.1) where Ω0is an open set whose boundary Γ0=∂Ω0is the position of the dislocation line at initial timet= 0.
In what follows, we will study this equation in the framework of discontinuous viscosity solutions (see Barles [7] for an introduction to this notion). To simplify the presentation we will state results in dimensionn= 2, assuming smooth (C∞) regularity of the initial position Γ0of the dislocation line, of the kernelc0, and of the velocityc1. We also assume the following behavior of the kernel at infinity (for some functiong)
c0(x) = 1
|x|3 g x
|x|
for |x| ≥1 (1.2)
which is a natural assumption for dislocations.
For considerably weakened assumptions and in any dimensions n, we refer the reader to the original articles cited in the references.
1.3. Organization of the paper
Although equation (1.1) seems very simple, general results of existence and unique- ness are unknown up to our knowledge. Technically, the main difficulty comes from the fact that we have no sign conditions on the kernelc0, and then that there is no inclusion principle for this evolution.
In this paper we present some partial results. In Section 2, we give a short time existence (and uniqueness) result for a smooth initial dislocation loop. In Section 3, we give a long time existence (and uniqueness) result for a smooth initial curve with non-negative velocity. Finally in Section 4, we consider the “monotone case”
where the kernel satisfies c0 ≥ 0. In this particular case, a Slepˇcev “level sets”
formulation of equation (1.1) is available. In this framework, we show that at large scales, the dislocation dynamics is asymptotically described by an (anisotropic) mean curvature motion related to the behavior of the kernelc0(x) as|x| →+∞.