From the Peierls-Nabarro model to the equation of motion of the dislocation continuum. (English) Zbl 1451.82057
Summary: We consider a semi-linear integro-differential equation in dimension one associated to the half Laplacian whose solution represents the atom dislocation in a crystal. The equation comprises the evolutive version of the classical Peierls-Nabarro model. We show that for a large number of dislocations, the solution, properly rescaled, converges to the solution of a well known equation called by Head (1972) “the equation of motion of the dislocation continuum”. The limit equation is a model for the macroscopic crystal plasticity with density of dislocations. In particular, we recover the so called Orowan’s law which states that dislocations move at a velocity proportional to the effective stress.
MSC:
| 82D25 | Statistical mechanics of crystals |
| 35R09 | Integro-partial differential equations |
| 74E15 | Crystalline structure |
| 35R11 | Fractional partial differential equations |
| 47G20 | Integro-differential operators |
Keywords:
Peierls-Nabarro model; nonlocal integro-differential equations; dislocation dynamics; fractional Allen-Cahn; homogenizationReferences:
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