## S. Dipierro - A. Figalli - E. Valdinoci

# Strongly nonlocal dislocation dynamics in crystals

created by figalli on 25 Apr 2014

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BibTeX]

*Accepted Paper*

**Inserted:** 25 apr 2014

**Last Updated:** 25 apr 2014

**Journal:** Comm. Partial Differential Equations

**Year:** 2014

**Abstract:**

We consider the equation
$v_t = L_s v-W'(v)+\sigma_\epsilon (t,x)$ in $(0,+\infty)\times\R,$
where $L_s$ is an integro-differential operator of order $2s$
with $s\in(0,1)$,
$W$ is a periodic potential, and $\sigma_\epsilon$ is a small external stress.
The solution $v$ represents the atomic dislocation
in the Peierls--Nabarro model
for crystals, and we specifically consider the case $s\in(0,1/2)$,
which takes into account a strongly nonlocal elastic term.

We study the evolution
of such dislocation function for macroscopic space and time scales,
namely we introduce the function
$v_{\epsilon}(t,x):=v\left(\frac{t}{\epsilon^{1+2s}}, \frac{x}{\epsilon}\right).$
We show that, for small $\epsilon$, the function $v_\epsilon$ approaches the sum of step
functions. {F}rom the physical point of view, this shows that
the dislocations have the tendency to concentrate
at single points of the crystal, where the size of the slip coincides with the
natural periodicity of the medium. We also show that the motion of
these dislocation points
is governed by an interior repulsive potential that is superposed
to an elastic reaction to
the external stress.

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