1,686 research outputs found
Homogenization of hexagonal lattices
International audienceWe characterize the macroscopic e ffective behavior of a graphene sheet modeled by a hexagonal lattice of elastic bars, using Gamma-convergence
Wave mechanics in media pinned at Bravais lattice points
The propagation of waves through microstructured media with periodically
arranged inclusions has applications in many areas of physics and engineering,
stretching from photonic crystals through to seismic metamaterials. In the
high-frequency regime, modelling such behaviour is complicated by multiple
scattering of the resulting short waves between the inclusions. Our aim is to
develop an asymptotic theory for modelling systems with arbitrarily-shaped
inclusions located on general Bravais lattices. We then consider the limit of
point-like inclusions, the advantage being that exact solutions can be obtained
using Fourier methods, and go on to derive effective medium equations using
asymptotic analysis. This approach allows us to explore the underlying reasons
for dynamic anisotropy, localisation of waves, and other properties typical of
such systems, and in particular their dependence upon geometry. Solutions of
the effective medium equations are compared with the exact solutions, shedding
further light on the underlying physics. We focus on examples that exhibit
dynamic anisotropy as these demonstrate the capability of the asymptotic theory
to pick up detailed qualitative and quantitative features
Auxetic two-dimensional lattice with Poisson's Ratio arbitrarily close to -1
In this paper we propose a new lattice structure having macroscopic Poisson's
ratio arbitrarily close to the stability limit -1. We tested experimentally the
effective Poisson's ratio of the micro-structured medium; the uniaxial test has
been performed on a thermoplastic lattice produced with a 3d printing
technology. A theoretical analysis of the effective properties has been
performed and the expression of the macroscopic constitutive properties is
given in full analytical form as a function of the constitutive properties of
the elements of the lattice and on the geometry of the microstructure. The
analysis has been performed on three micro-geometry leading to an isotropic
behaviour for the cases of three-fold and six-fold symmetry and to a cubic
behaviour for the case of four-fold symmetry.Comment: 26 pages, 12 figures (26 subfigures
Simplified modelling of chiral lattice materials with local resonators
A simplified model of periodic chiral beam-lattices containing local
resonators has been formulated to obtain a better understanding of the
influence of the chirality and of the dynamic characteristics of the local
resonators on the acoustic behavior. The simplified beam-lattices is made up of
a periodic array of rigid heavy rings, each one connected to the others through
elastic slender massless ligaments and containing an internal resonator made of
a rigid disk in a soft elastic annulus. The band structure and the occurrence
of low frequency band-gaps are analysed through a discrete Lagrangian model.
For both the hexa- and the tetrachiral lattice, two acoustic modes and four
optical modes are identified and the influence of the dynamic characteristics
of the resonator on those branches is analyzed together with some properties of
the band structure. By approximating the generalized displacements of the rings
of the discrete Lagrangian model as a continuum field and through an
application of the generalized macro-homogeneity condition, a generalized
micropolar equivalent continuum has been derived, together with the overall
equation of motion and the constitutive equation given in closed form. The
validity limits of the micropolar model with respect to the dispersion
functions are assessed by comparing the dispersion curves of this model in the
irreducible Brillouin domain with those obtained by the discrete model, which
are exact within the assumptions of the proposed simplified model
Anisotropic and dispersive wave propagation within strain-gradient framework
In this paper anisotropic and dispersive wave propagation within linear
strain-gradient elasticity is investigated. This analysis reveals significant
features of this extended theory of continuum elasticity. First, and contrarily
to classical elasticity, wave propagation in hexagonal (chiral or achiral)
lattices becomes anisotropic as the frequency increases. Second, since
strain-gradient elasticity is dispersive, group and energy velocities have to
be treated as different quantities. These points are first theoretically
derived, and then numerically experienced on hexagonal chiral and achiral
lattices. The use of a continuum model for the description of the high
frequency behavior of these microstructured materials can be of great interest
in engineering applications, allowing problems with complex geometries to be
more easily treated
High frequency homogenisation for elastic lattices
A complete methodology, based on a two-scale asymptotic approach, that
enables the homogenisation of elastic lattices at non-zero frequencies is
developed. Elastic lattices are distinguished from scalar lattices in that two
or more types of coupled waves exist, even at low frequencies. Such a theory
enables the determination of effective material properties at both low and high
frequencies. The theoretical framework is developed for the propagation of
waves through lattices of arbitrary geometry and dimension. The asymptotic
approach provides a method through which the dispersive properties of lattices
at frequencies near standing waves can be described; the theory accurately
describes both the dispersion curves and the response of the lattice near the
edges of the Brillouin zone. The leading order solution is expressed as a
product between the standing wave solution and long-scale envelope functions
that are eigensolutions of the homogenised partial differential equation. The
general theory is supplemented by a pair of illustrative examples for two
archetypal classes of two-dimensional elastic lattices. The efficiency of the
asymptotic approach in accurately describing several interesting phenomena is
demonstrated, including dynamic anisotropy and Dirac cones.Comment: 24 pages, 7 figure
Numerical Methods for Multilattices
Among the efficient numerical methods based on atomistic models, the
quasicontinuum (QC) method has attracted growing interest in recent years. The
QC method was first developed for crystalline materials with Bravais lattice
and was later extended to multilattices (Tadmor et al, 1999). Another existing
numerical approach to modeling multilattices is homogenization. In the present
paper we review the existing numerical methods for multilattices and propose
another concurrent macro-to-micro method in the numerical homogenization
framework. We give a unified mathematical formulation of the new and the
existing methods and show their equivalence. We then consider extensions of the
proposed method to time-dependent problems and to random materials.Comment: 31 page
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