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Adriana Garroni (University of Roma)
| Speaker: |
Adriana Garroni (University
of Roma)
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| Date: |
Wednesday May 4, 2010
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| Title: |
A variational derivation for a line tension model for
dislocations
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Abstract
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The main mechanism
for crystal plasticity is the formation and motion of a special class
of defects, the dislocations. These are topological defects in the
crystalline structure that can be identify with lines on which energy
concentrates. In recent years there has been a
considerable effort for the mathematical derivation of models that
describe these objects at different scales (from an energetic and a
dynamical point of view). The results obtained mainly concern special
geometries, as one dimensional models, reduction to straight
dislocations, the activation of only one slip system, etc. The
description of the problem is indeed extremely complex in its
generality.
We will restrict our analysis to the static case and consider a fully
three dimensional discrete variational model for crystal plasticity
that accounts for the presence of dislocations. The energy will be
essentially given by elastic interaction far from the dislocations,
with the condition that the curl of the elastic strain concentrates on
the dislocation curves. The model has many similarities with the
Ginzburg Landau model for superconductors and other variational model
in which concentration of energy occurs.
Under suitable scales we study the “variational limit” (by means of
Gamma-convergence) of this model and we deduce a line tension energy.
The characterization of the line tension energy density requires a
relaxation result for energies defined on curves.
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