70 research outputs found
Modeling of dislocations and relaxation of functionals on 1-currents with discrete multiplicity
In the modeling of dislocations one is lead naturally to energies
concentrated on lines, where the integrand depends on the orientation and on
the Burgers vector of the dislocation, which belongs to a discrete lattice. The
dislocations may be identified with divergence-free matrix-valued measures
supported on curves or with 1-currents with multiplicity in a lattice. In this
paper we develop the theory of relaxation for these energies and provide one
physically motivated example in which the relaxation for some Burgers vectors
is nontrivial and can be determined explicitly. From a technical viewpoint the
key ingredients are an approximation and a structure theorem for 1-currents
with multiplicity in a lattice
Duality arguments for linear elasticity problems with incompatible deformation fields
We prove existence and uniqueness for solutions to equilibrium problems for
free-standing, traction-free, non homogeneous crystals in the presence of
plastic slips. Moreover we prove that this class of problems is closed under
G-convergence of the operators. In particular the homogenization procedure,
valid for elliptic systems in linear elasticity, depicts the macroscopic
features of a composite material in the presence of plastic deformation
Homogenization of energies defined on -rectifiable currents
In this paper we study the homogenization of a class of energies concentrated
on lines. In dimension (i.e., in codimension ) the problem reduces to
the homogenization of partition energies studied by \cite{AB}. There, the key
tool is the representation of partitions in terms of functions with values
in a discrete set. In our general case the key ingredient is the representation
of closed loops with discrete multiplicity either as divergence-free
matrix-valued measures supported on curves or with -currents with
multiplicity in a lattice. In the dimensional case the main motivation for
the analysis of this class of energies is the study of line defects in
crystals, the so called dislocations
Boundary-layer analysis of a pile-up of walls of edge dislocations at a lock
In this paper we analyse the behaviour of a pile-up of vertically periodic
walls of edge dislocations at an obstacle, represented by a locked dislocation
wall. Starting from a continuum non-local energy modelling the
interactionsat a typical length-scale of of the walls subjected
to a constant shear stress, we derive a first-order approximation of the energy
in powers of by -convergence, in the limit
. While the zero-order term in the expansion, the
-limit of , captures the `bulk' profile of the density of
dislocation walls in the pile-up domain, the first-order term in the expansion
is a `boundary-layer' energy that captures the profile of the density in the
proximity of the lock.
This study is a first step towards a rigorous understanding of the behaviour
of dislocations at obstacles, defects, and grain boundaries.Comment: 25 page
Minimising movements for the motion of discrete screw dislocations along glide directions
In [3] a simple discrete scheme for the motion of screw dislocations toward
low energy configurations has been proposed. There, a formal limit of such a
scheme, as the lattice spacing and the time step tend to zero, has been
described. The limiting dynamics agrees with the maximal dissipation criterion
introduced in [8] and predicts motion along the glide directions of the
crystal. In this paper, we provide rigorous proofs of the results in [3], and
in particular of the passage from the discrete to the continuous dynamics. The
proofs are based on -convergence techniques
Asymptotic behavior of nonlinear elliptic systems on varying domains
We consider a monotone operator of the form Au = −div(a(x, Du)), with Ω ⊆ Rn and a : Ω×MM×N → MM×N , acting on W1,p 0 (Ω, RM). For every sequence (Ωh) of open subsets of Ω and for every f ∈ W−1,p0 (Ω, RM), 1/p+ 1/p0 = 1, we study the asymptotic behavior, as h → +∞, of the solutions uh ∈ W1 0 (Ωh, RM) of the systems Auh = f in W−1,p0 (Ωh, RM), and we determine the general form of the limit problem
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