413 research outputs found
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
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
Overall properties of a discrete membrane with randomly distributed defects
A prototype for variational percolation problems with surface energies is considered: the description of the macroscopic properties of a (two-dimensional) discrete membrane with randomly distributed defects in the spirit of the weak membrane model of Blake and Zisserman (quadratic springs that may break at a critical length of the elongation). After introducing energies depending on suitable independent identically distributed random variables, this is done by exhibiting an equivalent continuum energy computed using Delta-convergence, geometric measure theory, and percolation arguments. We show that below a percolation threshold the effect of the defects is negligible and the continuum description is given by the Dirichlet integral, while above that threshold an additional (Griffith) fracture term appears in the energy, which depends only on the defect probability through the chemical distance on the "weak cluster of defects"
Damage as Gamma-limit of microfractures in anti-plane linearized elasticity
A homogenization result is given for a material having brittle inclusions arranged in a periodic structure.
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According to the relation between the softness parameter and the size of the microstructure, three different limit models are deduced via Gamma-convergence.
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In particular, damage is obtained as limit of periodically distributed
microfractures
On the effect of interactions beyond nearest neighbours on non-convex lattice systems
We analyse the rigidity of non-convex discrete energies where at least nearest and next-to-nearest neighbour interactions are taken into account. Our purpose is to show that interactions beyond nearest neighbours have the role of penalising changes of orientation and, to some extent, they may replace the positive-determinant constraint that is usually required when only nearest neighbours are accounted for. In a discrete to continuum setting, we prove a compactness result for a family of surface-scaled energies and we give bounds on its possible Gamma-limit in terms of interfacial energies that penalise changes of orientation
Effective cohesive behavior of layers of interatomic planes
A simple model of cleavage in brittle crystals consists of a layer of material containing N atomic planes separating in accordance with an interplanar potential under the action of an opening displacement delta prescribed on the boundary of the layer. The problem addressed in this work concerns the characterization of the constrained minima of the energy E-N of the layer as a function of delta as N becomes large. These minima determine the effective or macroscopic cohesive law of the crystal. The main results presented in this communication are: (i) the computation of the Gamma limit E-0 of E-N as N -> infinity; (ii) the characterization of the minimum values of E-0 as a function of the macroscopic opening displacement; (iii) a proof of uniform convergence of the minima of E-N for the case of nearest-neighbor interactions; and (iv) a proof of uniform convergence of the derivatives of E-N, or tractions, in the same case. The scaling on which the present Gamma-convergence analysis is based has the effect of separating the bulk and surface contributions to the energy. It differs crucially from other scalings employed in the past in that it renders both contributions of the same order
Nematic elastomers: Gamma-limits for large bodies and small particles
We compute the large-body and the small-particle Gamma-limit of a family of energies for nematic elastomers. We work under the assumption of small deformations (linearized kinematics) and consider both compressible and incompressible materials. In the large-body asymptotics, even if we describe the local orientation of the liquid crystal molecules according to the model of perfect order (Frank theory), we prove that we obtain a fully biaxial nematic texture (that of the de Gennes theory) as a by-product of the relaxation phenomenon connected to Gamma-convergence. In the case of small particles, we show that formation of new microstructure is not possible, and we describe the map of minimizers of the Gamma-limit as the phase diagram of the mechanical model
Some homogenization and corrector results for nonlinear monotone operators
This paper deals with the limit behaviour of the solutions of quasi-linear
equations of the form \ \ds -\limfunc{div}\left(a\left(x, x/{\varepsilon
_h},Du_h\right)\right)=f_h on with Dirichlet boundary conditions.
The sequence tends to and the map is
periodic in , monotone in and satisfies suitable continuity
conditions. It is proved that weakly in , where is the solution of a homogenized problem \
-\limfunc{div}(b(x,Du))=f on . We also prove some corrector results,
i.e. we find such that in
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