# Dislocation dynamics

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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 == | + | == 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 | ||

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

- | 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 <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 $V\;'(y)=-\frac{1}{y^s}$. 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 <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 == | == Higher dimensions: dislocation lines == | ||

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\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<ref name="FLCM" /> |

\[ \left \{ \begin{array}{rll} | \[ \left \{ \begin{array}{rll} | ||

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{{reflist|refs= | {{reflist|refs= | ||

<ref name="BMK">{{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}}</ref> | <ref name="BMK">{{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}}</ref> | ||

+ | <ref name="FLCM">{{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}}</ref> | ||

}} | }} |

## 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

- ↑ 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 - ↑ 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