Strongly nonlocal dislocation dynamics in crystals
S Dipierro, A Figalli, E Valdinoci - Communications in Partial …, 2014 - Taylor & Francis
S Dipierro, A Figalli, E Valdinoci
Communications in Partial Differential Equations, 2014•Taylor & FrancisWe consider the equation where L s is an integro-differential operator of order 2 s, with
s∈(0, 1), W is a periodic potential, and σϵ 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∈(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 We show that, for small ϵ, the function v ϵ …
s∈(0, 1), W is a periodic potential, and σϵ 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∈(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 We show that, for small ϵ, the function v ϵ …
We consider the equation where L s is an integro-differential operator of order 2s, with s ∈ (0, 1), W is a periodic potential, and σϵ 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 ∈ (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
We show that, for small ϵ, the function v ϵ approaches the sum of step functions. From 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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