# Dislocation dynamics

(Difference between revisions)
 Revision as of 03:16, 24 January 2012 (view source)Nestor (Talk | contribs) (→One dimensional case)← Older edit Latest revision as of 03:28, 24 January 2012 (view source)Nestor (Talk | contribs) (3 intermediate revisions not shown) Line 1: Line 1: 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 == + == One dimensional case: dislocation densities == 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 Line 22: Line 22: \end{array} \right.\] \end{array} \right.\] - Then this characteristic function is expected to solve an Eikonal equation with a nonlocal velocity + Then this characteristic function is expected to solve an Eikonal equation with a nonlocal velocity $\left \{ \begin{array}{rll} \[ \left \{ \begin{array}{rll} Line 33: Line 33: {{reflist|refs= {{reflist|refs= {{Citation | last1=Biler | first1=Piotr | last2=Monneau | first2=Régis | last3=Karch | first3=Grzegorz | title=Nonlinear Diffusion of Dislocation Density and Self-Similar Solutions | doi=10.1007/s00220-009-0855-8 | year=2009 | journal=Communications in Mathematical Physics | issn=0010-3616 | volume=294 | issue=1 | pages=145–168}} {{Citation | last1=Biler | first1=Piotr | last2=Monneau | first2=Régis | last3=Karch | first3=Grzegorz | title=Nonlinear Diffusion of Dislocation Density and Self-Similar Solutions | doi=10.1007/s00220-009-0855-8 | year=2009 | journal=Communications in Mathematical Physics | issn=0010-3616 | volume=294 | issue=1 | pages=145–168}} + {{Citation | last1=Forcadel | first1=N. | last2=Lio | first2=F. | last3=Cardaliaguet | first3=P. | last4=Monneau | first4=Régis | title=Dislocation dynamics: a non-local moving boundary | publisher=[[Springer-Verlag]] | location=Berlin, New York | year=2007 | journal=Free boundary problems | pages=125–135}} }} }} ## Latest revision as of 03:28, 24 January 2012 Dislocations are microscopic defects in crystals that change over time (due for instance to shear stresses on the crystal). ## One dimensional case: dislocation densities 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 $V\;'(y)=-\frac{1}{y^s}$. We note that $\Lambda$ above denotes the Zygmund operator, also known was $(-\Delta)^{1/2}$. 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[2]

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

## 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
2. Forcadel, N.; Lio, F.; Cardaliaguet, P.; Monneau, Régis (2007), "Dislocation dynamics: a non-local moving boundary", Free boundary problems (Berlin, New York: Springer-Verlag): 125–135