Dislocation dynamics: Difference between revisions

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Dislocations are microscopic defects in crystals that change over time (due for instance to shear stresses on the crystal).  
Dislocations are microscopic defects in crystals that change over time (due for instance to shear stresses on the crystal).
 
== One dimensional case ==


If we have a finite number of parallel (horizontal) lines on a 2D crystal each given by the equation $y=y_i$ ($y_i \in \mathbb{R}$) then a simplified model for the evolution of these lines says that the positions of these lines evolve according to the system of ODEs
If we have a finite number of parallel (horizontal) lines on a 2D crystal each given by the equation $y=y_i$ ($y_i \in \mathbb{R}$) then a simplified model for the evolution of these lines says that the positions of these lines evolve according to the system of ODEs
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in the case where the interaction potential $V\;\;$ satisfies $V\;'(y)=-\frac{1}{y^s}$. For this one dimensional model (which enjoys a maximum principle) a complete theory in terms of viscosity solutions, including existence, uniqueness and regularity was recently developed <ref name="BMK" />. Further, one can go back between this model and the [[nonlocal porous medium equation]] in 1-d by integrating the solution $u$ with respect to $x$.  
in the case where the interaction potential $V\;\;$ satisfies $V\;'(y)=-\frac{1}{y^s}$. For this one dimensional model (which enjoys a maximum principle) a complete theory in terms of viscosity solutions, including existence, uniqueness and regularity was recently developed <ref name="BMK" />. Further, one can go back between this model and the [[nonlocal porous medium equation]] in 1-d by integrating the solution $u$ with respect to $x$.  
== Higher dimensions: dislocation lines ==
If we drop the assumption that the dislocations occur along parallel lines we can study a different regime of the problem, instead of considering a density of parallel dislocation lines we can focus on the evolution of the shape of a single dislocation line. If $\Gamma_t$ is a dislocation line and the boundary of  an open set $\Omega_t$,  and if we let
\[ \rho(x,t) = 1_{\Omega_t}(x):= \left \{ \begin{array}{rl}
1 & \mbox{ if } x \in \Omega_t \\
0 & \mbox{ if } x \not \in \Omega_t
\end{array} \right.\]
Then this characteristic function is expected to solve an Eikonal equation with a nonlocal velocity
\[ \left \{ \begin{array}{rll}
\rho_t & = (k\star \rho) \left |\nabla \rho \right | & \text{ in } \mathbb{R}^2\times (0,T)\\
\rho(.,0) & = 1_{\Omega_0} & \text{ in } \mathbb{R}^2
\end{array}\right.
\]


== References ==
== References ==

Revision as of 22:12, 23 January 2012

Dislocations are microscopic defects in crystals that change over time (due for instance to shear stresses on the crystal).

One dimensional case

If we have a finite number of parallel (horizontal) lines on a 2D crystal each given by the equation $y=y_i$ ($y_i \in \mathbb{R}$) then a simplified model for the evolution of these lines says that the positions of these lines evolve according to the system of ODEs

\[ \dot{y}_i=F-V\;'_0(y_i) - \sum \limits_{j \neq i} V\;'(y_i-y_j) \;\;\;\text{ for } i=1,...,N, \]

One can consider the case in which $N \to +\infty$ and consider the evolution of a density of dislocation lines. If $u(x,t)$ denotes the limiting density, then the it solves the integro-differential equation

\[ u_t +|u_x|\Lambda^s u = 0 \;\;\;\text{ for all } (x,t) \in \mathbb{R}\times\mathbb{R}_+ \]

in the case where the interaction potential $V\;\;$ satisfies $V\;'(y)=-\frac{1}{y^s}$. For this one dimensional model (which enjoys a maximum principle) a complete theory in terms of viscosity solutions, including existence, uniqueness and regularity was recently developed [1]. Further, one can go back between this model and the nonlocal porous medium equation in 1-d by integrating the solution $u$ with respect to $x$.

Higher dimensions: dislocation lines

If we drop the assumption that the dislocations occur along parallel lines we can study a different regime of the problem, instead of considering a density of parallel dislocation lines we can focus on the evolution of the shape of a single dislocation line. If $\Gamma_t$ is a dislocation line and the boundary of an open set $\Omega_t$, and if we let

\[ \rho(x,t) = 1_{\Omega_t}(x):= \left \{ \begin{array}{rl} 1 & \mbox{ if } x \in \Omega_t \\ 0 & \mbox{ if } x \not \in \Omega_t \end{array} \right.\]

Then this characteristic function is expected to solve an Eikonal equation with a nonlocal velocity

\[ \left \{ \begin{array}{rll} \rho_t & = (k\star \rho) \left |\nabla \rho \right | & \text{ in } \mathbb{R}^2\times (0,T)\\ \rho(.,0) & = 1_{\Omega_0} & \text{ in } \mathbb{R}^2 \end{array}\right. \]

References

  1. Biler, Piotr; Monneau, Régis; Karch, Grzegorz (2009), "Nonlinear Diffusion of Dislocation Density and Self-Similar Solutions", Communications in Mathematical Physics 294 (1): 145–168, doi:10.1007/s00220-009-0855-8, ISSN 0010-3616