771 research outputs found
Homogenization of oscillating boundaries and applications to thin films
We prove a homogenization result for integral functionals in domains with
oscillating boundaries, showing that the limit is defined on a degenerate
Sobolev space. We apply this result to the description of the asymptotic
behaviour of thin films with fast-oscillating profile, proving that they can be
described by first applying the homogenization result above and subsequently a
dimension-reduction technique.Comment: 31 pages, 7 figure
Motion of discrete interfaces in periodic media
We study the motion of discrete interfaces driven by ferromagnetic
interactions in a two-dimensional periodic environment by coupling the
minimizing movements approach by Almgren, Taylor and Wang and a
discrete-to-continuous analysis. The case of a homogeneous environment has been
recently treated by Braides, Gelli and Novaga, showing that the effective
continuous motion is a flat motion related to the crystalline perimeter
obtained by -convergence from the ferromagnetic energies, with an
additional discontinuous dependence on the curvature, giving in particular a
pinning threshold. In this paper we give an example showing that in general the
motion does not depend only on the -limit, but also on geometrical
features that are not detected in the static description. In particular we show
how the pinning threshold is influenced by the microstructure and that the
effective motion is described by a new homogenized velocity.Comment: arXiv admin note: substantial text overlap with arXiv:1407.694
Motion of discrete interfaces through mushy layers
We study the geometric motion of sets in the plane derived from the
homogenization of discrete ferromagnetic energies with weak inclusions. We show
that the discrete sets are composed by a `bulky' part and an external `mushy
region' composed only of weak inclusions. The relevant motion is that of the
bulky part, which asymptotically obeys to a motion by crystalline mean
curvature with a forcing term, due to the energetic contribution of the mushy
layers, and pinning effects, due to discreteness. From an analytical standpoint
it is interesting to note that the presence of the mushy layers imply only a
weak and not strong convergence of the discrete motions, so that the
convergence of the energies does not commute with the evolution. From a
mechanical standpoint it is interesting to note the geometrical similarity of
some phenomena in the cooling of binary melts.Comment: 20 pages, 3 figure
An integral-representation result for continuum limits of discrete energies with multi-body interactions
We prove a compactness and integral-representation theorem for sequences of
families of lattice energies describing atomistic interactions defined on
lattices with vanishing lattice spacing. The densities of these energies may
depend on interactions between all points of the corresponding lattice
contained in a reference set. We give conditions that ensure that the limit is
an integral defined on a Sobolev space. A homogenization theorem is also
proved. The result is applied to multibody interactions corresponding to
discrete Jacobian determinants and to linearizations of Lennard-Jones energies
with mixtures of convex and concave quadratic pair-potentials
Homogenization of cohesive fracture in masonry structures
We derive a homogenized mechanical model of a masonry-type structure
constituted by a periodic assemblage of blocks with interposed mortar joints.
The energy functionals in the model under investigation consist in (i) a linear
elastic contribution within the blocks, (ii) a Barenblatt's cohesive
contribution at contact surfaces between blocks and (iii) a suitable unilateral
condition on the strain across contact surfaces, and are governed by a small
parameter representing the typical ratio between the length of the blocks and
the dimension of the structure. Using the terminology of Gamma-convergence and
within the functional setting supplied by the functions of bounded deformation,
we analyze the asymptotic behavior of such energy functionals when the
parameter tends to zero, and derive a simple homogenization formula for the
limit energy. Furthermore, we highlight the main mathematical and mechanical
properties of the homogenized energy, including its non-standard growth
conditions under tension or compression. The key point in the limit process is
the definition of macroscopic tensile and compressive stresses, which are
determined by the unilateral conditions on contact surfaces and the geometry of
the blocks
A Relaxation result for energies defined on pairs set-function and applications
We consider, in an open subset Ω
of RN, energies depending on the perimeter of a subset
E С Ω
(or some equivalent surface integral) and on a function u which is defined only on
E. We compute the lower semicontinuous envelope of such energies. This relaxation has
to take into account the fact that in the limit, the “holes”
Ω \ E may collapse into a
discontinuity of u, whose surface will be counted twice in the relaxed energy. We discuss
some situations where such energies appear, and give, as an application, a new proof of
convergence for an extension of Ambrosio-Tortorelli’s approximation to the Mumford-Shah
functional
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