12 research outputs found
Homogenization of discrete high-contrast energies
Abstract. This paper focuses on deriving double-porosity models from simple high-contrast atomistic interactions. Using the variational approach and Γ-convergence techniques we derive the effective double-porosity type problem and prove the convergence. We also consider the dynamical case and study the asymptotic behavior of solutions for the gradient flow of the corresponding discrete functionals
Discrete double-porosity models for spin systems
We consider spin systems between a finite number of "species" or "phases"
partitioning a cubic lattice . We suppose that interactions
between points of the same phase are coercive, while between point of different
phases (or, possibly, between points of an additional "weak phase") are of
lower order. Following a discrete-to-continuum approach we characterize the
limit as a continuum energy defined on -tuples of sets (corresponding to the
strong phases) composed of a surface part, taking into account
homogenization at the interface of each strong phase, and a bulk part which
describes the combined effect of lower-order terms, weak interactions between
phases, and possible oscillations in the weak phase.Comment: arXiv admin note: text overlap with arXiv:1406.175
High contrast homogenisation in nonlinear elasticity under small loads
We study the homogenisation of geometrically nonlinear elastic composites
with high contrast. The composites we analyse consist of a perforated matrix
material, which we call the "stiff" material, and a "soft" material that fills
the pores. We assume that the pores are of size and are
periodically distributed with period . We also assume that the
stiffness of the soft material degenerates with rate
, so that the contrast between the two materials becomes infinite as
. We study the homogenisation limit in a
low energy regime, where the displacement of the stiff component is
infinitesimally small. We derive an effective two-scale model, which, depending
on the scaling of the energy, is either a quadratic functional or a partially
quadratic functional that still allows for large strains in the soft
inclusions. In the latter case, averaging out the small scale-term justifies a
single-scale model for high-contrast materials, which features a non-linear and
non-monotone effect describing a coupling between microscopic and the effective
macroscopic displacements.Comment: 31 page
An extension theorem from connected sets and homogenization of non-local functionals
We study the asymptotic behaviour of convolution-type functionals defined on
general periodic domains by proving an extension theore
Discrete double-porosity models for spin systems
We consider spin systems between a finite number N of “species” or “phases”
partitioning a cubic lattice Zd . We suppose that interactions between points of
the same phase are coercive while those between points of different phases (or
possibly between points of an additional “weak phase”) are of lower order. Following
a discrete-to-continuum approach, we characterize the limit as a continuum
energy defined on N-tuples of sets (corresponding to the N strong phases)
composed of a surface part, taking into account homogenization at the interface
of each strong phase, and a bulk part that describes the combined effect of lowerorder
terms, weak interactions between phases, and possible oscillations in the
weak phase
Homogenization of discrete thin structures
We consider graphs parameterized on a portion of a cylindrical subset of the lattice , and perform a discrete-to-continuum dimension-reduction process
for energies defined on of quadratic type. Our only assumptions are that
be connected as a graph and periodic in the first -directions. We show
that, upon scaling of the domain and of the energies by a small parameter
, the scaled energies converge to a -dimensional limit energy.
The main technical points are a dimension-lowering coarse-graining process and
a discrete version of the -connectedness approach by Zhikov
Homogenization of high-contrast Mumford-Shah energies
We prove a homogenization result for Mumford-Shah-type energies associated to
a brittle composite material with weak inclusions distributed periodically at a
scale . The matrix and the inclusions in the material have the
same elastic moduli but very different toughness moduli, with the ratio of the
toughness modulus in the matrix and in the inclusions being
, with small. We show that the
high-contrast behaviour of the composite leads to the emergence of interesting
effects in the limit: The volume and surface energy densities interact by
-convergence, and the limit volume energy is not a quadratic form in
the critical scaling , unlike the
-energies, and unlike the extremal limit cases